Fundamentals of the Radiolocation and Radionavigation 303110630X, 9783031106309

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Springer Aerospace Technology

Stanisław Rosłoniec

Fundamentals of the Radiolocation and Radionavigation

Springer Aerospace Technology Series Editors Sergio De Rosa, DII, University of Naples Federico II, Napoli, Italy Yao Zheng, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang, China Elena Popova, AirNavigation Bridge Russia, Chelyabinsk, Russia

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Stanisław Rosłoniec

Fundamentals of the Radiolocation and Radionavigation

Stanisław Rosłoniec Institute of Radioelectronics Warsaw University of Technology Warsaw, Poland

ISSN 1869-1730 ISSN 1869-1749 (electronic) Springer Aerospace Technology ISBN 978-3-031-10630-9 ISBN 978-3-031-10631-6 (eBook) https://doi.org/10.1007/978-3-031-10631-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my wife Wanda Krystyna

Preface

A long time ago, as a young boy, I read an interesting book entitled Summation of the Infinitely Small. Its author explained the basic concepts of differential and integral calculus in an extremely accessible way. I still clearly remember described in this book the method of calculating a volume of a bulgy beer barrel using simple mathematical formulas, including the formula for calculating the volume of a single cylinder. For this purpose, a barrel should be treated as a set of many cylinders of different diameters and the same small thickness. The sum of volumes of these cylinders will differ not much from the volume of the barrel if the thickness of the mentioned cylinders is sufficiently small. This extremely simple and useful method inspired me to many thoughts that turned out to be very helpful during my university studies. Undoubtedly, the interest in mathematics and technology that arose at that moment had a decisive influence on my choice of the direction of professional activity. Why am I writing about it here and how does it relate to this handbook on radiolocation and radionavigation? Well, Dear Reader, I would be glad if this handbook could fulfill a similar role as the aforementioned “wonderful” book from my school years. Naturally, at a different stage of your scientific and professional career. The handbook recommended to your attention has been prepared on the basis of the lecture notes “Fundamentals of the radiolocation and radionavigation” I have given many years for the students of the Warsaw University of Technology and to the engineers of the Telecommunications Research Institute (PIT). Thus, I wrote this book from my perspective as an academic teacher and a designer in industry. The title of the presented handbook suggests that it is devoted to the radiolocation and radionavigation. In fact, the issues of radiolocation are discussed in Part I, covering Chaps. 1–12. The remaining chapters, i.e., 13–18, are devoted to the issues of radionavigation. The book contains a systematic approach to the subject. Chapter 1 discusses various physical phenomena used in the modern radiolocation and radionavigation. Standard methods for measuring a distance to the detected objects and methods for determining their angular coordinates are presented. Detection and determining the position of an object by means of passive radiolocation, using DOA and TDOA

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methods, are discussed in Chap. 2. An important parameter determining the possibility of detecting an object and determining the range of a radiolocation system is the effective reflecting surface of this object, called RCS. The definition of this parameter and methods for its approximate determination, for monostatic and bistatic radiolocation systems, are presented in Chap. 3. The group of the discussed in Chap. 4 active radiolocation systems includes: active radiolocation with a passive response (primary radiolocation) and active radiolocation with an active response (secondary radiolocation). Bistatic and multistatic versions of these systems are described in Chaps. 5 and 6, respectively. Principles of operation of the systems under discussion are illustrated with relevant examples of calculations and applications. Chapter 7 is devoted to the issue of increasing the range of the radar systems in which the methods of additive and correlation reception are discussed. The theoretical basis of the correlation method of reception is matched filtering of the signal used in the system. This subject is discussed, step by step, in Chaps. 8 and 9. Various methods for eliminating interfering signals from radar echoes are subject of consideration in Chap. 10. The next, i.e., Chap. 11, presents various methods of searching 3D space in order to detect of objects and determine their positions. Chapter 12 is devoted to the amplitude and phase methods for determining the angular coordinates of an object observed with the mono-impulse radar. Part II of the handbook discusses the functional segments and principles of operation of terrestrial and satellite radionavigation systems. In order to make the discussed issues accessible to a wide range of readers, Chap. 13 defines the basic concepts and parameters (e.g., angles) used in the radionavigation. This chapter also provides examples of design solutions for various types of radio and microwave direction finders. The terrestrial systems are represented by the Loran C, Decca Navigator and Omega in Chap. 14. The TRANSIT system is discussed in Chap. 15 as an example of a hyperbolic satellite system. The group of the discussed stadiometric systems consists of GPS, GLONASS, GALILEO, BeiDou, IRNSS and QZSS. These systems are augmentated by appropriate differential satellite systems (SBAS), i.e., EGNOS, WAAS, MSAS, GAGAN, SDCM, BDSBAS and KASS. The above-mentioned stadiometric radionavigation systems and their differential versions are presented in the extensive Chap. 16. In Chap. 17, the Instrument Landing System (ILS), Microwave Landing System (MLS), and Transponder Landing System (TLS) supporting the landing of aircrafts are discussed in detail. The prospects for replacing these rather complex terrestrial systems with the aforementioned satellite systems augmentated by appropriate reference ground-based stations (GBAS) are also analyzed. Various radio beacons (VOR, D—VOR) and distance measuring devices (DME) used by these systems are described in last Chap. 18. Each individual chapter ends with a list of the relevant references. Appendices for some chapters are located at the end of the handbook similarly as the subject index. I also wish to present the main principles assumed at selection and presentation of the contents in order to make the handbook up to date and useful for engineering practice. The main of these principles are discussed below.

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The first is the principle of selection and ordering, according to which the growing quantity of information about a considered phenomenon or an engineering problem should be reasonably adjusted. The elements that are no longer up to date, for example, due to changes in the research methods or manufacturing technologies, should be removed. More modern, sufficiently verified, research methods and design solutions, useful for practice, should be introduced at the same time. The second principle relates to the physical nature of the phenomenon in question (or operating principle) and the various forms of its mathematical description which can be attributed to that phenomenon. It is obvious that knowing a mathematical description, more precisely the mathematical model, is useless without understanding the physical nature of the phenomenon they describe. Therefore, in this handbook, great emphasis was placed on explaining the physical essence of the discussed phenomena and the principles of operation of radio devices, using only the necessary mathematical formulas that constitute the basis of quantitative analysis. For teaching reasons, most of the mathematical dependencies were derived step by step, and their correctness was confirmed by many examples of calculations. The third principle deals with the regularity and transparency of the lecture. The reader should not be too frequently referred to the literature and not to be “surprised” by new, incomprehensible concepts or statements that are not explained in the work. Therefore, I referred only to some classical books and papers that I have found particularly helpful. Moreover, it should be pointed out that in the field of radiocommunication, some of 50-year-old (or older) publications are still useful and we should not reinvent fundamental methods for signal processing or constructional solutions. The ideas and design solutions described in them are often “jewels in the crown” in the fascinating development of radiolocation and radionavigation and therefore should be known to the widest possible group of well-educated scientists and engineers. Thus, this book is intended primarily for students and engineers interested in radar, radionavigation and aerospace engineering. Finally, I would like to express my gratitude to Mrs. Agata Szewczyk and Mr. Slawomir Sujecki from the University of Nottingham for the difficult-to-overestimate help they have provided in the preparation of the English-language manuscript of this book. Warsaw, Poland May 2022

Stanisław Rosłoniec

Contents

Part I 1

Basics Principles of the Radiolocation

Radiolocation and Its Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Standard Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Physical Phenomena Used in Modern Radiolocation . . . . . . . . . . . 1.3 Distance Measurement Method Using a Pulse Radar . . . . . . . . . . 1.4 Short Range Altimeter as an Example of Radars Using Frequency Modulation Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Standard Methods for Determining the Angular Coordinates of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 7 12

2

Determining the Object’s Position by Radiolocation Methods . . . . . . 2.1 The Direction of Arrival (DOA) Method . . . . . . . . . . . . . . . . . . . . . 2.2 The Time Difference of Arrival (TDOA) Method . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 28 31 43

3

Reflective Surface of the Detected Objects with Monostatic and Bistatic Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Reflective Surface Determined for a Monostatic Primary Radar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Reflection Surface of a Group Object . . . . . . . . . . . . . . . . . . . . 3.3 Monostatic and Bistatic Reflective Surfaces of the Conductive Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Radar Cross Section of an Object Determined FSR . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Range Equations of Primary and Secondary Radar Systems . . . . . . 4.1 Range Equation of the Primary Radar System . . . . . . . . . . . . . . . . 4.2 The Range Equation of the Secondary Radar System . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 21 24

45 48 56 59 61 63 65 65 72 76

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Bistatic Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Main Advantages and Disadvantages of the Bistatic System . . . . 5.2 Methods of Determining Object’s Position Using Bistatic Radar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Range Equation of the Bistatic Radar System . . . . . . . . . . . . . . . . . 5.4 Searching Space Using the Probe Signal Chasing Method . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 82 88 91

6

Multistatic Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1 The Method of Determining the object’s Position Using a Multistatic System with One Transmitter and Four Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.2 The Method of Determining the Velocity Vector of an Object in 3D Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 The Simulation Tests Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7

Standard Methods for Extending the Range of Radar Station . . . . . 7.1 Elements of the Radar Signals Theory . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Additive Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Correlation Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 120 121 125

8

Theoretical Basis of Matched Signal Filtration . . . . . . . . . . . . . . . . . . . 8.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Transmittance of a Matched Filter to a Given Signal . . . . . . . 8.3 Examples of Standard Signals Matched Filters . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 130 135 147

9

Filters Matched to the Typical Radar Signals . . . . . . . . . . . . . . . . . . . . 9.1 Filter Matched to a LFM Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Filters Matched to High Frequency Pulses with Bistate Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Introduction to a Digital Matched Filtration of Radar Signals . . . 9.4 Matched Filtration in Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 A Matched Filtration in the Frequency Domain . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149

10 Basic Methods for Eliminating Spurious Signals . . . . . . . . . . . . . . . . . 10.1 Basic Methods of Eliminating Signals Reflected from Terrain Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Moving Objects’ Reflections Elimination Methods . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

157 161 163 166 172

175 178 192

11 Searching the Three-Dimensional Space with Radar Devices . . . . . . 193 11.1 The Three-Dimensional Space Observation Methods . . . . . . . . . . 193 11.2 Observation of the Land and Sea Areas with Radar Devices Installed on Board of Aircrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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11.3 Tracking and Radar Homing Methods . . . . . . . . . . . . . . . . . . . . . . . 205 11.4 Autonomous Methods of Flying Objects Missile Homing . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 12 Methods of Determining the Angular Coordinates of an Object by Monopulse Radar Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Amplitude and Phase Methods of the Monopulse Radiolocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Discriminators of the Monopulse Radar Devices . . . . . . . . . . 12.3 Examples of Structural Solutions of the Monopulse Radar Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

223 223 228 233 254

Basic Principles of the Radionavigation

13 Basic Terms of Radionavigation and Object Position Determining Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Basic Terms, Parameters and Navigation Methods . . . . . . . . . . . . . 13.2 Classification Criterion of Radio Navigation Systems . . . . . . . . . . 13.3 Radio Direction-Finders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 259 266 266 281

14 Ground, Hiperbolic Radionavigation Systems . . . . . . . . . . . . . . . . . . . . 14.1 LORAN–C Pulse System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Decca Navigator and Omega Interference Systems . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 284 291 297

15 Satellite, Doppler Radionavigation Systems . . . . . . . . . . . . . . . . . . . . . . 15.1 A Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Navy Navigation System—TRANSIT . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 303 313

16 Satellite Navigation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 GPS—NAVSTAR System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 A System of Four Stadiometric Equations . . . . . . . . . . . . 16.1.2 Some Results of Simulation Calculations . . . . . . . . . . . . . 16.2 The System GLONASS and Its Functional Segments . . . . . . . . . . 16.2.1 The Space Segment and Transmitted Signals . . . . . . . . . . 16.2.2 The Control and User Segment . . . . . . . . . . . . . . . . . . . . . 16.3 The System GALILEO and Its Functional Segments . . . . . . . . . . . 16.3.1 The Space Segment, Transmitted Signals and Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 The Control and User Segments . . . . . . . . . . . . . . . . . . . . . 16.4 Other Stadiometric Satellite Navigation Systems . . . . . . . . . . . . . . 16.4.1 The Chinese Navigation System BeiDou . . . . . . . . . . . . . 16.4.2 The Indian Regional Navigation Satellite System (IRNSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 315 322 326 329 329 332 332 333 334 335 335 336

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16.4.3 The Japanese Regional Navigation Satellite System (QZSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 About Differential Versions of the Satellite Navigation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 A Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Examples of the Satellite Based Augmentation Systems (SBAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Aircraft Landing Aid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Instrument Landing System (ILS) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Microwave Landing System (MLS) . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Transponder Landing System (TLS) . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Ground and Satellite Based Augmentation Systems Used for a Precision Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Radio Beacons and Distance Measuring Equipment Supporting Flight and Landing of the Aircrafts . . . . . . . . . . . . . . . . . . 18.1 Phase Radio Beacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Doppler VOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Pulse Radio Beacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Distance Measuring Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 339 339 342 351 355 357 373 382 390 392 393 395 398 402 406 411

Appendix A: Altimeter Using a Signal with Sinusoidal Modulation of the Carrier Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Appendix B: Algebraic Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Appendix C: The Attenuation Factor Resulting from the Wave Absorption in Troposphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Appendix D: Microwave Signals Emitted by Ground and On-board Devices of the Secondary Surveillance Radar Systems Operating in the 3/A and 3/C Modes . . . . . 421 Appendix E: Determining the Dependence (7.41) . . . . . . . . . . . . . . . . . . . . . 425 Appendix F: Indeterminacy Function χ (τ, .) of a Single Rectangular Radio Pulse with Internal Linear Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Appendix G: Costas Loop as a BPSK Signal Demodulator . . . . . . . . . . . . . 433 Appendix H: Goniometric Radio Direction-Finder . . . . . . . . . . . . . . . . . . . . 437

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Appendix I: Time Calculation of a Satellite Radio Visibility Above the User’s Horizon Line . . . . . . . . . . . . . . . . . . . . . . . . . 441 Appendix J: Reproducing Methods of a Carrier Frequency Signal from a Phase Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . . 445 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

About the Author

Stanisław Rosłoniec was born on June 25, 1948, in Ostaszewo, Poland. He received his M.Sc. degree (in electronic engineering) from Warsaw University of Technology in 1972. After graduation, he joined the Department of Electronics (Institute of Radioelectronics) of Warsaw University of Technology where in 1976 he was granted with distinction his doctor’s degree (Ph.D.). The higher doctor’s degree (Dr. Sc.) he received in 1991 also from Warsaw University of Technology. Finally, he obtained the degree of professor of technical sciences from the President of the Republic of Poland in 2001. Since 1992, he is closely cooperating with the Telecommunications Research Institute (PIT), Poland. Therefore, his didactic and research interests are mainly focused on relevant practice problems of microwave and antenna techniques, especially on the design methods of various passive microwave devices and high-directivity radar antennas. Several array antennas designed by him operate in long-range surveillance radars of type TRD12, RST-12M, CAR-1100 and TRS-15 fabricated by PIT for the Polish Army and foreign contractors. Stanisław Rosłoniec is the author over 80 reviewed scientific papers, 25 technical reports, 3 patents and 10 books. A list of mentioned books includes 4 academic handbooks and 6 monographs. Two of them, viz. Algorithms for Computer—Aided Design of Linear Microwave Circuits (1990) and Fundamental Numerical Methods for Electrical Engineering (2008) have been published by top internationally recognized publishers, respectively by Artech House Inc., USA and xvii

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About the Author

Springer Verlag, Germany. The latest of his book entitled “Some design problems of contemporary antenna technique” (in Polish) has been published in 2022 by Publishing House of the Military University of Technology, Warsaw. He is a senior member of the Microwave Theory and Techniques Society (IEEE/MTT-S).

Acronyms

APV ARM ATC BPSK CPU CW/FM DBF DDC DDS DFT DGPS DME DOA DSP EGNOS FDOA FFT FSR GBAS GLONASS GPS HF ICAO IDFT IFF ILS LFM LORAN C MLS

Array propagation vector Anti-radiation missile Air traffic control Binary (0, π ) phase—shift keying Central processing unit Continuous wave/frequency modulation Digital beam forming Digital down converter [u(t) → (I (m), Q(m))] Direct digital synthesis Discrete Fourier transform Differential Global Positioning System Distance measuring equipment Direction of arrival Digital signal processing European Geostationary Navigation Overlay Service Frequency differences of arrival Fast Fourier transform Forward scattering radiolocation Ground-based augmentation system Russian: Globalnaja Nawigacjonnaja Sputnikowaja Sistiema (Global Satellite Navigation System) Global Positioning System—Navigation Signal Timing and Ranging, (also NAVSTAR) High frequency International Civil Aviation Organization Inverse discrete Fourier transform Identification friend—foe Instrument landing system Linear frequency modulation Long-range navigation—version C Microwave landing system xix

xx

MMICs MTD MTI PCL PPS PRS PSR QPSK RADAR RCS RSBN-4N SAR SAW SBAS SPASUR SPDT SPS SSR T/R TDOA TEM TLPSK TLS TRANSIT UHF UTC VHF VOR VSWR WGS’ 84

Acronyms

Monolithic microwave integrated circuits Moving target detector Moving target indicator Passive coherent location Precise positioning service Passive radar seeker Primary surveillance radar Quadrature (0, π/2, π, 3π /2) phase shift keying Radio detecting and ranging Radar cross section Russian: Radiosistiema bli˙zniej nawigacji (Short-range navigation system-version 4N) Synthetic aperture radar Surface acoustic waves Satellite-based augmentation system Space surveillance Single pole double throw Standard positioning service Secondary surveillance radar Transmit/receive module Time differences of arrival Transverse electromagnetic wave Three-level (π/3, 0, − π/3) phase shift keying Transponder landing system Navy Navigation Satellite System (also NAVSAT, NNSS) Ultra-high frequency Universal Time Coordinated Very high frequency VHF omni-directional range Voltage standing wave ratio World Geodetic System’ 84

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8

Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 1.13 Fig. 1.14 Fig. 1.15 Fig. 1.16 Fig. 1.17

Fig. 1.18

A local coordinate system used in the radiolocation . . . . . . . . . . Various types of the radiolocation . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of a bistatic radiolocation system . . . . . . . . The block diagram of a monostatic radiolocation using a two-channel duplexer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of a secondary radiolocation . . . . . . . . . . . . The spatial distribution of an electric field of TEM wave . . . . . The back scattering from a flat reflecting surface . . . . . . . . . . . . Graphical illustration of polarization matching the TEM wave to a rectilinear conductor, a the wave energy is not transferred to the conductor, b a part of the wave energy is transferred to the conductor . . . . . . . . . . . . . . . . . . . . . Illustration of a physical essence of the Doppler phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the primary radar system using a pulsed radio signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The pulse of a probing signal and its proper echo . . . . . . . . . . . The pulse of a probing signal and its incorrect (delayed over range) echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of a spatial cell including the pulse probing signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An electromagnetic beam covering two reflecting objects . . . . . An aircraft over a flat horizontal plane . . . . . . . . . . . . . . . . . . . . The block diagram of the on-board altimeter transceiver . . . . . . Frequencies of the transmitted ( f 1 ) and received ( f 2 ) signals without Doppler effect. The lower figure b shows the the difference | f 1 − f 2 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of saddle points of a signal of difference frequency where the difference frequency is equal to zero at the moment ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 5 6 6 8 9

10 11 12 13 14 14 15 15 16

16

17

xxi

xxii

Fig. 1.19

Fig. 1.20

Fig. 1.21 Fig. 1.22 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4

Fig. 2.5 Fig. 2.6 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15 Fig. 3.16 Fig. 3.17 Fig. 3.18

List of Figures

Frequencies of the transmitted ( f 1 ) and received signals. The receive signal ( f 2 ) has been changed by the Doppler effect. The lower figure b shows the difference | f 1 − f 2 | . . . . . Illustration of measuring the angular coordinate of an object using the “on maximum” method; a block diagram of the measuring system, b time waveforms of signals u 1 (t), u 2 (t), u 3 (t) and u 4 (t) . . . . . . . . . . . . . . . . . . . . Directivity characteristics of an antenna; a summation characteristic, b differential characteristic . . . . . . . . . . . . . . . . . . Graphic illustration of the angular resolution parameter . . . . . . Definition of sides and angles of the triangle ABC used in the triangulation method of navigation . . . . . . . . . . . . . . . . . . Illustration of the triangulation method for determining the coordinates of the O object in 3D space . . . . . . . . . . . . . . . . TDOA method for determining the coordinates of the O object in 2D space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric interpretation of the blur area (GDOP); a for triangulation method of navigation, b for the TDOA method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A passive radiolocation scenario with four receivers . . . . . . . . . Geometric illustration of a problem of occurrence of the second useless solution . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the bistatic angle β . . . . . . . . . . . . . . . . . . . . . . . . . Object classification used in this chapter . . . . . . . . . . . . . . . . . . . The rectilinear conductors; a the thin strip, b the thin wire . . . . The conductive cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The conductive conical surface . . . . . . . . . . . . . . . . . . . . . . . . . . The conductive cylindrical—conical surface . . . . . . . . . . . . . . . The circular conductive surface . . . . . . . . . . . . . . . . . . . . . . . . . . The square conductive surface . . . . . . . . . . . . . . . . . . . . . . . . . . . The triangular corner reflector a basic dimensions, b general view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The χ [dB] factor as a function of α and γ angles . . . . . . . . . . . Typical clusters of corner reflectors . . . . . . . . . . . . . . . . . . . . . . . The longitudinal section of the Luneburg lens . . . . . . . . . . . . . . Return power from a B-26 aircraft at 10 cm wavelength as a function of azimuth angle . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimentally determined RCS for a Boeing 727-100C aircraft with dimensions as shown in Fig. 3.15 . . . . . . . . . . . . . . Boeing 727-100C aircraft; a the side view, b the bottom view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A fairy-tale illustration of a ballistic missile surrounded by decoys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A group target consisting of two aircrafts . . . . . . . . . . . . . . . . . . The sum of two rotating vectors shifted by the angle ϕ . . . . . . .

18

22 23 24 28 30 32

34 34 43 46 47 48 49 50 50 51 51 52 53 53 54 54 55 55 57 57 58

List of Figures

Fig. 3.19

Fig. 3.20 Fig. 3.21

Fig. 3.22

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8

Fig. 5.9

Fig. 5.10 Fig. 6.1 Fig. 6.2 Fig. 6.3

xxiii

Characteristics σ (α)/(2σ0 ) calculated according to (3.18) for different values of d/λ; a fragment of the chart for d/λ = 10 and d/λ = 50, b full chart for d/λ = 50 . . . . . . . The back scattering from a metalic sphere . . . . . . . . . . . . . . . . . The plot of σ B (β)/σ M as a function of bistatic angle β obtained experimentally for an well conductive sphere with a radius r = 20λ/(2π ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The theoretical relationship σ B (β) determined for F-117 aircraft; a the model of F-117 under consideration, b the plot of the effective aera σ B (β) as a function of bistatic angle β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the primary radiolocation system; a with two separated antennas, b with common antenna . . . . . . Geometric illustration of a radar range limitation resulting from geometry of Earth surface . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric illustration of the electromagnetic wave refraction in the atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of effect of increasing the radar range due to refraction of electromagnetic wave in the atmosphere . . . . . . Basic functional blocks of the secondary radiolocation system (e.g. IFF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The functional blocks and geometry of a bistatic radiolocation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of a receiver unit . . . . . . . . . . . . . . . . . . . . . . Definition of angles α, ϕ R and γ R . . . . . . . . . . . . . . . . . . . . . . . . The bistatic radar system with variable base . . . . . . . . . . . . . . . . The bistatic radiolocation system geometry described by means of spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . The coverage area as a function of χ parameter . . . . . . . . . . . . . Geometric illustration of the radar range limitation due to the curvature of the Earth’s surface . . . . . . . . . . . . . . . . . . . . . Coverage areas of the bistatic radiolocation system; a without a “dead zone”, b with the dead zone marked by dashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection cells formed in an area of the intersection of the transmitting beam and the receiving beam; a completely filled with electromagnetic energy, b partially filled with electromagnetic energy . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of the “chasing probing impulse” method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The coverage (observation) area of the SPASUR system . . . . . . The geometry of a multistatic system with four receivers . . . . . The geometry of a multistatic system with four ground-based receivers and an aircraft-based transmitter . . . . .

59 60

61

62 66 69 70 71 72 78 79 79 81 82 85 87

88

89 90 94 96 96

xxiv

Fig. 6.4

Fig. 6.5 Fig. 7.1 Fig. 7.2 Fig. 7.3

Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7

Fig. 7.8 Fig. 7.9 Fig. 7.10 Fig. 7.11

Fig. 7.12

Fig. 7.13 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4

Fig. 8.5

Fig. 8.6 Fig. 8.7 Fig. 8.8

List of Figures

Graphic illustration of the spatial position of the velocity vector; a the general view of the vector position, b the velocity vector and its components . . . . . . . . . . . . . . . . . . . . The perpendicular projections of components of the velocity vector on the line di . . . . . . . . . . . . . . . . . . . . . . . A single rectangular pulse u(t) . . . . . . . . . . . . . . . . . . . . . . . . . . A plot of the χ = .u (ω)/(A2 ti2 ) function . . . . . . . . . . . . . . . . . The rectangular pulse and its autocorrelation function; a The rectangular pulse and its delayed copy, b the autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of the autocorrelation function (7.30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The rectangular u(t) and triangular ϑ(t) pulses . . . . . . . . . . . . . Normalized autocorrelation function calculated for the pulses shown in Fig. 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of the autocorrelation function (7.43) calculated for a single rectangular radio pulse of ti √ length and amplitude U m (t) = 1/ ti . . . . . . . . . . . . . . . . . . . . . Graphic illustration of an idea of the additive reception . . . . . . . Illustration of how a signal integrator works . . . . . . . . . . . . . . . . The illustration of an idea of the correlation reception . . . . . . . . The signal correlators; a the conventional solution, b the analog signal matched filter, c version of the digital signal matched filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Waveforms of signals s(t), x(t), s(t) · x(t) and of the correlation function ϕs,x (0) that illustrate a principle of operation of a correlator . . . . . . . . . . . . . . . . . . . . The block diagram of a multi-channel correlation receiver . . . . Graphical interpretation of a process described by the convolution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A plot of the output signal obtained in the optimal filtration through a two-port linear network . . . . . . . . . . . . . . . . The filter matched to impulse s(t) . . . . . . . . . . . . . . . . . . . . . . . . Relation between a signal and a pulse response of the filter matched with it; a a plot of the signal, b a plot of the pulse response of the matched filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical interpretation of the in-phase summation of the harmonic components of the output signal at the time moment t0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A rectangular pulse with an amplitude equal to 1 and a duration ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A pulse response of the filter under assumption that this filter is matched to the signal shown in Fig. 8.6 . . . . . . . . . . . . . The filter matched to the signal shown in Fig. 8.6 and its responses on s(t) ≡ δ(t) and s(t) = 1(t) − 1(t − ti ) . . . . . . . . .

99 99 112 112

114 115 116 116

119 120 121 122

122

124 125 128 130 131

132

134 135 136 136

List of Figures

Fig. 8.9

Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 8.13 Fig. 8.14

Fig. 8.15 Fig. 8.16 Fig. 8.17 Fig. 8.18 Fig. 8.19 Fig. 9.1

Fig. 9.2 Fig. 9.3 Fig. 9.4

Fig. 9.5

Fig. 9.6 Fig. 9.7

Fig. 9.8

xxv

Spectrum of the rectangular radio pulse with an amplitude equal to A and a duration ti ; a an amplitude spectrum, b a phase spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A sequence of N video pulses with the same duration τ0 and repetition frequency FP R = 1/TP R . . . . . . . . . . . . . . . . . . . The block diagram of the filter matched to the signal (8.30) . . . Time waveforms of signals at points A, B, C, and D of the filter shown in Fig. 8.11 when s(t) is the input signal . . . The plot of the transmittance |H. (ω)|/N , see (8.38), computed for N = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The signal s(t) and transmittances |H ( f )|, |Hc ( f )| of the filters matched to it; a the plot of the signal s(t), b the transmittance |H ( f )|, c the transmittance |Hc ( f )| of a comb filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of a comb filter whose transmittance |Hc ( f )| is shown in Fig. 8.14c . . . . . . . . . . . . . . . . . . . . . . . . . . . The time sequence of N radio pulses . . . . . . . . . . . . . . . . . . . . . . The radio signal waveforms which correspond to the respective signals shown in Fig. 8.12 . . . . . . . . . . . . . . . . The spectral characteristic of the signal s(t) described by formula (8.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the band comb filter matched to signal s(t) described by formula (8.40) . . . . . . . . . . . . . . . . . . A signal with linear frequency modulation (LFM); a a signal waveform in the time domain, b an instantaneous frequency waveform . . . . . . . . . . . . . . . . . . . A shape of amplitude spectrum of the LFM signal at sufficiently large frequency deviation . . . . . . . . . . . . . . . . . . . Graphical illustration of the LFM signal compression process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The LFM signal after compression and its ambiguity function; a a waveform of the compressed signal in the time domain, b an ambiguity function . . . . . . . . . . . . . . . A wobbulator as a simple source of LFM pulses; a block diagram of the wobbulator integrated with a power amplifier, b a waveform of the wobbulator signal frequency, c a waveform of the voltage supplying the power amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the passive transversal filter . . . . . . . . . . . A piezoelectric filter with interdigital transducers, a a geometric structure of the filter, b a moving impulse of a surface mechanical wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of geometrical structure of the SAW filter used for compression of the LFM signal . . . . . . . . . . . . . . . . . . .

137 139 141 141 142

143 144 145 145 146 146

150 151 152

153

154 154

155 156

xxvi

Fig. 9.9

Fig. 9.10

Fig. 9.11 Fig. 9.12 Fig. 9.13

Fig. 9.14 Fig. 9.15

Fig. 9.16 Fig. 9.17 Fig. 9.18 Fig. 10.1

Fig. 10.2

Fig. 10.3

Fig. 10.4

Fig. 10.5

List of Figures

A signal with the bistate phase modulation, a a code of the modulating signal, b a waveform of the modulated signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of a seven segment signal with the bistate phase modulation, a the waveform of the signal in the time domain, b the code of the modulating (control) signal . . . . . . . . The block diagram of the simple modulator circuit at the output of which the BPSK signal is obtained . . . . . . . . . . The block diagram of the filter matched to the BPSK signal composed of four segments (N = 4) . . . . . . . . . . . . . . . . The functional diagram of the two channel analog–digital converter at the outputs of which the orthogonal digital signals I (n) and Q(n) are obtained . . . . . . . . . . . . . . . . . . . . . . . Discrete values of input signal u c [n], filter response h[n] and output signal y[n] discussed in Example 9.2 . . . . . . . . . . . . An analog signal with limited spectrum and its digital equivalent; a the analog signal x(t), determined in a closed interval [0, T ], b the digital equivalent of x(t) as a segment of the periodic discrete function . . . . . . . . . . . . . . The waveform of the signal under analysis in Example 9.4 . . . . The response y(t) of the matched filter on an input signal similar to that shown in Fig. 9.16 . . . . . . . . . . . . . . . . . . . . . . . . . Discrete values of input signal u c [n], filter response h[n] and output signal y[n] discussed in Example 9.5 . . . . . . . . . . . . An example of classification of radiolocation interferences according to the criterion their source and course in the time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . The terrain obstacle against the background of the directivity characteristics of a radar system antennas; a single beam characteristic, b the characteristic composed of two independent beams . . . . . . . . . . . . . . . . . . . . . Echoes of moving objects; a an echo against the background of reflections from heavy clouds, b the echo imaging after partial eliminating reflections from clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A view of radar panoramic indicators during heavy snowfall; a a radar operating at carrier frequency f = 60 MHz, b the radar operating at carrier frequency f = 3 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radio pulses of radar signals; a the phase coherent pulses, b the phase incoherent pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157

158 158 160

161 165

167 170 170 172

176

176

177

177 179

List of Figures

Fig. 10.6

Fig. 10.7 Fig. 10.8

Fig. 10.9

Fig. 10.10

Fig. 10.11 Fig. 10.12

Fig. 10.13 Fig. 10.14 Fig. 10.15 Fig. 10.16

Fig. 10.17 Fig. 10.18 Fig. 11.1

Fig. 11.2

xxvii

Time courses of radio pulses and their frequency spectra. a A sequence of coherent radio pulses, b a frequency spectrum of the signal shown in Fig. 10.6a, c the frequency spectrum of the sequence of coherent radio pulses changed by the Doppler effect, d the sequence of coherent radio pulses changed by the Doppler effect . . . . . . . Block diagram of two-channel (I /Q) synchronous detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample output signals waveforms for two different Doppler pulsations and a solid constant phase shift; b output signal of the upper channel of the detector shown in Fig. 10.7a; c output signal of the lower channel of the detector shown in Fig. 10.7a. For ω D = 00 both output signals are the same and similar to that shown in Fig. 10.8a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of the relationship (ϑ) between FP(ϑ) R = 1/T P R and f D for which there are first blind speed and second blind speed effects; a and b effect of the first blind speed, c and d effect of the second blind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The single delay line canceller and its FIR filter counterpart; a functional diagram of the single delay line canceller, b functional diagram of the 2-pulse FIR filter . . . . . . | | The waveform of | H (1) ( f )| described by formula (10.18) . . . . The double delay line canceller and its FIR filter counterpart; a functional diagram of the double delay line canceller, b functional diagram of the 3-pulse FIR filter . . . . . . | | The waveform of | H (2) ( f )| described by formula (10.19) . . . . Graphical illustration of the process of filtering signal components in the frequency domain when ω D = 0 . . . . . . . . . Graphical illustration of the process of filtering signal components in the frequency domain when ω D /= 0 . . . . . . . . . The oppositive extreme values of amplitudes of two consecutive pulses, modulated by the Doppler effect, when f D = FP(ϑ) R /2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of successive sequences of radio pulses with a cyclic change of a repetition frequency . . . . . . . . . . . . . . The block diagram of a radar with MTI filter . . . . . . . . . . . . . . . Cross sections of the directivity characteristics of typical radar antennas; a, c, d elevation cross sections, b azimuth cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The elevation cross section of cosec2 directivity characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 182

183

184

185 186

186 187 187 188

188 189 191

194 194

xxviii

Fig. 11.3

Fig. 11.4 Fig. 11.5 Fig. 11.6 Fig. 11.7 Fig. 11.8 Fig. 11.9 Fig. 11.10 Fig. 11.11 Fig. 11.12 Fig. 11.13

Fig. 11.14 Fig. 11.15 Fig. 11.16 Fig. 11.17 Fig. 11.18

Fig. 11.19 Fig. 11.20 Fig. 11.21 Fig. 11.22 Fig. 11.23 Fig. 11.24 Fig. 11.25

Fig. 11.26

List of Figures

The receiving system with multi-beam antenna; a block diagram of the receiving system, b elevation cross sections of the formed beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of scanning a narrow sector of space . . . . . . . . . . . . . Observation of the terrain using airborne radar with two antennas placed on both sides of the fuselage . . . . . . . . . . . . . . . Positions of the electromagnetic beam at two different times; a initial time t0 = 0, b time t > t0 . . . . . . . . . . . . . . . . . . The echo signal; a signal waveform in the time domain, b course of the angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . A moving antenna as component of an equivalent linear array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of the Strip-map observation method . . . Definition of basic parameters of the Strip -map method . . . . . . Two-dimensional matrix in which samples of the echo signal are collected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conical scanning; a precession movement of the beam, b waveform of the received signal in the time domain . . . . . . . . . Conical scanning with radio pulse signals, a, b the beam axis coincides with the direction to the object (target), c, d the beam axis does not coincides with the direction to the object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A functional structure of the anti-aircraft defense missile systems with conical scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . The missile and its basic flight control blocks . . . . . . . . . . . . . . Graphical illustration of successive phases of missile flight control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the object position versus two perpendicular planes x and y; a front view, b side view . . . . . . . . . . . . . . . . . . Examples of amplitude waveforms of the probe signal and echo signal; a amplitude course of the probe signal, b amplitude course of the echo signal . . . . . . . . . . . . . . . . . . . . . A mechanical gyroscope; a basic components of the mechanical gyroscope, b general view . . . . . . . . . . . . . . . Graphical illustration of the three-point guidance method . . . . . Division of guidance methods into autonomous and non-autonomous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A passive homing system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A semi-active homing system . . . . . . . . . . . . . . . . . . . . . . . . . . . An active homing system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Definition of distances, angles and velocities used to describe the various methods of targeting a missile at an object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positions of a guided missile and a moving object at two different times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 196 197 198 199 201 202 203 205 206

206 207 208 208 209

209 211 211 212 212 213 213

214 215

List of Figures

Fig. 11.27

Fig. 11.28 Fig. 11.29 Fig. 11.30 Fig. 12.1 Fig. 12.2 Fig. 12.3 Fig. 12.4

Fig. 12.5 Fig. 12.6 Fig. 12.7 Fig. 12.8 Fig. 12.9 Fig. 12.10 Fig. 12.11 Fig. 12.12 Fig. 12.13 Fig. 12.14 Fig. 12.15 Fig. 12.16 Fig. 12.17 Fig. 12.18 Fig. 12.19 Fig. 12.20 Fig. 12.21

xxix

Pursuit homing method; a definition of the direction of the missile guidance, b positions of a missile and related to them an object (target) positions . . . . . . . . . . . . . Pursuit homing method with a constant angular lead . . . . . . . . . Definition of angles and velocities used in the method of parallel navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of a homing system operating according to the method of Pure Proportional Navigation . . . . . . . . . . . . . . Pelengation methods used in the radar monopulse systems . . . . The local coordinate system defined in 3D space . . . . . . . . . . . . Directivity patterns (beams) of a monopulse system using the amplitude method of pelengation . . . . . . . . . . . . . . . . . . . . . . Phase method of pelengation. a Directivity patterns (beams) of an antenna system, b reception of the plane electromagnetic wave by individual antennas . . . . . . . . . . . . . . . Definition of the azimuth angle ϕ and elevation angle γ in the assumed local coordinate system (x, y, z) . . . . . . . . . . . . The block diagram of the summation-differential amplitude discriminator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The direction finding characteristics of the discriminator shown in Fig. 12.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the summation-differential phase discriminator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of four receiving antenna beams used in a monopulse radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A distribution of four receiving antennas (beams) in a two-dimensional plane (x, y) . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the signals comparator . . . . . . . . . . . . . . . The block diagram of the attached, three-channel receiver . . . . The block diagram of a typical T/R modulus with a ferrite circulator acting as a duplexer (antenna switch) . . . . . . . . . . . . . A time domain waveform of a keying signal . . . . . . . . . . . . . . . . The block diagram of the analog–digital converter {u(t) → I [m], Q[m]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of a monopulse radar with T/R moduli and analog–digital converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . A linear antenna array with an analog divider/combiner of microwave signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The normalized group directivity pattern F(θ ) . . . . . . . . . . . . . . Reception of a flat electromagnetic wave by individual elements of a linear antenna array . . . . . . . . . . . . . . . . . . . . . . . . A linear antenna array with an analog divider/combiner of microwave signal and adjustable phase shifters . . . . . . . . . . . The block diagram of an antenna with digital forming of the directivity patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216 217 218 221 224 224 225

226 227 229 231 231 234 235 236 237 238 239 240 240 242 243 243 244 245

xxx

Fig. 12.22 Fig. 12.23 Fig. 12.24 Fig. 12.25 Fig. 12.26 Fig. 13.1 Fig. 13.2 Fig. 13.3 Fig. 13.4 Fig. 13.5 Fig. 13.6 Fig. 13.7 Fig. 13.8 Fig. 13.9 Fig. 13.10 Fig. 13.11 Fig. 13.12 Fig. 13.13 Fig. 13.14 Fig. 13.15 Fig. 13.16 Fig. 13.17 Fig. 13.18 Fig. 13.19 Fig. 13.20 Fig. 13.21 Fig. 13.22 Fig. 13.23 Fig. 13.24 Fig. 13.25

List of Figures

The normalized characteristics F(θ ) = f (θ )/ f max calculated for .ψ = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The normalized characteristic F(θ ) = f (θ )/ f max calculated for .ψ = −30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The normalized characteristic F(θ ) = f (θ )/ f max calculated for .ψ = 80◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The calculated characteristics F(θ ) and F. (θ ) . . . . . . . . . . . . . Interpolation of function d F(θ )/dθ with a straight line in the vicinity of its zero value . . . . . . . . . . . . . . . . . . . . . . . . . . . Geographic coordinates; equatorial plane (a) and meridian plane (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of course angle and bearing angle in 2D space . . . . . A navigation scenario in which angles β1 , β2 and the base b are known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A navigation scenario in which radii r1 , r2 and base b are known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A navigation scenario in which angles β1 , β2 and the base b are known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric interpretation of the orthodromic distance . . . . . . . . Geometric interpretation of the loxodrome line . . . . . . . . . . . . . Circles as position lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ellipses as position lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Branches of a hyperbola as position lines . . . . . . . . . . . . . . . . . . A conical surface with the same value of the Doppler frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A loop antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A multi-turn loop antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directivity diagram of a loop antenna . . . . . . . . . . . . . . . . . . . . . A shielded loop antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An antenna system and its directivity characteristics . . . . . . . . . A magnetic (ferrite) antenna: a outline of the structure, b method of inclusion in the input circuit of the receiver . . . . . . . The principle of operation of Doppler direction finder . . . . . . . . An angular Doppler frequency caused by the antenna orbiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Doppler direction finder with electronically commutated circular array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An angular Doppler frequency of signal at the output of the commutated antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A linear array with the .−. element . . . . . . . . . . . . . . . . . . . . . Uneven distribution of in-phase currents in individual elements of a linear antenna array . . . . . . . . . . . . . . . . . . . . . . . . Summation and differential directivity characteristics of a linear antenna array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The electrical scheme of the Wullenweber array antenna . . . . .

248 248 248 251 253 258 259 260 261 261 262 263 263 264 264 265 267 268 269 270 271 271 272 273 274 275 275 276 276 277

List of Figures

Fig. 13.26

Fig. 13.27 Fig. 13.28 Fig. 13.29 Fig. 14.1 Fig. 14.2

Fig. 14.3 Fig. 14.4 Fig. 14.5 Fig. 14.6 Fig. 14.7 Fig. 14.8 Fig. 14.9 Fig. 14.10 Fig. 15.1 Fig. 15.2 Fig. 15.3 Fig. 15.4 Fig. 15.5 Fig. 15.6 Fig. 15.7 Fig. 15.8 Fig. 15.9 Fig. 15.10 Fig. 15.11 Fig. 15.12

xxxi

The directivity characteristics of the circular array shown in Fig. 13.25; a summation characteristic, b differential characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The general view of Wullenweber direction finder . . . . . . . . . . . A functional block diagram of a radiocompass receiver . . . . . . Automatic direction finding by modulation of output signal . . . Branches of a hyperbola as position lines . . . . . . . . . . . . . . . . . . Examples of the spatial arrangement of radio stations in hyperbolic navigation systems a a chain with 4 stations, b a chain with 3 stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A typical Loran C chain diagram . . . . . . . . . . . . . . . . . . . . . . . . . The time sequence of signals in the chain of the Loran C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A shape of a radio pulse (signal) of the Loran C system . . . . . . A coverage area for the USA/9960 chain of the Loran C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A multipath transmission (propagation) due to reflections from the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions for distances and angles for the radionavigation systems Decca Navigator and Omega . . . . . . . . . . . . . . . . . . . . . A typical spatial configuration of the Decca Navigator chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geographical positions of radio stations of the Omega system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An image of the Sputnik satellite launched by the USSR . . . . . Definition of a radial velocity ϑr . . . . . . . . . . . . . . . . . . . . . . . . . Doppler shift of the frequency of the satellite signal received at point P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A conical surface with the same Doppler frequency . . . . . . . . . TRANSIT satellites; a Transit satellite b Nova satellite . . . . . . . A graphical illustration of satellite orbits of the TRANSIT system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The simplified functional diagram of a satellite’s transceiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signals with the phase modulation, bit (1) and bit (0), used to transfer data from the satellites . . . . . . . . . . . . . . . . . . . . The data format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The satellite t1(S) and terrestrial t (E) time scales shifted by an interval δt (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A flight of satellite from point S1 to point S2 , in time of t1(S) − t2(S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A typical waveform of the Doppler frequency signal u D (t (E) ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

278 278 279 280 284

284 285 286 287 290 291 291 293 296 300 300 301 302 304 305 305 306 307 308 308 309

xxxii

Fig. 15.13 Fig. 15.14

Fig. 15.15 Fig. 16.1 Fig. 16.2 Fig. 16.3 Fig. 16.4 Fig. 16.5 Fig. 16.6 Fig. 16.7 Fig. 16.8 Fig. 16.9

Fig. 16.10 Fig. 16.11 Fig. 16.12 Fig. 16.13 Fig. 16.14

Fig. 16.15

Fig. 16.16 Fig. 16.17 Fig. 16.18 Fig. 16.19 Fig. 16.20 Fig. 16.21 Fig. 16.22

List of Figures

The terrestrial and space functional segments of the TRANSIT system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial positions of the Transit satelline and related to them position lines, a three positions of the TRANSIT satellite estimated in time intervals (tk−1 − tk ) and (tk − tk+1 ), b the intersection point of the determined position lines, N k and N k+1 , defining the receiver’s position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A general view of AN/WRN–5 receiver used in military applications of TRANST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The space segment of GPS; a 6 almost circular orbits, b orbits of GPS versus BeiDou, GALILEO and GLONASS . . . A general image of satellites of different generations . . . . . . . . Comparing the G i (t) code with its replica generated by a GPS receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A typical plot of correlation function Rk (t − τ )/Rkmax calculated for the G i (t) code and its replica . . . . . . . . . . . . . . . . The time scales of user receiver t, GPS system T (G P S) and ith satellite T (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A distance to the ith satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . A coordinate system used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of position in 3D space by ranging to four satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The space segment of the own navigational systems, such as GLONASS; a three orbits of satellities, b a view of the satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The coverage area of GLONASS . . . . . . . . . . . . . . . . . . . . . . . . . The space segment of the GALILEO; a three orbits of satellities, b a view of the satellite . . . . . . . . . . . . . . . . . . . . . . Frequency bands used in the GALILEO system . . . . . . . . . . . . . The BeiDou Satellite Navigation System; a the orbits of satellities, b a view of the satellite . . . . . . . . . . . . . . . . . . . . . . The Japanese Regional Navigation Satellite System (QZSS); a three geosynchronous orbits, b The terrestrial tracks of QZSS satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different version of the satellite navigation system. a A scheme of the organizational structure, b a view of the reference station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the SBAS system . . . . . . . . . . . . . . . . . . . Coverage areas of SBAS systems used currently . . . . . . . . . . . . EGNOS terrestrial segment components . . . . . . . . . . . . . . . . . . . Terrestial and space components of WAAS . . . . . . . . . . . . . . . . Components of the terrestial segment of MSAS . . . . . . . . . . . . . Components of the GAGAN ground-based segment . . . . . . . . . Monitoring stations of the terrestial segment of BDSBAS . . . . .

311

312 313 316 316 320 321 321 323 327 327

330 330 333 333 336

338

340 343 343 345 347 348 349 350

List of Figures

Fig. 17.1 Fig. 17.2

Fig. 17.3 Fig. 17.4 Fig. 17.5 Fig. 17.6 Fig. 17.7 Fig. 17.8

Fig. 17.9 Fig. 17.10

Fig. 17.11

Fig. 17.12 Fig. 17.13 Fig. 17.14 Fig. 17.15 Fig. 17.16

Fig. 17.17 Fig. 17.18

Fig. 17.19 Fig. 17.20

xxxiii

Definition of landing, touchdown and approach zones around a runway; a top view, b side view . . . . . . . . . . . . . . . . . . Definitions of the CL—course line, GP—glide path, FP—flight path, .α—course deviation (azimuth) and .β—elevation deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antennas and distance marker beacons of ILS around a runway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic beams shaped by localizer array antennas A1 and A2 ; a top view, b side view . . . . . . . . . . . . . . . The amplitude spectra of received signals (17.1) and (17.2) . . . The block diagram of an on-board apparatus of the ILS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spectra of signals emitted by the particular electromagnetic beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spectra of signals; a described by (17.12), b described by (17.13), c simultaneously received by the airplane on-board receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The electromagnetic beams of the ILS system . . . . . . . . . . . . . . Elevation cross-sections of various electromagnetic beams formed by antennas A3 and A4 and spectra of the signals emitted by them; a beams of a simplest antenna solution, b beams of the antenna solution more useful for practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphic illustration of the reflection from an to terrain obstacle; a a side view, b the waviness of a glide path due to terrain reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The view of typical ILS antennas; a localizer array antenna, b glide path array antenna . . . . . . . . . . . . . . . . . . . . . . . Fundamental functional devices of the MLS located around a runway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The time structure of the MLS protocol . . . . . . . . . . . . . . . . . . . Illustration of scanning the aerospace area around the airport runway in the azimuth and elevation planes . . . . . . . Illustration of the MLS system operating principle; a scanning in the azimuth plane, b measurement of the time interval ta (.α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of an on-board device determining the deviation .α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the MLS system operating principle; a scanning in the elevation plane, b measurement of the time interval tβ (β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the on-board device determining the angle deviation .β − β0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . The azimuth sections of directivity patterns FA1 and FA2 shaped by the backlight antennas . . . . . . . . . . . . . . . . . . . . . . . . .

356

356 358 360 361 362 364

365 367

368

371 372 374 375 376

377 377

379 379 379

xxxiv

Fig. 17.21 Fig. 17.22 Fig. 17.23

Fig. 17.24 Fig. 17.25 Fig. 17.26

Fig. 17.27

Fig. 18.1 Fig. 18.2

Fig. 18.3 Fig. 18.4 Fig. 18.5 Fig. 18.6

Fig. 18.7 Fig. 18.8

List of Figures

Radio pulses of the backlight emitted by antennas A1 and A2 , see Fig. 7.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphic illustration of the principle of operation of the radio backlight system . . . . . . . . . . . . . . . . . . . . . . . . . . . . A general view of the microwave stations of MLS; a microwave station that scanning in the azimuth plane, b microwave station that scanning in the elevation plane . . . . . . . Fundamental functional devices of the TLS located around a runway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the three antenna interferometer . . . . . . . A view of fundamental components of the TLS. a the antenna system of the interrogator, b antennas of the azimuth interferometer, c the elevation interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local area augmentation system supporting landing of aircrafts; a a scheme of the organization structure, b a view of the reference station integrated with the VHF radio link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration of a slant (oblique) (range R ≡ |P V | between the radio beacon V and the aircraft P . . . . The radionavigation scenario with two radio beacons V1 and V2 . a Definition of oblique ranges and their of perpendicular projections, b definition of angles α1 , α2 , ϕ1 , ϕ2 and ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram illustrating the principle of operation of a radio phase beacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of an on-board receiver receiving the signals emitted by the radio phase beacon . . . . . . . . . . . . . . The diagram illustrating principle of operation of the Doppler navigation beacon . . . . . . . . . . . . . . . . . . . . . . . . Diagrams illustrating sensitivity of VOR and D-VOR one an influence of the interference signal reflected from terrain obstacle; a definition of angles α, α0 and αr , b diagrams of sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General view of a standard Doppler navigation beacon . . . . . . . Directivity characteristics F1 and F2 of antennas A1 and A2 , respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381 381

382 383 387

388

391 394

394 396 397 398

400 401 402

List of Figures

Fig. 18.9

Fig. 18.10 Fig. 18.11

Fig. 18.12 Fig. 18.13 Fig. 18.14 Fig. 18.15 Fig. 18.16 Fig. A.1

Fig. C.1 Fig. D.1 Fig. D.2

Fig. D.3 Fig. D.4 Fig. E.1 Fig. E.2

Fig. E.3 Fig. F.1 Fig. G.1

xxxv

Signals characterizing principle of operation of the impulse navigation beacon; a the first train of 35 identical pulses u 35 (t) emitted in time T by the omnidirectional antenna A2 , b the second train of 36 identical pulses u 36 (t) emitted in time T by the omidirectional antenna A2 , c the envelope of a signal received by on-board receiver from antenna A1 , d the negative pulse generated by “Flip-flop” of the electronic clock whose block diagram is shown in Fig. 18.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the electronic clock used to measure the time interval equal to (tα' − t0 ) = (tα − .tc ) − t0 . . . . . . . . a An influence of the interference voltage on the time being measured; a an aircraft position P, b illustration how the interference voltage u z increases the measured interval of time (tα − t0 ) by adding .tz . . . . . . . . . . . . . . . . . . . The terrestrial (transponder) and on-board (interrogator) segments of DME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The envelope of a signal transmitted towards a ground transponder by an interrogator . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of the time intervals tmax and .t . . . . . . . . . . . . . . . . Definition of the maximum distance Dmax on the ground surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphic illustration of the idea of synchronous receive . . . . . . . Plots of functions ψ r (t) and u out (t) illustrating principle of operation of the altimeter when a frequency of the carrier signal changes sinusoidally . . . . . . . . . . . . . . . . . . Attenuation caused by water vapor and oxygen . . . . . . . . . . . . . An envelope of the pulse radio signals used in the mode 3/A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The narrow interrogator beam and independently emitted wide masking beam with envelopes of radio pulses emitted by them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The characteristic (complete) code of the mode 3/A . . . . . . . . . The code representing the number 4317 . . . . . . . . . . . . . . . . . . . Graphic illustration of the ambiguity function described by formula (E.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross-sections of the ambiguity function that is presented in Fig. E.1. a cross-section at FD = 0, b cross-section at τ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal cross-sections of the ambiguity function that is presented in Fig. E.1 for various values of parameter q . . . . . . . Graphic illustration of the ambiguity function described by formula (F.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the Costas loop . . . . . . . . . . . . . . . . . . . . .

403 404

405 406 407 407 410 410

415 420 423

423 424 424 427

427 428 431 433

xxxvi

Fig. G.2 Fig. H.1 Fig. I.1 Fig. I.2 Fig. J.1 Fig. J.2

List of Figures

The signal obtained at the output of the Costas loop . . . . . . . . . The crossed-loop antenna with a goniometer . . . . . . . . . . . . . . . Illustration of the TRANSIT satellite flight in the area of radio visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The characteristic planes, ellipses, and poles of the system WGS 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the carrier recovery system from the N-state PSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the carrier recovery system from the BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

434 439 444 444 445 446

List of Tables

Table 1.1 Table 2.1 Table 2.2 Table 2.3

Table 2.4

Table 3.1 Table 5.1 Table 5.2 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 7.1 Table 8.1 Table 9.1 Table 9.2

Radio and Microwave bands of the electromagnetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation results of three positions of the object O . . . . . . . . The results obtained in several successive iterations of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object coordinates and number of iterations of calculations performed for different minimum values of |F(x, y, z)| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object coordinates and number of iterations of calculations performed for different values of a threshold F p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical experimentally evaluated mean values of σ M , m 2 for various types of aircrafts and ships . . . . . . . . . . . . . . . . . . . . Sample calculation results, performed according to the formulae (5.1)–(5.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √ Sample values of the function A B (b/ χ )/A M calculated √ for 0 ≤ b ≤ 3 χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample values of function E u(k) /E calculated for 1 ≤ k ≤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The consecutive partial input signals and corresponding output signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Barker codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Several exemplary of P S L-2 codes . . . . . . . . . . . . . . . . . . . . . .

4 33 39

40

42 56 81 85 102 102 102 103 104 104 105 113 128 159 161

xxxvii

xxxviii

Table 9.3

Table 10.1

Table 10.2

Table 11.1 Table 12.1 Table 12.2 Table 13.1 Table 14.1 Table 14.2 Table 14.3 Table 14.4 Table 14.5

Table 16.1 Table 16.2

Table 16.3

Table 16.4 Table 16.5 Table 17.1 Table 17.2 Table 17.3

List of Tables

Values of the spectral lines coefficients, of the input signal u c (n), filter pulse response h(n) and the output signal y(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The differences between the parameters of the probing signals and corresponding parameters of signals reflected from a moving object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of three consecutive blind speeds which were calculated for radars operating in different frequency ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The characteristic values of the angular frequency calculated according to the formula (11.10) . . . . . . . . . . . . . . . . The parameters of the linear array antenna determined in Example 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The calculation results that have been made in the close vicinity of θi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The characteristic values of .t, ω D2 (t) = (ω0 r ./c) sin(.t − θ0 ) and ϑr A . . . . . . . . . . . . . . . . . The courses of codes of the additional bistate (0, π ) phase modulation of transmitted signals . . . . . . . . . . . . . . . . . . . Names and geographic locations of the radio stations of the USA 9960 LORAN–C chain . . . . . . . . . . . . . . . . . . . . . . . Names stations and operation frequencies of the Decca Navigator system chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Names and geographical coordinates of 8 phase-synchronized radio stations of the Omega system . . . . . . The format that determines the emission of signals, in the time-domain, by individual radio stations of the Omega system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The main components and their locations of the GPS ground segment of control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The coordinates (xi , yi , z i ) of satellites and delay times corresponding to the measured pseudo-ranges di∗ = c · [t O(i ) − (TN(i) + δTS(i ) )] . . . . . . . . . . . . . . . . . . . . . . . . . . . The coordinates (xi , yi , z i ) of satellites and times t O(i) − (TN(i) + δTS(i) ) determined with two different precission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic parametres of 6 navigation signals used in the QZSS . . . Examples of systems augmenting the satellite navigation . . . . . The basic requirements imposed on airports of particular categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The center frequencies, f 0l and f 0g , of individual ILS channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The light and acoustic signals characterizing the individual distance markers . . . . . . . . . . . . . . . . . . . . . . . . . .

171

180

189 199 246 252 273 287 290 293 295

296 319

326

328 338 344 356 358 359

List of Tables

Table 18.1 Table D.1 Table I.1

xxxix

The values of basic parameters characterizing the X and Y operating modes of the DME . . . . . . . . . . . . . . . . . . . . . . Exemplary height values H and their corresponding ABCD codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primary parameters of the rotating ellipsoids WGS 84 and PZ-90.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409 422 443

Part I

Basics Principles of the Radiolocation

Chapter 1

Radiolocation and Its Basic Principles

The essence of radiolocation as a research field within telecommunications is explained by its name derived from the English words “radio” and “location”, [3, 4, 15, 18]. In the simplest terms, it can be defined as a research field dealing with the detection and location of objects distant in space, using radio waves. The word “radar” is closely associated with radiolocation, referring to most of the devices used to detect objects and determine their location. In the literature, it is often interpreted as an acronym for “radio detecting and ranging”. Information regarding the position of an object is useful only when the frame of reference, in which it was determined, is known. Most frequently the position of an object is defined in the Cartesian coordinate system (x, y, z), whose origin coincides with the receiver antenna phase centre “0” of the used radar, Fig. 1.1, and the axis “x” indicates the north direction (N ). The coordinate values measured by an active monostatic radar system with passive response (Primary Surveillance Radar) are the distance R to the object O, the azimuth angle ϕ and elevation angle γ . The derived quantities with regard to enlisted values are azimuth distance D = R cos(γ ), hight H = R sin(γ ) and coordinates x = R cos(ϕ) cos(γ ), y = R sin(ϕ) cos(γ ) and z = R sin(γ ). The origins of the radar technology date back to the 1930s whereby first radars were derived from radiocommunication systems operating in the HF and VHF bands. A typical example of devices stemming from that early period is the British chain of radar stations located on the North Sea coast. These radar stations emitted radio impulses with frequency of approximately 25 MHz, using antennas very similar to the radio communication antennas used at that time [6]. Undoubtedly, this made such antennas difficult to be detected in the initial phase of World War II. The development of microwave and antenna technology, as well as the theory of signals during the war, resulted in radiolocation becoming a separate, well-established field of technology. The second period of very dynamic radiolocation development took place during the last three decades and was stimulated by revolutionary changes in military aviation, rocketry and satellite technology. This fast progress was possible thanks to the achievements of modern microelectronics, instituting the methods of digital echo

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_1

3

4

1 Radiolocation and Its Basic Principles

Fig. 1.1 A local coordinate system used in the radiolocation

Table 1.1 Radio and Microwave bands of the electromagnetic spectrum VHF Range

Frequency [GHz]

Wavelength [mm]

Remarks

0.1–0.3

3000–1000

Meter waves Centimeter waves

Microwave bands L

1–2

300–150

S

2–4

150–75

C

4–8

75–37.5 37.5–24

X

8–12.5

Ku

12.5–18

24–16.7

K

18–26.5

16.7–11.3

Ka

26.5–40

11.3–7.5

V

40–75

7.5–4

W

75–110

4–2.7

Millimeter waves

signal processing and vast implementation of antenna systems in terms of multielement active arrays. In the latest generation of reconfigurable (multifunctional) radars, each transceiver of an antenna array is directly connected to a Transmit/ Receive module. When these modules are appropriately (software) controlled, the required directivity patterns, both transmitting and receiving, quickly moving in three-dimensional (3D) space, are obtained. Modern radars use complex signals with carrier frequencies lying in different bands of the VHF and microwave ranges. The list of frequency bands mostly used in these radars is presented in Table 1.1. Thus, one can conclude that modern radiolocation is a very broad interdisciplinary field of technology that, according to the author’s beliefs, cannot be comprehensively presented within one book. This introductory chapter presents only several standard types of radiolocation and discusses the principles of their operation. A brief description of the radiolocation methods presented in Fig. 1.2 is given in the following subsection.

1.1 Standard Radar Systems

5

Fig. 1.2 Various types of the radiolocation

1.1 Standard Radar Systems When discussing the radiolocation, it is essential to explain at the very beginning the meaning of the following basic terms: • • • •

active radar with passive response (Primary Surveillance Radar) active radar with active response (Secondary Surveillance Radar) and radioelectronic reconnaissance Passive Coherent Location (PCL).

In the initial phase of radiolocation development, the only primary radar system used in practice was a bistatic system, whose schematic diagram is illustrated in Fig. 1.3. The required separation between the transmitting and receiving trajectories in this system was achieved by placing the transmitting and receiving antennas at a sufficient distance, which undoubtedly complicated the structure of the system and made the omnidirectional observation more difficult. Further information regarding this topic is presented in Chap. 5. For these reasons, the bistatic system (Fig. 1.3) did not play a significant role in practice and was almost forgotten. In the last decade, however, it has become the subject of the renewed interest, due to the unique reflective properties of some objects, which resulted in creating a special version of the bistatic radar known as Forward Scattering Radiolocation [5, 7, 22].

Fig. 1.3 The block diagram of a bistatic radiolocation system

6

1 Radiolocation and Its Basic Principles

Fig. 1.4 The block diagram of a monostatic radiolocation using a two-channel duplexer

Invention of the duplexer system (T/R antenna switch) just before World War II was a milestone in the development of radiolocation and contributed to the rapid increase in the role of the monostatic radar system, with the configuration presented in Fig. 1.4. System presented in Fig. 1.4 is dominant to this day and works most frequently with time split, i.e. the same antenna connected to the transmitter and receiver through the duplexer D is used for transmitting and receiving. Due to this fact, the omnidirectional observation of three-dimensional space (3D) has become possible and can be easily accomplished by the mechanical antenna rotation in the azimuth plane. It is not difficult to imagine a primary radar system with only one transmitter and several receivers deployed in space, thus combining the advantages of monostatic and bistatic systems. In the literature, systems using such configuration are referred to as multistatic systems, [5, 7, 22]. The principles of a multistatic system with four receivers are described in Chap. 6 of this book. The secondary radar presented in Fig. 1.2 is in fact an active radar with an active response. Figure 1.5 depicts the principle of operation of this type of system. The components of the secondary radar are an interrogator and a transponder located on board of the identified object, e.g. an airplane. The interrogator (transmitter) sends a pulse interrogator signal with carrier frequency f 1 and a suitable structure (see Appendix D) towards the object. Depending on the interrogator signal structure, the transponder automatically sends the corresponding response signal to the interrogator with a carrier frequency f 2 which is near to f 1 . The response signal contains mostly data regarding the “nationality” of the object and its current position. A classic example of the discussed system is the Identification Friend Foe (IFF) system, widely used in military aviation, [3, 4, 21]. Systems operating similarly to secondary radar are extensively used in satellite technology as well, mainly for communication with artificial Earth satellites.

Fig. 1.5 The block diagram of a secondary radiolocation

1.2 Physical Phenomena Used in Modern Radiolocation

7

Radioelectronic reconnaissance, which is a type of passive radar, is based on the signal registration (electromagnetic waves) produced by various means and waves communication accompanied by multifarious physical phenomena. At this point, it is impossible to avoid mentioning the registration of waves, accompanying lightning discharges, by the first receivers, such as stormscopes designed by A. S. Popov. Another, more present-day example of radioelectronic reconnaissance is the registration of ballistic missile launch events. Streams of ionized gas ejected by jet engines are a source of a strong electromagnetic wave that propagates over very long distances. The maximum of the power spectral density for such waves corresponds to a frequency of 30 MHz. Recording the time lags for the wave front reaching several listening stations located around the globe, allows to precisely determine the launching ramp location. Currently, the radioelectronic reconnaissance is carried out in a very wide frequency range, e.g. from 300 kHz to 50 GHz and undoubtedly provides very important complement to thermal imaging reconnaissance, which uses the infrared radiation. The last type of radar listed in Fig. 1.2, is the passive (PCL) radar. A system of this type uses “occasional” transmitters as a source of an electromagnetic wave irradiating a selected section of (3D) space. The terrestrial digital television transmitters are frequently mentioned as an example of occasional transmitters. Due to the use of occasional transmitters, passive radar systems are relatively cheap and, most importantly, difficult to detect and destroy.

1.2 Physical Phenomena Used in Modern Radiolocation As mentioned before, the main tasks of the abovementioned radiolocation systems include the objects detection, determining their position and motion parameters and last but not least, identification of their nature and affiliation. In recent decades, a new important task has emerged, which consist in precise guidance of countermeasures, e.g. various types of missiles, at hostile objects. In order to accomplish these tasks, various systems are applied that take advantage of physical phenomena related to electromagnetic waves propagation and their “reflection” in the broad sense. According to author’s opinion, gaining an insight into the physical essence of these phenomena will facilitate the identification of potential possibilities and limitations of individual systems in the implementation of the abovementioned tasks. More specifically, the abovementioned types of radiolocation take advantage of the following physical phenomena: • • • •

propagation of TEM waves in free space along straight lines, preserving wave polarisation when propagating in free space, an aperiodic reflection from a homogenous, flat conductive surface, a resonant reflection of an electromagnetic wave from some fragments of an object with complex shape, • wave scattering by large, complex objects (e.g. ships),

8

1 Radiolocation and Its Basic Principles

Fig. 1.6 The spatial distribution of an electric field of TEM wave

• refraction of an electromagnetic wave occurring in a composite troposphere (formerly used in over-the-horizon radars) and • Doppler effect. Currently, much attention is paid to multi-layer materials that absorb or scatter electromagnetic energy in a wide frequency range. These materials are used to coat objects (mainly airplanes) in order to deteriorate their reflective features, namely reducing the effectiveness of their reflective surface. TEM wave propagates in a free and unlimited space, and is characterized by the mutual orthogonality of E and H vectors (respectively representing the magnetic field and electric field strength), which in turn - are perpendicular to the vector ϑ that indicates the direction of the energy transfer, Fig. 1.6. √ Vectors E and H are related by linear dependence, i.e. |E/H| = μ0 /ε0 = ξ0 , where ξ0 = 120π . is the wave impedance. The surface power density of TEM wave is determined by the absolute value of the Poynting vector defined as S = E × H∗ , [2, 8, 16]. This vector also indicates the direction of the wave energy transfer, which in free and unlimited space propagates along a straight line with the speed of light √ c = 1/ ε0 μ0 ≈ 3 × 108 m/s. TEM wave, which propagates in a free and unlimited space preserves its polarisation, i.e. spatial orientation E and H vectors. A specific case of TEM wave is a plane wave, for which a surface of the same phase is a plane perpendicular to the direction indicated by the Poynting vector. The TEM aperiodic reflection (nonresonant) is best depicted by reflection from a flat, metal surface which geometrical dimensions are much larger than the wavelength. Without restricting the general considerations, let us assume that the electric field vector E of the incident wave is parallel to the conductive surface, Fig. 1.7. The incident wave on this conductive surface induces at points A, A' , A'' , etc., currents oscillating with a frequency equal to the frequency of the incident wave. The wave phases at particular points will differ between each other (due to time delays that result from distance differences encountered by the wave front). These

1.2 Physical Phenomena Used in Modern Radiolocation

9

Fig. 1.7 The back scattering from a flat reflecting surface

currents constitute the source of the secondary wave (reflected), which is also a plane wave oscillating with the same frequency as the incident wave. The reflected wave will propagate into a direction inflected from the normal n, see Fig. 1.7, by an angle equaling the angle of incidence. In other words, the angle of reflection will equal the angle of incidence. In case of a perfectly conductive surface, the amplitudes of the reflected and incident waves are equal. This phenomenon is so-called a total internal reflection, (|.| = 1). In the remaining cases, i.e. when |.| < 1, the reflected wave power is lower than the incident wave power. The electromagnetic diffusive wave scattering occurs when the reflective surface with dimensions much larger than the wavelength, is not flat and is characterized by “undulations” (unevenness) comparable to the wavelength. This phenomenon is possible to be observed mostly in the case of millimetre and centimetre waves. Due to the different (random) orientation of the individual surface elements, the reflected energy propagates along different directions, i.e. it is scattered. For many complex surfaces, e.g. aircraft skins, the diffusive wave scattering is an appropriate description of this process. According to this principle, the directivity patterns of a reflective surface, for any electromagnetic wave’s angle of incidence, adopts the shape of a sphere tangent to this surface. The resonant reflection of an electromagnetic wave occurs when elements with dimensions commensurable to the wavelength λ, are present on the reflective surface, for instance wires with multiple λ/2 lengths. At the same time, the spatial orientation of these elements in relation to the electric field vector E of the incident wave is of a great importance, Fig. 1.8. Assume that the thin rectilinear conductors, presented in Fig. 1.8, are of a length l = λ/2. Moreover, presume that the first one receives an electromagnetic wave with a resonance frequency of f = c/(2l), whose electric field vector E is perpendicular to the side surface. This field generates a charge of a density ρ = ε ∇ ·E on the surface of the conductor, distributed evenly along the entire length l. In such environment, no standing current wave is activated in the conductor, which means that there is no transfer of wave energy to the conductor, which could potentially act as a halfwave antenna. In theory of antennas, this is known as a polarization incompatibility. Otherwise, when the electric field E falls parallel to the surface of the conductor, causes a standing current wave in it, which is the source of a new electromagnetic wave, see Fig. 1.8b. In other words, the conductor is used then as a relay half-wave

10

1 Radiolocation and Its Basic Principles

Fig. 1.8 Graphical illustration of polarization matching the TEM wave to a rectilinear conductor, a the wave energy is not transferred to the conductor, b a part of the wave energy is transferred to the conductor

antenna and the radiated wave is received by the radar station as an echo signal. This signal is strongest when the resonance condition l = λ/2 = c/(2 f ), assumed in the introduction, is fulfilled. The resonant reflection of an electromagnetic wave is frequently used in radiolocation, including the helicopters’ identification. According to Fig. 1.8a, when the electric field is perpendicularly directed, the induced current in the conductor, which consist in the rotor blade, will be minimal, so the secondary wave (reflected) will be minimal. Conversely, with a parallel electric field presented in Fig. 1.8b, the induced current in rotor blade reaches its maximum value and creates a strong secondary wave. Basing on the resonance frequencies, with the strongest echo signal and wave polarization appropriately selected, it is possible to deduce the length of the rotor blades, and thus the type of the detected helicopter. This conclusion is confirmed by numerous observations conducted by radars operating in the VHF, UHF and L ranges. It has also been confirmed that in these frequency ranges, many other aircraft units have similar dimensions to the wavelength. The refraction of an electromagnetic wave can be described as the curvature of its trajectory caused by the center inhomogeneity, which consist in the atmosphere, and especially its lower layer (troposphere) with an average thickness of 10–12 km, dependent on the latitude and season. The influence of refraction on the range of radar station is discussed in the following chapters. In order to explain the essence of Doppler effect, an electromagnetic wave falling on a moving object (Fig. 1.9), should be considered. For instance, the object like an airplane, moves at the velocity ϑ. This value can be divided into two components, i.e. radial ϑr and perpendicular ϑ p , whereby ϑr corresponds to the velocity at which the object, represented by the center of gravity O, moves towards the radar station R. The wavelength corresponding to the frequency f of the wave radiated towards the object (leaving the antenna of the RS transmitter) is equal to λ=

c f

(1.1)

1.2 Physical Phenomena Used in Modern Radiolocation

11

Fig. 1.9 Illustration of a physical essence of the Doppler phenomenon

where c ≈ 3 × 108 m/s. In the case of a stationary object, in one second, a wave consisting of the f periods, will be recorded. The number of periods recorded for an object moving with velocity, ϑr /= 0, towards the radar station, will increase by . f , where .f =

f .l ϑr · 1 = = ϑr · λ λ c

(1.2)

Thus, the total number of wave periods recorded in 1 s for an object moving at radial velocity of ϑr is equal to (

f Ob

ϑr = f + .f = f 1 + c

) (1.3)

So far, the process of electromagnetic wave propagation from the radar station RS to the moving object O has been considered, Fig. 1.9. Now we will consider the trajectory of the reflected wave from the object O to the radar station RS. In this case, the object can be treated as a stationary wave source with the frequency (1.3) irradiating the radar (its receiver) approaching it with radial velocity ϑr . Thus, according to the abovementioned reasoning, the frequency of the wave reaching the receiver of the radar station is equal to ( ) ( ) ) ( ϑr ϑr 2 2ϑr = f 1+ f R = f Ob 1 + ≈ f 1+ c c c

(1.4)

In practice, it is assumed that f R = f (1 + 2ϑr /c) because the component (ϑr /c) 2 fulfils the condition (ϑr /c) 2 TP(2)R + ti , it means that the echo visible other way round. If tdel

14

1 Radiolocation and Its Basic Principles

Fig. 1.12 The pulse of a probing signal and its incorrect (delayed over range) echo

on the monitor will receive unequally two different positions towards the beginning (∗1) (∗2) ' ' = tdel − TP(1)R and tdel = tdel − TP(2)R . This of the time base defined by delays tdel “leaping” echo should be eliminated as a reflection out of the maximum observation range. The task of determining the distance to an object is inextricably connected to the problems of measurement accuracy and distance distinguishability. According to the fundamental dependence R = tdel c/2, the accuracy of delay time measurement has a significant impact on the accuracy of distance measurement, in which the electromagnetic wave relocates from the antenna phase center to the object and back. The problem of distance distinguishability should be comprehended as follows. Figure 1.13 presents an electromagnetic wave beam with angular widths θ A and θ E . Within this beam a radio pulse of duration ti is transmitted. For these parameters the term resolution cell is defined. If two objects are inside this cell, they are not distinguished then, which means they form the so-called group object. The cross-section of this cell (perpendicular towards the direction of wave propagation) alters simultaneously with the distance change to the object R, however, the cell thickness is constant and equals .R = c ti /2. The dependence .R = c ti /2, which describes the theoretical distance-resolution of radar, can be derived as follows. Assume that presented objects O1 and O2 in Fig. 1.14 are inside the electromagnetic beam with angular width θ . The distances of these objects from the antenna of the radar station RS are respectively equal to R = ctdel /2 and R + .R = c(tdel + .t)/2. According to the comparison of these distances, we can conclude that

Fig. 1.13 Graphical illustration of a spatial cell including the pulse probing signal

1.4 Short Range Altimeter as an Example of Radars …

15

Fig. 1.14 An electromagnetic beam covering two reflecting objects

.t =

2.R c

(1.9)

If .t > ti are the echoes of objects O1 and O2 are perceived as separable. In the remaining cases, i.e. when .t = ti or .t < ti these echoes adhere together adequately or mask each other. Taking advantage of the above reasoning, it can be assumed that the radar station provides the distinguishability of two objects at distance, if .t = 2.R/c ≥ ti . Whilst the above dependence is remodified, .R ≥ c ti /2 is obtained and this required argumentation. In the radiolocation system that uses relatively long pulses with internal Linear Frequency Modulation, the theoretical distance-resolution of radar is determined by a spectrum width B of these pulses and equals .R ≥ c /(2B).

1.4 Short Range Altimeter as an Example of Radars Using Frequency Modulation Signals One of the main navigational parameters used in the aircraft landing is flight altitude H , which is the shortest distance between the phase center of the radar altimeter antenna and the ground’s surface. The altitude measurement is performed according to the basic principle of primary radar, which takes the advantage of information contained in the echo signal reflected from the surface of the ground, Fig. 1.15. The transmitter of the radar altimeter generates a microwave signal which is emitted towards the ground through a transmitting antenna. When the signal is reflected from the ground, a part of it reaches the receiver via the receiving antenna.

Fig. 1.15 An aircraft over a flat horizontal plane

16

1 Radiolocation and Its Basic Principles

Fig. 1.16 The block diagram of the on-board altimeter transceiver

Then, a small part of the transmitted signal is fed to the receiver as a reference signal. As a result of combining the received and reference signals, the signal containing a differential frequency component is obtained which is proportional to delay time of the wave in the following pattern: transmitting antenna–ground–receiving antenna. Assuming that r1 = r2 ≈ H , see Fig. 1.15, the time is equal to t H = 2H/c where c ≈ 3 × 108 m/s. In order to measure low altitudes Hmax ≤ 1500 m, the CW / FM altimeters are usually used. Figure 1.16 demonstrates the altimeter’s functional diagram of the indicated type. The generator of the modulating signal u m (t) constitute the first block in the transmission path, frequently with a sawtooth waveform. The mentioned signal controls the microwave generator, F M also known as wobbulator, whose frequency f 1 (t) oscillates with the modulating signal in the range of ( f 0 − . f /2) − ( f 0 + . f /2) where f 0 = 4.3 GHz and . f ≈ 150 MHz, Fig. 1.17. The power of the radiated microwave signal correspond the range 0.1 W [9, 14]. The microwave signal with a constant amplitude and changeable frequency f 1 (t) (Fig. 1.17), is transmitted via directional coupler to antenna A1 that radiates this signal towards the ground. Part of this signal, reflected from the ground’s surface, is received by the antenna A2 and fed to a balanced microwave mixer. Afterwards, the copy of probe signal is also fed to the mentioned microwave mixer. At this point,

Fig. 1.17 Frequencies of the transmitted ( f 1 ) and received ( f 2 ) signals without Doppler effect. The lower figure b shows the the difference | f 1 − f 2 |

1.4 Short Range Altimeter as an Example of Radars …

17

it should be emphasized that antenna A2 should receive the signal only reflected from the ground’s surface and the separation between antennas A1 and A2 should be sufficiently large, for example greater than 80 dB. The easiest way to separate these antennas is to place them at a sufficient distance, no less than 1 m, for example, on each side of the fuselage. To begin with, assume that a plane flying at a constant altitude H , fulfils the condition r1 = r2 ≈ H , which means that the plane’s settling velocity totals υdes ≡ υr = 0. The above assumption, though, does not take the account of frequency change of the reflected signal triggered by the Doppler effect. Due to the delay time t H = 2H/c, the temporary frequency of the reflected signal (determined in the antenna A2 ) changes according to the course f 2 (t) presented in Fig. 1.17a with a dashed line. Signals that are fed to balanced microwave mixer, are processed. As a result, signal with a differential frequency component fr (t) = | f 1 (t) − f 2 (t) | is obtained and its course is presented in Fig. 1.17b. The second signal component, obtained in, with a frequency equaling the sum of the processed signals frequencies (also known as summative component) is filtered out with a low pass filter (LPF) /Amplifier. There is a modulus of the differential frequency f 1 (t) − f 2 (t) in the mentioned frequency fr (t), since a negative value of the difference does not make physical sense. In fact, we deal with a signal frequency fr (t) ≥ 0 which contains some saddle points. The position of these points is determined on the timeline by ti = t0 + i T /2 when fr (ti ) = 0, where i = 1, 2, 3, ..., Figure 1.18 presents the examples of difference signal courses u r (t) in the vicinity of saddle point ti . The difference frequency signal is amplified in a low frequency amplifier W (integrated with LPF) and then limited on both sides by a diode amplitude limiter. Due to this process, the generated rectangular pulses are fed to a digital counter that counts the maximum number of pulses in a one-second time gate, i.e. the maximum difference signal frequency fr max ≡ f H . This frequency contains information regarding the flight altitude H . Taking advantage of Figs. 1.17a, b, the following equation can be formulated: 1 2H fH tH = · = 0.25T 0.25T c 0.5. f

(1.10)

Fig. 1.18 Examples of saddle points of a signal of difference frequency where the difference frequency is equal to zero at the moment ti

18

1 Radiolocation and Its Basic Principles

Correspondingly, the following dependence stems from the above equation H=

cT · fH 4. f

(1.11)

which is correct for t H ≤ T /4. It follows from (1.11) that the maximum value of the measured altitude Hmax ≈ 1500 m corresponds the maximum value of the difference frequency f H max , which determines the required transmitted frequency band of the applied low pass filter (LPF) presented in Fig. 1.16. In case of losing height υdes = υr > 0, the frequency of the reflected signal increases due to the Doppler effect. The Doppler frequency is added to its frequency change caused by the delay time t H = 2H/c. Thus, the resultant, referred to as the instantaneous signal frequency f 2 reaching the receiving antenna, has a course which is near to the one presented in Fig. 1.19 with a dashed line. According to Fig. 1.19, the frequency of the transmitted signal f 1 (t) is described by the dependence | | f 0 + αt | )] ( f 1 (t) = | [ | f 0 + .2f − α t − T4

for 0 ≤ t ≤ T /4 for t H + T /4 ≤ t ≤ t H + T /2

(1.12)

where α = 2. f /T . It is possible to similarly inscribe the frequency of the echo signal changed due to delay time t H = 2H (t)/c and Doppler effect, i.e. f 2(D) (t)

| ) ( | | [ f 0 + α(t − t H )] · 1 + 2ϑc r = || [ )] ( ( .f | f 0 + 2 − α t − t H − T4 · 1 +

2ϑr c

)

for t H ≤ t ≤ t H + T /4 for t H + T /4 ≤ t ≤ t H + T /2

(1.13) In the range t H ≤ t ≤ t H + T /4 the difference frequency fr 1 is equal to

Fig. 1.19 Frequencies of the transmitted ( f 1 ) and received signals. The receive signal ( f 2 ) has been changed by the Doppler effect. The lower figure b shows the difference | f 1 − f 2 |

1.4 Short Range Altimeter as an Example of Radars …

19

) ( 2ϑr fr 1 (t) = f 1 (t) − f 2(D) (t) = f 0 + αt − [ f 0 + α(t − t H )] · 1 + c 4. f H 2ϑr − [ f 0 + α(t − t H )] (1.14) = T c c In the second range i.e. t H + T /4 ≤ t ≤ t H + T /2 ( )] [ .f T − α t − tH − fr 2 = f 2(D) (t) − f 1 (t) = f 0 + 2 4 ) [ )] ( ( 2ϑr .f T − f0 + × 1+ −α t − c 2 4 [ ( )] 2ϑr 4. f H .f T + f0 + − α t − tH − = T c 2 4 c

(1.15)

The sum of difference frequencies (1.14) and (1.15) is [ ] 2ϑr 4t 8H 4. f H + .f 1 − + fr 1 (t) + fr 2 (t) = 2 T c c T Tc

(1.16)

The second component present in the dependency (1.16) can be omitted because it takes very small values due to the condition ϑr fr 1 4 f 0 [1 + . f /(2 f 0 )]

(1.20)

In case of a standard radar altimeter . f /(2 f 0 ) = 150/(2 · 4300) ≈ 0.017441, which argues the acceptability of the approximation used in practice

20

1 Radiolocation and Its Basic Principles

ϑr =

c [ fr 2 (t) − fr 1 (t)] when fr 2 > fr 1 4 f0

(1.21)

Thus, the discussed radar altimeter allows for the simultaneous determination of flight altitude H and settling velocity υdes = υr > 0. Calculations H and υr are performed according to formulae (1.18) and (1.21) with a processor (Fig. 1.16). Appropriate digital codes of these values are sent to the graphic indication system or straightforwardly to a digital display. If the altitude H is lower than a critical value, e.g. Hmin = 200 ft, then an appropriate warning acoustic signal is also generated. The condition fr 2 > fr 1 presented in formula (1.20) is always fulfilled when the landing is correctly performed, namely when υdes = υr ≤ 2.3 m/s. However, one can imagine the situation in which f D ≈ f 0 2ϑr /c ≥ f H . In this hypothetical situation, which is possible to occur only at very high settling velocity, the following approximate dependences can be used fr 1 = f D − f H , f r 2 = f H + f D which indicate that H=

cT cT fH = ( f r 2 − fr 1 ) 4. f 8. f

υr ≡ υdes =

c ( f r 2 + fr 1 ) 4 f0

(1.22) (1.23)

In the case of using a CW/FM radar altimeter with standard parameters f 0 = 4.3 GHz, . f ≈ 150 MHz and T = 6 ms, such situation could occur while measuring the altitude H = 60 m, if the settling (descent) velocity υdes was greater than 2511.628 km/godz, i.e. during the crash. The CW/FM radar altimeter works similarly to the one described above. The corresponding description is given in Appendix A. The main purpose of the considerations presented above was to explain the principle of operation of the altimeter which uses two mutually separated antennas. Due to this solution, there is no so-called dead zone in the range of altitude measurement 0 − Hmax , which is very important for the landing process. The dead zone does not play very significant role in case of high altitudes measurements, which are most often performed with pulse radar altimeters with one, switched antenna, [13, 14]. So far, it has been tacitly assumed that the incident and reflected waves propagate along single radiuses r1 and r 2 which are perpendicular to the ground’s surface and its reflection is point-like (Fig. 1.15). In reality, the directivity pattern of both antennas have a conical shape with an angular width θ3 d B = 40°–50°. This means that the transmitting antenna irradiates a relatively large fragment of the ground’s surface, whose individual and elementary sections are placed at different distances from the receiving antenna. Moreover, the complex reflectances .i can vary significantly from these elementary sections both, in amplitude and phase. At this point, it should be concluded that in order to limit the influence of reflections from distant elementary sections of the ground, the angular widths θ3 d B of the antenna pattern beams should be reduced. Assuming that the flight of plane takes place exactly in the horizontal plane, the following reasoning is valid. When the antenna beams are too narrow, any banks of the plane will cause noticeable errors in determining the altitude, due to the

1.5 Standard Methods for Determining the Angular …

21

slanting incidence of the dominant part of the wave energy on the ground’s surface. Following the plane bank, not the actual altitude H would be measured though, but the slant distance [31, 56]. The uneven, in amplitude and phase, reflection from a relatively large part of the ground results in accidental distortion of the signal reaching the receiving antenna. The mentioned distortion can be considered as the parasitic modulation of the amplitude and phase. Unfortunately, phase distortions of this signal impact the low-frequency differential signal, used to determine the flight altitude H and settling velocity υdes . The fluctuations of differential frequencies fr 1 (t/T ) and fr 2 (t/T ) presented in Fig. 1.19, are the direct consequences of these distortions’ influence. Due to this situation there is a problem of defining the mentioned frequencies’ estimates that most accurately determine the real, temporary values H and υdes . In modern CW / FM radar altimeters, this problem is solved by using a properly retuned band–pass filter (BPF), instead of a stationary low–pass filter (LPF) with a relatively wide transmitted frequency band 0 − f H max . Both, in literature and further considerations, this filter is referred to as a narrow-band-pass tracking filter in the frequency domain. In fact, it is a multifunctional, complex electronic system with a transmitted frequency band 2. f S , approximately equal to the spectrum width of the filtered differential signal. The monitored value is considered as the entire amplitude spectrum of the differential signal, which changes its position on the frequency axis simultaneously with flight altitude H . When the tracking filter is tunned, its transmitted frequency band should exactly match the current spectrum of the filtered differential signal. Further, this spectrum is analyzed and as a result the component with the largest spectral density power is separated. For this purpose, a specialized spectrum analyzer (frequency discriminator) with high resolution is used. As a general rule, there is a selected spectrum component in its initial range and is related to the reflection of the wave from the elementary sections of the ground’s surface, closest to the receiving antenna. The spectral density power of this component is the largest due to the shortest distances, constituting the best approximation of the determined altitude H , which undoubtedly facilitates its isolation. Presented in Fig. 1.16, both, the tracking filter and the frequency discriminator are controlled by a multitasking processor. Due to this solution, the features of CW / FM radar altimeter include the increased accuracy in determining the altitude and, undoubtedly, the greater resistance to various spurious responses.

1.5 Standard Methods for Determining the Angular Coordinates of Objects The determination of the object’s directional coordinates (of the following angles: azimuth ϕ and elevation γ , Fig. 1.1) is in general indirect, which means that the direction of radiating the probe signal by the antenna as well as receiving the echo signal, is determined. It is obvious that in order to determine this direction precisely, the main beam of the antenna’s directivity pattern should be possibly the narrowest.

22

1 Radiolocation and Its Basic Principles

Unfortunately, this situation is not easy to be achieved due to specific requirements, namely, the ratio of the antenna aperture dimensions should be sufficiently large to the wavelength, [2, 8, 16]. Moreover, the axis of this beam should match the designated direction, which occurs when the value of the received signal reaches maximum. The fulfillment of this condition is ensured when the measurements are carried out in the manner illustrated in Fig. 1.20. Assume that the received signal (proportional to | f [θ (t)]|) exceeds the threshold voltage UT hr for t p1 ≤ t ≤ t p2 . Then, within the same timeslot, the “Pulse generator” system generates N = entir e [(t p2 − t p1 )/TG ] of the identical pulses with a constant interval TG . These pulses with their copies, delayed by TG , are sent to inputs of XOR logic gates. At the outputs of these gates, signals u 3 (t) and u 4 (t) are obtained. They determine, respectively, the beginning t p = t p1 and the end tk = t p1 + TG N of the measured interval [1]. The approximate dependence t0 = (t p + t k )/2 = t p1 + TG N /2 determines the moment when the axis of antenna pattern | f [θ (t)]| matches the designated direction. The moment t0 corresponds to the angular position of the beam

Fig. 1.20 Illustration of measuring the angular coordinate of an object using the “on maximum” method; a block diagram of the measuring system, b time waveforms of signals u 1 (t), u 2 (t), u 3 (t) and u 4 (t)

1.5 Standard Methods for Determining the Angular …

23

Fig. 1.21 Directivity characteristics of an antenna; a summation characteristic, b differential characteristic

θ0 ≡ θ (t0 ) = . t0 = θ (t p1 ) + .TG N /2,

(1.24)

where θ (t p1 ) = . t p1 refers to the position of the beam designated at the | f [θ (t)]|. of displacement initial moment t p and . refers to the angular velocity | | Figure 1.20b and dependence (1.24) highlight that |tk − t p2 | < TG and the error of the angle θ0 does not exceed .TG /2. For instance, for . = 6 · (2π ) ≈ 37.7 rad/s and TG = 10−5 s this error indicates less than 0.000377 rad. The linear displacement corresponding to this angular error is of the range 19 m when the object is located 100 km in distance. In radar devices that use multi-element antenna arrays, the designation of the direction is somewhat more realistic to succeed due to the possibility of simultaneous shaping of two directivity patterns, i.e. sum f . (θ ) and difference f . (θ ), on the receiving side, [11, 17]. The typical cross-sections of both mentioned characteristics are presented in Fig. 1.21. In order to designate the direction, from which the reflected wave is received, a difference characteristic f . (θ ) is particularly useful. It takes a zero value for θ = 0 and features a very sheer slopes (large angle of arrival) in close vicinity θ = 0, i.e. around the normal direction to the antenna aperture. In this direction f . (θ ) reaches a global maximum value. The signal u . (θ ) related to f . (θ ) is used in the angle discriminator as a reference signal. In order to explicitly determine the direction of > 0 and reception, it is necessary to fulfill the conditions u . (θ → 0) → u (max) . u . (θ → 0) → 0, where u . (θ ) is the signal related to f . (θ ). These conditions must be met simultaneously, because the difference signal u . (θ ) reaches zero value in the absence of a reflected wave as well. Examples of structural solutions of the antenna arrays that form f . (θ ) and f . (θ ) in the analogue way, are presented in Sect. 13.3. In the presented solutions, a microwave sum-difference element (. − .) is used for this purpose. Another extensively used forming method f . (θ ) is a numerical differentiation f . (θ ) towards the angle θ (Example 12.1). The angular widths, azimuth and elevation, of the summation directivity pattern of the used antenna, have a decisive influence on the so-called angular resolution of

24

1 Radiolocation and Its Basic Principles

Fig. 1.22 Graphic illustration of the angular resolution parameter

the radar system. In order to clarify the essence of this term, a scenario similar to the one presented in Fig. 1.22 should be considered. Two airplanes specified in this figure, located at the same distance R from the antenna, will be perceived separately when the distance S between them will be greater than Sθ = 2R sin(θ/2)

(1.25)

referred to as angular resolution in the angle plane θ . For most antennas, the angular resolution is determined in two mutually perpendicular planes, most often azimuth and elevation. Furthermore, it should also be emphasized that with the parallel flight of these aircrafts, the condition S > Sθ = 2R sin(θ/2) will be fulfilled for R < Rmin =

S 2 sin(θ/2)

(1.26)

According to the dependence (1.26) it is recommended that two or more airplanes (a group) should try to fly as close as possible to each other, so that they can be identified as separate objects possibly at the latest moment (the closest to the radar station).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Bakuljev PA (2004) Radiolocation systems (in Russian). “Radio-Engineering”, Moscow Balanis CA (1997) Antenna theory, analysis and design, 2nd edn. Wiley, New York Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood Barton DK (2005) Radar system analysis and modeling. Artech House Inc., Norwood Chernyak VS (1998) Fundamentals of multisite radar systems. Gordon and Breach Science Publishers, Amsterdam Cook CHE, Bernfeld M (1967) Radar signals, an introduction to theory and applications. Academic, New York Hanle E (1986) Survey of bistatic and multistatic radars. IEE Proc 133(7):587–595 Johnson RC (ed) (1993) Antenna engineering handbook, 3rd edn. McGraw-Hill, New York Kayton M, Fried WR (1996) Avionic navigation systems, 2nd edn. Wiley, New York Komarov WM et al (1991) Height and oblique distance radar meters with continuous frequency modulated wave (in Russian). Foreign Radioelectron 12:s.52–s.70 Leonov AI, Fomichev KJ (1986) Monopulse radar. Artech House Inc., Norwood Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken

References

25

13. Maloratsky LG (2002) Single antenna FM radio altimeter operating in a continous mode and in interrupted continous wave mode. US Patent, No. 6.426,117, July 2002 14. Maloratsky LG (2003) An aircraft single—antenna FM radio altimeter. Microwave J 46 15. Peebles PZ Jr (1998) Radar principles. Wiley, New York 16. Rosłoniec S (2006) Fundamentals of the antenna technique (in Polish). Publishing house of the Warsaw University of Technology, Warsaw 17. Sherman SL (1984) Monopulse principles and techniques. Artech House Inc., Norwood 18. Skolnik MI (ed) Radar handbook, 2nd edn. McGraw-Hill, New York 19. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 20. Sosnovskij AA (ed) (1990) Aviation radionavigation (in Russian). “Transport”, Moscow 21. Stevens MC (1988) Secondary surveillance radar. Artech House Inc., Boston 22. Willis NJ (2005) Bistatic radar, 2nd edn. Scitech Publishing Inc., Releigh

Chapter 2

Determining the Object’s Position by Radiolocation Methods

Despite the fact that electromagnetic phenomena have accompanied mankind since the beginning of time, it was not until the second half of the nineteenth century that people became aware of their presence and the theoretical fundamentals of electromagnetism were established. On the basis of the equations formulated in 1861 by James Clerk Maxwell, a second order partial differential equation was derived, describing the propagation of electromagnetic interference in various media, including free space. This equation, also known as the Helmholtz equation, describes, i.a. the propagation of the electromagnetic TEM waves, which were briefly described in the previous chapter. The first physicist who experimentally confirmed the validity of this equation was Heinrich Rudolf Hertz. In Russia, the beginnings of radio engineering have been associated with the works of Alexander Stepanovich Popov, including, among others, so-called stormscopes, i.e. the receivers of electromagnetic waves caused by strong atmospheric discharges which frequently accompany storms. Undoubtedly, the works of Hertz initiated the intense development of radio engineering and radio communication, whose consequences can be observed to this day. Due to the fact that TEM waves propagate along straight lines in free space, they are broadly used in radiolocation and radio navigation. In case of passive radar, the TEM waves enable detection and determination of the position of electromagnetic emission sources. These may include the transmitters of various services’ types or physical phenomena during atmospheric discharges, various explosions or rocket launches. The methods used to achieve this purpose can be divided into two basic groups, i.e. triangulation and time methods. The present chapter describes these two groups with the Direction Of Arrival (DOA) and Time Differences Of Arrival (TDOA) methods, respectively.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_2

27

28

2 Determining the Object’s Position by Radiolocation Methods

2.1 The Direction of Arrival (DOA) Method Assume that the object at point C acts as an electromagnetic wave source reaching two listening stations located at points A and B, Fig. 2.1. The position of this object is determined by the point of intersection of two straight lines, namely: • a straight line intersecting the base AB at point A at the azimuth ϕ1 and • a straight line intersecting the base AB at point B at the azimuth π − ϕ2 . In this case, the angles ϕ1 and ϕ2 are the measurands. Therefore, the method of determining the object’s position, basing on these angles, is called the Direction Of Arrival method (DOA) [2, 8, 9]. Presented in Fig. 2.1. object’s position regarding base AB in two-dimensional (2D) space, is explicitly defined by the segments AC ≡ a and BC ≡ b, which are computed as follows a=c

sin(ϕ1 ) sin(ϕ2 ) b=c sin(ϕ1 + ϕ2 ) sin(ϕ1 + ϕ2 )

(2.1)

Inverse formulas with respect to (2.1) are ] ] [ 2 c + b2 − a 2 a 2 + c2 − b2 , ϕ2 = arccos ϕ1 = arccos 2a c 2b c [

(2.2)

Due to the above formulae it is possible to compute the angles of triangle ϕ1 and ϕ2 , when its sides a, b and c are known. Derivation of formulas (2.1) and (2.2)

Fig. 2.1 Definition of sides and angles of the triangle ABC used in the triangulation method of navigation

2.1 The Direction of Arrival (DOA) Method

29

The first formula (2.1) can be obtained using the following dependencies h = d1 tan(ϕ1 ), h = d2 tan(ϕ2 )

(2.3)

h = a sin(ϕ1 ), h = b sin(ϕ2 )

(2.4)

The below proportion stems from the dependence (2.4) b=a

sin(ϕ1 ) sin(ϕ2 )

(2.5)

According to Fig. 2.1 and dependencies (2.3)–(2.5) h a sin(ϕ1 ) b sin(ϕ2 ) h + = + tan(ϕ1 ) tan(ϕ2 ) tan(ϕ1 ) tan(ϕ2 ) ] [ sin(ϕ 1 ) sin(ϕ2 ) sin(ϕ1 ) cos(ϕ1 ) cos(ϕ2 ) +a = a sin(ϕ1 ) + =a tan(ϕ1 ) sin(ϕ 2 ) tan(ϕ2 ) sin(ϕ 1 ) sin(ϕ 2 ) ] [ sin(ϕ1 + ϕ2 ) cos(ϕ1 ) sin(ϕ 2 ) + cos(ϕ2 ) sin(ϕ 1 ) =a = a sin(ϕ1 ) sin(ϕ 1 ) sin(ϕ 2 ) sin(ϕ2 )

c = d1 + d2 =

(2.6)

The dependence (2.6) can be inscribed in the form a=c

sin(ϕ2 ) sin(ϕ1 + ϕ2 )

which is identical to the first formula (2.1). The validity of the second formula (2.1) is proved analogically. The validity of formulae (2.2) can be shown as follows. According to Fig. 2.1 c = d 1 + d2 = a cos(ϕ1 ) + b cos(ϕ2 )

(2.7)

The dependence (2.5) demonstrates that sin(ϕ1 ) = (b/a) sin(ϕ2 ). Therefore, cos(ϕ1 ) =

/ / 1 − sin2 (ϕ1 ) = 1 − (b2 /a 2 ) sin2 (ϕ 2 )

(2.8)

After considering the dependence (2.8), the expression (2.7) can be inscribed as / a 1 − (b2 /a 2 ) sin2 (ϕ 2 ) = c − b cos(ϕ2 )

(2.9)

When the both sides of Eq. (2.9) are squared, we obtain a 2 − b2 sin2 (ϕ2 ) = c2 − 2b c cos(ϕ2 ) + b2 cos2 (ϕ 2 )

(2.10)

30

2 Determining the Object’s Position by Radiolocation Methods

Fig. 2.2 Illustration of the triangulation method for determining the coordinates of the O object in 3D space 2 2 After ordering, the expression (2.10) the form c2 −a [ receives ] +b = 2b c cos(ϕ2 ) 2 2 2 and directly results in ϕ 2 = arccos (c + b − a )/(2bc) , which in the normal course of events had to be proved. The first formula (2.2) is analogically derived.

The Triangulation Formulas Describing the Position of an Object in Threedimensional (3D) Space Assume that the object located at point O is visible from the point (0, 0, 0), i.e. the origin of the coordinate system (x, y, z), at the angles: azimuth α1 and elevation γ1 (Fig. 2.2). The azimuth α1 and elevation γ1 angles determine the direction of the electromagnetic wave reception, emitted by the identified object. Similarly, the position of this object observed from point A is determined by the azimuth α2 and elevation γ2 angles. According to Fig. 2.2, the following two equations can be formulated correspondingly b = b1 + b2 = r cos(γ1 ) cos(α1 ) + r A cos(γ2 ) cos(180 − α2 ) r cos(γ1 ) sin(α1 ) = r A cos(γ2 ) sin(180 − α2 )

(2.11) (2.12)

Equation (2.12) demonstrates that r A cos(γ2 ) = r

cos(γ1 ) sin(α1 ) sin(180 − α2 )

(2.13)

After considering the dependence (2.13) in the expression (2.11), we receive

2.2 The Time Difference of Arrival (TDOA) Method

r=

b [ ] cos(γ1 ) cos(α1 ) − sin(α1 )ctg(α2 )

31

(2.14)

Thus, the object’s position, in regard to the origin (0, 0, 0) of the assumed coordinate system (x, y, z), is explicitly defined by the angles α1 , γ1 and the position vector r . In case when γ1 → 0, the dependence (2.14) is simplified to the form b cos(α1 ) − sin(α1 )ctg(α2 ) b = cos(α1 ) + sin(α1 )cos(180 − α2 )/sin(180 − α 2 ) sin(180 − α2 ) =b cos(α1 ) sin(180 − α2 ) + sin(α1 )cos(180 − α2 ) sin(180 − α2 ) =b sin(α1 + 180 − α2 ))

r=

which is identical to the first Eq. (2.1) because 180 − α2 = ϕ 2 and b ≡ c. Example 2.1 Determine the position (x, y, z) of the object O in the coordinate system as shown in Fig. 2.2 for the following data: b = 10,3812.330 m, α1 = 0.628088 rad and γ1 = 0.153543 rad. Calculate the altitude of the object in regard to the horizontal plane (x0y). The length of the position vector calculated according to the formula (2.14) is. r=

10,3812.330 ≈ 85,000 m. 0.988235 [0.809152 − 0.587599(−0.726190)]

Therefore, x = r cos(γ1 ) cos(α1 ) = 85,000 · 0.988253 · 0.809152 ≈ 67,968 m y = r cos(γ1 ) sin(α1 ) = 85,000 · 0.988253 · 0.587599 ≈ 4,9358 m z ≡ h = r sin(γ1 ) = 85000 · 0.152941 ≈ 13,000 m

2.2 The Time Difference of Arrival (TDOA) Method In order to clarify the essence of the TDOA method, its variant relating to the navigation function defined in (2D) space, i.e. on the plane, should be considered (Fig. 2.3).

32

2 Determining the Object’s Position by Radiolocation Methods

Fig. 2.3 TDOA method for determining the coordinates of the O object in 2D space

In this case, the differences of signal reception time .t A = t A − t0 and .t B = t B − t0 (emitted by the object O) are used as measuring data by three listening stations located at points A, 0 ≡ (0, 0) and B of the rectilinear base AB. The international SOS signal (which is the abbreviation standing for “Save Our Souls”) given by a sinking individual, constitutes a good example of such signal. Taking the differences .t A and .t B into account, the distance differences .r A = r A − r = c .t A and .r B = r B − r = c .t B are calculated, where c ≈ 2.99713 × 108 m/s corresponds to the electromagnetic wave velocity in free space. The position lines corresponding to the distance differences .r A and .r B in the two-dimensional space, are the branches of two hyperbolas with an object lying at their intersection, [2, 7, 10]. In the calculation process, the distance r and angle α, which are the polar coordinates of the object, are determined. The object’s position is determined by the radius r=

2b2 − (c.t A )2 − (c.t B )2 2(c.t A + c.t B )

(2.15)

and angle [ α = ± arccos

(c.t A )2 − (c.t B )2 + 2r (c.t A − c.t B ) 4br

] (2.16)

In order to decide whether the angle α takes positive or negative value, some additional information about the object’s position is required. Derivation of Formulas (2.15) and (2.16) Formulae r A and r B of the right-angled triangles specified in Fig. 2.3. r A2 = (c.t A + r )2 = [b + r cos(α)] 2 + r 2 sin2 (α)

(2.17)

2.2 The Time Difference of Arrival (TDOA) Method

33

r B2 = (c.t B + r )2 = [b − r cos(α)] 2 + r 2 sin2 (α)

(2.18)

After ordering, Eqs. (2.17) and (2.18) can be inscribed as follows b2 + 2br cos(α) = (c.t A )2 + 2r c.t A

(2.19)

b2 − 2br cos(α) = (c.t B )2 + 2r c.t B

(2.20)

When Eqs. (2.19) and (2.20) are added, the following dependence is obtained 2b2 = (c.t A )2 + (c.t B )2 + 2r (c.t A + c.t B )

(2.21)

which is identical to (2.15). Similarly, in case of subtracting Eq. (2.20) from (2.19), the obtained expression 4br cos(α) = (c.t A )2 − (c.t B )2 + 2r (c.t A − c.t B ) is identical to the dependence (2.16). Example 2.2 Table 2.1 presents the exemplary calculation results of three positions of the object O (Fig. 2.3) which were obtained according to formulae (2.15) and (2.16). The calculations were performed for time differences .t A and .t B , given with different accuracy. Using any of the above methods, DOA or TDOA, the accuracy of determining the object’s position depends on the measuring accuracy of the angles ϕ1 and ϕ2 or time differences .t A and .t A . Thus, assume that the angles ϕ1 = ϕ1 ± .ϕ1 and ϕ2 = ϕ2 ± .ϕ2 were measured with a known accuracy represented by .ϕ 1 and .ϕ2 , see Fig. 2.4(a). In this relatively simple two-dimensional (2D) scenario, the point determining the object’s position O lies within the blur area marked in Fig. 2.4a by hatching. Often this area is called Geometrical Dilution of Precision (GDOP).The size of this area is directly proportional to the angular deviations .ϕ 1 and .ϕ2 , and to the object’s distance from the base line AB. In the case of using the TDOA method, the two hyperbolic trajectories correspond to the time differences .t A = .t A ± δ.t A and .t B = .t B ± δ.t B , whose intersection (the common area) defines the blur area, Fig. 2.4b. The point where the Table 2.1 Calculation results of three positions of the object O b = 30, 000 m, c ≈ 3 × 108 m/s .t A , µs

.t B , µs

r, m

α, rad

90.534793 90.53

−67.480270 −67.48

47,169.905 47,611.636

0.558599 0.557737

41.421356 41.42

41.421356 41.42

30,000.000 30,001.389

1.570796 1.570796

−19.745306 −19.74

85.224590 85.22

28,284.271 28,286.234

2.356194 2.356085

34

2 Determining the Object’s Position by Radiolocation Methods

Fig. 2.4 Geometric interpretation of the blur area (GDOP); a for triangulation method of navigation, b for the TDOA method

object O is located, lies within this area. In this case, the size of the blur area depends mainly on the accuracy (δ.t A ,δ.t B ) of how the time differences .t A and .t B are measured. The task considered in the Example 2.2 can be treated as a special case of the more general task illustrated in Fig. 2.5. This three-dimensional task contains the data with the coordinates (xi , yi , z i ), where i = 0, 1, 2 . . . 3, of the receivers of four listening stations Ri , while the unknowns refer to the coordinates (x, y, z) of the object O emitting or reflecting the radio signal. On the basis of the measured reception time ti of this signal by individual stations, the differences .ti0 = ti − t0 and the corresponding distance differences .di = c(ti −t0 ) are estimated. The determined three distance differences, are related to the searched coordinates (x, y, z) by the following equations: .d 1 (x, y, z) = d1 (x, y, z) − d0 (x, y, z) = c(t1 − t0 )

Fig. 2.5 A passive radiolocation scenario with four receivers

2.2 The Time Difference of Arrival (TDOA) Method

35

.d 2 (x, y, z) = d2 (x, y, z) − d0 (x, y, z) = c(t2 − t0 ) .d 3 (x, y, z) = d3 (x, y, z) − d0 (x, y, z) = c(t3 − t0 )

(2.22)

. where: di (x, y, z) = (xi − x)2 + (yi − y)2 + (z i − z)2 for i = 0, 1, 2 . . . 3. For a fixed value c(t1 − t0 ), the first equation of the system (2.22) describes one of the hyperboloid sheets. The rotation axis of the hyperboloid sheet passes through its focal points lying at the same points as the receivers R0 and R1 . Also, the two remaining equations describe similar hyperboloid sheets whose axes pass through R0 and R2 , along with R0 and R3 , respectively. If the system of Eqs. (2.22) gives real solutions, then the set of points belonging to two intersecting sheets, creates a curve that intersects the third of the sheets at two points. One of these points constitutes the searched solution, i.e. it determines the position of the detected object. The system of Eqs. (2.22) can be solved by various methods, both iterative and the Chan-Ho method, [3, 4], described in Sect. 6.1. For the didactic reasons, the essence of the above-mentioned iterative methods is discussed in this section. The most popular ones include the Least Squares, Newton’s method, Steepest Descent Method and their combinations, often called Hybrid methods or Levenberg–Marquardt algorithms, [5, 7]. Least Squares Method The task solution of determining the position (x, y, z) of the detected object is equivalent to determining the global minimum point of the following function F ≡ F(x, y, z) =

3 .

[.dk (x, y, z) − c.tk0 ] 2

(2.23)

k=1

where .tk0 = tk − t0 and c correspond the speed of light. The expressions .dk (x, y, z) used in (2.23), refer to distance differences which are described by Eqs. (2.22). It is obvious that the function (2.23) takes the minimum value equal to zero at the point (x ∗ , y ∗ , z ∗ ), constituting the searched solution. At this point, the condition of the maxima and minima expressed by the following equations must be satisfied 3 . ∂F ∂ =2 [.dk (x, y, z) − c.tk0 ] [.dk (x, y, z)] = 0 ∂x ∂ x k=1 3 . ∂ ∂F =2 [.dk (x, y, z) − c.tk0 ] [.dk (x, y, z)] = 0 ∂y ∂ y k=1 3 . ∂ ∂F =2 [.dk (x, y, z) − c.tk0 ] [.dk (x, y, z)] = 0 ∂z ∂z k=1

(2.24)

36

2 Determining the Object’s Position by Radiolocation Methods

The partial derivatives occurring in Eqs. (2.24) can be computed by using the formulae ∂ x − xk x − x0 − [.dk (x, y, z)] = ∂x dk (x.y, z) d0 (x, y, z) y − y0 y − yk ∂ − [.dk (x, y, z)] = ∂y dk (x.y, z) d0 (x, y, z) z − z0 z − zk ∂ − [.dk (x, y, z)] = ∂z dk (x.y, z) d0 (x, y, z)

(2.25)

After taking the above dependencies in Eqs. (2.24) into account and performing elementary conversions, the following is obtained ∂F = 12x − Sx ∂x ] 3 [ . (x − x0 )[c.tk0 − dk (x, y, z)] (x − xk )[d0 (x, y, z) + c.t k0 ] − +2 d0 (x, y, z) dk (x.y, z) k=1 =0 ∂F = 12y − S y ∂y ] 3 [ . (y − y0 )[c.tk0 − dk (x, y, z)] (y − yk )[d0 (x, y, z) + c.t k0 ] +2 − d0 (x, y, z) dk (x.y, z) k=1 =0 ∂F = 12z − Sz ∂z ] 3 [ . (z − z 0 )[c.tk0 − dk (x, y, z)] (z − z k )[d0 (x, y, z) + c.t k0 ] − +2 d0 (x, y, z) dk (x.y, z) k=1 =0

(2.26)

where: Sx = 6x0 + 2x1 + 2x2 + 2x3 , S y = 6y0 + 2y1 + 2y2 + 2y3 and Sz = 6z 0 + 2z 1 + 2z 2 + 2z 3 . The system of nonlinear equations (2.26) can be solved iteratively with Newton’s method, described in many books on numerical methods, for instance in [6]. Therefore, assume that the approximation of the object’s position (x (n) , y (n) , z (n) ), determined while solving n−th iteration of solving the system of Eqs. (2.26), is known. In the next iteration, i.e. (n + 1), the system of Eqs. (2.26) is solved again, using the distances d0 (x, y, z) and dk (x, y, z) calculated for (x (n) , y (n) , z (n) ). Once the solution is calculated, the variables’ increments .x,.y and .z, are obtained. When added to the values determined in the previous iteration, these variables give a new (n + 1)— approximation, i.e. x (n+1) = x (n) + .x, y (n+1) = y (n) + .y and z (n+1) = z (n) + .z.

2.2 The Time Difference of Arrival (TDOA) Method

37

If the objective function (2.23) calculated for this approximation takes an acceptably small value, the obtained approximation can be treated sufficiently close in comparison to the searched solution (x ∗ , y ∗ , z ∗ ). The solution convergence condition understood this way, expresses the following inequality |F(x, y, z)| ≤ ε

(2.27)

where ε is a required, sufficiently small positive number. Otherwise, i.e. when |F(x, y, z)| > ε, the calculations continue according to the algorithm described above. Newton’s Method In order to recall the essence of Newton’s method, assume that for x = x (n) + .x, y = y (n) + .y and z = z (n) + .z Eqs. (2.26) are satisfied, which can be presented in the form of expansions into Taylor series around the point (x (n) , y (n) , z (n) ), i.e. ] ∂2 F ∂ [ ∂F =0= F(x (n) , y (n) , z (n) ) + .x ∂x ∂x ∂x2 ∂2 F ∂2 F .y + .z + · · · + ∂ x∂ y ∂ x∂ z ] ∂F ∂2 F ∂ [ F(x (n) , y (n) , z (n) ) + =0= .x ∂y ∂y ∂ y∂ x ∂2 F ∂2 F .z + · · · .y + + ∂ y2 ∂ y∂z ] ∂F ∂ [ ∂2 F =0= F(x (n) , y (n) , z (n) ) + .x ∂z ∂z ∂ z∂ x ∂2 F ∂2 F .y + 2 .z + · · · + ∂z∂ y ∂z

(2.28)

All partial derivatives occurring in the expansion (2.28) are computed at point (x (n) , y (n) , z (n) ). If components with second and higher derivatives are omitted, the system of Eqs. (2.28) can be inscribed in the following matrix form ⎡ ⎢ ⎣

∂2 F ∂2 F ∂ x 2 ∂ x∂ y ∂2 F ∂2 F ∂ y∂ x ∂ y 2 ∂2 F ∂2 F ∂z∂ x ∂ z∂ y

∂2 F ∂ x∂ z ∂2 F ∂ y∂ z ∂2 F ∂ z2

⎡ [ ]⎤ ⎤ − ∂∂x F(x (n) , y (n) , z (n) ) .x ] ⎥⎣ ⎢ ∂[ (n) (n) (n) ⎥ ⎦ .y ⎦ = ⎣ − ∂ y [ F(x , y , z )] ⎦ .z − ∂∂z F(x (n) , y (n) , z (n) ) ⎤⎡

(2.29)

Multiplying both sides of the system of Eqs. (2.29) by the inverse Hessian matrix, the searched variable increments are obtained

38

2 Determining the Object’s Position by Radiolocation Methods

⎤⎡ ⎤ ⎡ ⎤ ⎡ ∂2 F 100 .x .x ∂x2 ∂2 F ⎣ 0 1 0 ⎦⎣ .y ⎦ = ⎣ .y ⎦ = ⎢ ⎣ ∂ y∂ x ∂2 F 001 .z .z ∂ z∂ x ⎡

∂2 F ∂ x∂ y ∂2 F ∂ y2 ∂2 F ∂z∂ y

∂2 F ∂ x∂ z ∂2 F ∂ y∂z ∂2 F ∂ z2

⎤ −1 ⎡ ⎥ ⎦

[ ]⎤ − ∂∂x F(x (n) , y (n) , z (n) ) [ ] ⎢ − ∂ F(x (n) , y (n) , z (n) ) ⎥ ⎣ ∂y [ ]⎦ ∂ F(x (n) , y (n) , z (n) ) − ∂z (2.30)

The elements of the inverse Hessian matrix can be easy computed according to the formulae given in Appendix B. The increase of the variables (2.30) added respectively to x (n) , y (n) i z (n) form the successive (n + 1)—approximation of the searched solution, i.e. ⎤ ⎡ (n) ⎤ ⎡ ∂ 2 F x x (n+1) ∂x2 ∂2 F ⎣ y (n+1) ⎦ = ⎣ y (n) ⎦ − ⎢ ⎣ ∂ y∂ x ∂2 F z (n+1) z (n) ∂z∂ x ⎡

∂2 F ∂ x∂ y ∂2 F ∂ y2 ∂2 F ∂ z∂ y

∂2 F ∂ x∂ z ∂2 F ∂ y∂ z ∂2 F ∂ z2

⎤ −1 ⎡ ⎥ ⎦

⎢ ⎣

]⎤

[

∂ F(x (n) , y (n) , z (n) ) ∂x [ ] ∂ F(x (n) , y (n) , z (n) ) ⎥ ∂y [ ]⎦ ∂ (n) (n) (n) F(x , y , z ) ∂z

(2.31)

By introducing the standard notation, i.e [

∂ F(w) ∂ F(w) ∂ F(w) , , ∂x ∂y ∂z ⎤ ∂ 2 F(w) ∂ 2 F(w)

]T

w = [x, y, z]T , ∇[F(w)] = ⎡ ∂ 2 F(w) ∂x2

∂ x∂ y ∂ x∂ z ∂ 2 F(w) ∂ 2 F(w) 2 ∂y ∂ y∂ z ∂ 2 F(w) ∂ 2 F(w) ∂ 2 F(w) ∂ z∂ x ∂ z∂ y ∂ z2

⎢ 2 and [H (w)] = ⎣ ∂ ∂ F(w) y∂ x

⎥ ⎦

(2.32)

Equation (2.31) can be expressed in a symbolic (vector) form w(n+1) = w(n) − [H (w(n) )]−1 · ∇[F(w(n) )]

(2.33)

The process of determining successive approximations (2.33) continues as long as the condition (2.27) is not satisfied. Example 2.3 According to dependences (2.23)–(2.33), a computer program “XYZ_N” was developed, which enabled performing control calculations for the following data: R0 R1 R2 R3

: : : :

(x0 = 0, y0 = 0, z 0 = 0), (x1 = −30,000, y1 = −20,000, z 1 = 30), (x2 = −18, 000, y2 = 40,000, z 2 = −100), ( x3 = 25, 000, y3 = 15,000, z 3 = 70),

t0 t1 t2 t3

= 169.129283854 µs = 266.747447916 µs = 101.229023437 µs = 128.155833333 µs

All the above-mentioned coordinates of the receivers R0 ÷ R3 are expressed in meters. The results obtained in several successive iterations of calculations are presented in Table 2.2.

2.2 The Time Difference of Arrival (TDOA) Method

39

Table 2.2 The results obtained in several successive iterations of calculations n

x, [m]

y, [m]

z, [m]

F(x, y, z)

1

6067.467

41,736.954

1779.328

78,727,136

2

8051.028

45,646.326

1531.718

13,894,433

3

8942.885

47,449.771

1706.364

1,478,663.5

…

…

…

…

…

7

9509.845

49,019.771

917,520.968

4189.9

…

…

…

…

…

10

9500.394

49,001.121

9123.345

F(x (n) ,y (n) , z (n) ), it then returns to the point (x (n) , y (n) , z (n) ) where the new minimization direction is calculated and the search for the minimum continues, most frequently with a decreased step h. In case of functions F(x, y, z), the easiest way to complete calculations is to use the condition (2.27). (n+1)

Example 2.4 The program ‘XYZ_G’ developed on the basis of the dependence (2.34)–(2.36), turned out to be very effective in minimizing the objective function (2.23). The results of several control calculations made according to this program in sixteen-digit arithmetic are illustrated in Table 2.3. These calculations were made for the same input data as presented in Example 2.3. In the ‘XYZ_G’ program, the search for the minimum of the objective function F(x, y, z) in kth direction is carried out with a constant step equal to h k = 100/k. Thanks to this, in the initial iterations, the search for the minimum is carried out with Table 2.3 Object coordinates and number of iterations of calculations performed for different minimum values of |F(x, y, z)| Starting point (x (0) , y (0) , z (0 ) ≡ (1, 1, 1) Coordinates and N (G)

|F(x, y, z)| ≤ 10

|F(x, y, z)| ≤ 1

|F(x, y, z)| ≤ 0.1

x (N ) , [m]

9504.637

9501.527

9500.274

y (N ) , [m]

49,011.337

49,002.632

48,999.921

z (N ) , [m]

9163.404

9130.987

9120.170

N (G)

560

729

1506

2.2 The Time Difference of Arrival (TDOA) Method

41

relatively large steps, which are gradually decreased while approaching the searched solution. Indirectly, this leads to a significant reduction in the total computing time. In case of results compiled in Table 2.3 the mentioned time corresponded one second. The results compiled in Tables 2.2 and 2.3 confirm the opinion established in the literature that the Newton’s method used to minimize the objective function (2.23) demonstrates a better convergence in comparison to the steepest decrease method but, unfortunately, it can be unreliable, especially when the starting point is far from the searched solution. On the contrary, the steepest descent method always reduces the value of the objective function (2.23), and thus enables approaching the area which contains the solution. In this situation, it is reasonable to use both methods mentioned in sequence, i.e. the steepest descent method in the initial one and the Newton’s method in the final stages of the calculation process. In other words, the steepest descent method is used to determine the starting point, appropriate for Newton’s method, providing calculation convergence to the searched solution. The above idea is reflected in the hybrid method discussed below and in the Levenberg–Marquardt method [5]. Hybrid Method When using the previously derived Eqs. (2.34) and (2.37), a similar equation can be inscribed for the simplest hybrid method created from the steepest descent and Newton’s methods . . h ]|| [I ] + (1 − λ(n+1) )[H (w(n) )]−1 w(n+1) = w(n) − λ(n+1) || [ || ∇ F(w(n) ) || × ∇[F(w(n) )]

(2.37)

The matrix [I ] appearing in the expression (2.37) is the identity matrix of the third degree and λ(n+1) corresponds the weighting factor (the key) assuming the value 1 or 0. If λ(n+1) = 1, then the successive approximations of the searched solution are calculated according to the steepest descent method. In case of λ(n+1) = 0, only Newton’s method is used. Thus, the question arises regarding the method of selecting the switching moment (λ(n+1) : 1 → 0), for which the discussed hybrid method will find an approximate solution with the lowest numerical cost within the shortest possible time. According to author’s opinion, the most appropriate approach seems to be linking this moment with the value of the minimized function (2.23) or the gradient modulus (2.34). Therefore, in the XYZ_H computer program, compiled with the previously developed programs “XYZ_N” and “XYZ_G”, it was assumed that the switching criterion (λ(n+1) : 1 → 0) constitutes the condition |F(x, y, z)| ≤ F p where F p is a threshold value set. This value should be slightly lower than F p max , which is easier to be determined experimentally for a specific arrangement of listening stations Rk , where k = 0, 1, 2 i 3, i.e. by appropriately planned simulation

42

2 Determining the Object’s Position by Radiolocation Methods

calculations. A similar condition, i.e. |F(x, y, z)| ≤ ε = 0.1 is used as a criterion of completing the calculations. A step change of the weighting factor (λ(n+1) : 1 → 0) when |F(w)| ≤ F p is a distinctive feature of the hybrid method. This means that the steepest descent and Newton’s methods are used separately, i.e. with time distribution. A smooth change of λ(n+1) is also possible, and the before-mentioned methods will be used together in each iteration. However, in this case it is necessary to determine the optimal course λ(n+1) as a function of the iterations’ number n or |F(w)|. The discussion regarding the significance of the problem of hybrid optimization methods is beyond the scope of this book. Example 2.5 According to the XYZ_H program, the results of several testing calculations made in sixteen-digit arithmetic are compiled in Table 2.4. These calculations were made for the same input data as in the Examples 2.3 and 2.4. The last row in Table 2.4 shows the number of iterations (namely changes in the searching direction) performed while using the steepest descent and Newton’s methods, respectively. These numbers are significantly smaller than their counterpart N (G) = 1506, presented in Table 2.3. This fact confirms the validity of using gradient methods to determine the starting point appropriate for Newton’s method, providing the convergence of the objective function minimization process (2.23). All the calculation results presented in Tables 2.2, 2.3 and 2.4 were obtained for identical reception times t0 , t1 , t2 and t3 , set with the accuracy of approximately 10−15 s. It is obvious that providing this high timing accuracy in practice is very difficult, if not impossible. As a consequence of an inaccurate reception time measurement, the obtained approximations of the object’s position will be placed in the larger, more spacious blur area, which is illustrated in Fig. 2.4. Another, equally significant problem though, is the uniqueness of the obtained solution describing the actual object’s position. Assume hypothetically that the receivers of listening stations Rk , where k = 0, 1, 2 . . . 3, are located in the same horizontal plane which is described by the equation z = 0. In this radio navigation scenario, the presented hybrid method may trigger one of two equal solutions, i.e. (x ∗ , y ∗ , z ∗ ) or (x ∗ , y ∗ , −z ∗ ) (Fig. 2.6). Table 2.4 Object coordinates and number of iterations of calculations performed for different values of a threshold F p Starting point (x (0) , y (0) , z (0 ) ≡ (1, 1, 1) F p > 3,300,000 Coordinates and number of iterations ε = 0.1 x, [m]

No result obtained ! 9499.959

y, [m] z, [m] N (G)

+

F p = 3,000,000 F p = 1,000,000 F p = 100,000 ε = 0.1 ε = 0.1 ε = 0.1

N (N )

9499.884

9500.147

48,999.796

48,999.660

49,000.526

9119.621

9119.380

9121.198

46 + 7

73 + 5

147 + 4

References

43

Fig. 2.6 Geometric illustration of a problem of occurrence of the second useless solution

The above-mentioned conclusion can be validated as follows. The values of the expressions (z k − z)2 in the formulae (2.22), where (z k − z)2 , for z k = 0 are equal to (0 − z)2 = [0 −(−z)] 2 . Thus, in this specific case, the value of the objective function (2.23) does not depend on the sign of the coordinate z of the object. The condition z > z k min = −100 m, which stems from the physical nature of the radio navigation task considered in Examples 2.3, 2.4 and 2.5, is easily met when assuming z = −z, if z < z k min . Such instruction of substitution was inscribed in lines 650 and 1410 of the "XYZ_H" program, whose BASIC listing was included in Appendix 2.2 of [7].

References 1. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Soviet Radio, Moscow (in Russian) 2. Grigorin-Rjabow WW (ed) (1970) Radar devices, theory and principles of construction. Soviet Radio, Moscow (in Russian) 3. Hill D, Galloway P (2008) Multi-static primary surveillance radar. Report No: 72/07/376/U written for Eurocontrol, issue 1.2, pp 1–183. Roke Manor Research Ltd., Hampshire (UK), July 2008. https://www-test.eurocontrol.int/sites/default/files/2019-05/surveillance-reportmulti-static-primary-surveillance-radar-an-examination-of-altervative-frequency-bands-200 807.pdf 4. Ho KC, Xu W (2004) An accurate algebraic solution for moving source location using TDOA and FDOA measurements. IEEE Trans Signal Process 52:2435–2463 5. Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Quart Appl Math 2:164–168 6. Rosłoniec S (2008) Fundamental numerical methods for electrical engineering. Springer, Heidelberg 7. Rosloniec S (2020) Fundamentals of the radiolocaction and radionavigation, 2nd edn. Military University of Technology, Warsaw (in Polish)

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2 Determining the Object’s Position by Radiolocation Methods

8. Shen G et al (2008) Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment. In: Proceedings of the 5th workshop on positioning, navigation and communication (WPNC’08), pp 71–78 9. Stoica P, Li J (2006) Source localization from range – difference measurements. IEEE Signal Process Mag 23(6):63–69 10. Szirman JD (ed) (1970) Theoretical basis of radiolocation. Soviet Radio, Moscow (in Russian)

Chapter 3

Reflective Surface of the Detected Objects with Monostatic and Bistatic Radar Systems

At the core of active radiolocation with a passive response is the phenomenon of secondary electromagnetic wave radiation [2, 16, 18, 23]. This phenomenon occurs whenever an electromagnetic wave encounters an obstacle (object) on its path with electrical and magnetic properties different from the corresponding properties of free space expressed by ε0 , μ0 and σ0 = 0. The wave falling on the obstacle is called the primary wave, and the wave radiated by the obstacle is called the scattered or reflected wave. Thus, the obstacle is often treated as a source of secondary radiation. Under the influence of a wave falling on the conductive surface of the obstacle, conduction currents are created, which are the source of a new wave radiated in different directions. In the case of a dielectric obstruction, displacement currents arise in it, which, like the conduction currents, are a source of secondary radiation. From the point of view of electrodynamics, the geometric shape of the object and the ratio of the wavelength λ0 to its dimensions have a great influence on the process of secondary radiation (retransmission). Equally important is the geometric position of the electric field vector E of the primary wave in relation to the obstacle illuminated by this wave, in which the aforementioned conduction currents or displacements are excited. For example, when the electric field vector E is directed parallel to a rectilinear thin metal conductor with a length L close to λ0 /2 the same, strong conduction currents are induced in the conductor, which are the source of the secondary wave. We are talking then about strong scattering or reflection. Such a scenario is called matching the polarization of the wave to the illuminated wire. If the electric field vector E of the primary wave is directed perpendicularly to the above-mentioned metal conductor, electric charges are induced on its surface with a uniform spatial distribution, which means that no conduction currents flow on this surface. In other words, the energy of the primary wave is minimally “absorbed” by the conductor, which means that the energy radiated re-radiation through this conductor reaches the minimum value. In this case, we speak of a polarization mismatch of the wave to the illuminated wire. The retransmission properties of the illuminated object in a given direction are determined by a parameter called the effective reflection surface. In the Englishlanguage literature, this parameter is called Radar Cross Section and is marked with © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_3

45

46

3 Reflective Surface of the Detected Objects with Monostatic …

Fig. 3.1 Definition of the bistatic angle β

the acronym RCS. The mathematical of this parameter is discussed below. | | meaning |Ei |2 /ξ0 , denotes the power density of the Let us assume that |Si | = |Ei × Hi∗ | = √ incident wave at the object where ξ0 = μ0 /ε0 = 120π . is the free space wave impedance (Fig. 3.1). Part of the energy of this wave after scattering (reflection) reaches |the receiver | of the radar station distant from the object by a distance R. Let |S R | = |E R × H∗R | = |E R |2 /ξ0 denote the wave power density of the received echo signal. The term reflecting surface of an object (RCS) is understood as a surface σ for which the following equation is satisfied 4π R 2 |S R | = σ |Si |

(3.1)

| | 2 ∗| | | (3.1)2 and the densities |Si | = Ei × Hi = |Ei | /ξ0 , |S R | = | From ∗Eq. |E R × H | = |E R | /ξ0 formulated above it follows R | |2 | |2 | | | | |S R | 2 | ER | 2 | HR | σ = 4π R = 4π R | | = 4π R | |Si | Ei Hi | 2

(3.2)

which results from the preliminary assumption that the object is located in the far zone. In other words, the wave reaching the object and the secondary wave reaching the receiver are plane waves. In the literature, the above condition with relation (3.2) is written in the following general form ( ) |E R |2 4π R 2 R→∞ |Ei |2

σ B = lim

(3.3)

where Ei is the incident intensity vector at the plane (object), E R stands for the electric field vector of the reflected wave reaching the antenna of the receiving unit and R corresponds to the distance between the object and the receiving unit. The area σ B describes the object’s ability to reflect energy of an incident electromagnetic wave in the direction differing from the direction of the wave incidence by an angle β,

3 Reflective Surface of the Detected Objects with Monostatic …

47

which is further described in Chap. 5. In case of the monostatic system, bistatic angle β = 0 and respectively σ B → σ M . The term “electromagnetic wave reflection” is a synonym of the retransmission phenomenon, understood as follows. The electromagnetic incident wave causes conduction and displacement currents on the surface of the object, which depend on the type of surface (conductor, dielectric, etc.), as well as the shape and size of the object. These currents, on the other hand, are a source of secondary, scattered radiation. In case of an object with a complex shape, the term reflection is the sum of many partial reflections that interfere mutually. These reflections, due to the size and shape of the reflecting surface, may demonstrate different physical nature. For example, in case of a helicopter, one part of energy is reflected from the fuselage and the other one from its rotors. The fuselage’s reflection is usually aperiodic, and the rotors’ reflection–resonant [6, 8]. The resonant (strong) reflection effect occurs when the length of the rotor blades is approximately equal to λ/2 or to its multiplicity, where λ refers to the wavelength. One of these reflections may be dominant, depending on the direction of the object’s observation. Similarly, in case of a rocket, the reflection from its front conical surface is generally weaker in comparison to its cylindrical side surface. The above examples show a fact of practical importance, namely that σ B depends on the directions of illumination and observation of the object (separated by a bistatic angle β), as well as on the polarization and frequency of the illuminating wave [8, 15, 17]. In other words, every object with a set frequency and polarization of the illuminating wave, and the illuminance direction set, has a specific directivity pattern. Objects detected by radiolocation methods are conveniently classified according to the criterion of determining their effective reflection surface. Due to this criterion, they can be divided into elementary and complex objects (Fig. 3.2).

Fig. 3.2 Object classification used in this chapter

48

3 Reflective Surface of the Detected Objects with Monostatic …

3.1 The Reflective Surface Determined for a Monostatic Primary Radar System In recent decades, the monostatic radar system has been playing a key role which can be noticed nowadays as well. The radar cross-section σ M used in its range equation has been the subject of numerous theoretical and experimental works, whose results are included in many broadly available publications, for example in [2, 6, 8]. Therefore, only some of them, referring to typical and most common objects, are presented below. 1. Rectilinear Wire Antenna An example of the simplest object reflecting the energy of an electromagnetic wave is narrow strip of conductive foil with a length l close to l ≈ n · λ0 /2, where n = 1, 2, 3, ... and λ0 /2 is a length of the wave scattered (Fig. 3.3a). Thus, σ (l/λ0 ) of this strip has a resonance character. The wave is scattered in a similar manner by a rectilinear thin conductor with a diameter d > λ, where λ = c/ f

60

3 Reflective Surface of the Detected Objects with Monostatic …

corresponds to the wavelength in free space. The dependence σ M /(π r 2 ) versus the ratio 2π r/λ, calculated with this assumption, is presented in Fig. 3.20 [22, 25]. Figure 3.20 presents three basic zones, namely the refractive zone (also known as Rayleigh scattering), the resonance scattering zone and the stable scattering zone. If R >> λ and 0.05λ/π ≤ r ≤ 0.5λ/π then the wave refraction plays a dominant role, which decreases with the increase of the sphere radius. A strong (resonant) reflection effect is observed when r ≈ λ/(2π ). The extreme value of the sphere’s reflective surface in this range takes 4π r 2 (marked by point A in the diagram above) and is 4 times greater than the corresponding determined surface in the far zone. The range limited by R >> λ and 0.5λ/π ≤ r ≤ 5λ/π is called the resonance scattering range, in which the minimum value, illustrated by point B in the diagram shown in Fig. 3.20, of the reflective surface is equal to 0.26π r 2 . In the far zone defined by R >> λ and r > 10λ/(2π ), the resonant reflection effect disappears and the surface of the mirror image of the sphere equals σ M = π r 2 which is also weakly dependent on the wavelength λ. Furthermore, it should be emphasized that the foregoing considerations refer to the monostatic surface σ M determined when the illuminance direction is identical to the direction of observation. For bistatic radar, the knowledge of dependence σ B (β) (defined in the far zone) as a function of the bistatic angle is crucial. So far, a large number of theoretical contributions and experiments has been devoted to the task of determining the dependencies between surfaces σ M and σ B (β). The nature of this dependence is illustrated by the graphs σ B (β)/σ M , and σ M ≡ σ B (β = 0) presented in Fig. 3.21, which were created on the basis of experimental data obtained for an ideally conductive sphere with a radius r = 20λ/(2π ) [1]. The dependencies σ B (β)/σ M presented in Fig. 3.21, were prepared on the basis of experimental data obtained for: (1) → 2π r/λ = 2.36 and (2) → 2π r/λ = 20. The solid lines show changes σ B (β)/σ M which were caused by changes in the bistatic angle β in the electric field’s plane. The dashed lines demonstrate the course σ B (β)/σ M in the function of the angle β, changed in the magnetic field’s plane. The significant changes σ B (β)/σ M visible in the above diagram are justified as follows. With the determined illuminance direction, the incident plane wave induces currents (conduction and displacement) on the sphere’s surface, which affect its shaded side, creating so-called creeping currents or creeping waves. In this situation, the wave propagating in the direction differing

Fig. 3.20 The back scattering from a metalic sphere

3.4 Radar Cross Section of an Object Determined FSR

61

Fig. 3.21 The plot of σ B (β)/σ M as a function of bistatic angle β obtained experimentally for an well conductive sphere with a radius r = 20λ/(2π )

by the bistatic angle β from the incidence direction is generally the result of the interference of two waves, namely the wave reflected from the sphere according to the principle of geometrical optics and the wave generated by currents affecting its shaded side. Depending on the phase relations either constructive or destructive wave interference occurs, along the selected observation direction. The influence of the phenomenon discussed above is minimal at small bistatic angles β. The results of the ◦ analysis carried out in [14] indicate that for β ≤ 5 the reflective surface σ B (β) of a complex object is close to its monostatic surface σ M measured in the direction indicated by the bistatic angle bisector at the frequency decreased by a factor cos(β/2). The case of incident wave scattering when β → 180 0 is particularly interesting. ◦ According to [1, 25], when β = 180 and r >> λ, the spheres’ bistatic reflective surface with radius r is equal to σ B ≡ σ BF = 4π A2 /λ2

(3.19)

where A = π r 2 denotes the aperture area (shadow) of the considered sphere. This particular case of bistatic radar, known as Forward Scattering Radiolocation (FSR), should be understood as radiolocation with clearance illumination. The dependence σ BF = 4π A2 /λ2 can be inscribed as σ BF = D · A, where D = 4π A/λ2 corresponds to the maximum value of the directive gain of an evenly evoked flat surface A [3, 13].

3.4 Radar Cross Section of an Object Determined FSR In order to emphasize the significance of the discussed case for bistatic radar, let us determine the surface σ BF , known as Forward Bistatic Radar Cross Section (FBRCS), for the sphere with the radius r = 0.5 m, assuming that λ = 0.2 m. According to (3.19) σ BF = 4π

4 [π(0.5)2 ]2 3 (0.5) = 4π ≈ 193.7892 m2 (0.2)2 (0.2)2

62

3 Reflective Surface of the Detected Objects with Monostatic …

The calculated area σ BF is 246.74 times greater than the surface σ M )≡ σ B (β = 0) ( totaling π(0.5)2 . Thus, the obtained “increment” equals 10 log σ BF /σ M ≈ 23.92 dB. The results of numerous experimental studies confirm that the dependence (3.19) determined by the Babinet’s principle is valid for an object of any shape and does not correspond to the material type covering its surface when the following conditions are met. ◦

1. System bistatic angle β → 180 . 2. The object’s geometrical dimensions exceed the wavelength λ of the probe signal. 3. The object is located in the far zone with regards to transmitter and the receiver. These studies also show that in case of complex objects, their effective areas ◦ σ B (β) reach the maximum values for β ≈ 165 [1, 11]. For example, Fig. 3.22b shows the theoretical relationship σ B determined for F-117 aircraft, the model of which is shown in Fig. 3.22a. According to [11], the airplane model, with its parallel wings to the ground’s plane, was illuminated by a plane wave with a frequency of f = 0.9 GHz, directed ◦ in the elevation plane at an angle of 30 towards the ground’s surface. The diagrams presented in Fig. 3.22b illustrate the dependence of the bistatic ◦ effective area σ B of the F-117 airplane model from the bistatic angle 0 ≤ β ≤ 180 for two different polarizations of the incident wave, i.e. when both angles β in the plane of the electric and magnetic fields are changed. The discussed effect of the effective area’s σ BF rapid increase, has been known since the sixties of the previous century [1, 25, 26]. Nevertheless, the FSR does not belong to the group of standard ◦ and broadly used radar systems. This is because the condition β → 180 from which it follows that the object must be located either near to or exactly on the base line, connecting the transmitter with the receiver. This condition significantly limits the size of the detection area. Basing on the Fig. 5.1 and the dependencies ◦ (5.1)–(5.1) presented in Chap. 5, it is easy to justify that for β = 180 the angle ◦ α = 0 and the distance difference .L = 0 In this situation it is not possible to determine uniquely the distance R R , i.e. the distance between the detected object and the receiver. Moreover, it is even difficult to imagine an FSR system which would detect aircraft and rockets coming from different directions.

Fig. 3.22 The theoretical relationship σ B (β) determined for F-117 aircraft; a the model of F-117 under consideration, b the plot of the effective aera σ B (β) as a function of bistatic angle β

References

63

References 1. Averianov V (1978) Multistatic radar stations and systems (in Russian). Science and Technology, Minsk (Belarus) 2. Bakuljev PA (2004) Radiolocation systems (in Russian). Radio-Engineering, Moscow 3. Balanis CA (1997) Antenna theory, analysis and design, 2nd edn. Wiley, New York 4. Barton DK (2005) Modern radar system analysis and modelling. Artech House Inc., Norwood 5. Bird D (2004) Design and manufacture of a low-profile radar retro-reflector. In: RTO SCI Symposium on sensors and sensor denial by camouflage, concealment and deception, Brussels, Belgium, April 2004. Published in RTO-MP-SCI-145, pp 1–12 6. Crisprin JW, Siegel KM (eds) (1968) Methods of radar cross section analysis. Academic, New York 7. Doerry AW, Brock B (2009) Radar cross section of triangular trihedral reflector with extendet bottom plate’. Sandia report SAND2009-2993. https://prod-ng.sandia.gov/techlib-noauth/acc ess-control.cgi/2009/092993.pdf 8. Ewell GW (1989) Bistatic radar cross section measurements. In: Curie NC (ed) Techniques of radar reflectivity measurement. Artech House Inc., Norwood 9. Nathanson FE, Reilly JP, Cohen MN (1991) Radar design principles, signal processing and the environment, 2nd edn. McGraw-Hill, New York 10. Grigorin-Rjabow WW (ed) Radar devices, theory and principles of construction (in Russian). Soviet Radio, Moscow 11. Hill D, Galloway P (2008) Multi-static primary surveillance radar. Report No: 72/07/376/U written for Eurocontrol, issue 1.2, pp–183. Roke Manor Research Ltd., Hampshire (UK), July 2008. https://www-test.eurocontrol.int/sites/default/files/2019-05/surveillance-report-multistatic-primary-surveillance-radar-an-examination-of-altervative-frequency-bands-200807.pdf 12. Jackson MC (1986) The geometry of bistatic radar systems. IEE Proc. 133(Pt. E., No.7):604– 612 13. Johnson RC (ed) (1993) Antenna engineering handbook, 3rd edn. McGraw-Hill , New York 14. Kell RE (1965) On the deriviation of bistatic RCS from monostatic measurements. Proc IEEE 53:983–988 15. Knott EF (2006) Radar cross section measurement. Scitech Publishing Inc., Releigh 16. Levanon N (1988) Radar principles. Wiley, New York 17. Mahafza BR (2000) Radar systems analysis and design using MATLAB. CRC Press, Boca Raton 18. Peebles PZ Jr (1998) Radar principles. Wiley, New York 19. Rosłoniec S (2006) Fundamentals of the antenna technique (in Polish). Publishing House of the Warsaw University of Technology, Warsaw 20. Sharp E, Diab M (1960) Van Atta reflector array. IRE Trans. Antennas Propaga 8(4):436–438 21. Skolnik MI (ed) (1990) Radar handbook, 2nd edn. McGraw-Hill, New York 22. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 23. Szirman JD (ed) (1970) Theoretical basis of radiolocation. Soviet Radio, Moscow (in Russian) 24. Tseng WJ et al (2000) A planar Van Atta array reflector with retrodirectivity in both E-plane and H-plane. IEEE Trans. Antennas Propag 48:173–175 25. Willis NJ (2005) Bistatic radar, 2nd edn. Scitech Publishing Inc., Releigh 26. Willis NJ, Griffiths HD (eds) Advances in bistatics radar. Scitech Publishing Inc., Releigh

Chapter 4

Range Equations of Primary and Secondary Radar Systems

The concept of the range equation is understood as a mathematical dependence describing the influence of various parameters of the radar system (its energy potential), the RCS of the object and propagation conditions of the maximum distance (range) Rmax between the radar station and the object, at which the object is still detectable, Fig. 4.1. The energy potential of the radar station is usually expressed by the following parameters: • • • •

PT —power of the probe signal fed to the transmitting antenna, G T —the energy gain of the transmitting antenna, (e f f ) A R —the effective aperture of the receiving antenna, PR(min) —the minimum power of the echo signal for which the receiver is able to distinguish this signal from the background of simultaneously penetrating interferences.

The attenuation of the probe signal (electromagnetic TEM wave) in the troposphere is most frequently determined by the total attenuation rate γ [dB/km], whose physical meaning is discussed in Appendix C. The ability of the detected object to reflect the coming probing signal (electromagnetic wave) is marked by the RCS equal to σ M(Ob) .

4.1 Range Equation of the Primary Radar System Using the data presented above, the range equation of the primary radar system can be determined as follows, [5, 9, 11]. Assume that PT corresponds to the power of the probe signal fed to the transmitting antenna. Then, the surface power density of the signal reaching the object is equal to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_4

65

66

4 Range Equations of Primary and Secondary Radar Systems

Fig. 4.1 The block diagram of the primary radiolocation system; a with two separated antennas, b with common antenna

S0b =

PT G T 4π R 2

(4.1)

where R refers to the object’s distance, see Fig. 4.1. If the RCS of the object is equal to σ M(Ob) , the power of the signal "captured” by the object totals P0b = SOb σ M(Ob) =

P T G T (Ob) σ 4π R 2 M

(4.2)

The echo signal with the power (4.2) is isotropically scattered, so the surface power density of the echo signal reaching the receiving antenna aperture of the radar station is equal to SR =

POb P T G T (Ob) 1 P T G T (Ob) = σ = σ 4π R 2 4π R 2 M 4π R 2 (4π )2 R 4 M

(4.3)

Thus, the power of the echo signal transmitted through the receiving antenna to the receiver of the radar station totals (e f f )

PR = S R A R

· η(R) A =

P T G T (Ob) (e f f ) (R) σ AR ηA (4π )2 R 4 M

(4.4)

4.1 Range Equation of the Primary Radar System

67

where η(R) A denotes the coefficient determining the power efficiency of the receiving antenna. According to the above notation, the power (4.4) receives the minimum value equal to PR(min) , if the detected object is at the edge of the range, i.e. at Rmax . In other words, PR(min) =

PT GT (e f f ) σ M(Ob) A R η(R) A 4 (4π )2 Rmax

(4.5)

In the general case, the power obtained from formula (4.5) may be lower than the power of noise and various types of non-coherent interference penetrating the receiver through the receiving antenna along with the echo signal. By using an appropriate reception technique (e.g. additive or correlation reception), the echo signal is "pulled" above the level of noise and of the mentioned interference.The range Eq. (4.5) can be expressed in the following equivalent form

Rmax

1 =√ 4π

[

]0.25

PT

(e f f ) G T σ M(Ob) A R η(R) A PR(min)

(4.6)

which is often referred to as the classic form of the range equation. Another form of the range equation, frequently mentioned in the literature, is (e f f ) obtained by replacing A R with an expression (e f f )

AR

=

G R λ2 4π η(R) A

(4.7)

where: G R —power gain of the receiving antenna, η(R) A —power efficiency of the receiving antenna and λ—the wavelength of the probe signal. The expression (4.7) follows from the dependencies: G = D · η A and D = 4π · A/λ2 , linking the power gain of the antenna G with its effective aperture area A and power efficiency η A . Derivation of these dependencies can be found in the books on antenna theory, for example in [1, 7]. After substituting the expression (4.7) in Eq. (4.6), one obtains

Rmax

[ ]0.25 √ PT λ (Ob) G T G R · σM = (4π )0.75 PR(min)

(4.8)

In the primary radar system with one switched antenna, see Fig. 4.1b, G T ≡ G R = G A , which allows the Eq. (4.8) to be expressed as [ ]0.25 PT λ 2 (Ob) G σ = (4π )0.75 PR(min) A M √

Rmax

(4.9)

68

4 Range Equations of Primary and Secondary Radar Systems

Equations (4.8) and (4.9) have been derived on the assumption that the attenuation (γ ) of the troposphere is negligible. The influence of this factor on the range Rmax can be easily taken into account, assuming that the signal power reaching the receiver is [ ]2 (γ ) lower than the power obtained from (4.5) by exp( − 0.23 γ Rmax ) times. The physical sense of the multiplier exp(−0.23 γ R) is discussed in Appendix C. Accordingly, taking the attenuation of the troposphere into account (e f f )

PR(min) = S R A R

( =

1 (γ ) Rmax

)4

]2 P T G T (Ob) (e f f ) (R) [ (γ ) σ M A R η A · exp(−0.23 γ Rmax ) 2 (4π ) (4.10)

Finally, equation (4.10) can be recast into the form: (γ ) Rmax

1 =√ 4π =

[

]0.25

PT

(e f f ) G σ (Ob) A R η(R) A (min) T M PR (γ ) Rmax exp(−0.115γ Rmax )

] [ (γ ) 0.5 · exp(− 0.23γ Rmax (4.11)

This relatively simple nonlinear equation of one variable can be easily solved numerically. (γ )

Example 4.1 In the following example, the maximum range Rmax of the primary radar system (cf. Figure 4.1b) was determined assuming the following data: Radar station G A = 36[dB] PT = 5 × 105 [W] PR(min) = 10−12 [W] λ = 0.224[m]

Object σ M(Ob) = 3[m2 ]

T r opospher e γ = 7 · 10−6 [dB/m]

Calculation results: G A = 1036/10 = 3981.071706 [ ]0.25 √ T According to dependency (4.9) Rmax = (4π)λ0.75 P(min) G 2A σ M(Ob) ≈ 156581 [m]. (γ )

PR

(γ )

(γ )

Equation (4.11), e.g. Rmax = Rmax exp(−0.115γ Rmax ) is satisfied for Rmax = 139,903 [m]. Another factor limiting the range is the curvature of the Earth’s surface. The essence of this limitation, assuming that the electromagnetic wave propagates along a straight line (no refraction effect), is illustrated in Fig. 4.2. Assume that the radar station’s antenna is located at height h and the detected object is flying at height H . Consequently, the object will still be visible when the wave radius of the probe signal (as well as the echo signal) is tangent to the Earth’s surface at a point C. The geometric (horizontal) range can be described as the distance between the points A and B, in which the antenna of the radar station and the object

4.1 Range Equation of the Primary Radar System

69

Fig. 4.2 Geometric illustration of a radar range limitation resulting from geometry of Earth surface

are located respectively. According to Fig. 4.2 |AB| = |AC| + |C B|

(4.12)

where: |AC| = |C B| =

/ . (R E + h)2 − R 2E = h 2 + 2h R E ,

/ . (R E + H )2 − R 2E = H 2 + 2H R E

The ground radar meets the following conditions: h 0, significantly reduces the range of the radar station. The negative refraction occurs most frequently in the polar (arctic) regions due to the inversion of temperature and air humidity distribution in the troposphere. Under these adverse conditions, the dn dh

4.1 Range Equation of the Primary Radar System

71

Fig. 4.4 Illustration of effect of increasing the radar range due to refraction of electromagnetic wave in the atmosphere

range may be reduced to a small value of the order of single kilometers, which causes a huge danger of collision with another ship or iceberg. The positive normal refraction is the most significant for radiolocation, which dn < 0. When this refraction takes the value −15.7 × 10−8 [1/m] < dh | ' |stage occurs, under conditions of the normal troposphere, the increased range | AB | presented in Fig. 4.4, is determined by the following approximate dependence | '| (√ √ ) . | | h+ H | AB | ≈ 2Re f f

(4.15)

The normal troposphere can be described as the troposphere which is marked by T o = 291K , p = 1015 mb, the relative air humidity totals 75% and the relative ' permittivity is equal to ε = 1.000676. The effective radius length Re f in formula (4.15), is calculated according to the formula Re f f =

RE · ρ ρ − RE

(4.16)

where ρ—is the radius of curvature of the electromagnetic wave path. The positive normal refraction is marked by R E ≤ ρ < ∞. For the normal dn ≈ − 4 × 10−8 [1/m]. troposphere ρ ≈ 25,500 × 103 m and dh Example 4.3 For the following data: h = 20 m, H = 10,000 m, R E = 6368 × range of the radar station calculated 103 m and ρ = 25,500 × 103 m, the |maximum '| according to formula (4.15), totals | AB | ≈ 430, 434 × 103 m, which accounts approximately for 115% of the range calculated in Example 4.2. dn = −15.7 × 10−8 [1/m], a positive critical refraction takes place, for which If dh ρ ≈ R Z and consequently Re f f → ∞. In these conditions, the electromagnetic wave orbits the Earth just like an artificial satellite circles around the circular orbit. In the last decades of the previous century, numerous attempts were made in order to use the discussed refraction in, so-called, long-range over-the-horizon radar.

72

4 Range Equations of Primary and Secondary Radar Systems

4.2 The Range Equation of the Secondary Radar System The principle of secondary radar operation was invented by Robert Watson Watt in 1935 for the purposes of British air defense, [6, 9]. That is why it is often referred to as a system for identifying friends or foes and denoted by the acronym IFF. Soon after, similar works were carried out in the United States Navy Research Laboratory (NRL) and laboratories of German air forces (Luftwaffe). Of course, each project was carried out with great national secrecy, as a result of which all three projects remained in complete isolation from each other. In case of active radar with an active response, commonly known as secondary radar, two subsidiary ranges are distinguished. Correspondingly, these include the range R 1 from the interrogator to the transponder’s receiver located on the identified object and the range R 2 from the transponder’s transmitter to the interrogator’s receiver, [4, 6]. Assume that the interrogation signal has been received by the transponder’s receiver. Then, the transponder automatically sends to the interrogator a response signal, which corresponds to the nature of the interrogation signal, see Appendix D. The system works correctly when the distance R between the interrogator and the transponder meets the condition R ≤ min(R1 , R2 ) presented in Fig. 4.5. A typical example of a secondary radar system, widely used in the air force and navy, is the Identification Friend—Foe (IFF) system, [4, 10]. The interrogation signal of IFF is transmitted on a carrier frequency f 1 = 1030 MHz and the transponder response is transmitted on the carrier frequency f 2 = 1090 MHz. Due to the small interval between the frequencies f 1 and f 2 , the same antenna is used for reception and transmission and its directivity pattern has the value of 3 dB width of the order of 30◦ in the horizontal plane. The power density of the electromagnetic energy flux (transmitting the interrogation signal) at the transponder’s location (of the identified object) is equal to S1 = PI

GI 4π R 2

Fig. 4.5 Basic functional blocks of the secondary radiolocation system (e.g. IFF)

(4.17)

4.2 The Range Equation of the Secondary Radar System

73

where: PI —refers to the power of interrogator’s transmitter, G I —interrogator’s antenna power gain and R—distance between interrogator and transponder. The power of the signal reaching the transponder’s receiver is directly proportional to the surface density (4.17) as well as the effective aperture area A T and power efficiency ) η(T A of its antenna, i.e. ) P1 = S1 · A T η(T A = PI

GI ) A T η(T A 4π R 2

(4.18)

If the sensitivity of the transponder’s receiver is determined by the minimum power value P1 equalling P1(min) then according to the Eq. (4.18) it is possible to determine / PI 1 (max) R1 =√ G A η(T ) (4.19) (min) I T A 4π P1 The maximum value of the range from the transponder to the interrogator’s receiver is determined in a similar way to the one presented above, i.e. R2(max)

1 =√ 4π

/

PT P2(min)

G T A I η(IA )

(4.20)

where: A I denotes the antenna effective aperture area of the integrator and η(IA ) is a coefficient determining antenna power efficiency. Similarly: PT − refers to the power of the transponder’s transmitter, G T —the power gain of the transponder’s antenna and P2(min) —the signal power determining the threshold sensitivity of the interrogator’s receiver. In dependencies (4.19) and (4.20), the interrogator and transponder’s antennas are represented by power gains G I , G T and their effective surfaces A I and A T respectively. The above-mentioned parameters are combined together by dependencies G I = 4π

G I λ2 A I (I ) η A or A I = 2 λ 4π η(IA )

(4.21)

G T = 4π

A T (T ) G T λ2 η or A = T ) λ2 A 4π η(T A

(4.22)

where: λ—stands for the length of the propagated wavelength in free space, η(IA ) —the ) power efficiency of the interrogator’s antenna and η(T A − the power efficiency of the transponder’s antenna, [1, 7]. It should be emphasized that for the interrogation and response signals the wavelengths are different and equal to λ1 = c/ f 1 and λ2 = c/ f 2 , where c ≈ 3 × 108 m/s refers to the phase velocity of the TEM wave in free space. Once the dependencies (4.21) and (4.22) in expressions (4.19) and (4.20) are taken

74

4 Range Equations of Primary and Secondary Radar Systems

into account, one obtains / R1(max)

= /

R2(max)

=

PI G I 1 A η(T ) = (min) 4π T A P1 PT G T 1 A I η(IA ) = P2(min) 4π

/

/

/

λ1 PI G I 1 G T λ12 = (min) 4π 4π 4π P1 PT G T 1 G I λ22 λ2 = (min) 4π 4π 4π P2

P I G I GT P1(min)

/

PT G T G I P2(min)

(4.23)

(4.24)

As mentioned before, the system will operate properly, if the distance R between the interrogator and transponder meets the following condition R (max) ≤ min(R1(max) , R(max ) 2

(4.25)

Due to the limited energy potentials of the interrogator and transponder, one should strive to equalize the ranges R1(max) and R2(max) when designing the system, which is expressed by the equality R1(max) = R2(max)

(4.26)

After taking into account Eqs. (4.19) and (4.20), the dependence (4.26) can be written as [ ] ] [ ) (min) PI G I P2(min) A T η(T A I η(IA ) = P G P (4.27) T T 1 A The parts of Eq. (4.27) included in square brackets are called energy potentials of interrogator and transponder, respectively. The foregoing considerations have been carried out with the assumption that the attenuation of the troposphere, in which the interrogator and response signals propagate, is negligibly small. In order to determine how this phenomenon influences the range (4.25), assume that the attenuation is expressed by a factor γ [dB/m], see Appendix C. According to (4.18) and the dependencies presented in Appendix C, the signal power reaching the transponder’s receiver from the interrogator, taking into account the attenuation of the troposphere, is (γ )

P1

=

PI GI A T ηA(T) · exp(−0.23 γ R) 4π R 2

(4.28)

The range equation resulting from the dependencies (4.19) and (4.24) is equal to / (max,γ ) R1

=

PI G I 1 (max,γ ) A η(T ) exp(−0.23γ R) = R1(max) exp(−0.115γ R1 ) (min) 4π T A P1 (4.29)

4.2 The Range Equation of the Secondary Radar System

75

Proceeding analogously one obtains (max,γ )

(max,γ )

= R2(max) exp(−0.115γ R2

)

(4.30)

R (max,γ ) = R (max) exp(−0.115γ R (max,γ ) )

(4.31)

R2 Finally,

The range R (max,γ ) is a solution of the nonlinear Eq. (4.31) for which the parameters are R (max) and γ . This solution is usually determined numerically. Example 4.4 In the following example, the maximum range R (max,γ ) of the secondary radar system was determined by adopting the following data for the interrogator, transponder and troposphere.

I nterr ogator G I [d B] = 20 [dB] PI = 500 [W] P2(min) = 10−9 [W]

T ransponder G T = 4 [dB] PT = 50 [W] P1(min) = 10−8 [W]

T r opospher e γ = 7 × 10−6 [dB/m]

3×108 [m/s] 1030×106 [1/s] 82,068[m] and R2(max)

The calculation results: G I = 100, G T ≈ 2.512, λ1 ≡ λ I = 3×108 [m/s] 1090×106 [1/s]

R1(max)

≈

0.291 [m], λ2 ≡ λT = ≈ 0.275 [m], ≈ = 77,556[m] Recording the results obtained, it can be assumed that R (max) [m] = 77,500 ≤ ). Ultimately, R (max,γ ) determined from the equation R (max,γ ) = min(R1(max) , R(max 2 (max) exp(−0.115 γ R (maz,γ ) ) is equal to R (max,γ ) ≈ 73,073[m]. R In summary, IFF system is a two channel radiolocation system, using the 1030 MHz band for an interrogation and the 1090 MHz band for response signals. It can typically operate in four standard modes, two for military use and two for military and civilian use. A brief specification of these modes is given below and in Appendix D. (1) Mode 1 is used in military air traffic control to determine what type of aircraft or what kind of mission it is performing. (2) Mod 2 is only used for military purposes. (3) Mode 3/C is used to obtain information about the flight current altitude of the aircraft. (4) Mode 3/A is the standard air traffic control mode. It is used automatically with mode 3/C to ensure the identification and control of all aircraft operating under instrument flight rules. According to the ICAO standard ground based interrogators typically alternate modes, sending alternately signals of mode 3/A and mode 3/C to the aircraft, receiving continuous identification and elevation data about the aircraft under recognization.

76

4 Range Equations of Primary and Secondary Radar Systems

Currently, in addition to the above mentioned modes special modes are used also, denoted by numbers 4 and 5 and the letter S.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Balanis CA (1997) Antenna theory, analysis and design, 2nd edn. Wiley, New York Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood (MA) Barton DK (2013) Radar equations for modern radar. Artech House Inc., Norwood (MA) Kayton M, Fried WR (1996) Avionic navigation systems, 2nd edn. Wiley, New York Mahafza BR (2013) Radar systems analysis and design using MATLAB, 3rd edn. CRC Press, Boca Raton (FL) Moir I, Allan S (2006) Military avionics systems. Wiley, New York Rosłoniec S (2006) Fundamentals of the antenna technique (in Polish). Publishing house of the Warsaw University of Technology, Warsaw Rosloniec (2020) Fundamentals of the radiolocaction and radionavigation (in Polish), 2nd edn. Publishing house of the Military University of Technology, Warsaw Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York Stevens MC (1988) Secondary surveillance radar. Artech House Inc., Boston (MA) Szirman JD (ed) (1970) Theoretical basis of the radiolocation, (in Russian). Soviet radio, Moscow

Chapter 5

Bistatic Radar Systems

A monostatic (single-positioned) radar system is a primary radar system (active with passive response) with transmitting and receiving antennas placed in the same location. In other words, the distance between the phase centers of these antennas is disproportionately small in comparison to the distance determination accuracy, [3, 6, 7]. According to this criterion, a bistatic (two-positioned) radar system is a primary radar system, with transmitting and receiving antennas placed in two significantly distant locations. An example of the simplest bistatic system can be derived from a monostatic system by displacing its receiving unit over a considerable distance. A multistatic system has a more complex structure and consists of the monostatic radar and an additional receiving unit (antenna with a receiver) located at a sufficiently large distance.

5.1 Main Advantages and Disadvantages of the Bistatic System In the initial period of the radiolocation development, the role of the bistatic system was dominant. However, in the 1940s it was replaced by a pulsed monostatic system with a joint transmitting and receiving antenna [3, 10]. Undoubtedly, the invention of the duplexer system in 1936, through which the transmitter and receiver are connected to a common antenna at appropriate time periods, contributed to the development of the monostatic system. The second important reason was the ease of 3D omnidirectional observation processing with the use of a monostatic radar, by the mechanical rotation of its antenna in the azimuthal plane. Unfortunately, bistatic radar did not provide such observation possibility. The renaissance of bistatic and multistatic systems took place in the last decades of the previous century, due to their unique properties (advantages). First of all, the receiving units of these systems are passive, which undoubtedly makes their detection and possible destruction more difficult. It should be also emphasized that the bistatic system plays a significant part © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_5

77

78

5 Bistatic Radar Systems

in the broadly understood rocket artillery and detection systems of artificial Earth satellites. Considering the available literature, it can be concluded that the bistatic system is marked by the following advantages, in comparison to the corresponding monostatic system, [1, 6, 12]. 1. Increased probability of detecting objects with a small reflective surface. 2. The possibility of ensuring the covert operation of so-called electromagnetic silence, by using the signals emitted by other systems’ transmitters, for example transmitters of VHF/UHF television stations or microwave transmission, as illuminating signals. 3. The possibility of shaping (for instance expanding) the spatial detection area, which is particularly significant for early detection of rockets and low-flying objects. 4. Increased resistance to active jamming, as well as detection and destruction, e.g. by Anti-Radiation Missile (ARM). The unquestioned disadvantages of the bistatic system include: 1. More extensive technical structure in comparison to the monostatic system, and thus the costs of its implementation and exploitation are increased. 2. The necessity of ensuring the accurate and reliable work synchronization of individual system units, taking into account the signals’ propagation times between the units. 3. The 3D space observation is more difficult, which results in limiting the observation area to a segment with an angular width much smaller than 180°.

5.2 Methods of Determining Object’s Position Using Bistatic Radar System A typical arrangement of the transmitter T , receiving unit RU and object O of the bistatic system in 3D space is presented in Fig. 5.1.

Fig. 5.1 The functional blocks and geometry of a bistatic radiolocation system

5.2 Methods of Determining Object’s Position Using Bistatic Radar System

79

The apex angle of the triangle (T , O, RU ), i.e. the angle between the radii RT and R R , is called the bistatic angle and is denoted by the symbol β. Figure 5.2 illustrates the functional diagram of the receiving unit RU used in this system. In order to discuss the nature of this system in the tracking mode, assume that the transmitter T sends the probe signal s(t) towards the object O. The origin of this signal (in the transmitter antenna) determines the pulsed synchronizing signal i (t) that is transmitted along the base b to the antenna Ai of the receiving unit RU . At the same time, the interrogator beam of the antenna directivity pattern A x of the receiving unit RU is directed towards the object, and the interrogator’s beam position is determined by the azimuth ϕ R and elevation angle γ R , see Fig. 5.3. The probe signal s(t), reflected from the object O, reaches the antenna A x with a delay of .ts = (R R + RT )/c, where c ≈ 2.998 × 108 m/s. This delay, counted

Fig. 5.2 The block diagram of a receiver unit

Fig. 5.3 Definition of angles α, ϕ R and γ R

80

5 Bistatic Radar Systems

from the moment of sending by the transmitting antenna, is passed from the receiver to the Control Processor Unit (CPU), along with the measured direction angles ϕ R and γ R . In the processing system, a time receiving difference of the echo signal e(t) and the synchronizing signal i (t), e.g. .t = .ts − b/c, are determined. Determining the position of the object O is equivalent to calculation of the following parameters: α ≡ ∠(T , RU, O) and .L = c.t = R R + RT − b

(5.1)

The path difference .L = c.t = R R + RT − b is calculated using the time difference .t determined by CPU. The angle α is related to the angles ϕ R and γ R by the following dependence α = 180◦ − δ = 180◦ − arccos[cos(ϕ R ) cos(γ R )]

(5.2)

The dependence (5.2) can be derived from the following equations: |RU, B| = |RU, A| cos (γR ), |RU, C| = |RU, B| cos (ϕR ) = |RU, A| cos (γR ) cos (ϕR ) |RU, C| = |RU, A| cos (δ) → cos(δ) = cos(ϕ R ) cos(γ R )

(5.3)

The above equations have been formulaed considering the (A, B, C, RU ) presented in Fig. 5.3. According to [1], the distances RT and R R , respectively, from the transmitter T to the object O and from the object O to the receiving unit RU , can be calculated according to the following formulae RR =

.L(2b + .L) 1 , RT = .L + b − R R , when β < 180◦ 2 .L + b[1 − cos(α)]

(5.4)

The angles ζ and β of the bistatic triangle (T , O, RU ), presented in Fig. 5.1, are respectively equal to: [ ζ = arccos

] RT2 + b2 − R 2R , β = 180◦ − α − ζ 2b RT

(5.5)

A simple method of determining the dependence (5.5) is given in [9]. Example 5.1 Sample calculation results, performed according to the formulae (5.1)– (5.5), are presented in Table 5.1. A special case of the discussed radiolocation is a bistatic radar with variable base b and angle β → 0 [11, 12]. In this type of radiolocation, (Fig. 5.4), the angle α → 180◦ , and thus the dependence (5.4), is reduced to R R = .L/2 = RT − b,

5.2 Methods of Determining Object’s Position Using Bistatic Radar System

81

Table 5.1 Sample calculation results, performed according to the formulae (5.1)–(5.5) b = 20,000 m, γ R = 0 rad, .t = 240 μs, c ≈ 2.998 × 108 m/s, .L = c.t ≈ 71952 m ϕ R rad

0

π/4

π/2

3π/4

π

α rad

3.141592654

2.356194490

1.570796372

0.785398163

0

ζ rad

0

0.520364494

1.142457850

1.996614782

3.141592654

β rad

0

0.265033669

0.428338476

0.359579708

0

RR m

35,976.000

37,962.372

43,800.952

51,761.884

55,976.000

RT m

55,976.000

53,989.627

48,151.048

40,190.116

35,976.000

Fig. 5.4 The bistatic radar system with variable base

which is confirmed by the calculation results visible in the second column of Table 5.1. In case of a fixed base b and angle β → 180◦ , we, thus, deal with a radar defined by the acronym FSR, described in Chap. 3.3. The above-described method of determining the object’s position uses measurement data such as direction angles ϕ R and γ R (Fig. 5.3), and the time difference .t = .L/c = (R R + RT − b)/c. A direct measurement of this difference is only possible if the probe signal accounts for a sequence of high frequency radio pulses in succession. Moreover, thanks to the bistatic radar system with a continuous and harmonic probe signal, it is possible to determine the position of a moving object. A modified version of the mentioned method was proposed in [4], in which the Doppler effect frequency of the echo signal is used to determine the sum of the distances L(t) = RT (t) + R R (t). However, due to a large number of limitations and, consequently, restricted practicality, the method of determining object’s position is not discussed in this chapter.

82

5 Bistatic Radar Systems

5.3 Range Equation of the Bistatic Radar System One of the fundamental parameters of any radar system, both monostatic and bistatic, is its range. As the name suggests, the maximum distance Rmax is defined by the range, for which the system is able to detect an object with a precisely defined Bistatic Radar Cross Section σ B (BRCS). In the case of a bistatic system, with the structure shown in Fig. 5.5, the range is defined as the distance between the object O and the receiving unit RU . With no restrictions to generality of considerations, one can assume that the transmitter T and the receiving unit RU are located at points (−b/2, 0, 0) and (+b/2, 0, 0) of the coordinate system (x, y, z), respectively. To simplify further considerations, a polar coordinate system (r, θ ) has been additionally introduced in Fig. 5.5. The squares of distances RT and R R , expressed in the polar coordinate system (r, θ ), are respectively equal to RT2 = (r 2 + b2 /4) + r b cos(θ ),

R 2R = (r 2 + b2 /4) − r b cos(θ )

(5.6)

The formulae (5.6) were attained by using the following dependencies: r 2 = |O Q|2 + [r cos(θ )] 2 , R 2R = |O Q|2 + [r cos(θ ) − b/2] 2 , R 2R − r 2 = b2 /4 − r b cos(θ ), RT2 = |O Q|2 + [r cos(θ ) + b/2] 2 , RT2 − r 2 = b2 /4 + r b cos(θ ) which were obtained using Fig. 5.5. The range equation of the bistatic system combines the distances RT and R R to its energy potential K and the bistatic radar cross section σ B . In order to solve this equation, assume that the transmitter power PT and the power gain of the transmitting antenna G T , are known. For these data, the surface power density in the object’s area is described by the dependence SOb =

PT G T 4π RT2

(5.7)

Fig. 5.5 The bistatic radiolocation system geometry described by means of spherical coordinates

5.3 Range Equation of the Bistatic Radar System

83

Thus, the energy power captured by an object with bistatic radar cross section σ B , is equal to POb = SOb · σB =

PT G T σB 4π RT2

(5.8)

The surface power density which is sent from the object to the receiving unit, totals S RU =

POb 1 PT G T = σB 4π R 2R 4π R 2R 4π RT2

(5.9)

Accordingly, the energy power reaching the receiver is the product of power ) of the density (5.9) by effective aperture area A RU and power efficiency η(RU A receiving antenna, which is expressed by the equation PRU =

1 PT G T 1 PT G T ) σ · (A RU · η RU ) = σ B A RU η(RU A 2 2 B 4π R R 4π RT (4π )2 RT2 R 2R

(5.10)

The effective area A RU of the receiving antenna is related to its power gain G RU as follows A RU =

G RU λ2 ) 4π η(RU A

(5.11)

where λ corresponds to the length of the received wave, [2, 8]. After considering the dependence (5.11), in the expression (5.10) we obtain PRU =

1 PT G T λ2 G RU σB RT2 R 2R (4π )3

(5.12)

Assume that the minimum power value (5.12) of the detected echo signal, limited by the power of noise and various types of interference in the receiver system, is (min) (min) . The echo signal power PRU reaches the receiver RU when the equal to PRU object O is at the edge of the range. Therefore, in this borderline case (min) PRU =

1 PT G T λ2 G RU ) ) σB (4π )3 RT2 R 2R max

(5.13)

The Eq. (5.13) expressed as )

RT2 R 2R

) max

)2 ( ≡ RT(max) R (max) = R

1 PT G T λ2 G RU σB (min) (4π )3 PRU

(5.14)

84

5 Bistatic Radar Systems

describes the contour of a closed area, hereinafter referred to as the coverage area. According to the foregoing considerations, the bistatic radar system will not detect an object with a surface σ B , if it is outside of area determined by Eq. (5.14). Using the concept of energy potential K =

1 PT G T G RU λ2 (min) (4π )3 PRU

(5.15)

and dependence (5.6), the Eq. (5.14) can be expressed in the coordinate system (r, θ ), e.g. (

RT(max) R (max) R

)2

[ ] = (r 2 + b2 /4)2 − r 2 b2 cos2 (θ ) max = K · σ B = χ 2

(5.16)

The Eq. (5.16) demonstrates that the value K and, independent of the angle θ , bistatic radar cross section σ B correspond to the object’s positions (points), which form in the general case, the oval of Cassini, described by the equation (RT R R )max = χ

(5.17)

The √ edge’s shape of the coverage area depends on the size of the parameter χ = K σ B normalized with regard to the base length b. The figures discussed below account for special cases of the oval of Cassini. 1. If b = 0, then we deal with a monostatic radar system, whose coverage area √ is limited by a circle with radius equal to r ≡ R M = RT = R R = χ and coverage surface A M = π R 2R = π χ . 2. For χ > b2 /4, the oval of Cassini accounts for the contour of the coverage area, which is narrower as the parameter χ decreases, Fig. 5.6. The surface of this ellipse-like area may be calculated according to the following approximate formula A B2

)( 8 )] [ ( )( 4 ) ( b b 3 1 − ≈ πχ 1 − 64 χ2 16384 χ4

(5.18)

3. If χ = b2 /4, then the figure described by the Eq. (5.17) corresponds to Bernoulli’s lemniscate, whose vertex is located at the origin of the polar system (r, θ ). According to Eq. (5.16), r = 0 at the vertex point. In this particular case, the coverage area is equal to A B3 = 2χ . 4. If χ < b2 /4, then the coverage area consists of two disjoint and symmetrical to each other subdomains, comprising the transmitter T and the receiving unit RU , respectively. In compliance with [12], the total area of the mentioned subdomains can be calculated according to the formula

5.3 Range Equation of the Bistatic Radar System

85

Fig. 5.6 The coverage area as a function of χ parameter √ √ Table 5.2 Sample values of the function A B (b/ χ)/A M calculated for 0 ≤ b ≤ 3 χ √ √ √ χ b 0 2 χ 3 χ √ 1 ≈ 0.9842 2/π ≈ 0.6366 ≈ 0.2281 A B (b/ χ)/ A M

A B4 ≈

[ ] 2χ 2 2π χ 2 12χ 4 100χ 6 1 + + + b2 b4 b8 b12

(5.19)

The dependencies (5.18) and (5.19) show that the surface of the coverage area A B of the bistatic system is explicitly dependent on the ratio of the base length b to the √ √ parameter χ. Data presented in Table 5.2, illustrate the function A B (b/ χ )/A M , where A M = π χ denotes the surface of the coverage area of the monostatic system (b = 0) with the same energy potential. Data included in Table 5.2, confirm a conclusion established in the literature √ that the area A B (b/ χ) is always smaller than the area A M . However, it should be √ emphasized that this difference between both areas is small when 0 < b ≤ χ . In the foregoing considerations, the only factor limiting the coverage area was the minimum level of the echo signal, which enabled object’s detection. In terrestrial and marine long-range bistatic systems, the coverage area may be additionally limited by the phenomenon of negative electromagnetic wave refraction in the troposphere, attenuation of the troposphere, as well as shadow effect following from the Earth’s curvature.

86

5 Bistatic Radar Systems

The attenuation of the troposphere can be taken into account in a similar manner as for the monostatic radiolocation described in the previous chapter. In this case, the following expressions are the equivalents of dependencies (5.7), (5.8), (5.9) and (5.13): (γ )

SOb = (γ )

PT G T (γ ) 4π (RT )2

(γ )

POb = SOb · σB = (γ )

(γ )

S RU = = (min) PRU =

POb

(γ ) 4π (R R )2

(γ )

exp(−0.23 γ RT )

PT G T σ B (γ ) 4π (RT )2

(5.20) (γ )

exp(−0.23γ RT )

(5.21)

(γ )

exp(−0.23γ R R )

1

PT G T σ B

(γ ) 4π (R R )2

(γ ) 4π (RT )2

(γ )

(γ )

exp(−0.23γ R R ) exp(−0.23γ RT )

(5.22)

PT G T λ2 G RU σ B 1 (max,γ ) (max,γ ) ) exp(−0.23γ RT ) ) exp(−0.23γ R R 3 ( (4π ) R (max,γ ) R (max,γ ) 2 T R (5.23)

The dependencies (5.23) and (5.14) yield the following equation (

(max,γ )

RT

(max,γ )

RR

)

( ) [ ] (max,γ ) (max,γ ) = RT(max) R (max) exp −0.115γ (R + R ) (5.24) T R R

The monostatic radar discussed in Chap. 4 can be treated as a borderline case (max,γ ) of bistatic radar, for which b = 0, β = 0, σ B ≡ σ M and consequently, RT = (max,γ ) (max,γ ) RR = RM . In this borderline case, the following dependence is obtained from the general Eq. (5.24) (max,γ )

RM

) ( (max,γ ) = R (max) exp −0.115γ R M M

(5.25)

which is( identical to Eq. ) (4.11). When solving Eq. (5.24) numerically, the (max,γ ) (max,γ ) RR is attained which is always smaller than the product product RT ) ( (max) (max) RT R R , because the troposphere’s attenuation rate γ is greater than zero. Thus, the system coverage area, taking into consideration tropospheric attenuation, is always smaller than the corresponding coverage area, from which the product of the energy potential K and the area σ B follow. The second of the mentioned limits is related to the object’s direct visibility condition, both by the transmitter and the receiving unit. The essence of this limitation is presented in Fig. 5.7. The longest segment of direct visibility (along a straight line) of the object O through the transmitter T , is the segment |T O| = |T PS | + |Ps O| visible in Fig. 5.7,

5.3 Range Equation of the Bistatic Radar System

87

Fig. 5.7 Geometric illustration of the radar range limitation due to the curvature of the Earth’s surface

where: |T Ps | = |Ps O| =

/

(R E + h T )2 − R 2E =

/

2R E h T + h 2T ,

/ / (R E + HO )2 − R 2E = 2R E HO + HO2 .

In English-language based literature, the longest segment of direct visibility is referred to as Line Of Sight (LOS). The heights h T and HO which correspond to transmitter and object vertical location, respectively, are disproportionately small in comparison to the radius √ of√the | |T ≈ P 2R E h T Earth. Thanks√to this, the following approximations are valid: s √ and |Ps O| ≈ 2R E HO . The direct visibility area of the object O by the transmitter T accounts for a circle with radius (. . . ) (5.26) L T = |T O| ≈ 2R E h T + HO and the centre at the location point of this transmitter. Frequently, this circle is referred to as the geometric range circle of the transmitter. The direct visibility area of the object O by the receiving unit RU , determined by a similar method, is a circle with a radius (. . . ) (5.27) L RU ≈ 2R E h RU + HO and the center coincident with the location point of RU. In order to guarantee the correct operation of the bistatic system, the observed object must be located within the direct visibility range, both, by the transmitter and the receiving unit. The following condition leads to conclusion that the coverage area

88

5 Bistatic Radar Systems

Fig. 5.8 Coverage areas of the bistatic radiolocation system; a without a “dead zone”, b with the dead zone marked by dashes

limited by the Earth’s curvature, is an intersection of two areas of direct visibility discussed above. The area AC , which is significant from the practical viewpoint, may be both larger or smaller than the coverage area A B . The Fig. 5.8 depicts two typical scenarios for this problem. Figure 5.8a illustrates the situation where the area AC covers excessively the entire area A B . In this case, the curvature of the Earth does not limit the coverage area A B . The opposite situation is illustrated in Fig. 5.8b, where the left flank and right flank of the area A B were cut off by circles with radii L RU and L T , see Eqs. (5.27) and (5.26). Thus, two dead zones were created, which are marked by hatching in Fig. 5.8b. It is not difficult to imagine even more disadvantageous situation, namely, when the radii L RU and L T are shorter than the base b. Recording the sketches presented in Fig. 5.8, it can be concluded that the most appropriate system configuration is marked by the area AC , which covers the entire area A B with a slight excess. According to the dependencies derived before, the surface and shape of the area A B depend on the potential K , area σ B and base length b. In turn, the area AC is proportional to the height h T , h RU and HO , which the transmitter antenna T , the receiver antenna RU and the object O are respectively located on. Thus, when the parameters K , σ B , b and HO(min) are determined, the optimization of the system spatial configuration indicates selecting the appropriate values h T and h RU .

5.4 Searching Space Using the Probe Signal Chasing Method The available literature sources suggest that bistatic systems are currently mainly used to search the narrow 3D space sectors and to track the detected object. In mentioned applications, the following conditions must be satisfied simultaneously:

5.4 Searching Space Using the Probe Signal Chasing Method

89

Fig. 5.9 Detection cells formed in an area of the intersection of the transmitting beam and the receiving beam; a completely filled with electromagnetic energy, b partially filled with electromagnetic energy

• the detection cell obtained due to the intersection of the angularly narrow beams of transmitting and receiving antennas, must be placed within the coverage area, • in the designated cell there is an object (with a reflecting surface σ B ), • the incident electromagnetic energy of the probe signal on this object. The selected 3D space sector search is most frequently conducted according to the probe pulse chasing method, whereby the receiving beam follows the probe signal [6], while the position of the transmitting beam is fixed. Assume that the axes of the transmitting beam BT and receiving beam BR intersect the detection cell placed in the coverage area, Fig. 5.9. Depending on the duration, the probe signal may completely or partially fill this cell. It is not difficult to verify that the length of the cell presented in Fig. 5.9b totals . R = c ti where ti denotes the duration of the probe signal. In this situation, time ti limits the length of the detection cell. As mentioned before, in this cell detection can occur if at a given time instant the cell contains an object with a sufficiently large bistatic radar cross section σ B illuminated by the probe signal. The essence of the described space chasing method, is such change in the angular position of both, transmitting and receiving beams, i.e. their direction angles ψT (t) and ψ R (t), so that the above conditions are satisfied, with the maximum use of the energy potential of the entire system. The position of both mentioned beams is presented in Fig. 5.10. Assume that the axis of the transmitting beam BT overlaps with the fixed direction of observation defined by the angle ψT . The observation starts when the transmitter T sends to the receiving unit RU at the time t0 the synchronization pulse u s , together with the preamble signal containing the current value of angle ψT . The pulse u s will reach the receiving unit after b/c time, i.e. at the moment t s = t0 + b/c where c ≈ 3 × 108 m/s. After a strictly determined code delay .tk > b/c, the transmitter sends a probe signal that reaches the spatial cell V1 after time |V1 T |/c, i.e. at t1 = t0 +.tk +|V1 T |/c. Assume that there is an object positioned in cell V1 , which reflects an electromagnetic energy. Thus, the signal reflected from this object will reach the receiving unit antenna

90

5 Bistatic Radar Systems

Fig. 5.10 Graphical illustration of the “chasing probing impulse” method

after time period |V1 RU |/c, i.e. at the time instant t1(RU ) = t0 + .tk + |V1 T |/c + |V1 RU |/c. This means that at the moment t1(RU ) the directivity pattern main lobe of the receiving antenna should be directed at the angle ψ R and its intersection with the main lobe of the transmitting antenna pattern determines the cell V1 . The above condition is satisfied when [ ] |V1 T | sin(ψT ) ψ R (t0 + .tk + |V1 T |/c) + |V1 RU|/c) = arctg (5.28) b − |V1 T | cos(ψT ) where: / |V1 RU | = |V1 T |2 sin2 (ψT ) + [b − |V1 T | cos(ψT )]2 / = |V1 T |2 + b2 − 2b|V1 T | cos(ψT ). The above dependence determination is described in [9]. The distance | V1 T | of the dependence, can be replaced by the product c · (t1 − t0 − .t k ), where t1 corresponds to time counted from the moment of sending the probe signal. Dependence (5.28) should be satisfied for each different cell Vi position (i.e. V0 , V1 , V2 , V3 . . .). In other words, the chasing condition of the cell Vi after the moving probe signal, should be satisfied in the range 0 ≤ ψ R ≤ ψ R max (ψ T ), whose upper limit ψ R max (ψ T ) is limited by the maximum range, represented in Fig. 5.10 by the arc of the oval of Cassini. Once the observation along a given direction ψT is performed, the observation direction changes by an angle .ψT (approximately equal to 3 dB of the transmitting beam width) and the entire search procedure is repeated again. When the object is detected, the position of this object is determined using the current value of the direction angle ψT and distance |V1 RU |, presented in Eq. (5.28). In order to illustrate and evaluate the applicability of the above-described search method, assume that b = 150 km, ψT = 80◦ , .tk = 10−3 s and the maximum range for direction

References

91

ψT = 80◦ equals |V T |max = 270 km. In this case, b/c = 5 × 10−4 s thus, the condition b/c < .tk is satisfied. The maximum range value of the probe signal will be reached after time |V T |max /c = 0.9 × 10−3 s. In this case |V RU |max = 285.102 km, the transition time of the reflected signal from the object to the receiving unit is equal to |V RU |max /c ≈ 0.95 × 10−3 s. The total transition time of the probe signal from the transmitting antenna to the receiving antenna is equal to ( |V RU |max + |V T |max )/c ≈ 1.85 × 10−3 s. During this relatively short time interval, the receiving beam should change its angular position from ψ R min = 0◦ to ψ R max (ψT ) ≈ 68◦ . In order to satisfy the above requirement, the multi-element array, with phase—phase or digital directivity pattern forming control, needs to be used as a receiving antenna. For the antennas mentioned above, the computed scan range 0–68° can be considered as equivalent to the range − 34 to 34°, which was defined with respect to their normal aperture. This approach simplifies the scanning process and reduces the receiving beam deformation resulting from its inflection. The above-mentioned dependencies demonstrate that the angular velocity of the receiving beam changed position, i.e. dψ R (t)/dt, is a nonlinear time function, dependent on the observation angle ψT , which is varied within the range ψTmin ≤ ψT ≤ ψT max .

References 1. Averianov V (1978) Multistatic radar stations and systems (in Russian). Science and technology, Minsk (Belarus) 2. Balanis CA (1997) Antenna theory, analysis and design, 2nd edn. Wiley, New York 3. Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood (MA) 4. Bljachman AB, Kowaljev FN, Ryndyk AG (2001) A method of determining the coordinates of a moving target using bistatic RLS (in Russian). (A.S.Popov) Radio—engineering, No. 1, c. 4–9 5. Chernyak VS (1998) Fundamentals of multisite radar systems. Gordon and Breach Science Publishers, Amsterdam 6. Hanle E (1986) Survey of bistatic and multistatic radars. In: IEE proceedings, vol 133, Pt. F, No 7, pp 587–595 7. Jackson MC (1986) The geometry of bistatic radar systems. In: IEE proceeding, vol 133, Pt. E., No 7, pp 604–612 8. Rosłoniec S (2006) Fundamentals of the antenna technique (in Polish). Publishing house of the Warsaw University of Technology, Warsaw 9. Rosloniec S (2020) Fundamentals of the radiolocaction and radionavigation (in Polish), 2nd edn . Publishing house of the Military University of Technology, Warsaw 10. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 11. Willis NJ (2005) Bistatic radar, 2nd edn. Scitech Publishing Inc., Releigh (NC) 12. Willis NJ, Griffiths HD (eds) (2007) Advances in bistatics radar. Scitech Publishing Inc., Releigh (NC)

Chapter 6

Multistatic Radar Systems

One of the fundamental principles of modern radiolocation is the use of possibly all properties of an echo signal containing information about an object from which the echo signal was reflected. In case of the object observed by a multistatic system, this information may be contained in: the object’s distance from the transmitter, the distance differences of the object from the receivers distant from each other and the frequency differences of the echo signals (following from the Doppler effect) received by individual receivers, [1, 4, 17]. Both the distance differences of the reception time and the frequency differences, can be used to determine the object’s position and its velocity vector, applying the Time Differences Of Arrival method (TDOA) and Frequency Differences Of Arrival method (FDOA), [4, 7, 12]. From the practical point of view, the most interesting and important scenario corresponds to the situation in which an object moves in 3D space. In order to determine the object’s position and object’s velocity vector, various multistatic systems may be used, provided they consist of at least one transmitter and four receivers. The necessity of this condition can be demonstrated as follows. When attempting to determine the coordinates (x, y, z) of an object, the application of TDOA method in essence comes down to determining the three time differences or the corresponding three distance differences. In order to determine these differences, at least four measurements of the echo signals’ receiption time by four mutually distant receivers, are necessary. In other words, the solution of the system of four nonlinear equations formulated for the system consisting of four receivers, gives the three object’s coordinates and the object’s distance from the reference receiver, relative to which the distance differences are determined.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_6

93

94

6 Multistatic Radar Systems

The presented justification also applies to the measurement condition of at least four echo signal frequencies, which can be used to determine three frequency differences, and then three components (ϑx , ϑ y , ϑz ) of the velocity vector and the incident wave frequency by applying the FDOA method. The Main Advantages of the Multistatic Radar Systems The research results presented so far in the available literature show that in the case of the multistatic systems with several transmitters, it is possible to operate in the so-called ring, which is marked by a random selection of the transmitter illuminating the detected objects. As a result, the probability of detecting the emission source and its subsequent possible destruction is reduced. The multistatic system with MTD radars enables determining the position and temporary velocity vector value of the observed object, using four radial velocities, or four Doppler frequencies, measured by four receiving units deployed in 3D space. This property is especially useful for various types of rockets identification. The Application Example of the Multistatic System The Space Surveillance system (SPASUR), [1, 16] ís an example of the multistatic system that has been used for over five decades to detect and track artificial Earth satellites and various types of “space debris”. The active segment of this system consists of three transmitters located in Jordan Lake (AL), Lake Kickopoo (TX) and Gila River (AZ). They emit continuous-time signals with a carrier frequency f = 216.98 MHz ± 1 kHz. The strongest transmitter, with 1 MW power, is located in Lake Kickopoo (TX). These transmitters feed multi-element linear antenna arrays of approximately 3.3 km length, forming a fan of three beams which are directed towards sky. The width of these beams, determined at the half power level in the meridional (north–south) plane, does not exceed 0.5°. A fan, presented in Fig. 6.1, with a span of approximately 1500 nautical miles, extends from San Diego (CA) to Savannah (GA). The passive segment of the system consists of six receiving stations located in Tattnal (GA), Hawkinsvile (Ga), Silver Lake (MS), Red River (AR), Elephant Buttle (NM), and San Diego (CA).

Fig. 6.1 The coverage (observation) area of the SPASUR system

6.1 The Method of Determining the object’s Position Using …

95

Antennas of the receiving stations are also expressed in the form of multi-element, linear arrays of a length close to 1 km. By using the phase interferometry method, the antennas determine the reception directions of signals which are reflected from the observed object. Thus, the object’s position is determined by a triangulation method using the echo signals, received by at least two receiving stations [1, 17]. The reception of these relatively weak signals, is carried out using the correlation methods, which involves ensuring the synchronization (in real time) of its individual stations. The control center of the entire system, as well as the data processing center obtained from the receiving stations (SPASUR Operations Center), is located in Dahlgren (VA). The results obtained over 50 years of the system exploitation, confirm its high efficiency. According to the specialists’ statements exploiting the mentioned system, 99.9% of all objects penetrating through its observation zone (Fig. 6.1) was detected within 3 years, during its first passage. Detecting small objects, such as fragments of rocket engine casing, various types of metal connectors and electric wires, prove that the discussed system is highly distinguishable [1, 17].

6.1 The Method of Determining the object’s Position Using a Multistatic System with One Transmitter and Four Receivers The first part of this chapter describes a multistatic system with one transmitter and four receivers. The organizational structures of two versions of the considered system, i.e. with a stationary transmitter and with a moving transmitter, are shown in Figs. 6.2 and 6.3, respectively. At this point, it should be emphasized that the version of the system presented in Fig. 6.3 is particularly interesting for military applications, due to its high resistance to various countermeasures, including self homing ARM missiles. Each of these structures is marked by one independent transmitter T and four receivers, hereinafter referred to as passive receiving units R0 , R1 , R2 and R3 , whence R0 additionally serves as a master station. As a general rule, the master station controls the work of other three stations, called slave stations, and processes measurement data obtained throughout the system. In further considerations, it is assumed that the coordinates (xi , yi , z i ) of the receiving units Ri , where i = 0, 1, 2, 3, are known. The measured time instant ti , at which the echo signals reflected from the object O reach these units, are also known. As mentioned earlier, these time instants are determined in the correlation reception process. The reference signals, which are essential for the reception (copies of the probe signal), are delivered from the transmitter T to individual units Ri via special microwave transmission lines or optoelectronic links. Thus the only unknowns for the considered problem are the coordinates (x, y, z) of the object O, which can be determined by the TDOA method. According to the idea of the TDOA method, the

96

6 Multistatic Radar Systems

Fig. 6.2 The geometry of a multistatic system with four receivers

Fig. 6.3 The geometry of a multistatic system with four ground-based receivers and an aircraft-based transmitter

distance squares of the receiving units R0 , R1 , R2 and R3 from the object O are equal to di2 = (x − xi )2 + (y − yi )2 + (z − z i )2

where

i = 0, 1, 2, 3

(6.1)

Taking into consideration (6.1), the differences of the distance squares di2 and d02 are determined, e.g. di2 − d02 = (x − xi )2 + (y − yi )2 + (z − z i )2 − (x − x0 )2 − (y − y0 )2 − (z − z 0 )2 = x2(x0 − xi ) + y2(y0 − yi ) + z2(z 0 − z i ) − (x02 + y02 + z 02 ) + (xi2 + yi2 + z i2 ) = (di − d0 )(di + d0 ) = .di (di + d0 ) = .di (.di + 2d0 )

(6.2)

6.1 The Method of Determining the object’s Position Using …

97

where .di = di − d0 for i = 1, 2, 3. Using (6.1) and (6.2) one obtains the following system of algebraic equations [A] [X ] = [B]

(6.3)

where ⎤ ⎡ ⎤ x x0 − x1 y0 − y1 z 0 − z 1 [A] = ⎣ x0 − x2 y0 − y2 z 0 − z 2 ⎦, [X ] = ⎣ y ⎦ x0 − x3 y0 − y3 z 0 − z 3 z ⎤ ⎤ ⎡ ⎡ d0 .d1 + k1 2d .d + .d12 + (x02 + y02 + z 02 ) − (x12 + y12 + z 12 ) 1⎣ 0 1 [B] = 2d0 .d2 + .d22 + (x02 + y02 + z 02 ) − (x22 + y22 + z 22 ) ⎦ = ⎣ d0 .d2 + k2 ⎦ 2 d0 .d3 + k3 2d0 .d3 + .d32 + (x02 + y02 + z 02 ) − (x32 + y32 + z 32 ) ⎡

ki =

] 1[ 2 .di + (x02 + y02 + z 02 ) − (xi2 + yi2 + z i2 ) for i = 1, 2, 3 2

The distance differences .di are calculated using the measured time differences .ti0 = ti − t0 of the receiving echo signals and are respectively equal to .di = di − d0 = c.ti0 . The system of Eqs. (6.3) can be solved calculating an inverse matrix method with respect to the matrix of coefficients [A], [3, 10]. Assume that the inverse matrix is known with respect to the matrix [A], i.e. ⎤ a11 a12 a13 = ⎣ a21 a22 a23 ⎦ a31 a32 a33 ⎡

[A]−1

(6.4)

The formulae for calculating the elements ai j of the inverse matrix using the elements of matrix [A], are given in Appendix B. Due to a premultiplication of both sides of the Eq. (6.3) by the matrix (6.4), one obtains [A]−1 [A] [X ] = [X ] = [A]−1 [B]

(6.5)

In order to ensure the transparency of the further considerations, the Eq. (6.5) should be expressed in the following expanded form ⎡ ⎤ ⎡ ⎤⎡ ⎤ x d0 .d1 + k1 a11 a12 a13 ⎣ y ⎦ = ⎣ a21 a22 a23 ⎦ ⎣ d0 .d2 + k2 ⎦ a31 a32 a33 d0 .d3 + k3 z The above-mentioned Eqs. (6.6) demonstrate that x = a11 (d0 .d1 + k1 ) + a12 (d0 .d2 + k2 ) + a13 (d0 .d3 + k3 ) = d0 n 1 + m 1 y = a21 (d0 .d1 + k1 ) + a22 (d0 .d2 + k2 ) + a23 (d0 .d3 + k3 ) = d0 n 2 + m 2

(6.6)

98

6 Multistatic Radar Systems

z = a31 (d0 .d1 + k1 ) + a 32 (d0 .d2 + k2 ) + a33 (d0 .d3 + k3 ) = d0 n 3 + m 3 (6.7) . . where: m i = 3j=1 ai j k j , n i = 3j=1 ai j .d j for i = 1, 2, 3. After considering the Eqs. (6.7) in the expression d02 = (x − x0 )2 + (y − y0 )2 + (z − z 0 )2 , one obtains d02 = [(m 1 − x0 ) + d0 n 1 ]2 + [(m 2 − y0 ) + d0 n 2 ] 2 + [(m 3 − z 0 ) + d0 n 3 ] 2 (6.8) The Eq. (6.8) can be expressed as a standard quadratic equation αd02 + βd0 + γ = 0

(6.9)

where α = n 21 + n 22 + n 23 − 1 β = 2[(m 1 − x0 )n 1 + (m 2 − y0 )n 2 + (m 3 − z 0 )n 3 ] γ = (m 1 − x0 )2 + (m 2 − y0 )2 + (m 3 − z 0 )2 . When solving this equation, the root d0 = (−β− β 2 − 4αγ )/(2α) is determined and then, according to the formulae (6.7), the coordinates (x, y, z) of the detected object O are calculated.

6.2 The Method of Determining the Velocity Vector of an Object in 3D Space The position of the object’s velocity vector in the coordinate system (x, y, z) is shown in Fig. 6.4. In accordance with the agreed indications, the vector ϑ can be represented by its components ϑx = ϑ cos(γo ) cos(ϕo ), ϑ y = ϑ cos(γo ) sin(ϕo ), ϑz = ϑ sin(γo )

(6.10)

Conversely, when the components ϑx , ϑ y and ϑz are known, a modulus of the vector ϑ and its direction angles can be calculated according to the following formulae ϑ=

/

ϑx2 + ϑ y2 + ϑz2 ,

γo = arcsin(ϑz /ϑ),

ϕo = arctg(ϑ y /ϑx )

(6.11)

In order to determine an orthogonal projection of the vector ϑ onto the straight line di passing through the object O and ith of the receiving unit Ri , the orthogonal projection of its individual components ϑx , ϑ y and ϑz onto the straight line should be determined first, Fig. 6.5.

6.2 The Method of Determining the Velocity Vector …

99

Fig. 6.4 Graphic illustration of the spatial position of the velocity vector; a the general view of the vector position, b the velocity vector and its components Fig. 6.5 The perpendicular projections of components of the velocity vector on the line di

100

6 Multistatic Radar Systems

The projection of the vector ϑ onto the straight line di which is shown in the Fig. 6.5, yields the dependence ϑdi = ϑi x + ϑi y + ϑi z = −ϑx cos(ψi ) cos(γi ) + ϑ y sin(ψi ) cos(γi ) + ϑz sin(γi ) (6.12) As shown in Fig. 6.5 the vector ϑdi is oppositely directed towards the radial velocity ϑri , which is used to determine the Doppler frequency translation. In this case, the dependence f i = f Ob (1 + ϑri /c) = f Ob (1 − ϑdi /c) is valid, where f Ob corresponds to the frequency of the incident electromagnetic wave on the object O and f i denotes the frequency of the electromagnetic wave (of the echo), reaching ith receiving unit. The above-mentioned dependence shows that ϑdi = c(1 − f i / f Ob ). Considering the general dependence (6.12), for the individual receiving units (i = 0, 1, 2, 3), the following equations can be written − ϑx cos(ψ0 ) cos(γ0 ) + ϑ y sin(ψ0 ) cos(γ0 ) + ϑz sin(γ0 ) = ϑd0 = (1 − f 0 / f Ob ) c − ϑx cos(ψ1 ) cos(γ1 ) + ϑ y sin(ψ1 ) cos(γ1 ) + ϑz sin(γ1 ) = ϑd1 = (1 − f 1 / f Ob ) c − ϑx cos(ψ2 ) cos(γ2 ) + ϑ y sin(ψ2 ) cos(γ2 ) + ϑz sin(γ2 ) = ϑd2 = (1 − f 2 / f Ob ) c − ϑx cos(ψ3 ) cos(γ3 ) + ϑ y sin(ψ3 ) cos(γ3 ) + ϑz sin(γ3 ) = ϑd3 = (1 − f 3 / f Ob ) c (6.13) Then considering the Eq (6.13), the following system of three equations is formulated ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A11 A12 A13 ϑx ϑd1 − ϑd0 −(c/ f Ob ). f 10 ⎣ A21 A22 A23 ⎦⎣ ϑ y ⎦ = ⎣ ϑd2 − ϑd0 ⎦ = ⎣ −(c/ f Ob ). f 20 ⎦ (6.14) A31 A32 A33 ϑz ϑd3 − ϑd0 −(c/ f Ob ). f 30 where Ak1 = − cos(ψk ) cos(γk ) + cos(ψ0 ) cos(γ0 ) for k for k Ak2 = sin(ψk ) cos(γk ) − sin(ψ0 ) cos(γ0 ) for k Ak3 = sin(γk ) − sin(γ0 ) for k . f i0 = f i − f 0

= 1, 2, 3 = 1, 2, 3 = 1, 2, 3 = 1, 2, 3

Assume that the inverse matrix with regard to the matrix [A] of the system of Eq. (6.14) is known, i.e. ⎡

[A]−1

⎤ a11 a12 a13 = ⎣ a21 a22 a23 ⎦ a31 a32 a33

(6.15)

The formulae for calculating the elements ai j of this matrix using the elements of the matrix [A], are given in Appendix B. Consequently, due to the premultiplication

6.3 The Simulation Tests Results

101

of both sides of the system (6.14) by the matrix (6.15), one obtains ⎡

⎤ ⎡ ⎤⎡ ⎤ ϑx a11 a12 a13 −(c/ f Ob ). f 10 ⎣ ϑ y ⎦ = ⎣ a21 a22 a23 ⎦ ⎣ −(c/ f Ob ). f 20 ⎦ ϑz a31 a32 a33 −(c/ f Ob ). f 30 ⎤ ⎡ ( ) a . f + a12 . f 20 + a13 . f 30 c ⎣ 11 10 =− a21 . f 10 + a22 . f 20 + a23 . f 30 ⎦ f Ob a31 . f 10 + a32 . f 20 + a33 . f 30

(6.16)

After substituting components ϑx , ϑ y and ϑz , which are described by Eq. (6.16), to the first of Eq. (6.13) and implementing elementary transformations, the following dependence is obtained f Ob = (a11 . f 10 + a12 . f 20 + a13 . f 30 ) cos(ψ0 ) cos(γ0 ) − (a21 . f 10 + a22 . f 20 + a23 . f 30 ) sin(ψ0 ) cos(γ0 ) − (a31 . f 10 + a32 . f 20 + a33 . f 30 ) sin(γ0 ) + f 0

(6.17)

After substituting the calculated unknown f Ob to Eq. (6.16), the component values ϑx , ϑ y and ϑz are determined. Afterwards, according to the formulae (6.11), the modulus and direction angles of the sought vector ϑ of the object’s instantaneous velocity are calculated.

6.3 The Simulation Tests Results In order to verify the correctness of the methods for determining the position of the object and velocity vector, presented above, the simulation tests were conducted, whose results are given in the two examples presented below. Example 6.1 The subject considered in this example corresponds to a multistatic radar system, whose four receiving units are placed at the points with the coordinates (xi , yi , z i ) shown in Table 6.1. The time differences .ti0 = .di0 /c, which are essential to perform the calculations, were determined assuming initially that the coordinates (x O , y O , z O ) of the object O flying at high altitude are equal to: x O = 9500 m, y O = 49, 000 m, z O = 9120 m. Considering these data, the determined distances di , times ti = di /c and time differences .ti0 = .di0 /c, are presented in the fifth, sixth and seventh columns of Table 6.1, respectively. In the second, fundamental stage, considering the differences .ti0 , the object’s distances di∗ and the coordinates (x O∗ , y O∗ , z ∗O ) were calculated using the method described in Sect. 6.1. The obtained results are shown in Table 6.2. The table also includes the previously shown in Fig. 6.4a values of the direction angles ϕi and γi , at which the object O is visible considering the individual receiving units O, where i = 0, 1, 2, 3.

102

6 Multistatic Radar Systems

Table 6.1 Input data i

xi , m

yi , m

zi , m

di , m

ti , μs

.ti0 , μs

0

0

0

0

50,738.785

169.129283854

0

1

− 30,000

− 20,000

30

80,024.234

266.747447916

97.618164062

2

− 18,000

40,000

− 100

30,368.707

101.229023437

− 67.900260416

3

25,000

15,000

70

38,446.750

128.155833333

− 40.973450520

Table 6.2 Results of calculations ∗ = 9499.996 m, y ∗ = 48999.992 m,z ∗ = 9119.983 m Object’s coordinates: x O O O

i

di∗ , m

δdi = di∗ − di , m

ϕi , rad

γi , rad

0

50,738.773

− 0.012

1.379294

0.177844

1

80,024.226

− 0.008

1.050870

0.113105

2

30,368.697

− 0.010

0.316285

0.294757

3

38,446.738

− 0.012

− 1.143061

0.231181

The direction angles were calculated according to the following formulae ( ) ) z O − zi y O − yi , γi = arctan . ϕi = arctan x O − xi (x O − xi )2 + (y O − yi )2 (

(6.18)

In order to verify the correctness of the method of determining the velocity vector described in Sect. 6.2, assume that the observed object flies with the velocity ϑ = 200 m/s towards the direction specified by the angles ϕ O = 0.68 rad and γ O = 0.35 rad, see Fig. 6.4b. Moreover, assume that the frequency of the object’s incident wave is equal to f Ob = 1010 Hz. The frequencies of the echo signals reaching the individual receiving units will be changed due to the Doppler effect according to dependencies (6.13). The values of the angles ψi occurring in these dependencies are related to the angles ϕi and equality ψi = π − ϕi which are listed in Table 6.2. The frequencies of echo signals reaching the individual receiving units, calculated according to dependencies (6.13), and the differences . f i0 = f i − f 0 , are listed respectively in the second and third columns of Table 6.3. Table 6.3 Input data i

f i , Hz

. f i0 , Hz

ψi , rad

γi , rad

0

9,999,994,878.526102

0

1.762298

0.177844

1

9,999,993,942.575457

− 935.950644

2.090722

0.113105

2

9,999,993,735.511324

− 1143.014777

2.825306

0.294757

3

10,000,000,997.73308

6119.206980

4.284653

0.231181

6.3 The Simulation Tests Results

103

The frequency differences . f i0 and direction angles ψi and γi , compiled in Table 6.3, considering that i = 0, 1, 2, 3, constitute an input data determining the instantaneous velocity vector ϑ using the FDOA method which was described before. Due to the calculations the ϑx = 146,086 m/s, ϑ y = 118,134 m/s and ϑz = 68,579 m/s, are obtained. According to (6.11), the modulus and the direction angles of this vector are equal to ϑ = 200 m/s, ϕ O = 0.68 rad and γ O = 0.35 rad. The obtained solution is consistent with assumptions used to determine the input data shown in Table 6.3. The impact of the measuring errors for time .ti0 and frequency . f i0 differences on the object’s coordinates and the object’s velocity vector, are illustrated by the results summarized in Table 6.4. The first calculations were made assuming that . f i0 are determined with an error less than 10−3 Hz(shown in the second column of Table 6.3) and for the differences .ti0 the error growths due to numerical truncation errors. For example, the notation δti0 = 10−8 s corresponds to the calculations assuming that t0 = 169.12 µs, t1 = 266.74 µs, t2 = 101.22 µs and t3 = 128.15 µs, instead of the accurate values presented in the sixth column of Table 6.1. Thus, the notation δ f i = 101 s indicates that the calculations were performed for f 0 = 9, 999, 994, 870 Hz, f 1 = 9, 999, 993, 940 Hz, f 2 = 9, 999, 993, 730 Hz and f 3 = 10, 000, 000, 990 Hz, shown in the second column of Table 6.3. The values of the deviations of the object’s Table 6.4 Results of calculations Detailed object’s coordinates: x O = 9500 m, y O = 49,000 m, z O = 9120 m δti , s

δ f i , Hz

x, m

y, m

z, m

. p [m]

|.ϑ|[m/s]

δ f i = 10−3 Hz ,10−12 ≤ δti ≤ 10−6 s 10−12

10−3

9499.996

48,999.992

9120.000

0.019

< 10−4

10−9

10−3

9500.344

49,000.529

9121.429

1.626

< 2.8 · 10−3

10−8

10−3

9497.805

48,995.429

9108.651

12.477

24.4 · 10−3

10−7

10−3

9518.105

49,038.841

9225.030

113.402

187 · 10−3

10−6

10−3

9657.590

49,678.621

11,390.825

2375.266

1.104

δti = 10−8 s, 10−2 ≤ δ f i ≤ 102 Hz 10−8

10−2

9497.805

48,995.428

9108.650

12.430

24.2 · 10−3

10−8

10−1

9497.805

48,995.428

9108.650

12.430

19.9 · 10−3

10−8

100

9497.805

48,995.428

9108.650

12.430

13.8 · 10−3

10−8

101

9497.805

48,995.428

9108.650

12.430

0.242

10−8

102

9497.805

48,995.428

9108.650

12.430

1.025

δti = 10−7 s, 10−2 ≤ δ f i ≤ 102 Hz 10−7

10−2

9518.107

49,038.847

9225.057

113.46

0.187

10−7

10−1

9518.107

49,038.847

9225.057

113.46

0.191

10−7

100

9518.107

49,038.847

9225.057

113.46

0.197

10−7

101

9518.107

49,038.847

9225.057

113.46

0.023

10−7

102

9518.107

49,038.847

9225.057

113.46

1.262

104

6 Multistatic Radar Systems

Table 6.5 Input data i

xi , m

yi , m

zi , m

di , m

ti , µs

.ti0 , µs

0

0

0

0

168,195.046

560.650156250

0

1

−30,000

−20,000

30

191,816.921

639.389739583

78.739583333

2

−18,000

40,000

−100

130,630.156

435.433854166

− 125.216302084

3

25,000

15,000

70

153,935.000

513.116666666

− 47.533489958

.p[m] position, shown . in Table 6.4, were calculated according to the following (x − x O )2 + (y − y O )2 + (z − z O )2 , where (x O , y O , z O ) formula .p[m] = correspond to the accurate values of the object’s coordinates, adopted for the simulation/calculations. Similarly, the velocity differences |.ϑ|[m/s] should be perceived as ϑx2 + ϑ y2 + ϑz2 − ϑ, where ϑ = 200 m/s denotes the object’s velocity, assumed for the purpose of the calculation. Example 6.2 One of the potential applications for the considered system, corresponds to the earliest possible detection of objects flying at low altitudes. Therefore, using the example 6.2, the calculations analogous to the calculations described in example 6.1 were performed, changing only the object’s coordinates (x O , y O , z O ), accordingly. The time differences .ti0 used in the present example, were determined assuming initially that x O = 8070 m, y O = 168, 000 m and z O = 670 m. The distances di , times ti = di /c and time differences .ti0 = .di0 /c determined for the object’s position, are expressed in the fifth, sixth and seventh columns of Table 6.5. The coordinates (x O∗ , y O∗ , z ∗O ) and the distances di∗ of the detected object, calculated using the differences .ti0 , are shown in Table 6.6. The Table 6.6 also includes the values of the direction angles ϕi and γi , Fig. 6.4a, at which the object O is visible from the individual receiving units Ri , where i = 0, 1, 2, 3. The mentioned angles were calculated according to the formulae (6.18). Also the example 6.2 assumes that the object’s velocity is equal to ϑ = 200 m/s and the spatial position of the object’s vector is determined by the angles ϕ O = 0.68 rad and γ O = 0.35 rad, Fig. 6.4b. With the above assumption and with f Ob = 1010 Hz, the frequency differences of the received echo signals were determined and their values are visible in the second column of Table 6.7. Table 6.6 Results of calculations ∗ = 8070.001 m,, y ∗ = 168,000.327 m,z ∗ = 672,947 m Object’s coordinates: x O O O

i

di∗ , m

δdi = di∗ − di , m

ϕi , rad

γi , rad

0

168,195.390

0.344

1.522797

0.004001

1

191,817.250

0.329

1.370998

0.003352

2

130,630.492

0.336

1.369873

0.005917

3

153,935.343

0.343

− 1.460591

0.003917

References

105

Table 6.7 Input data i

f i , Hz

. f i0 , Hz

ψi , rad

γi , rad

0

9,999,995,823.973245

0

1.618795

0.004001

1

9,999,995,166.429604

− 657.543640

1.770594

0.003352

2

9,999,995,156.135200

− 667.838045

1.771719

0.005917

3

10,000,003,369.37665

7545.403405

4.602183

0.003917

Considering presented in Table 6.7 frequency differences . f i0 and direction angles ψi = π − ϕi and γi , where i = 0, 1, 2, 3, the calculated components of the velocity vector are approximately equal to: ϑx = 146,086 m/s, ϑ y = 118,134 m/s and ϑz = 68,579 m/s. Thus, according to the dependencies (6.11), the modulus and direction angles of this vector have the values ϑ = 200 m/s, ϕ O = 0.68 rad and γ O = 0.35 rad. As in the previous example, the obtained solution is consistent with the data adopted in order to determine the input data shown in Table 6.7. Also, in the present example, the calculations were performed with the additional assumption, namely that the time differences .ti0 included in Table 6.5, are specified with an accuracy up to 10−3 µs, and the frequency differences . f i0 presented in Table 6.7 are specified with the accuracy up to 1 Hz. Such ‘’limited” data corresponds to the following solution: (x O∗ = 8,070,541 m, y O∗ = 168, 014.415 m, z ∗O = 660,532 m) and (ϑx∗ = 145,972 m/s, ϑ y∗ = 118,138 m/s, ϑz∗ = 66,224 m/s). In this case δϑ = |ϑ ∗ − ϑ| < 0.9 m/s, which is a completely acceptable result. The calculation results, which are presented above, confirm the usefulness of the TDOA and FDOA methods when applied to determine the object’s position and the object’s velocity vector in three—dimensional (3D) space. Moreover, the results also indicate the required measuring accuracy of the time .ti0 and frequency . f i0 differences, so that the object’s position and object’s velocity vector can be determined with sufficient accuracy. In the case of the radiolocation scenario, which was analyzed in example 6.1, the time differences should be determined with an accuracy better than 10−8 s, which will limit a position deviation .p[m] to 10– 20 m. Simultaneously, the frequency differences . f i0 should be measured with the accuracy better less than 10 Hz (Table 6.4) which corresponds to the accuracy of determining the velocity better than 1 m/s(3.6 km/h).

References 1. Averianov V (1978) Multistatic radar stations and systems (in Russian). Science and technology, Minsk (Belarus) 2. Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood (MA)

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6 Multistatic Radar Systems

3. Bljachman AB, Kowaljev FN, Ryndyk AG (2001) A method of determining the coordinates of a moving target using bistatic RLS (in Russian). (A.S. Popov) Radio—Engineering, No 1, c 4–9 4. Yang BJ, Li J, Zhou Y (2008) The target location study of multistatic radar base on noncooperative emitter illuminating. In: Proceeding of the international conference on information and automation, June 2008, Zhangjiajie, China, pp 1433–1436 5. Chernyak VS (1988) Fundamentals of multisite radar systems. Gordon and Breach Science Publishers, Amsterdam 6. Hanle E (1986) Survey of bistatic and multistatic radars. IEE Proc 133(7):587–595 7. Ho KC, Wenwei Xu (2004) An accurate algebraic solution for moving source location using TDOA and FDOA measurements. IEEE Trans Sig Proc SP–52(9):2453–2463 8. Milne K (1977) Principles and concepts of multistatic surveillance radars. Proc IEE Conf Radar 77(155):46–52 9. Real Time Location Systems (RTLS) - version 1.02, A white paper from nanotron technologies GmbH, Germany (2007). www.nanotron.com/EN/pdf/WP_RTLS.pdf 10. Rosłoniec S (2008) Fundamental numerical methods for electrical engineering. Springer, Heidelberg/Berlin 11. Sammartino PF et al (2008) Moving target location with multistatic radar systems. In: Proceeding of the IEEE radar conferences, May 2008, pp 1–6 12. Schau HC, Robinson AZ (1987) Passive source localization employing intersecting spherical surfaces from time-of-arrival differences. IEEE Trans Acoust Speech Sig Process 35(8):1223– 1225 13. Shen G at all (2008) Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment. In: Proceeding of the 5-th workshop on positioning, navigation and communication (WPNC’08), pp 71–78 14. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 15. Stoica P, Li J (2006) Source localization from range-difference measurements. IEEE Sig Process Mag 23(6):63–69 16. Willis NJ (2005) Bistatic radar, 2nd edn. Scitech Publishing Inc., Releigh (NC) 17. Willis NJ, Griffiths HD (eds) (2007) Advances in bistatics radar. Scitech Publishing Inc., Releigh (NC)

Chapter 7

Standard Methods for Extending the Range of Radar Station

One of the effective methods of extending the range of a radar station includes increasing the ratio of the echo signal power s(t) to the interference (noise) power n(t), which enables to extract a desired signal. This can be achieved either by an appropriate selection of receiver bandwidth or by exploiting the statistical properties of the white noise and the cross-correlation function for the received signal and a copy of the probe signal. There are three methods, which are relevant here: • the optimization of the receiver’s transmitted bandwidth with respect to the spectral width of the received signal, • additive reception, • correlation reception. In order to facilitate understanding of the above-mentioned methods, some fundamental terms relating to the theory of signals are recalled first [3, 8].

7.1 Elements of the Radar Signals Theory 1. An average value of the real signal u(t) in the range [T1 , T2 ] 1 u(t) = |T2 − T1 |

.T2 u(t)dt

(7.1)

T1

2. The scalar product of two determined real signals u(t) and ϑ(t). .∞ (u, ϑ) =

u(t) · ϑ(t)dt

(7.2)

−∞

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_7

107

108

7 Standard Methods for Extending the Range of Radar Station

In the case u(t) ≡ ϑ(t) .∞ (u, u) =

u 2 (t)dt = E u

(7.3)

−∞

The integral (7.3) refers to as an intermediate energy of the signal. 3. Fourier series and integral Fourier transform Assume that a periodic function f (t) = f (t +n T ) satisfying the Dirichlet conditions in the considered domain 0 ≤ t ≤ T , is introduced. The function is implicitly integrable and has a finite number of local extrema and discontinuity points of type one (also known as the removable discontinuities). The periodic function that satisfies the above conditions can be approximated by a series f (t) =

where:

and

ω1 =

bi =

2 T

2π , T T./2

−T /2

∞ . a0 + [ai cos (i · ω1 t) + bi sin (i · ω1 t)] 2 i= 1

a0 =

2 T

T./2 −T /2

f (t)dt,

ai =

2 T

T./2 −T /2

(7.4)

f (t) cos( iω1 t)dt

f (t) sin(i ω1 t)dt

A large number of mathematical monographs, including [10], is devoted to the derivation of the above-mentioned dependencies. The series (7.4) is frequently expressed in the following form ∞

f (t) =

a0 . + [m i cos(i · ω1 t + ϕi )] 2 i= 1

(7.5)

/ where |m i | = ai2 + bi2 and tg(ϕi ) = −bi /ai . In the case of an aperiodic function f (t) approximation, i.e. a function of a period T → ∞, the series (7.4), converts into the following integral 1 f (t) = 2π

.∞ F(ω)e jωt dω

(7.6)

−∞

where: .∞ F(ω) = −∞

f (t)e− jωt dt, j =

√

−1

(7.7)

7.1 Elements of the Radar Signals Theory

109

corresponds to the F—Fourier transform (spectrum) of a function f (t). In order to convert the series (7.4) into the integral (7.6), one should express functions cos(i ω1 t), sin(i ω1 t) and the coefficients a i , bi , where i = 1, 2, 3, . . ., as a combination of the following exponential functions: cos(iω1 t) =

1 jiω1 t 1 ji ω1 t + e− jiω1 t ), sin(i ω1 t) = − e− jiω1 t ) (e (e 2 2j

(7.8)

Taking into consideration (7.8) one obtains: 2 ai = T 2 bi = T

.T /2 −T /2

.T /2 −T /2

1 f (t) cos( i ω1 t)dt = T

.T /2

f (t)[(e jiω1 t + e− ji ω1 t )] dt

−T /2

1 f (t) sin( i ω1 t)dt = jT

.T /2

f (t)[(e ji ω1 t − e− jiω1 t )] dt

(7.9)

−T /2

Substituting the expansions (7.8) and (7.9) into the series (7.4), yields ] ) ) ( ∞ [( . ai bi bi a0 ai ji ω1 t − ji ω1 t + + e − e f (t) = + 2 2 2j 2 2j i= 1 ⎡ ⎤ T /2 T /2 . . ∞ . ⎢1 a0 1 ⎥ = + f (t)e− ji ω1 t dt · e ji ω1 t + f (t)e jiω1 t dt · e− ji ω1 t ⎦ ⎣ 2 T T i= 1 ⎡

=

∞ . ⎢1 ⎣ T i=−∞

−T /2

.T /2

−T /2

⎤

∞ . [ ] ⎥ f (t)e− jiω1 t dt · e ji ω1 t ⎦ = ci · e ji ω1 t

(7.10)

i=−∞

−T /2

where 1 ci = T

.T /2

f (t)e− ji ω1 t dt

(7.11)

−T /2

corresponds to ith coefficient of the complex Fourier series which is determined for −∞ < i < ∞. The spectral representation of the signal defined by the formula (7.10) is mirror-symmetrical with respect to ω = 0 and contains components with negative angular frequencies. The components with positive and negative angular frequencies, occurring in the series (7.10), form coupled pairs the sum of which gives a real number. Naturally, negative angular frequencies are only a mathematical

110

7 Standard Methods for Extending the Range of Radar Station

artefact and do not have a physical meaning. Summing the components of the series (7.10) with respect to i when T → ∞ converts sum into an integration with respect to pulsation ω = i · ω1 . After considering the dependences: ω = i · ω1 , dω = di · ω1 , di = dω/ω1 , the function f (t) when T → ∞ can be written as .∞ f (t) = −∞

⎡ ⎣ 1 T ω1

.∞

⎤ f (t)e− j ωt dt ⎦ e jωt dω

(7.12)

−∞

Finally, .∞ f (t) = −∞

⎡ ⎣ 1 2π

.∞ −∞

⎤ 1 f (t)e− jωt dt ⎦ e jωt dω = 2π

.∞

⎡ ⎣

−∞

.∞

⎤ f (t)e− jωt dt ⎦ e j ωt dω

−∞

(7.13) 4. Generalized Rayleigh formula. Assume that both signals u(t) and ϑ(t) occurring in the formula (7.2), are represented by their spectra .∞ U(ω) =

u(t)e

− jωt

.∞ dt, V(ω) =

−∞

ϑ(t)e− jωt dt

(7.14)

−∞

According to the integral Fourier transform (the formulae 7.6 and 7.7) 1 u(t) = 2π

.∞ U (ω)e

j ωt

dω,

−∞

1 ϑ(t) = 2π

.∞ V (ω)e jωt dω

(7.15)

−∞

After substituting the signal ϑ(t) expressed by the formula (7.15) to the expression (7.2), one obtains 1 (u, ϑ) = 2π

.∞

.∞ u(t) ·

−∞

V(ω)e j ωt dω dt

(7.16)

−∞

The integral (7.16) can be expressed in the form, which is obtained by swapping the order of the integration with respect to time and pulsation 1 (u, ϑ) = 2π

.∞ −∞

⎡ V (ω) · ⎣

.∞

−∞

⎤ u(t)e j ωt dt ⎦ dω

(7.17)

7.1 Elements of the Radar Signals Theory

111

Undoubtedly, the internal integral occurring in formula (7.17), corresponds to the signal spectrum u(t), which is calculated for the angular frequency ω with a changed sign, i.e. .∞ u(t)e jωt dt = U(−ω)

(7.18)

−∞

Assuming that the signals u(t) and ϑ(t) correspond to the real time functions, one can write U(−ω) = U∗ (ω)

(7.19)

where * indicates the complex conjugate value. Ultimately, one obtains 1 (u, ϑ) = 2π

.∞

V(ω) · U∗ (ω)dω

(7.20)

−∞

The dependence (7.20) is referred to as the general Rayleigh formula, [3, 5]. According to this formula, the scalar product of two real signals is proportional to the scalar product of their spectra. In a special case of u(t) ≡ ϑ(t) (u, u) =

1 2π

.∞

U(ω) · U∗ (ω)dω =

−∞

1 2π

.∞ |U(ω)|2 dω

(7.21)

−∞

After introducing the variable f = ω/(2π ) in the expression (7.21) and comparing with the expression (7.3), the following dependence is obtained .∞ (u, u) =

1 u (t)dt = 2π

.∞

−∞

.∞ |U(ω)| dω =

|U( f )|2 d f

2

2

−∞

(7.22)

−∞

The dependence (7.22) is known as Parseval’s identity. The function |U(ω)|2 occurring in the integral (7.21) is the spectral energy density of the signal and is marked with the symbol .(ω). .∞ Thus, the dependence (7.21) can be expressed as E u = −∞ u 2 (t)dt = . ∞ 1 .(ω)dω which is presented in the literature as the special case of Rayleigh 2π −∞ formula. This formula confirms that the energy of any signal can be expressed as the sum of the partial energies deriving from the individual components of its spectrum, [3, 4, 8, 9].

112

7 Standard Methods for Extending the Range of Radar Station

Fig. 7.1 A single rectangular pulse u(t)

Example 7.1 Consider a single rectangular pulse as shown in Fig. 7.1. The spectrum U(ω) and spectral energy density .(ω) of this signal is described by the dependencies; .ti /2 U(ω) = A

e− jωt dt =

−ti /2

] sin(ωti /2) A [ exp(− j ωti /2) − exp( j ωti /2) = Ati − jω ωti /2

.u (ω) = |U(ω)|2 = A2 · ti2 ·

sin2 (ωti /2) (ωti /2)2

(7.23)

The normalized (χ = .u (ω)/(A2 ti2 )) plot of (7.23) is shown in Fig. 7.2. In many tasks it is important to determine what part of the total signal energy is contained in the limited band, Fig. 7.2.

E u(k)

1 = 2π

2πk/t . i

−2πk/ti

1 .(ω)dω = π

2π . k/ti

0

A2 ti2 .(ω)dω = π

2π . k/ti

0

sin2 (ωti /2) dω (ωti /2)2

After introducing an auxiliary variable α = 0.5ω · t i , dω = 2dα/ti , one obtains

E u(k)

Fig. 7.2 A plot of the χ = .u (ω)/( A2 ti2 ) function

2 A2 ti2 = π

.π k 0

sin2 (α) dα (α)2

(7.24)

7.1 Elements of the Radar Signals Theory Table 7.1 Sample values of function E u(k) /E calculated for 1 ≤ k ≤ 3

113

k

1

2

3

E u(k) /E

0.902

0.950

0.973

Table 7.1 gives the values of the energy E u(k) /E which were numerically calculated, taking into consideration the first, second and third lobes of the function plotted in Fig. 7.2. Table 7.1 shows that the doubling of the filter’s transmission bandwidth, through which the square pulse passes, results in the signal energy increase by 4.8%. Simultaneously, the power of the transmitted white noise doubles. Thus, there is a certain (optimal) filter transmitted bandwidth, for which the ratio of the signal power to the white noise power (with a regular spectral density N ) reaches a maximum. Considering the analyzed example, the optimal filter bandwidth is equal to . f opt = 1.37/ti . 5. Signal autocorrelation function. In order to perform the quantitative assessment of the signal dissimilarity between u(t) and its displaced copy u(t − τ ), the autocorrelation function is introduced .∞ ϕu (τ ) =

u(t) · u(t − τ )dt

(7.25)

−∞

The Eq. (7.25) shows that when the displacement equals τ = 0, the autocorrelation function has the maximum value equaling the average signal energy ϕu (0) = E u

(7.26)

For any τ /= 0, the autocorrelation function is an even function, i.e. ϕu (τ ) = ϕu (−τ ) The definition (7.25) shows that the autocorrelation function is equal to the scalar product of signals u(t) and u τ (t) = u(t −τ ), which, according to the general formula of Rayleigh’s theorem, can be expressed as follows 1 (u, u τ ) = 2π

.∞

U(ω) · Uτ∗ (ω)dω

(7.27)

−∞

According to the displacement theorem, the spectrum Uτ (ω) of a signal u τ (t) = u(t − τ ) is equal to U (ω)e− jωτ , where U (ω) corresponds to the spectrum of the signal u(t). After taking into consideration the above dependencies of the expression (7.27), the following formula is obtained 1 (u, u τ ) = 2π

.∞

∗

U(ω) · U (ω)e −∞

jωτ

1 dω = 2π

.∞ |U(ω)|2 e jωτ dω −∞

(7.27a)

114

7 Standard Methods for Extending the Range of Radar Station

Fig. 7.3 The rectangular pulse and its autocorrelation function; a The rectangular pulse and its delayed copy, b the autocorrelation function

The formula (7.27)a indicates that the autocorrelation function and the energy spectrum are related to each other by the Fourier transform. Example 7.2 In the present example 7.2 the autocorrelation function of a rectangular signal, shown in Fig. 7.3a, is determined. For the signal u(t) considered in this example, the autocorrelation function is determined by the following formula: | | A2 · t (1 − | i ϕu = | |0

|τ | ) ti

for |τ | < ti for |τ | ≥ ti

(7.28)

Example 7.3 One should determine the autocorrelation function of the radio signal described by the dependence: | | A cos(ωt) u(t) = || 0

for |t| < ti /2 for |t| ≥ ti /2

(7.29)

Assume that 0 < τ < ti , which does not restrict the generality of considerations because the autocorrelation function is an even function. .ti /2 ϕu (τ ) = A

cos(ωt) cos[ω(t − τ )]dt =

2 −ti /2+τ

.ti /2 1 2 cos(ωτ )dt + A cos(2ωt − ωτ )]dt = 2 −ti /2+τ −ti /2+τ ] [ sin[ω(ti − |τ |)] 1 (7.30) = A2 (ti − |τ |) cos(ωτ ) + 2 ω(ti − |τ |)

1 = A2 2

.ti /2

For τ = 0 the autocorrelation function ϕu (0) is equal to the energy expressed by the formula

7.1 Elements of the Radar Signals Theory

115

Fig. 7.4 Graphical illustration of the autocorrelation function (7.30)

Eu =

[ ] sin(ωti ) 1 2 A ti 1 + 2 (ωti )

The function (7.30) is plotted in Fig. 7.4. For τ = 0 and t i → 0 the signal energy E u → A2 ti . The autocorrelation function shown in Fig. 7.4 has zero values when |τ | ≥ ti as well as at the several discrete points included in the range [−ti , ti ]. At these discrete points, the correlated radio pulses are mutually orthogonal. In other words, the phase angles of the radio pulses are shifted by π/2 rad. In radio engineering, such signals are referred to as quadrature signals. 6. The cross-correlation function of two signals. The cross-correlation function of two real signals u(t) and ϑ(t) is expressed by the following integral .∞ ϕu,ϑ (τ ) =

u(t) · ϑ(t − τ )dt

(7.31)

−∞

.∞ If signals u(t) and ϑ(t) are orthogonal then −∞ u(t) · ϑ(t)dt = ϕu,ϑ (0) = 0. The cross-correlation function of two signals is correct when ϕu,ϑ (τ ) = ϕϑ,u (−τ )

(7.32)

which can be justified as follows. According to the definition (7.31) .∞ ϕϑ,u (−τ ) =

ϑ(t) · u(t + τ )dt −∞

the auxiliary .variable t ' = t + τ , for which dt = dt ' , ϕϑ,u (−τ ) = . ∞Substituting ∞ ' ' ' = −∞ u(t ' )ϑ(t ' − τ ))dt ' is obtained for the integral −∞ ϑ(t − τ )u(t )dt presented above. The obtained dependence is identical to (7.31), which confirms the correctness of the dependence (7.32). Unlike the autocorrelation function, the

116

7 Standard Methods for Extending the Range of Radar Station

correlation function of two signals is not an even function of the displacement τ what expres the inequality ϕu,ϑ (τ ) /= ϕϑ,u (−τ )

(7.33)

If u(t) and ϑ(t) correspond to the signals of a finite energy, the cross-correlation function is also limited, whereby for τ = 0 this function does not have to reach the maximum value. Example 7.4 The example 7.4 is devoted to determining the cross-correlation function ϕu,ϑ (τ ) of signals u(t) and ϑ(t), shown in Fig. 7.5. If τ > 0 then the signal ϑ(t) is time-delayed with respect to the signal u(t). Then, T ] [ 2 . ϕuϑ (τ ) = UT (t − τ )dt = U 2 T 21 − α + 21 α 2 where α = τ/T . In the case of τ

τ < 0 ϕuϑ (α) =

U2 T

T −|αT . | 0

(t + |αT |)dt =

U2T 2

(1 − α 2 ). The plot of a function

χ = ϕu,ϑ (α)/(U 2 T ) for signals u(t) and ϑ(t) is shown in Fig. 7.6. 7. The analytical signal Assume that u(t) corresponds to a real signal which is considered to be limited energy (magnitude) and time limited. Fig. 7.5 The rectangular u(t) and triangular ϑ(t) pulses

Fig. 7.6 Normalized autocorrelation function calculated for the pulses shown in Fig. 7.5

7.1 Elements of the Radar Signals Theory

117

For such signal, a corresponding analytical signal z(t) = u(t) + j ϑ(t) can be constructed, using the Hilbert transform to calculate ϑ(t). 1 ϑ(t) = π

.∞ −∞

u(τ ) dτ t −τ

(7.34)

The complex signal z(t) obtained from (7.34), despite of its abstract nature, significantly simplifies the analysis of properties for u(t), especially in the case of complex radar signals, [5, 18]. Each analytical signal can be expressed in exponential form z(t) = A(t) exp[ j ϕ(t)]

(7.35)

. √ where: A(t) = u 2 (t) + ϑ 2 (t), ϕ(t) = arctg[ϑ(t)/u(t)], j = −1. For example, for a signal u(t) = U m (t) cos[ψ(t)], the component ϑ(t) = U m (t) sin[ψ(t)] and the analytical signal formed from u(t) and ϑ(t), describes the dependence z(t) = U m (t) exp[ jψ(t)]. If the signal phase (7.35) is equal to ϕ(t) = ω0 t + ϕ0 then z(t) = A(t) exp[ j (ω0 t)] where A(t) = A(t) exp( j ϕ0 ) denotes a complex modulation function, also referred to as the complex envelope, [4, 8, 12]. 8. The two-dimensional autocorrelation function. The autocorrelation function, which is defined by the dependence (7.16), is a measure of similarites (correspondence) between the signal u(t) and its copy displaced in time by τ . Analogically, one can define a function which gives a measure of correspondence between a narrowband radio signal u(t) and its copy displaced in the frequency domain by an interval .. An actual radar signal, after reflection from a moving object for instance, experiences in free space a shift both in time and frequency domain. Therefore, a two-dimensional autocorrelation function was introduced in the theory of radar signals, which incorporates both above-mentioned displacements simultaneously. In order to guarantee the concreteness and transparency of the further narrative, one should assume that u(t) = U m (t) cos[ω0 t + ϕ(t)], whereby the amplitude function U m (t) changes very slowly in comparison to the phase ψ(t) = ω0 t + ϕ(t). The counterpart of this narrowband signal, displaced both in time and in frequency, is the signal u τ,. (t) = U m (t + τ ) cos[(ω0 + .)(t + τ ) + ϕ(t + τ )]

(7.36)

The analytical signals z(t) and z τ,. (t), can be assigned to signals u(t) and u τ,. (t), respectively, using the dependencies (7.34) and (7.35). Once the indicated transformations are performed, one obtains z(t) = Um (t) exp[ jϕ(t)] · exp( jω0 t) = Um (t) exp( jω0 t)

(7.37)

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7 Standard Methods for Extending the Range of Radar Station

z τ,. (t) = U m (t + τ ) exp[ j ϕ(t + τ )] · exp[ j (ω0 + .)(t + τ )] = Um (t + τ ) · exp[ j (ω0 + .)τ ] · exp[ j (ω0 + .)t]

(7.38)

where Um (t) = Um (t) exp[ j ϕ(t)] corresponds to the complex modulation function of the signal u(t). Analogically, Um (t +τ ) = U m (t +τ ) exp[ j ϕ(t + τ )] indicates the complex modulation function of the signal u τ,. (t). The general correlation function of the analytical signals z(t) and z τ,. (t) is determined by the following dependence, [3, 5] .∞ Bz (τ, .) =

∗ z(t) · z τ,. (t)dt

−∞

.∞ = exp[− j (ω0 + .)τ ] −∞ .∞

= exp[− j (ω0 + .)τ ]

Um (t) · Um∗ (t + τ ) · exp( j ω0 t) · exp[− j (ω0 + .t]dt

Um (t) · Um∗ (t + τ ) · exp(− j.t)dt

(7.39)

−∞

The function Um∗ (t + τ ), occurring in this dependence, is a complex modulation function conjugated with respect to Um (t + τ ). The expression module (7.39), i.e. | ∞ | |. | | | ∗ | R(τ, .) = | Um (t)· Um (t + τ ) exp(− j.t)dt || | |

(7.40)

−∞

is referred to as the two-dimensional correlation function [4, 8, 12]. Thus, this function is frequently expressed as | ∞ | |. | | | ∗ | R(τ, .) = | S (ω)· S(ω + .) exp(− jωτ )dω|| | |

(7.41)

−∞

As shown in the Appendix E S(ω) and S(ω + .)) correspond to the spectra of the complex envelopes Um (t) and Um (t + τ ), respectively. For τ = 0 and . = 0, the function (7.40) has the maximum value .∞ R(0, 0) = Rmax = −∞

Um (t) · Um∗ (t)dt =

.∞ |Um (t)|2 dt = 2E

(7.42)

−∞

where E denotes the energy of the real signal u(t). On the other hand, the absolute value of the function (7.40) normalized with respect to its maximum value, i.e.

7.1 Elements of the Radar Signals Theory

119

| | | R(τ, .) | | | | R(0, 0) | = χ (τ, .)

(7.43)

is referred to as the ambiguity function. The ambiguity function has the following basic properties: 1. Function χ (τ, .) → 1 for τ → 0 and . → 0 2. Function χ (τ, .) is directly symmetric, e.g. χ (τ, .) = χ (−τ, −.) 3. In the cartesian coordinate system [τ, ., χ (τ, .)], the function (7.43) can be illustrated by a “folded” surface. A piece of such surface, determined √ for a single rectangular radio pulse of a ti length and amplitude U m (t) = 1/ ti , is shown in Fig. 7.7. The surface shown in Fig. 7.7 is a spatial representation of the function | |( || 1 − || χ (τ, .) = | |0

|τ | ti

)

|

sin[π FD ti (1−|τ |/ti )] | π FD ti (1−|τ |/ti ) |

for |τ | < ti for |τ | > ti

(7.44)

where: FD corresponds to the Doppler frequency which is related to the angular frequency (pulsation) . by the dependence FD = ./(2π ), see Appendix F. It has been shown in the available literature the surface χ (τ, .) and the plane χ (τ, .) = 0 limit an area (the field of ambiguity) of a constant volume V, which is equal to V =

1 2π

¨ [χ (τ, .)] 2 dτ d. =

2π 2π

¨ [χ (τ, .)] 2 dτ d FD = 1

(7.45)

The dependence (7.45) leads to an important conclusion namely that when the field of ambiguity is reduced (compressed) along the axis τ , it expands along the axis .. Simultaneously, when the field of ambiguity narrows along the axis ., it widens

Fig. 7.7 Graphical illustration of the autocorrelation function √ (7.43) calculated for a single rectangular radio pulse of ti length and amplitude U m (t) = 1/ ti

120

7 Standard Methods for Extending the Range of Radar Station

along the axis τ . The indicated property shows that it is not possible to increase the resolving power of the signal in the time and frequency domain simultaneously. 4. The function (7.44) is reduced to a one-dimensional autocorrelation function (7.16) defined in the time domain when the frequency translation equals zero (FD = 0). The plot of the √ function is illustrated in Fig. 7.3b, since in this particular case A2 ti = (1/ ti )2 t i = 1. 5. The plot of of χ (τ, .) allows for an easy evaluation of the probe signal’s properties subject to optimal processing, i.e. filtration using a matched filter. For example, the presence of only one (central) lobe with a high and sharp peak, indicates the possibility of an accurate measurement of both, the object’s velocity and object’s distance, or good distinguishability from nearby objects. In the light of the above considerations, one should aim to select a probe signal for which the function χ (τ, .) has one possibly high and sharp peak or a significant and narrow central lobe (ridge), especially in sections (τ = 0) or (. = 0).

7.2 The Additive Reception The essence of the additive reception method, using the probe signal consisting of the n identical pulses, is illustrated by Figs. 7.8 and 7.9. In radar systems, the most common interference occurs as an uncorrelated white noise. The theoretical evidence and experimental confirmations show that the average

Fig. 7.8 Graphic illustration of an idea of the additive reception

7.3 The Correlation Reception

121

Fig. 7.9 Illustration of how a signal integrator works

value of the sum corresponding the uncorrelated noise signals ei (t), is close to zero, if the number of summed signals ei (t) is sufficiently large. The device which stores the sum of the signals ei (t), containing the echo signal, is referred to as an integrator, Fig. 7.9. The presented integrator cyclically generates the output signal which is a sum of the n signals ei (t).

7.3 The Correlation Reception The cross-correlation function of two real signals is used in modern radiolocation and radionavigation systems for two main purposes, namely: • measuring the delay time of the received signal with respect to a probe signal copy, • optimization of the reception process for pulse signal corrupted by noise. Both of the above-mentioned tasks are performed simultaneously in the multichannel correlation system of the receiver with the functional structure, as shown in Fig. 7.10. According to the properties of the cross-correlation and autocorrelation functions, the maximum signal will appear at the output of the kth comparison system when both signals, at theirs inputs, are near to each other in terms of the shape and time of occurrence. Knowing the number of the channel in which the described situation occurs, the delay time of the received signal x(t − τ ) with respect to the copy s(t) and thus the distance to the detected object is known. A more precise value of the delay time of the received signal x(t − τ ) with respect to the copy s(t) is determined during the signals processing, appearing at the output of the kth comparison system and the neighboring outputs, [6, 12]. The multiplication and integration of the signals s(t) and x(t − τ ) = n(t) + α · s(t − τ ) is performed in the comparison systems illustrated in Fig. 7.10, which is described by the formula .t2 s(t − τk )[α · s(t − τ + n(t)] dt t1

(7.46)

122

7 Standard Methods for Extending the Range of Radar Station

Fig. 7.10 The illustration of an idea of the correlation reception

where: k = 1, 2, 3, . . . , N , t1 —the beginning of an observation, t2 —the end of an observation period, n(t)—interference (noise), α—the coefficient of an attenuation, (α < 1). According to the properties of the cross-correlation function, the voltage at the output of the comparison system is the highest when the received signal x(t − τ ) coincides with the copy of the signal s(t). This voltage is relatively small when the received signal consists only of noise. The calculation (7.46) is performed using one of the systems presented in Fig. 7.11, which respectively include a correlator, a signal matched filter and its digital version. The system shown in Fig. 7.11a performs the operation of multiplication and integration, similarly as it was performed when determining the cross-correlation function. The second of the presented systems, shown in the Fig. 7.11b, consists of a passive band-pass filter with a transfer function H(ω) related to the spectrum

Fig. 7.11 The signal correlators; a the conventional solution, b the analog signal matched filter, c version of the digital signal matched filter

7.3 The Correlation Reception

123

S(ω) of the signal s(t) with appropriate dependencies derived in Chap. 8.2. Generally, the filter of this type is referred to as a matched filter s(t). In the third of the presented systems, the signal x(t) is initially processed with an analogue-digital converter x(t) → (I, Q), which gives two discrete digital signals I (tn ) ≡ I (n) and Q(tn ) ≡ Q(n) at the output, containing a comprehensive information regarding its amplitude and phase. The Chap. 9.3 extensively covers the principle of operation for a converter of this type. Then, the signals I (tn ) and Q(tn ) are processed with an appropriate computer program. Information about the signal s(t) with a matched filter is contained in the structure of this program and can be easily modified. Thanks to these advantages, this solution is broadly used in most modern radar and radio navigation devices. For teaching reasons, let’s consider again the correlator system with the functional pattern, as shown in Fig. 7.11a, where: s(t)—a copy of the desired signal (delivered from the transmitter), x(t) = n(t) + α · s(t)—the received signal, n(t)—the spurious response (noise) and α—a coefficient of attenuation, (α < 1). At the output of the multiplication system one obtains x(t) · s(t) = [n(t) + α · s(t)]s(t) = n(t) · s(t) + α s 2 (t)

(7.47)

The signal (7.47) equals zero when the signal s(t) is absent. The integrating element of the correlator converts the signal (7.47) in time from t0 to t0 +T presenting a proportional output voltage to the function t. 0 +T

ϕx,s (τ = 0) = Rmax =

t. 0 +T

n(t) · s(t)dt + t0

α · s 2 (t)dt

(7.48)

t0

As an example, assume that signal s(t) corresponds to a radio signal of a duration equal to 2T , where T denotes the signal period which along with the noise, is fed to one of the correlator’s inputs. The voltage courses induced by this signal and by the copy of the signal s(t) in the individual correlator nodes, are shown in Fig. 7.12. If the voltage, proportional to the value of the function ϕx,s (0), at the correlator’s output exceeds the threshold value ϕx,s (0)T hr , the received signal x(t) is considered to contain the part of the energy of the probe signal s(t). The correlator system shown in Fig. 7.11a constitutes the basic functional block of the multi-channel correlation receiver with the structure shown in Fig. 7.13. The reference signals (copies of the signal emitted towards the detected object) are fed to the individual correlators from the successive taps of the delay line which the receiver’s signal is fed to. Assume that the signal which is fed from the delay line to the kth correlator, coincides with the signal α · s(t − τ ) which is contained in the received signal x(t − τ ), see the Eq. (7.45). Then the voltage at the output of this correlator obtains a maximum value, greater than the threshold voltage ϕx,s (0)T hr , which is considered as an echo signal detection s(t) and thus an object detection. The above considerations show that the multi-channel receivers with the structures shown in Figs. 7.10 and

124

7 Standard Methods for Extending the Range of Radar Station

Fig. 7.12 Waveforms of signals s(t), x(t), s(t) · x(t) and of the correlation function ϕs,x (0) that illustrate a principle of operation of a correlator

7.13, do not differ fundamentally in terms of the principle of operation, but significant differences pertain to the specific structural solutions. The Fig. 7.10 shows that the required delays (τ1 , τ2 , τ2 , . . . , τ N ) of the probe signal copy s(t) are obtained at the outputs of the set N of the parallel delay lines. In the receiver with a structure shown in Fig. 7.13, one delay line with a series-arranged taps is used for this purpose. Furthermore, a common component for all correlators (matched filters) is referred to as a ‘decisive device’, containing an appropriate threshold system. Exceeding the threshold value by the output signal is considered as the detection of an echo signal that was occurred due the reflection of the probe signal from the target object. The structural receiver, as shown in Fig. 7.13, is considered to be a complicated device, thus its application area is limited. Moreover, the probe pulse should be sufficiently long to enable the correct operation of the correlator, which additionally limits the usage scope of this reception technique. As mentioned before, considering the majority of radar devices, it is more convenient to use the filters that are matched to the received, desired signal. At present, i.e. in the era of digital signal processing, the transfer function of the matched filter is “contained” in a special computer program processing the appropriately recorded digital sample I, Q of the received signal, [8, 11].

References

125

Fig. 7.13 The block diagram of a multi-channel correlation receiver

References 1. Abramowitz M, Stegun IA (1960) Handbook of mathematical tables. In: National bureau of standards, applied mathematics series, vol 55. US Goverment Printing Office, Washington 2. Barton DK (2013) Radar equations for modern radar. Artech House Inc., Boston, London 3. Baskakov SI (1988) Signals and electronic circuits (in Russian), 2nd edn. “Higher school”, Moscow 4. Cook CHE, Bernfeld M (1967) Radar signals, an introduction to theory and applications. Academic Press, New York, London 5. Gonorowskij IS (1977) Radio circuits and signals, (in Russian) 3rd edn. “Soviet radio”, Moscow 6. Grigorin-Rjabow WW (ed) (1970) Radar devices, theory and principles of construction (in Russian). “Soviet radio”, Moscow 7. Levanon N (1988) Radar principles. Wiley, New York 8. Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken, New Jersey 9. Richards MA (2014) Fundamentals of radar signal processing, 2nd edn. McGraw-Hill Education, New York 10. Rosłoniec S (2008) Fundamental numerical methods for electrical engineering. Springer, Heidelberg, Berlin 11. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 12. Szirman JD (ed) (1970) Theoretical basis of radiolocation, (in Russian). “Soviet radio”, Moscow

Chapter 8

Theoretical Basis of Matched Signal Filtration

This chapter discusses the basic concepts and mathematical forms of describing the problem of matched filtering of signals used in radiolocation and in radionavigation.

8.1 Convolution The present chapter covers the analysis of a two-port network, which is linear (following the principle of superposition) and time invariant, activated by signal x(t). Any signal x(t) can be assumed as a set of infinitely short (∆t → 0) and consecutive pulses. { xi (t) =

for ti ≤ t ≤ ti + ∆t x(ti ) 0, for t < ti and t > ti + ∆t

(8.1)

Pulses (8.1) have the same width (duration) ∆t and are shifted by a multiple ∆t, Fig. 8.1. Assume that the input of the analyzed two-port network is fed by a pulse with a unit amplitude 

s (t) =

{

1, 0,

for 0 ≤ t ≤ ∆t for t < 0 and t > ∆t

(8.2)



The pulse (8.2) gives the response h (t) at the output for t ≥ 0.  Subject to (8.2), the signal y0 (t) = x(t0 ) h (t − t0 ) is the response of the two-port  network to the first pulse x(t) = x(t0 ) s (t − t0 ), see Fig. 8.1. Similarly, the response  of the two-port network to the second pulse x(t1 ) s (t − t1 ), occurring at the time 

instant t1 = t0 + ∆t, is y1 (t) = x(t1 ) h (t − t1 ). Analogically, one can express the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_8

127

128

8 Theoretical Basis of Matched Signal Filtration

Fig. 8.1 Graphical interpretation of a process described by the convolution function

consecutive input partial signals and corresponding output signals, as shown in Table 8.1 [2, 3, 6, 10]. Table 8.1 The consecutive partial input signals and corresponding output signals Time, t

Input signal 

Output signal 

x(t0 ) h (t − t0 ) for t ≥ t0

t0

x(t0 ) s (t − t0 ) for t0 ≤ t ≤ t0 + ∆t = t1

t1

x(t1 ) s (t − t1 ) for t1 ≤ t ≤ t1 + ∆t = t2

t2

x(t2 ) s (t − t2 ) for t2 ≤ t ≤ t2 + ∆t = t3

t3

x(t3 ) s (t − t3 ) for t3 ≤ t ≤ t3 + ∆t = t4

x(t3 ) h (t − t3 ) for t ≥ t3

…

…

…









x(t1 ) h (t − t1 ) for t ≥ t1 

x(t2 ) h (t − t2 ) for t ≥ t2 

8.1 Convolution

129

Considering the partial signals, shown in the second column of Table 8.1, an approximation of the input signal x(t) is obtained 

x (t) =

∞ ∑



x(ti ) s (t − ti )

(8.3)

i=0

The approximation (8.3) is correct within the considered domain. Analogically, using the partial signals, which are listed in the third column in Table 8.1, one can express the output signals 

y (t) =

∞ ∑



x(ti ) h (t − ti )

(8.4)

i=0 



The smaller is the length ∆t of the unit pulse s (t), the better the signal x (t)  approximates the input signal x(t). For ∆t → 0, the unit pulse s (t) becomes identical to the Dirac delta function δ(t). In this limiting case. 

h (t − t i ) → h(t − ti ),

{∞



x (t) → x(t) =

x(ti )δ(t − ti )dti 0



{∞ x(ti )h(t − ti )dti

and y (t) → y(t) = 0

where h(t) corresponds to the two-port pulse response. Due to the fact that h(t) is the response of a linear system to the Dirac delta signal δ(t), the following condition is satisfied: h(t) = 0, for t < 0

(8.5)

According to the condition (8.5), the function h(t − ti ) = 0 for t < ti

(8.6)

Once the meaning τ ≡ ti is clarified including the condition (8.6), one can derive the response of the analyzed system to the input signal x(t): {t y(t) ≡ y(t) ∗ h(t) =

x(τ )h(t − τ )dτ

(8.7)

0

Formula (8.7) is referred to in the literature as the convolution. The upper limit of the integral (8.8) is equal to τ = t, because for τ > t the integrand has zero

130

8 Theoretical Basis of Matched Signal Filtration

values. The equivalent of the convolution (8.7) in the frequency domain (pulsation) corresponds to the following formula Y(ω) = X(ω) · H(ω)

(8.8)

where {∞ Y(ω) =

y(t)e

− jωt

{∞ dt, X(ω) =

−∞

x(t)e− jωt dt

−∞

{∞ and H(ω) =

h(t)e− jωt dt.

−∞

Formula (8.8) can be easily derived by substituting function y(t), which is expressed by Eq. (8.7), in the transform Y(ω) and performing an appropriate mathematical transformation [2, 4, 6, 7].

8.2 The Transmittance of a Matched Filter to a Given Signal A matched filter is the linear part of the receiver transmittance (placed in front of the detector) having a band-pass filter transmittance form, which provides an optimal filtration of the received signal. Due to the optimal filtration, the maximum value of the output peak power ratio to the average noise power n 2 , is obtained in the specific time instant t0 , see Fig. 8.2. In the section that follows it will be shown that with an optimal signal filtration of the white noise, the ratio of the signal power to the average noise power, which is determined at the output of the matched filter, depends only on the energy E of the signal and on the spectral density of the noise power N . The signal to noise ratio

Fig. 8.2 A plot of the output signal obtained in the optimal filtration through a two-port linear network

8.2 The Transmittance of a Matched Filter to a Given Signal

131

Fig. 8.3 The filter matched to impulse s(t)

does not depend on the shape of the signal. However, one should emphasize that the shape of the signal has a significant impact on the construction and complexity of the matched filter. Formula (8.9) shows that the filter matched to the signal s(t) has a coupled frequency transmittance H(ω) with respect to the spectrum S(ω) of this signal, i.e. H(ω) = h 0 · S∗ (ω)

(8.9)

where h 0 corresponds to a solid coefficient [3, 6, 7]. The correctness of this conclusion will be described in two stages. Firstly, the pulse response h(t) of the filter matched to the signal s(t) will be determined and secondly, considering h(t), the corresponding transmittance H(ω) will be obtained. Therefore, a linear two-port network with a pulse response h(t) should be considered, to which an input signal x(t), specified in the time interval 0 ≤ t ≤ T0 ≤ t0 , is fed, Fig. 8.3. An output signal y(t) obtains at the time instant t0 ≥ T0 the following value {t0 y(t0 ) =

x(τ )h(t0 − τ )dτ

(8.10)

0

Formula (8.10) has been derived in Sect. 8.1. Introducing t ≡ τ , one can express formula (8.19) correspondingly {t0 y(t0 ) =

x(t)h(t0 − t)dt

(8.11)

0

The following condition should be fulfilled for the optimal reception with a filter matched to the signal s(t) {t0 y(t0 ) =

{∞ x(t)h(t0 − t)dt = h 0 , ϕxs (τ = 0) = h 0

0

x(t)s(t)dt

(8.12)

0

where h 0 is a solid and fixed coefficient and ϕxs (τ ) corresponds to a cross-correlation function between signals x(t) and s(t). According to the presumed assumption, the values of the signal x(t) are different than zero only for 0 ≤ t ≤ T0 , which facilitates expressing formula (8.12) as follows

132

8 Theoretical Basis of Matched Signal Filtration

{t0 y(t0 ) =

{t0 x(t)h(t0 − t)dt = h 0

0

x(t)s(t)dt

(8.13)

0

Formula (8.13) is satisfied when h(t0 − t) = h 0 s(t)

(8.14)

Using a subsidiary variable t ' = t0 − t, one can easily show that the condition (8.14) can also be written as follows h(t) = h 0 s(t0 − t)

(8.15)

Formulae (8.14) and (8.15) describe the fundamental relation between the filter pulse response h(t) and the signal s(t) to which this filter is matched. The essence of this dependence is also illustrated in Fig. 8.4. The transmittance H(ω) of the analyzed filter is related to the pulse response h(t) with the integral Fourier transform, i.e. {∞ H(ω) = −∞

h(t)e− jωt dt = h 0

{∞

s(t0 − t)e− jωt dt

(8.16)

−∞

This can be shown in the following way. The input signal of the Dirac delta pulse δ(t) corresponds to a continuous spectrum with an even distribution F[δ(t)] = 1. This spectrum multiplied by the transmittance H(ω) of the filter is equal to the output signal spectrum which in turn corresponds to the pulse response h(t) in the time domain. By introducing a subsidiary variable t ' = t0 − t, for which dt ' = −dt, formula (8.16) can be written as

Fig. 8.4 Relation between a signal and a pulse response of the filter matched with it; a a plot of the signal, b a plot of the pulse response of the matched filter

8.2 The Transmittance of a Matched Filter to a Given Signal

{∞ H(ω) = h 0

s(t0 − t)e

− jωt

{∞ dt = −h 0

−∞

= −h 0 e− jωt0

133

s(t ' ) exp[− j ω(t 0 − t ' )]dt '

−∞

{∞

'

s(t ' )e j ωt ) dt ' = −h 0 e− j ωt0 S(−ω)

(8.17)

−∞

By assumption, s(t) is a real signal, for which S(−ω) = S∗ (ω). In other words S (ω) is complex conjugate to S(ω). Considering formula (8.17), one obtains ∗

H(ω) = −h 0 e− j ωt0 S∗ (ω)

(8.18)

Formulae (8.13) and (8.18) show that the time instant t0 , in which the output signal y(t) reaches the maximum value, should not be shorter than the duration T0 of the input signal. In other words, a filter that is matched to the signal “uses” the entire energy of the input signal. The choice of t0 < T0 is unacceptable. The correctness of this limitation can be demonstrated as follows. In the case x(t) ≡ δ(t), the input signal has a nonzero value only when t = 0. The response of the matched filter cannot occur earlier, e.g. for t < 0. When the signal δ(t) is activated, the response can occur once the input signal δ(t) is ended, namely when t0 ≥ T0 = 0. Despite the fact that the role of the matched filter is similar to the correlation system, the principle of its operation is different. The correlator starts working at the instant t = 0 and stops at the instant T0 (operating only during the signals’ duration), obtaining the function value ϕx,s (τ ) at the output. In order to restart the correlator’s operation, its indication should be reset. The signal matched filter does not require resetting and consequently the signals processing with any delay, is facilitated. One should also emphasize that the matched filter is insensitive to changes in the amplitude of the received signal. Formula (8.18) shows that the amplitude-frequency response of the filter matched to the signal s(t) is uneven. The high-energy spectral components are attenuated to the least extent, whereas the low-energy components are most strongly attenuated, thus the ratio of the signal to the noise is improved. Due to an unevenness of transmittance (8.18), the shape of a signal obtained at the output of the matched filter y(t) differs from the input signal x(t). The phase characteristic of the matched filter has the maximum (extreme) signal value at the instant t = t0 ≥ T0 . In order to demonstrate this property, one should consider any component of the input signal spectrum with a frequency f . The phase of this component is equal to (2π f )t + ψ( f ) where ψ( f ) corresponds to the spectrum phase S( f ) = |S( f )| · exp[ j ψ( f )]. The component passing through the matched filter with a transmittance (8.18) changes the phase by a value −(2π f )t 0 − ψ( f ). Thus, the convolution component phase with the frequency f is equal to (2π f )t + ψ( f ) + [−(2π f )t0 − ψ( f )] = 2π f (t − t0 ) at the output of the matched filter and reaches 0 value at the instant t = t0 regardless of the frequency f . In other words, all the harmonic components at the output of the matched filter at the instant t = t 0

134

8 Theoretical Basis of Matched Signal Filtration

are in the phase and sum of them reaches an extreme value. The following property is shown in Fig. 8.5, using an example of three harmonic components. The below equation shows the reaction of the signal s(t) matched filter to signal x(t) ≡ s(t). According to formulae (8.10) and (8.15) {t y(t) =

{∞ s(τ ) · h(t − τ )dτ = h 0

−∞

s(τ ) · s[τ − (t − t0 )]dτ

−∞

because h(t − τ ) = h 0 s[t0 − (t − τ )] = h 0 s[τ − (t − t0 )]. At the instant t = t 0 , the above equation is simplified to {∞ y(t0 ) = h 0

s(τ ) · s(τ )dτ = h 0 · E

(8.19)

−∞

where E denotes a signal energy. The signal value y(t0 ), expressed by formula (8.19), corresponds to the maximum value of the cross-correlation function ϕs,s (0). To conclude the general considerations regarding the matched filter, the ratio of the signal peak power, obtained at the output of the matched filter, to the average power of white noise, will be determined. The average value of the squared noise voltage at the output of the matched filter to the signal s(t), is expressed by the formula:

Fig. 8.5 Graphical interpretation of the in-phase summation of the harmonic components of the output signal at the time moment t0

8.3 Examples of Standard Signals Matched Filters

n2

N = 2π

{∞

135

h2 N |H(ω)| dω = 0 2π

{∞ |S(ω)|2 dω

2

0

(8.20)

0

where{ N denotes a spectral the Rayleigh formula { ∞ density2 of a white { noise. Using ∞ 1 ∞ 1 2 E = −∞ s 2 (t)dt = 2π −∞ |S(ω)| dω = π 0 |S(ω)| dω one can express {∞ |S(ω)|2 dω = E · π

(8.21)

0

Considering formula (8.21) in the expression (8.20), one obtains n2 =

h2 N · E h 20 N · E ·π = 0 2π 2

(8.22)

The ratio of the signal peak power to the average power of white noise can be derived from formulae (8.19) and (8.22) and is equal to |y(t0 )|2 n2

=

h 20 E 2 h 20 N E 2

=2

E . N

(8.23)

It follows from (8.19) that the above ratio is independent of the signal shape.

8.3 Examples of Standard Signals Matched Filters Example 8.1 The present section illustrates a design example for the filter matched to a pulse shown in Fig. 8.6 . The pulse shown in Fig. 8.6 can be expressed as a linear combination of two Heaviside step functions, e.g. u(t) = 1(t) − 1(t − ti ) where ti is a pulse duration and the Heaviside step function is defined as: Fig. 8.6 A rectangular pulse with an amplitude equal to 1 and a duration ti

136

8 Theoretical Basis of Matched Signal Filtration

| | 0 for t < 0 | 1(t) = || 0.5 for t = 0 | 1 for t > 0 According to formula (8.15), i.e. h(t) = h 0 s(t0 − t), the pulse response of the designed filter must be a “mirror reflection” of the input signal u(t) multiplied by a constant factor, see Fig. 8.7. Such response h(t) corresponds to a circuit (filter) whose scheme is shown in Fig. 8.8 If a rectangular signal u(t) is fed at the input of this circuit (filter), then a triangular signal with a maximum value 1 · ti and base 2ti appears at its output, Fig. 8.8. Example 8.2 Example 8.2 shows a filter which is matched to a single radio pulse, expressed by the following formula

Fig. 8.7 A pulse response of the filter under assumption that this filter is matched to the signal shown in Fig. 8.6

Fig. 8.8 The filter matched to the signal shown in Fig. 8.6 and its responses on s(t) ≡ δ(t) and s(t) = 1(t) − 1(t − ti )

8.3 Examples of Standard Signals Matched Filters

137

| | A cos(ω0 t) for |t| ≤ ti /2 s(t) = || 0 for |t| > ti /2 A complex spectrum of this single radio pulse is determined using the integral Fourier transform (7.7) and Euler substitution cos(ω0 t) = [exp( j ω0 t) + exp(− j ω0 t)]/2. Consequently, due to these operations one obtains A S(ω) = 2 A = 2

{ti /2 [exp( j ω0 t) + exp(− j ω0 t)] exp(− j ωt)]dt −ti /2

{ti /2 −ti /2

A [exp[ j (ω0 − ω)t]dt + 2

{ti /2 [exp[− j(ω0 + ω)t]dt −ti /2

[ ] Ati sin[(ω0 − ω)ti /2 sin[(ω0 + ω)ti /2 = + 2 (ω0 − ω)ti /2 (ω0 + ω)ti /2

(8.24)

As indicated in Fig. 8.6, for ω0 = 0, the considered signal transforms into a video pulse and its spectrum (8.24) is reduced to the form [ ] sin(ωti /2) Ati 2 sin(ωti /2) = Ati S(ω) = 2 ωti /2 ωti /2 The spectrum distribution for positive angular frequencies is described by S(ω) =

Ati sin[(ω − ω0 ) · ti /2] · 2 (ω − ω0 ) · ti /2

(8.25)

Equation (8.25) is illustrated in Fig. 8.9. It results from (8.25) that For ω → ω0

Fig. 8.9 Spectrum of the rectangular radio pulse with an amplitude equal to A and a duration ti ; a an amplitude spectrum, b a phase spectrum

138

8 Theoretical Basis of Matched Signal Filtration

the spectrum |S(ω)| → Ati /2. The angular frequencies ω(0) = ω0 ± n∆ω determine the zeros of a spectrum (8.25), where ∆ω = (2π )/ti and n is a natural number. According to formula (8.18), the transmittance of the signal matched filter should indicate H(ω) = h 0 · e− jωt0 · S∗ (ω). Unfortunately, from the technical viewpoint, the implementation of such filter is highly difficult. Therefore, a band-pass filter with a limited filter’s transmission bandwidth and equal to 2B, is used as a replacement. The signal at the output of an ideal band-pass filter with a bandwidth equaling 2B, is determined by the formula 1 s0 (t) = 2π

ω0 { +2π ·B

1 · e j (ω−ω0 )τ0 S(ω)e jωt dω

(8.26)

ω0 −2π ·B

where τ0 corresponds to a time delay determined by this filter. Substituting formula (8.25) to the integral (8.26) and integrating, one obtains so (t) = Ao (t) cos(ω0 t)

(8.27)

where )] [ ( )]} { [ ( 1 1 A − Si 2π B t − τ0 − ti , Si 2π B t − τ0 + ti π 2 2 {x sin(t) Si (x) = dt t Ao (t) =

0

At the time instant t = τ0 , the output voltage amplitude reaches a maximum Ao max =

2A Si (π · B · ti ) π

(8.28)

The average value of the white noise voltage square is equal to n 2 = N · 2B. Thus, the ratio of the peak signal power to the average noise power equals A2o max n2

=

4 A2 Si 2 (π · B · ti ) π 2 N · 2B

(8.29)

The ratio (8.29) reaches a maximum value, equal to 0.82 A2 ti /N for 2Bti = 1.37 [6, 8]. The previous considerations derived from formula (8.23) show that the ratio (8.29) of the matched filter is equal to 2E/N . Thus, the energy of the signal s(t) equals E = A2 ti /2. Considering the formulae derived above, one can determine the deterioration coefficient of the signal to the noise ratio, which in this case equals

8.3 Examples of Standard Signals Matched Filters

139

Fig. 8.10 A sequence of N video pulses with the same duration τ0 and repetition frequency FP R = 1/TP R

ρ=

0.82 · A2 ti /N 0.82 · A2 ti /N = = 0.82 2E/N 2 · A2 ti /(2N )

This results (ρ = 0.82) shows that in the analyzed case an ideal band-pass filter with an optimal bandwidth 2B = 1.37/ti deteriorates the ratio of the signal power to the white noise only by 18%, in comparison to the value of the filter ideally matched to the considered signal. Example 8.3 Example 8.3 presents a filter which is matched to a sequence of video pulses, shown in Fig. 8.10. The signal s(t) which occurs as a sequence of N pulses, can be expressed as follows s(t) =

N −1 ∑

Uk · u(t − k · TP R )

(8.30)

k=0

where u(t) corresponds to the function describing single pulses with an amplitude equaling 1 and Uk denoting an amplitude factor. In order to determine the transmittance H(ω) of the signal (8.30) matched filter, it is necessary to determine the spectrum of the signal S(ω), which can be expressed as follows S(ω) =

N −1 ∑ k=0

{∞ Uk

u(t − k · TP R )e− j ωt dt

(8.31)

−∞

When the spectrum of the kth pulse u(t − k · TP R ) is computed, a shift in time domain corresponds to multiplication by an exponential function in the frequency domain: Sk (ω) = S0 (ω)e− j ωkTP R

(8.32)

140

8 Theoretical Basis of Matched Signal Filtration

where S0 (ω) is the spectrum of the initial pulse (t = 0) which is subsequently referred to as the reference pulse. The index k refers to the delayed pulse k · TP R with respect to the initial pulse, i.e. the pulse occurring at the instant t = k · TP R . Considering Eq. (8.32), from the expression (8.31) one obtains S(ω) = S0 (ω)

N −1 ∑

Uk e− jωkTP R

(8.33)

k=0

Formula (8.33) demonstrates that the overall duration of the signal s(t) (packets of N pulses) is equal to T0 = τ0 + (N − 1)TP R . The transmittance H(ω) of a filter matched to the signal s(t) considered in this example, can be expressed as follows H(ω) = h 0 e− jωT0 · e j ω(N −1)TP R · S∗0 (ω) ·

N −1 ∑

Uk e j ωkTP R · e− jω(N −1)TP R

k=0

The above formula yields the following equation H(ω) = h 0 e

− jωτ0

·

S∗0 (ω)

·

N −1 ∑

Uk e− jωTP R [(N −1)−k] = H1 (ω) · H (ω)

(8.34)

k=0

where H1 (ω) corresponds to a single pulse matched filter and the formula H (ω) =

N −1 ∑

Uk e− jωTP R [(N −1)−k]

(8.35)

k=0

denotes the transmittance of the integrator. According to formula (8.35), the integrator achieves a weight summation in the reversed time order to the order of pulses included in the packet. Accordingly, the weighting factor U0 is multiplied by the function exp[− j ωTP R (N − 1)] determining the position of the last pulse in the packet. Similarly, the coefficient U1 is multiplied by the function exp[− j ωTP R (N − 2)] which determines the position of the penultimate pulse in the packet. The last coefficient U N −1 is multiplied by the function exp[− j ωTP R (N − 1 − N + 1)] = 1, determining the position of the first, non-delayed pulse. The following reasoning explains the principle of operation of the matched filter, the functional diagram of which is shown in Fig. 8.11. The delay line with taps performs a function of a time-delay circuit. The weighting factors U0 , U1 , U2 , . . . U N −1 are set by the broadband attenuators or amplifiers. Figure 8.12 shows the time plot in the individual nodes of the considered filter, assuming that the input signal consists of three identical pulses, N = 2, U0 = U1 = U2 . For a packet of the identical pulses with an amplitude Uk = 1, where k = 0, 1, 2, . . . N − 1, the expression (8.35) is simplified to:

8.3 Examples of Standard Signals Matched Filters

141

Fig. 8.11 The block diagram of the filter matched to the signal (8.30)

Fig. 8.12 Time waveforms of signals at points A, B, C, and D of the filter shown in Fig. 8.11 when s(t) is the input signal

142

8 Theoretical Basis of Matched Signal Filtration

H (ω) = e− jω(N −1)TP R [1 + e jωTP R + e jω2TP R + · · · + e j ω(N −1)TP R ]

(8.36)

The multiplier which is included in square brackets of formula (8.36), is the sum of the geometric progress with base a = 1 and the exponent q = exp( j ωTP R ). The corresponding sum can be calculated according to the following formula 1 + q + q 2 + q 3 + · · · + q N −1 =

1 − qN 1−q

Accordingly, the transmittance (8.45) can be defined as follows H (ω) = e− jω(N −1)TP R /2

1 − e j ωN TP R sin(N ωTP R /2) = e− j ω(N −1)TP R /2 1 − e jωTP R sin(ωTP R /2)

(8.37)

The transmittance modulus of an interrogator H (ω) which is defined by Eq. (8.36), is equal to | | | sin(N π TP R f ) | | |H ( f )| = || sin(π TP R f ) |

(8.38)

The analysis of formula (8.38) shows that the transmittance H (ω) has maximum values which are equal to N , for the frequency f 0 = 0, f 1 =

1 2 3 , f2 = , f3 = ,... TP R TP R TP R

(8.39)

The transmittance modulus (8.38) takes zero values for frequencies f k = k/(N TP R ), which are different from frequencies (8.39). The transmittance |H (ω)|/N plot, which is normalized to the maximum value and computed for N = 11, is shown in Fig. 8.13.

Fig. 8.13 The plot of the transmittance |H (ω)|/N , see (8.38), computed for N = 11

8.3 Examples of Standard Signals Matched Filters

143

According to Fig. 8.13, the width of individual lobes of the presented characteristic decreases with the increase of the number of pulses N. In the limiting case N → ∞, the side lobes disappear and the width of the main lobes with an amplitude equaling 1, approaches zero. Then, the transmittance H1 (ω) · H (ω) corresponds to transmittance of the matched filter to the infinite pulse train of T p period. Multiplying the transmittance H (ω), which is described by formula (8.35), by the transmittance H1 (ω) = h 0 e− j ωτ0 · S∗0 (ω), see the Eq. 8.34 gives the transmittance H(ω) of the filter matched to the pulse packet with the length τ0 and repetition time TP R , Fig. 8.14a. Figure 8.14b shows an example plot of the transmittance H( f ) modulus in the range from f = 0, Hz to f = 1/τ0 , Hz. Constructing a filter with such transmittance, as shown in Fig. 8.14b, is remarkably difficult. Therefore, in the past, filters with a similar transmittance waveform, for example as in Fig. 8.14c, were used. The outline of this type of filter, commonly referred to as a comb filter, is shown in Fig. 8.15.

Fig. 8.14 The signal s(t) and transmittances |H ( f )|, |Hc ( f )| of the filters matched to it; a the plot of the signal s(t), b the transmittance |H ( f )|, c the transmittance |Hc ( f )| of a comb filter

144

8 Theoretical Basis of Matched Signal Filtration

Fig. 8.15 The block diagram of a comb filter whose transmittance |Hc ( f )| is shown in Fig. 8.14c

Each narrow-band channel of the comb filter has a delay line with an appropriate time delay τi and a narrow-band filter with a mid-band frequency i · F p , where i = 0, 1, 2, 3, . . . , k. The time delay τi corresponds to the complete delay related to the transmittance H1 (ω) = h 0 e− jωτ0 · S∗0 (ω) of the single pulse matched filter and the transmittance H (ω) = e− jω(N −1)TP R /2 sin(N ωTP R /2)/sin(ωTP R /2) of the integrator. The following calculation example shows the constructional complexity of such filter. For N = 20, F P R = 1/TP R = 500 Hz and τ0 = 1 μs, k = entire [1/(τ0 · FP R ] + 1 = entire [1/10−6 · 500] + 1 = 2001. The transmission bandwidth 2∆ f (at zero level) of each constitutive filter is equal to 2∆ f = 2FP R /N = 2 · 500/20 Hz = 50 Hz. Undoubtedly, the implementation of such complex comb filter is very demanding if not practically impossible. Example 8.4 This example determines the frequency transmittance of the filter matched to the N packet of radio pulses described by the formula s(t) =

N −1 ∑

Uk · u(t − k · TP R ) cos(ω0 t)

(8.40)

k=0

for which the function u(t − k · TP R ) and the amplitude coefficients Uk are identical to those described in Example 8.3. The signal plot in the time domain is shown in Fig. 8.16. Also in this case, the transmittance H(ω) of the filter matched to signal (8.40) is the product of the transmittance H1 (ω) of the single radio pulse matched filter and the transmittance H (ω) of the radio pulses integrator, implemented similarly as in Example 8.3. Figure 8.17 shows the radio signal waveforms which correspond to the respective pulses shown in Fig. 8.12. In the case when the amplitudes Ui of all N radio pulses are equal, the modulus of the spectral-response characteristic of the analyzed signal has a shape similar to the one shown in Fig. 8.18.

8.3 Examples of Standard Signals Matched Filters

145

Fig. 8.16 The time sequence of N radio pulses

Fig. 8.17 The radio signal waveforms which correspond to the respective signals shown in Fig. 8.12

The signal in Fig. 8.18 can be assigned to a band comb filter (approximating the integrator circuit) with the structure shown in Fig. 8.19. In comparison to the filter shown in Fig. 8.15, the band comb filter contains almost twice as many narrowband channels. Consequently, the filter is more complex,

146

8 Theoretical Basis of Matched Signal Filtration

Fig. 8.18 The spectral characteristic of the signal s(t) described by formula (8.40)

Fig. 8.19 The block diagram of the band comb filter matched to signal s(t) described by formula (8.40)

expensive and difficult to implement. The nature of the considerations regarding the comb filters introduced in the present chapter is exclusively cognitive. Due to the development of computer technology and digital signal processing methods, matched filtering is currently implemented numerically, as described at the end of Chap. 9.

References

147

References 1. Abramowitz M, Stegun IA (1960) Handbook of mathematical tables. National Bureau of Standards, Applied mathematics series 55. US Government Printing Office, Washington 2. Baskakov SI (1988) Signals and electronic circuits, 2nd edn. Publishing House “Higher School”, Moscow (in Russian) 3. Cook CHE, Bernfeld M (1967) Radar signals, an introduction to theory and applications. Academic Press, New York 4. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “Soviet Radio”, Moscow (in Russian) 5. Grigorin-Rjabow WW (ed) (1970) Radar devices, theory and principles of construction. Publishing House “Soviet Radio”, Moscow (in Russian) 6. Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken 7. Richards MA (2014) Fundamentals of radar signal processing, 2nd edn. McGraw-Hill Education, New York 8. Rosloniec S (2020) Fundamentals of the radiolocation and radionavigation, 2nd edn. Publishing House of the Military University of Technology, Warsaw (in Polish) 9. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 10. Yoshifumi A, Sato Y (1999) Introduction to signal management, 2nd edn. Ohmsha, Japan. The Russian translation of this book was published in 2002 by Dodeka–XXI, Moscow

Chapter 9

Filters Matched to the Typical Radar Signals

Radar signals which occur as a sequence of high frequency N pulses do not allow for a large detection range with sufficient resolution by a radar station. According to a general principle, in order to obtain large range, one should use high energy probe signals, for instance with a long duration T0 . On the other hand, in order to provide a sufficient resolution at a distance, the probe signal should be as short as possible. Such contradictory requirements can be fulfilled simultaneously however, by using more complex signals, e.g. high frequency (high energy) long pulses with internal frequency or phase modulation. An example of such signal is considered in the following subsection.

9.1 Filter Matched to a LFM Signal First, a signal with linear frequency modulation (LFM) is considered. This high frequency signal is defined given by the following formula | [ ] | A cos ω0 t + 1 μt 2 for 0 ≤ t ≤ T0 2 | s(t) = | 0 for 0 > t > T0

(9.1)

where T0 is a duration and μ corresponds to a fixed coefficient. The instantaneous frequency ] [ d 1 2 ω0 t + μt = ω0 + μt ω(t) = dt 2

(9.2)

of this signal corresponds to a linear time function for 0 ≤ t ≤ T0 . A signal waveform with a linear frequency modulation is shown in Fig. 9.1. A matched filter should receive such high frequency signal against a white noise background, which means that the transmittance H(ω) should be expressed by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_9

149

150

9 Filters Matched to the Typical Radar Signals

Fig. 9.1 A signal with linear frequency modulation (LFM); a a signal waveform in the time domain, b an instantaneous frequency waveform

formula H(ω) = h 0 · e− jωt0 S∗ (ω)

(9.3)

where: h 0 is an amplitude coefficient, t0 > T0 is the given time delay and S∗ (ω) is the complex conjugate spectrum with respect to the signal spectrum, i.e. S(ω) = |S(ω)|e jϕS (ω)

(9.4)

According to (9.3), the plot of the filter’s transmittance |H(ω)| should be similar to the complex spectrum magnitude |S(ω)|. On the contrary, the phase characteristic of the filter’s transmittance should be expressed by the formula ψ(ω) = −ωt0 − ϕs (ω)

(9.5)

Applying the Fourier transform to the signal (9.1), one obtains .T0 S(ω) = 0

] [ 1 cos ω0 t + μt 2 e− j ωt dt 2

(9.6)

9.1 Filter Matched to a LFM Signal

151

Due to integration, the following formula is derived [3, 8]. 1 S(ω) = 2

/

] [ ] π j (ω0 − ω)2 [ + · C + C − + j (S + + S − ) exp μ 2μ

(9.7)

where: ] ] [ μT0 − 2(ω0 − ω) μT0 + 2(ω0 − ω) − , C =C C =C √ √ 2 πμ 2 πμ ] ] [ [ μT0 − 2(ω0 − ω) μT0 + 2(ω0 − ω) , S− = S S+ = S √ √ 2 πμ 2 πμ +

[

( 2) .x ( 2) .x Functions C(x) = 0 cos π2t dt and S(x) = 0 sin π2t dt are the Fresnel integrals [3, 8, 10]. If the frequency deviation . f L F M = f max − f min of the signal (9.1) is sufficiently large, assuming approximately that the shape of amplitude-frequency response |H(ω)| ≈ |S(ω)| is similar to the one shown in Fig. 9.2, one can derive its phase characteristic: ϕs (ω) =

(ω0 − ω)2 2μ

(9.8)

In this situation, one can assume that the filter intended for an optimal signal reception with a linear frequency modulation has a rectangular amplitude-frequency response with a width equal to B ≈ μT0 /(2π ) and a phase characteristic expressed by the formula ψ(ω) = −ωt0 −

(ω0 − ω) 2 2μ

(9.9)

The delay of the any four-terminal network totals tdel = − ∂ψ(ω) , therefore the ∂ω delay of the considered filter, is equal to Fig. 9.2 A shape of amplitude spectrum of the LFM signal at sufficiently large frequency deviation

152

9 Filters Matched to the Typical Radar Signals

tdel = −

∂ψ(ω) (ω0 − ω) = t0 − ∂ω μ

(9.10)

Formula (9.10) shows that the filter matched to the signal (9.1) introduces a time delay that changes linearly together with the frequency. Depending on the parameter μ (its magnitude and sign) and the frequency range of the filter’s operation [ωmin , ωmax ], one should assume such value of t0 > T0 , that the filter delay is not negative for ωmin ≤ ω ≤ ωmax . Due to the changing delay time, the pulse at the output of the matched filter is significantly shorter than the input pulse, but its amplitude is correspondingly larger. This property is referred to as a pulse width compression. The discussed compression process can be easily explained by treating the continuous frequency modulation pulse as a limiting case of a “step modulation” pulse, i.e. as a sequence of consecutive short pulses with a width T0 /N , where N corresponds to the number of discrete frequency values, Fig. 9.3. The frequency transmission of each short pulse differs, whereby the frequency within such short pulse is constant. The pulses within a filter matched to the signal are delayed to a different extent, whereby the lowest frequency pulses are delayed most. Thus, it is possible to select the filter’s length and the frequency range [ f min , f max ] (when the fixed coefficient μ is fixed) so that all component pulses (with a T0 /N length) reach the end of the filter at the same time instant. Thus, as shown in Fig. 9.3c, a pulse with duration T0 /N and amplitude N · A occurs due to the coincidence of the component pulses. Consequently, the output signal with a spectrum expressed by formula (9.7) and shape similar to the one shown in Fig. 9.4a is obtained, when Fig. 9.3 Graphical illustration of the LFM signal compression process

9.1 Filter Matched to a LFM Signal

153

the continuous linear frequency modulation (N → ∞) is applied. The duration ratio of the input pulse T0 to the “main lobe” duration of the output signal TC ≈ 1/B is referred to as the compression coefficient kC = T0 /TC ≈ T0 B. In practice, the compression ratio reaches values of the order of 1000 when analog compression filters are used. This conclusion does not apply directly to numerically implemented compression, i.e. the digital matched filtration. The envelope of the output signal shown in Fig. 9.4a is similar to a cross section χ (τ, FD = 0) of the indeterminacy function, Fig. 9.4b, which was derived in Appendix G. The LFM signal with a frequency changed due to the Doppler effect, still remains nearly matched to a filter constructed for the original pulse, which is very important from the practical viewpoint. In the initial period of radiolocation, an electronically tuned generator, commonly referred to as wobbulator, was considered as a source of LFM pulses, Fig. 9.5.

Fig. 9.4 The LFM signal after compression and its ambiguity function; a a waveform of the compressed signal in the time domain, b an ambiguity function

154

9 Filters Matched to the Typical Radar Signals

Fig. 9.5 A wobbulator as a simple source of LFM pulses; a block diagram of the wobbulator integrated with a power amplifier, b a waveform of the wobbulator signal frequency, c a waveform of the voltage supplying the power amplifier

The signal generated by the wobbulator was amplitude stabilized and amplified by a broadband amplifier. This solution, which can be found in the contemporary radio technology, unfortunately had a number of limitations, e.g. related to the implementation of a suitable microwave wobbulator. Therefore, in the improved version of the LFM signal generator, the wobbulator was replaced with a passive transversal filter which is shown in Fig. 9.6. Such filter consists of a lossless and non-dispersive delay line L O with taps [1, 5, 11]. The components, which occur as a delayed input signal u in (t), are fed through the taps with weighting elements wi to the adder .. Obtaining a signal to similar LFM signal is possible by selecting appropriate spacing (delays) between taps and weighting factors wi , changing both, the amplitude

Fig. 9.6 The block diagram of the passive transversal filter

9.1 Filter Matched to a LFM Signal

155

Fig. 9.7 A piezoelectric filter with interdigital transducers, a a geometric structure of the filter, b a moving impulse of a surface mechanical wave

and the phase of the transmitted signal. Depending on the operating frequency range, various types of waveguides, including mechanical wave guides, were used as delay lines. For example, the wire lines operating in the range of 1–10 MHz, in which the torsional vibration are induced by magnetostriction transducers, are included in [8]. The significant breakthrough in the field of transversal filters development, which are suitable for modern radiolocation, was possible due to the piezoelectric materials and photolithographic techniques. The piezoelectric filters, made on quartz (SiO2 ) or lithium niobate (LiNbO3 ) substrates, have been used for many years both, for the formation and optimal filtration of LFM radio pulses [6, 7]. The construction outline and the principle of operation of this filters type are shown in Fig. 9.7. The main elements of the discussed filter include two IDT interdigital transducers. Each transducer, in turn, consists of metal electrodes applied to the upper surface of the piezoelectric plate by the photolithographic method. The alternating voltage applied to the input transducer generates an alternating electric field in its area, which leads, in turn, to vibrations of the surface part of the substrate. These vibrations propagate in the form of a mechanical wave to both sides, i.e. towards the input terminals and the output transducer. The average velocity of the Surface Acoustic Waves (SAW) propagation in the quartz filter is relatively low and is equal to ϑ = 3100 m/s. Thus, it is possible to obtain sufficiently large time delays. The mechanical (acoustic) wave reaching the individual electrodes of the output transducer, generates a voltage on its terminals, the amplitude and frequency of which depend on the wave propagation in the piezoelectric substrate and the output converter geometry. Assume that a short and high amplitude video pulse is fed to the terminals 1−1' of the filter shown in Fig. 9.7a. The broadband input transducer, with a small number

156

9 Filters Matched to the Typical Radar Signals

of electrodes, induces a surface wave as a short pulse similar to the input electric pulse, Fig. 9.7b. Such wave, passing through the output transducer of a length l, induces an electric signal of a variable frequency on its terminals 2−2' , which is dependent on the distance di ≡ d(xi ) between the successive electrodes. As a general rule, this distance decreases linearly, which results in the increase of the output signal frequency f (x i ) ≈ ϑ/(2di ), similar to the linear one. The overall length of the generated LFM signal is approximately equal to T = l/ϑ. For example for l = 62 mm T = 60 × 10−3 l/3100 ≈ 20 µs. At both ends of the piezoelectric filter’s substrate, the Surface Acoustic Waves attenuator is located, which eliminates wave reflections. One should also emphasize that the discussed filter operation range equals several dozen MHz at most and at a low power level mW. Thus, in order to generate a radar probe signal, the signal obtained at the output of the piezoelectric filter should be amplified and shifted to the appropriate frequency range. Currently, the LFM signal is synthesized mainly by Direct Digital Synthesis (DDS). Then, such signal is shifted to the appropriate range of microwave frequencies and amplified [9, 12]. As mentioned before, SAW filters were also used for the optimal signal LFM reception. The received echo signal was initially shifted to the lower frequency range of the order of several dozen MHz. Such LFM signal was fed to the input transducer. The surface acoustic wave, which is generated by the input transducer, reaching the ' output transducer as shown in Fig. 9.8, induced at its terminal 2−2 a chirp signal with a duration Tw = 1/B, where B corresponds to the effective spectral width of the input signal. The passage time of the input signal’s initial fragment through the filter, which has the lowest filling frequency/ frequency of the pulse duration modulation, is the

Fig. 9.8 An example of geometrical structure of the SAW filter used for compression of the LFM signal

9.2 Filters Matched to High Frequency Pulses with Bistate …

157

largest, because it is “transmitted” by the last (the farthest distanced from the receiver electrodes) electrodes of the output transducer. Conversely, the closing fragment of the input signal with the highest frequency of the pulse duration modulation is least delayed since it is transmitted by the closest electrodes. If the difference in delay times of these signal fragments is equal to the duration of the input signal, i.e. T ≈ l/ϑ, they ' reach the terminals 2−2 simultaneously. The propagation of remaining fragments of the signal, distributed symmetrically with respect to its beginning and end, should be considered in a similar fashion. As a result of overlapping the signal fragments, an output signal of a very short duration TC is obtained. In the last two decades, piezoelectric filter technology, which is considered as a fairly “fickle” and expensive one, has been replaced by appropriate algorithms of digital, optimal filtration. The theory elements of this modern and more universal method of digital signal processing are described at the end of this chapter.

9.2 Filters Matched to High Frequency Pulses with Bistate Phase Modulation Another group of signals used in modern radiolocation and radionavigation consists of long pulses of high frequency (high energy) with bistate (0, π ) phase modulation referred as Binary Phase-Shift Keying (BPSK), Fig. 9.9. As an example let us consider a waveform of the seven segment signal, similar to that shown in Fig. 9.10. Figure 9.10b symbolically shows the plot of the signal phase as a code (−1, 1, −1, −1, 1, 1, 1). Such signals are desirable due to the rectangular envelope of the entire pulse, and ease of generation and matched filter implementation. The signal shown in Fig. 9.10a can be considered as one composed of seven (N = 7) short pulses of equal lengths τ0 = T 0 /7. The phases of individual short radio pulses are switched between 0 and π radians according to a control code, which is most frequently implemented using a circuit similar to the one shown in Fig. 9.11. The autocorrelation function of the BPSK signal should have one maximum with possibly the highest value in order to secure the detection uniqueness. This condition is satisfied by special coding signals, including the Barker codes [2, 8]. The mathematical models of these codes (that are known so far) are shown in Table 9.1. Fig. 9.9 A signal with the bistate phase modulation, a a code of the modulating signal, b a waveform of the modulated signal

158

9 Filters Matched to the Typical Radar Signals

Fig. 9.10 An example of a seven segment signal with the bistate phase modulation, a the waveform of the signal in the time domain, b the code of the modulating (control) signal

Fig. 9.11 The block diagram of the simple modulator circuit at the output of which the BPSK signal is obtained

The autocorrelation function of the codes from Table 9.1, is calculated according to the formula, ϕ(k) =

∞ .

u j u j−k , k ≤ N − 1

(9.11)

j=−∞

The autocorrelation function assumes the values of −1, 0 or 1 for all k /= 0 and has the maximum value equal to N for k = 0.

9.2 Filters Matched to High Frequency Pulses with Bistate …

159

Table 9.1 Barker codes N

Code (u 1 , u 2 , u 3 , . . . , u N )

Code −(u 1 , u 2 , u 3 , . . . , u N )

2

1, −1

−1, 1

3

1, 1, −1

−1, −1, 1

4

1, 1, 1, −1 or 1, 1, −1, 1

−1, −1, −1, 1 or −1, −1, 1, −1

5

1, 1, 1, −1, 1

−1, −1, −1, 1, −1

7

1, 1, 1, −1, −1, 1, −1

−1, −1, −1, 1, 1, −1, 1

11

1, 1, 1, −1, −1, −1, 1, −1, −1, 1, −1

−1, −1, −1, 1, 1, 1, −1, 1, 1, −1, 1

13

1, 1, 1, 1, 1, −1, −1, 1, 1, −1, 1, −1, 1

−1, −1, −1, −1, −1, 1, 1, −1, −1, 1, −1, 1, −1

Example 9.1 As an example illustrating (9.11) one should consider a four-element code. . . 0, 1, 1, 1, −1, 0, . . . that, along with its copies shifted by k position, where k = 1, 2, 3, can be expressed correspondingly Reference → . . . k=0 ... k=1 ... k=2 ... k=3 ...

0, 0, 0, 0, 0,

1, 1, 0, 0, 0,

1, 1, 1, 0, 0,

1, 1, 1, 1, 0,

−1, 0, 0, 0, −1, 0, 0, 0, 1, −1, 0, 0, 1, 1, −1, 0, 1, 1, 1, −1,

0, 0, 0, 0, 0,

... ... ... ... ...

The autocorrelation function of the code calculated for individual shifts has the following values: ϕ(k = 0) = 1 · 1 + 1 · 1 + 1 · 1 + (−1) · (−1) = 4, ϕ(k = 1) = 1 · 0 + 1 · 1 + 1 · 1 + (−1) · 1 = 1, ϕ(k = 2) = 1 · 0 + 1 · 0 + 1 · 1 + (−1) · 1 = 0 ϕ(k = 3) = 1 · 0 + 1 · 0 + 1 · 0 + (−1) · 1 = −1. A functional structure of the filter matched to the BPSK signal, modulated according to the four-element code with overall length T0 = 4τ0 , is shown in Fig. 9.12. The phase SPD demodulator (synchronous phase detector) corresponds to a first functional block of the matched filter and does not deteriorate the ratio of a signal to a noise. At the output of this detector, a pulse signal mapping the phase waveform of the received signal s(t), is obtained. This signal is processed in the interrogator circuit. The time-shifted copies of this signal are summed up with weighting factors inserted in the reversed order to their occurrence in the processed signal, see Fig. 9.12. In order to reproduce the phase signal, a harmonic reference signal with a frequency equal to a carrier frequency of the received signal is required. The SPD system reproduces such signal considering the initial waveform of the signal s(t). The processes of carrier wave reproduction signal s(t) detection, are most frequently implemented

160

9 Filters Matched to the Typical Radar Signals

Fig. 9.12 The block diagram of the filter matched to the BPSK signal composed of four segments (N = 4)

together, using a phase lock loop, referred to in the literature to as the Costas loop, see Appendix H. Optimal Binary Codes Except the Barker codes discussed in the present section, radiolocation uses the so-called optimal binary codes for which most extreme side lobes of the autocorrelation function reach the minimum level. In other words, these codes are determined according to the minimax criterion [8]. In the literature, such codes are marked with symbols P S L-1, P S L-2, P S L-3 …, where the digit standing next to the symbol’s dash corresponds to the maximum level of side lobes. In the case of the codes shown in Table 9.1, the ratio of the autocorrelation function’s maximum value to the maximum value of the side lobes is equal to |±1|/13. According to [8], the corresponding groups of codes include: 1. P S L-1; 7 codes (N = 2, 3, 4, 4, 5, 7, 11, 13). 2. P S L-2; 10 codes (13 < N ≤ 21, N = 25 and N = 28). 3. P S L-3; 25 codes (22 ≤ N ≤ 48, excluding N = 25, 28 and N = 51). A several exemplary P S L-2 codes are shown in Table 9.2. The P S L, dB values in the third column are calculated according to the following formula: P S L [dB] = 20 log(2/N ). Similarly, for the thirteen-position (N = 13) Barker code, the values can be expressed by the following formula: P S L [dB] = 20 log(1/13) = −22.27.

9.3 Introduction to a Digital Matched Filtration of Radar Signals

161

Table 9.2 Several exemplary of P S L-2 codes Length N of code

Codes number

P S L, dB

Exemplary plot of the code P S L-2

14

18

−16.90

0, 1, 0, 1, 0, 0.1, 0, 0, 0, 0, 0, 1, 1

16

20

−18.06

0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1

21

6

−20.42

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1

9.3 Introduction to a Digital Matched Filtration of Radar Signals The examples of the previously demonstrated implementation of matched filters, indicate their complexity in terms of construction and the application limited area. In other words, a filter matched to a particular signal is not useful for matched filtration of different signals. These limitations do not occur in the case of implementation of digital matched filtration, which has been developed in the recent decades. The following operations are distinguished in the digital matched filtration algorithm: 1. Shifting the received radar signal into the lower (video) frequencies and creating two analog signals u (I ) (t) and u (Q) (t) which are orthogonal to each other. This task can be performed using an analog converter segment I /Q with a functional diagram as in Fig. 9.13. 2. Converting the analog signals u (I ) (t) and u (Q) (t) into appropriate digital signals by sampling, quantizing and coding. These operations are implemented by sampling circuits U P and analog-to-digital converters A/D.

Fig. 9.13 The functional diagram of the two channel analog–digital converter at the outputs of which the orthogonal digital signals I (n) and Q(n) are obtained

162

9 Filters Matched to the Typical Radar Signals

3. Numerical computation of the digital filter’s response matched to the activation by the digital signal obtained in the previous step. This task can be solved both in time and frequency domain. Both of these approaches are the subject of further considerations. Assume that the following signal s(t) = S(t) cos[ω0 t + φ S (t)]

(9.12)

is fed to the analog converter segment I /Q. According to Fig. 9.13, such signal together with the heterodyne signal u h (t) = Uh cos(ω0 t + 0) activates the mixer M I which obtains a signal proportional to the product s(t) · u h (t) at the output, i.e. u (IM) (t) = k M S(t)Uh cos[ω0 t + φ S (t)] cos(ω0 t) 1 1 = k M S(t)Uh cos[2ω0 t + φ S (t)] + k M S(t)Uh cos[φ S (t)] 2 2

(9.13)

When the signal (9.13) is filtered out of the high-frequency component (with a pulsation equal to ≈ 2ω0 ) with a low-pass filter L P F, one obtains u (I ) (t) =

1 k M k F S(t)Uh cos[φ S (t)] 2

(9.14)

where k M and k F correspond to the coefficients of a mixer conversion M I and filter attenuation L P F, respectively. The signal s(t) and the heterodyne signal delayed by the angle π/2 rad, are fed to the mixer M Q of the discussed converter. Similarly to the upper channel, the signal appearing at the output of the mixer M Q is proportional to the product of the fed signals, i.e. u (Q) M (t) = k M S(t)Uh cos[ω0 t + φ S (t)] cos(ω0 t − π/2) 1 1 = − k M S(t)Uh sin[2ω0 t + φ S (t)] − k M S(t)Uh sin[φ S (t)] 2 2

(9.15)

When the high-frequency component is filtered out, the following signal is obtained [ 1 π] 1 (9.16) u (Q) (t) = − k M k F S(t)Uh sin[φ S (t)] = k M k F S(t)Uh cos φ S (t) + 2 2 2 The signal (9.16) is placed in the quadrature with respect to (9.14). Both real signals (9.14) and (9.16) contain full information regarding the plot of amplitude S(t) and phase φ S (t) of the signal (9.12), i.e. 2 S(t) = k M k F Uh

/ [ ]2 [ ]2 u (I ) (t) + u (Q) (t) and

9.4 Matched Filtration in Time Domain

163

[

u (Q) (t) φ S (t) = −arctg (I ) u (t)

] (9.17)

Assume that the spectra of signals (9.14) and (9.16) are accumulated in the frequency range [0, f m ]. In other words, the frequencies of all components of these signals are lower than the frequencies f m . The low-pass filters (L P F), which are used in the converter, should also have this corner frequency. In the digital segment of the converter shown in Fig. 9.13, the signals u (I ) (t) and u (Q) (t) are evenly sampled at a frequency f S ≥ 2 f m , i.e. according to the Kotelnikov-Shannon sampling theorem. Then, the analog samples of signals are quantized in order to assign to them the corresponding digital values. Thus, the appropriate multi-bit A/D converters are used. The process in which the quantized sample values is replaced with the corresponding digits, is referred to as a coding. Consequently, a set of number pairs written in the binary number system is obtained, corresponding to individual complex samples √ u(tn ) = u (I ) (tn ) + ju (Q) (t n ), where j = −1. The argument of each sample corresponds to collection time, which can be represented by its n number when sampled at a constant frequency (time interval .t). The determination of the described complex samples facilitate the digital matched filtration, which can be performed both in time and frequency domains.

9.4 Matched Filtration in Time Domain According to the theory discussed in the previous section, a matched filter implements a convolution of the signal fed and its √ pulse response h(t). For a discrete signal u(ti ) = u (I ) (ti ) + ju (Q) (t i ), where j = −1, the convolution function is described by y(tn ) = h 0

n .

u(ti )h(tn − ti )

(9.18)

i=−∞

where h(tn − ti ) = 0 for tn − ti < 0. The pulse response of the filter matched to the signal containing N + 1 of samples u(ti ) = u (I ) (ti ) + ju (Q) (ti ), is related to this signal with the dependence h(tk ) = u (I ) (N · .t − tk ) − ju (Q) (N · .t − tk )

(9.19)

where .t = tk+1 − tk for k = 0, 1, 2, 3, . . . N − 1. The dependence (9.19) confirms the general principle that the pulse response of the filter matched to the discrete signal is proportional to the time inverted coupling of this complex signal. For tk = tn − ti the pulse response (9.19) is expressed as follows h(tn − ti ) = u (I ) (N · .t − tn + ti ) − ju (Q) (N · .t − t n + ti )

(9.20)

164

9 Filters Matched to the Typical Radar Signals

When u(ti ) = u (I ) (ti ) + ju (Q) (t i ) and (9.20) is substituted to formula (9.18), one obtains n .

y(tn ) = h 0

n .

u(ti )h(tn − ti ) = h 0

i=−∞

[X (tn , ti ) + jY (tn , ti )]

(9.21)

i=−∞

where: X (tn , ti ) = u (I ) (ti ) · u (I ) (N · .t − tn + ti ) + u (Q) (ti ) · u (Q) (N · .t − t n + ti ) Y (tn , ti ) = u (Q) (ti ) · u (I ) (N · .t − tn + ti ) − u (I ) (ti ) · u (Q) (N · .t − t n + ti ) The time instants tn , ti and duration of the input signal N · .t are explicitly defined by their indexes, i.e. n, i and N , respectively. When using these indexes, formula (9.21) can be expressed in the following “symbolic” form [2] y[n] = h 0

n .

u[i] · h[n − i] = h 0

i=−∞

n .

[X [n, i] + jY [n, i]]

(9.22)

i=−∞

where: X [n, i] = u (I ) [i] · u (I ) [N − n + i] + u (Q) [i] · u (Q) [N − n + i] Y [n, i] = u (Q) [i] · u (I ) [N − n + i] − u (I ) [i] · u (Q) [N − n + i] In the case of a signal consisting of real samples u(ti ) ≡ u(ti ), formula (9.22) is simplified to y[n] = h 0

n .

u[i] · h[n − i] = h 0

i=−∞

n .

u[i] · u[N − n + i]

(9.23)

i=−∞

For n = N , the filter response to the overall signal is obtained y[n = N ] = h 0

N .

u[i] · u[i] = h 0

i=−∞

N .

|u[i]|2 = h 0 E

(9.24)

i=−∞

where E corresponds to the signal energy of u[n]. Example 9.2 Example 9.2 shows discrete values of a signal y[n], which are calculated according to formula (9.20) for signal u c [n] = δ(n) + δ(n − 1) + δ(n − 2) + 0 · [δ(n − 3) + δ(n − 4) + δ(n − 5)], response h[n] = 0 · [δ(n) + δ(n − 1) + δ(n − 2)] + δ(n − 3) + δ(n − 4) + δ(n − 5) and h 0 = 1, see Fig. 9.14. According to formula (9.23)

9.4 Matched Filtration in Time Domain

165

Fig. 9.14 Discrete values of input signal u c [n], filter response h[n] and output signal y[n] discussed in Example 9.2

y[n = 0] =

0 .

u c [i] · h[0 − i] = 1 · 0 = 0

i=0

y[n = 1] =

1 .

u c [i] · h[1 − i] = 1 · 0 + 1 · 0 = 0

i=0

y[n = 2] =

2 .

u c [i] · h[2 − i] = 1 · 0 + 1 · 0 + 1 · 0 = 0

i=0

y[n = 3] =

3 . i=0

u c [i] · h[3 − i] = 1 · 1 + 1 · 0 + 1 · 0 + 0 · 0 = 1

166

9 Filters Matched to the Typical Radar Signals

y[n = 4] =

4 .

u c [i] · h[4 − i] = 1 · 1 + 1 · 1 + 1 · 0 + 0 · 0 + 0 · 0 = 2

i=0

y[n = 5] =

5 .

u c [i] · h[5 − i] = 1 · 1 + 1 · 1 + 1 · 1 + 0 · 0 + 0 · 0 + 0 · 0 = 3

i=0

y[n = 6] =

6 .

u c [i] · h[6 − i] = 1 · 0 + 1 · 1 + 1 · 1 + 0 · 1 + 0 · 0 + 0 · 0 + 0 · 0

i=0

=2 y[n = 7] =

7 .

u c [i] · h[7 − i] = 1 · 0 + 1 · 0 + 1 · 1 + 0 · 1 + 0 · 1 + 0 · 0 + 0 · 0

i=0

+0·0=1 y[n = 8] =

8 .

u c [i] · h[8 − i] = 1 · 0 + 1 · 0 + 1 · 0 + 0 · 1 + 0 · 1 + 0 · 1 + 0 · 0

i=0

+0·0+0·0=0 y[n > 8] = 0 The waveform of the output signal y[n] is shown in the lower part of Fig. 9.14, maintaining the proper (time) coincidence with respect to u[n] and h[n].

9.5 A Matched Filtration in the Frequency Domain There are several basic computational operations of digital matched filtration, which are relevant in the frequency domain: 1. Calculation of the spectrum X(ω) of a given discrete (digital) input signal x(t). 2. Determination of the spectrum H(ω), which is identical to the filter’s frequency transmittance, of the impulse response h(t) of the filter matched to this signal. 3. Multiplying the input signal spectrum X(ω) by the filter’s transmittance H(ω), in order to obtain the spectrum Y(ω) of the discrete output signal. 4. Recreating the output signal y(t) considering the spectrum Y(ω), which is calculated in step 3. In order to implement the described operations, the knowledge of the following mathematical dependencies and transformations is crucial. The Fourier Transform for a Discrete Function in the Time Domain (DFT) Consider an analog signal x(t), determined in a closed interval [0, T ], Fig. 9.15a Such signal can be represented by a set N of discrete values (samples) [x0 , x1 , x2 , . . . , x N −1 ], which are defined at time instants

9.5 A Matched Filtration in the Frequency Domain

167

Fig. 9.15 An analog signal with limited spectrum and its digital equivalent; a the analog signal x(t), determined in a closed interval [0, T ], b the digital equivalent of x(t) as a segment of the periodic discrete function

[0, .t, 2.t, . . . , (N − 1).t], where .t = T /N . Consequently, the set consisting of discrete values (real or complex), which are determined in accordance to a sampling theorem, forms a discrete (still analog) signal xd (t) =

N −1 .

x(k.t) · δ(t − k.t)

(9.25)

k=0

where δ(t −k.t) corresponds to a function which has value 1 for t = k.t or 0, when t /= k.t. The signal (9.25) repeated (thinkingly) the endless amount of time, forms the periodic discrete signal with a plot as shown in Fig. 9.15b. The signal (9.25) in the segment [0, T ] can be expressed as the Fourier series. xd (t) =

∞ 1 . cn exp( j2π nt/T ) T n=−∞

(9.26)

The coefficients cn of the series (9.26) can be calculated according to the formula .T cn =

xd (t) exp(− j2π nt/T )dt

(9.27)

0

When formula (9.25) is substituted to (9.27) and a non-dimensional variable ς = t/.t is introduced, one obtains

168

9 Filters Matched to the Typical Radar Signals

cn =

.N .t . N −1 0

=

.N . N −1 0

=

xk · δ(ς − k) exp(− j2π nς/N )dς

k=0

N −1 . k=0

=

xk · δ(t − k.t) exp(− j2π nt/T ).tdς

k=0

N −1 .

.N δ(ς − k) exp(− j2π nς/N )dς

xk 0

xk exp(− j2π nk/N )

(9.28)

k=0

The derived formula cn =

N −1 .

xk exp(− j2π nk/N ) =

k=0

N −1 .

xk · [cos(2π nk/N ) − j sin(2π nk/N )]

k=0

(9.29) is referred to as the discrete Fourier transform, or colloquially called the discrete transform, with the following properties. 1. The transform (9.29) is a linear transformation with respect to xk , which means that the sum of the discrete signals is equal to the sum of their discrete transforms. 2. The number of different coefficients c0 , c1 , c2 , . . . , c N −1 , which are calculated according to formula (9.29), is equal to the number of discretization points N . 3. The coefficient c0 (constant . N −1 component) is a arithmetic average of all discrete xk . values xk , i.e. c0 = k=0 . N −1 4. If the number of discretization points N is even, then c N /2 = k=0 xk (−1)k . 5. If the discrete values xk are real numbers, then the coefficients cn with symmetrical indexes with respect to N /2, form complex and coupled pairs c N −n =

N −1 . k=0

xk exp[− j2π(N − n)k/N ] =

N −1 .

xk exp( j2π nk/N ) = cn∗

k=0

Example 9.3 A discrete signal consists of a set of six (N = 6) numbers {xk } = (1, 1, 1, 0, 0, 0) determined at equidistant points of the period T , i.e. (0, T /6,T /3,T /2, 2T /3, 5T /6). The coefficients c0 , c1 , c2 , c3 , which are calculated according to formula (9.28), are respectively equal to: c0 = 3, √ ) ) ( ( c1 = 1 + 1 · e− jπ/3 + 1 · e− j2π/3 = 1 − j 3 ,

9.5 A Matched Filtration in the Frequency Domain

169

) ( c2 = 1 + 1 · e− j2π/3 + 1 · e− j4π/3 = 0 ) ( c3 = 1 + 1 · e jπ + 1 · e j2π = 1. Considering the conclusion which stems from point 5, (the remaining coefficients √ ) ∗ ∗ are correspondingly equal to c4 = c2 = 0 and c5 = c1 = 1 + j 3 . The Inverse Fourier Transform for a Discrete Function (IDFT) In case, the discrete values [x0 , x1 , x2 , . . . , x N −1 ] of the function xd (t) = . Nthis −1 x(k.t) · δ(t − k.t) are recreated, considering the coefficients cn which are k=0 determined by formula (9.29). When t = k · .t and T = N · .t are substituted to Eq. (9.26) and taking into account that only terms of the series with coefficients cn ⊂ [0, N − 1] are summed up, one obtains xd (k · .t) ≡ xk =

N −1 1 . cn ·[cos(2π nk/N ) + j sin(2π nk/N )] N n=0

(9.30)

Recreating an Analog Signal x(t) Considering a Discrete Transform (9.29) Considering the coefficients c0 , c1 , c2 , . . . , c N /2 of the real and discrete Fourier transform [x0 , x1 , x2 , . . . , x N −1 ], one can recreate a real analog signal x(t), see Fig. 9.15a. The Fourier series of this signal with an even number of samples N , corresponds to the following sum ) ) | c | ( 4π t | c | ( 2π t c0 | 2| | 1| + 2| | cos + ϕ1 + 2| | cos + ϕ2 N N T N T ) |c | ( Nπt | N /2 | + ϕ N /2 + ··· + | | cos N T

x(t) =

(9.31)

where ϕi is an argument (phase angle) of the corresponding coefficient ci . Example 9.4 Figure 9.16 shows a waveform of the analog signal x(t) = 0.5 + (2/3) cos(ωt − π/3) + (1/6) cos(3ωt), where ω = 2π/T , which is recreated according to the formula (9.31), considering the coefficients c0 , c1 , c2 = 0 and c3 calculated in the Example 9.3. At this point one should emphasize that the reproduced signal is identical to the analog sampling signal, see Fig. 9.15a, which can demonstrated as follows. The maximum spectrum frequency of the recreated signal is equal to f max = f 3 = 3 · 2π t/T and is twice lower than the sampling frequency f p = 2π t/(T /6). In other words, the Kotelnikov-Shanon condition is fulfilled. The signal subjected to optimal filtration gives a response y(t) at the output of the matched filter, with the plot as shown in Fig. 9.17. The presented plot y(t) is calculated according to the following formula

170

9 Filters Matched to the Typical Radar Signals

Fig. 9.16 The waveform of the signal under analysis in Example 9.4

Fig. 9.17 The response y(t) of the matched filter on an input signal similar to that shown in Fig. 9.16

y(t) =

2N −1 . n=0

y(n) ·

sin[π(t/.t − n)] π(t/.t − n)

(9.32)

where N = 6, .t = T /(N − 1) and y(n) ≡ y(n · .t) are the discrete values of a response signal, which are calculated in Example 9.2. Example 9.5 The second, third and fourth columns of Table 9.3 includes the values of the spectral lines coefficients, of the input signal u c (n), filter pulse response h(n) and the output signal y(n), respectively. Time courses u c (n), h(n) and y(n) are shown in Fig. 9.18. The calculations of the input signal spectral lines and the pulse response were made according to formula (9.29), assuming that N = 12. However, the spectral lines of the output signal are the products of the corresponding spectral lines of the input signal and the pulse response. Due to the matched filtration, the lengths u c (n) and h(n) need to be doubled by adding zero values from n = 6 to n = 11, where the length of the output signal is twice as long as the length of the input signal equaling Ns = 6, see Figs. 9.14 and 9.18. On the other hand, the signal spectrum in the form of N = 12 samples evenly distributed along the time axis, consists of N = 12 spectral lines distributed evenly along the frequency axis. One should also emphasize that the implemented “extension” of u c (n) and h(n) lengths, does not change their shape and mutual position and thus the waveform obtained at the output of the filter matched to u c (n).

9.5 A Matched Filtration in the Frequency Domain Table 9.3 Values of the spectral lines coefficients, of the input signal u c (n), filter pulse response h(n) and the output signal y(n)

171

n

u c (n)

h(n)

y(n)

0

3.000000 + j0.000000

3.000000 + j0.000000

9.000000 + j0.000000

1

2.366025 − j1.366025

−1.366025 − j2.366025

−6.464102 − j3.732051

2

1.000000 − j1.732051

−1.000000 + j1.732051

2.000000 + j3.464102

3

0.000000 − j1.000000

1.000000 + j0.000000

0.000000 − j1.000000

4

0.000000 + j0.000000

0.000000 + j0.000000

0.000000 + j0.000000

5

0.633975 + j0.366025

0.366025 − j0.633975

0.464102 − j0.267949

6

1.000000 + j0.000000

−1.000000 + j0.000000

−1.000000 + j0.000000

7

0.633975 − j0.366025

0.366025 + j0.633975

0.464102 + j0.267949

8

0.000000 + j0.000000

0.000000 + j0.000000

0.000000 + j0.000000

9

0.000000 + j1.000000

1.000000 + j0.000000

0.000000 + j1.000000

10

1.000000 + j1.732051

−1.000000 − j1.732051

2.000000 − j3.464102

11

2.366025 + j1.366025

−1.366025 + j2.366025

−6.464102 + j3.732051

The discrete output signal which is recreated according to formula (9.30) considering the coefficients in the fourth column of Table 9.3 is equal to y(n) ≡ [0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0]. This signal is identical to the signal y(n) shown in Example 9.2, see Fig. 9.14, which illustrates the time-domain matched filtration algorithm. Thus, this signal corresponds to the analog signal y(t), which is shown in Fig. 9.17.

172

9 Filters Matched to the Typical Radar Signals

Fig. 9.18 Discrete values of input signal u c [n], filter response h[n] and output signal y[n] discussed in Example 9.5

References 1. Bakuljev PA (2004) Radiolocation systems. Publishing House “Radio-Engineering”, Moscow (in Russian) 2. Baskakov SI (1988) Signals and electronic circuits, 2nd edn. Publishing House “Higher School”, Moscow (in Russian) 3. Cook CHE, Bernfeld M (1967) Radar signals, an introduction to theory and applications. Academic Press, New York 4. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “Soviet Radio”, Moscow (in Russian) 5. Grigorin-Rjabow WW (ed) Radar devices, theory and principles of construction. Publishing House “Soviet Radio”, Moscow (in Russian) 6. Introduction to SAW filter theory and design techniques. Application note of API Technologies, Corp. https://www.rfcafe.com/references/app-notes-copyrighted/SAW-Filter-Theory-Whitep aper-API-Technologies.pdf 7. Kawalec A, Pasternak M (2018) Acustoelectronics from fundamentals, e-book. Publishing House of the Military University of Technology, Warsaw (in Polish) 8. Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken

References

173

9. Peebels PZ Jr (1998) Radar principles. Wiley, New York 10. Richards MA (2014) Fundamentals of radar signal processing, 2nd edn. McGraw-Hill Education, New York 11. Rosloniec S (2020) Fundamentals of the radiolocation and radionavigation, 2nd edn. Publishing House of the Military University of Technology, Warsaw (in Polish) 12. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 13. Yoshifumi A, Sato Y (1999) Introduction to signal management, 2nd edn. Ohmsha, Japan. The Russian translation of this book was published in 2002 by Dodeka–XXI, Moscow

Chapter 10

Basic Methods for Eliminating Spurious Signals

During the reception of a radar signal, a number of spurious signals usually appear and their impact on the quality of a desired signal reception is difficult to determine by a single coefficient. The energy criterion which determines the ratio of the spurious power to the power of the desired signal (at the receiver’s input) that facilitates a regular operation of the radar station is most frequently used for this purpose. Nevertheless, radiolocation interferences are diverse to such an extent that it seems impossible to classify them unambiguously and comprehensively. However, even an incomplete classification of the interferences due to the origin, the production method and the influence on the received desired radar signal, is of a practical importance. Therefore, Fig. 10.1 shows an example of such a classification [11].

10.1 Basic Methods of Eliminating Signals Reflected from Terrain Obstacles Firstly, one should consider the task of eliminating signals reflected from terrain obstacles and slowly moving objects, such as, thermal air turbulences, clouds and precipitation. Signals which are reflected from such objects are colloquially referred to as the passive permanent echoes or shortly as a clutter. A typical example of passive interferences corresponds to reflection from foil metal strips (dipoles), intentionally dropped from airplanes, which shade the detected objects. All methods, which attenuate the echoes of passive interferences, rely on the differences between signals produced by these echoes and signals reflected from detected objects. Methods used to attenuate echoes of passive interferences are briefly discussed below. The simplest method of eliminating reflections from terrain obstacles (ground clutter), shown in Fig. 10.2a, relies on gating the signal reflected from this obstacle. Such procedure is referred to as direction and distance distinguishing because usually a spatial position (distance, azimuth angle) of an obstacle is known. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_10

175

176

10 Basic Methods for Eliminating Spurious Signals

Fig. 10.1 An example of classification of radiolocation interferences according to the criterion their source and course in the time domain

Fig. 10.2 The terrain obstacle against the background of the directivity characteristics of a radar system antennas; a single beam characteristic, b the characteristic composed of two independent beams

The impact of terrain obstacles reflections, located at a very close distance to the radar station, is usually lowered by reducing the receiver’s sensitivity. This type of procedure is referred to as a time gain compensation (TGC). The TGC is used, among others, in marine radars, in which the sea waves reflections are approximately the same for each direction in the azimuth plane. Furthermore, complex antenna systems are used in order to achieve a significant attenuation of reflections from terrain objects, for example a two-beam system, in which the lower beam is used to detect objects at further distances and lower heights, Fig. 10.2b. The receiver which cooperates with a beam 1 includes the TGC and gaiting system which reflects signals from strictly defined terrain obstacles. The second beam-2, omits the terrain obstacles. Sometimes objects which are the source of passive interference are marked by specific reflective properties. The most characteristic example includes raindrops, with a spherical-like shape. Circularly polarized electromagnetic wave is “reflected” from raindrops, because the relative electric permeability of water is equal to εr ≈ 81. The reflected wave also has a circular polarization, but with an opposite helicity. Since the same antenna is usually used for transmitting

10.1 Basic Methods of Eliminating Signals Reflected …

177

and receiving in the radiolocation, theoretically the signals reflected from raindrops will be detected as weaker than the circularly polarized probe signal. In practice, the improvement of the ratio of the signal reflected from an airplane flying in the storm cloud to the signal reflected from the precipitation, using circular polarization oscillates from 8 to 25 dB. One should emphasize that advantages of using the circular polarization depend on the radar operating wavelength. Many years of observations confirm that various precipitations reflect the electromagnetic energy of the probe signal the less, the lower its carrier frequency is. Both of the mentioned effects are illustrated by the radar screen picture shown in Figs. 10.3 and 10.4. Figure 10.3a shows the panoramic gauge radar screen picture obtained during heavy rainfall while probing with a signal of a carrier frequency f = 10 GHz and linear polarization. Such reflection has been significantly eliminated by changing the wave polarization to the circular polarization, as shown in Fig. 10.3b. Figure 10.4 shows presentation on panoramic indicators of two different Marconi radars, obtained in identical and difficult weather conditions, i.e. during heavy snowfall. The first of the radars gave the indication as shown in Fig. 10.4a at the carrier

Fig. 10.3 Echoes of moving objects; a an echo against the background of reflections from heavy clouds, b the echo imaging after partial eliminating reflections from clouds

Fig. 10.4 A view of radar panoramic indicators during heavy snowfall; a a radar operating at carrier frequency f = 60 MHz, b the radar operating at carrier frequency f = 3 GHz

178

10 Basic Methods for Eliminating Spurious Signals

frequency f = 60 MHz. The second indication, see Fig. 10.4b, was obtained by a radar whose carrier frequency was equal to f = 3 GHz [13].

10.2 Moving Objects’ Reflections Elimination Methods The most effective method of distinguishing the signals reflected from moving objects against the permanent echoes corresponds to the time–frequency method which uses probe signal changes caused by the Doppler effect. The frequency of the echo signal reflected from the object moving towards the station at radial velocity ϑr , differs from the probe signal frequency by the Doppler frequency. f D = f0

2ϑr c

(10.1)

where c ≈ 3 × 108 m/s and f 0 corresponds to the carrier frequency of the probe signal. Objects generating “permanent echoes” are in most cases stationary or move significantly slower than detected objects, thus their effective recognition and suppression of the echoes is possible. Moving objects can be classified into appropriate groups (e.g. piston engine airplanes, jet airplanes, rockets, etc.) by applying correct Doppler frequency filtration. The set of filters, used for this particular purpose, is commonly referred to as a bank. Pulse radar stations that use the Doppler effect in order to eliminate permanent echoes, are called pulse-Doppler radar stations. Depending on the ratio of the probe signal length T0 to the repetition time TP R , those stations are divided as follows: • pulse-Doppler radar stations with a high pulse-duty factor (T 0 /TP R > 0.1). • pulse-Doppler radar stations with a low pulse-duty factor (T0 /TP R < 0.001). Signals that are used in both stations consist of high-frequency N pulses of the same frequency f 0 and of the identical initial phases. Such signals are referred to as coherent signals, see Fig. 10.5. In stationary radar stations, the coherent probe signal is formed by a harmonic (continuous) signal u n (t) = Um cos(ω0 t) which is generated by a highly stable generator. At this point one should consider in more detail the properties of the signal reflected from a moving object. The spectrum of the signal reflected from the moving object consists of harmonic components shifted in frequency by f D and determined by formula (10.1), where ϑr corresponds to the radial velocity of the object with respect to the radar station. Considering the mentioned principle, the kth spectrum component of the received echo signal is marked by the following frequency ) ( 2ϑr f k(ϑ) = f k 1 + c

(10.2)

10.2 Moving Objects’ Reflections Elimination Methods

179

Fig. 10.5 Radio pulses of radar signals; a the phase coherent pulses, b the phase incoherent pulses

where f k is the frequency of the probe signal k-component (sent by the radar station). The frequency interval between the approximal spectral lines is equal to FP(ϑ) R

( ) ) ( 2ϑr 2ϑr − ( f k − FP R ) 1 + = − = fk 1 + c c ( ) 2ϑr = FP R 1 + c f k(ϑ)

(ϑ) f k−1

(10.3)

The frequency interval between the center frequency f 0 and the first spectrum zero, as well as the successive ones, is equal to 1 τ0(ϑ)

( =

f0 +

1 τ0

) ) ) )( ( ( 1 2ϑr 2ϑr 2ϑr − f0 1 + = 1+ 1+ c c τ0 c

(10.4)

The obtained dependencies show that the spectrum of the signal reflected from the object approaching the receiver, ϑr > 0, is extended and shifted towards higher frequencies. Thus the spectrum, which has been changed respectively, corresponds to the shifted time-domain signal. Consequently, due to the Doppler effect, the pulse length of this signal is reduced to τ0(ϑ) =

τ0 1 + 2ϑr /c

(10.5)

According to (10.3), among the properties of a pulse train of the reflected signal, one may also include a reduced repetition time

180

10 Basic Methods for Eliminating Spurious Signals

TP(ϑ) R =

1 FP(ϑ) R

=

TP R 1 + 2ϑr /c

(10.6)

On the contrary, as shown in Table 10.1, both, the length of a single pulse and the pulse repetition time increase as the moving object recedes from the radar receiver. The differences between the parameters of the probe signals and signals reflected from a moving object, are also shown in Fig. 10.6, assuming that ϑr ≡ ϑ. In order to distinguish moving objects from stationary ones, one of the following differences may be used: .ω = ω D − ω0 , .τ 0 = τ0(ϑ) − τ0 , .T P R = TP(ϑ) R − T P R or all of them together. However, the relative changes .ω/ω0 and .τ 0 /τ0 are very small and difficult to measure. Thus, the repetition times difference .TP R is frequently used applying the two-channel (I /Q) synchronous detector, the functional diagram of which is shown in Fig. 10.7. Assume that a radar station sends probe signals s(t) =

N .

Um u(t − kTP R ) cos(ω0 t + ϕ0 )

(10.7)

k=1

where u(t − kTP R ) corresponds to a function describing the envelope of a single pulse with an amplitude equaling 1, TP R is the pulse repetition time, and k is a natural number 1, 2, 3, . . . , N . Moreover, assume that the signal (10.7) is coherent with a reference signal u h (t) = Uh cos(ω0 t + ϕ0 )

(10.8)

Part of the signal (10.7) energy, after reflection from the moving object, returns to the radar receiver in the following form u s (t) =

N .

αUm u (ϑ) (t − kTP(ϑ) R ) cos(ωϑ t + ϕ0 + ϕr e f )

(10.9)

k=1

Table 10.1 The differences between the parameters of the probing signals and corresponding parameters of signals reflected from a moving object Received signal parameter

Object approaching the radar station, ϑr > 0

Object deflecting the radar station, ϑr < 0

ω0

ω D = ω0 2ϑr /c

ω D = −ω0 2ϑr /c

(ϑ) FP R (ϑ) TP R (ϑ) τ0

FP R (1 + 2ϑr /c)

FP R (1 − 2ϑr /c)

TP R /(1 + 2ϑr /c)

TP R /(1 − 2ϑr /c)

τ0 /(1 + 2ϑr /c)

τ0 /(1 − 2ϑr /c)

(D)

10.2 Moving Objects’ Reflections Elimination Methods

181

Fig. 10.6 Time courses of radio pulses and their frequency spectra. a A sequence of coherent radio pulses, b a frequency spectrum of the signal shown in Fig. 10.6a, c the frequency spectrum of the sequence of coherent radio pulses changed by the Doppler effect, d the sequence of coherent radio pulses changed by the Doppler effect

where α is the attenuation coefficient along the transmitter-object-receiver path, u (ϑ) (t − kTP(ϑ) R ) corresponds to the function modified due to the Doppler effect which describes a single video pulse, and ωϑ is the Doppler-modified pulsation. If the object moves towards the monostatic radar at radial velocity ϑr , then ( ) 2ϑr 4π ϑr f 0 = ω0 + ω D where ω D = ωϑ = ω0 1 + c c

(10.10)

182

10 Basic Methods for Eliminating Spurious Signals

Fig. 10.7 Block diagram of two-channel (I /Q) synchronous detector

As it is shown in Fig. 10.7 signals (10.8) and (10.9) are multiplied in the first block of the upper (I—in phase) detector channel. The product of these signals is equal to u h (t) · u s (t) = 0.5αUh Um

N .

u (ϑ) (t − kTP(ϑ) R ) cos(ω D t + ϕr e f )

k=1

+ 0.5αUh Um

N .

u (ϑ) (t − kTP(ϑ) R ) cos[(2ω0 + ω D )t + 2ϕ0 + ϕr e f ]

k=1

(10.11) Then the low-frequency component of (10.11) is filtered out by means of the low-pass filter L P F. According to (10.11) this low-frequency component is given by ) u (I out (t) = 0.5kd αUh Um

N .

u (ϑ) (t − kTP(ϑ) R ) cos(ω D t + ϕr e f )

(10.12)

k=1

where kd denotes the filter attenuation coefficient. The low-frequency component of the lower (Q—in quadrature) detector channel, derived in a similar way, is equal to u (Q) out (t) = 0.5kd αUh Um

N .

u (ϑ) (t − kTP(ϑ) R ) sin(ω D t + ϕr e f )

(10.13)

k=1

From (10.12) and (10.13) results the following formula /[

) u (I out (t)

]2

N [ ]2 . (Q) + u out (t) = 0.5kd αUh Um u (ϑ) (t − kTP(ϑ) R ) ≡ Uout (t) k=1

10.2 Moving Objects’ Reflections Elimination Methods

183

It describes the principle of operation of an amplitude detector insensitive to ω D and ϕr e f , see Fig. 10.7. Sample output signals waveforms for two different Doppler pulsations and a solid constant phase shift ϕr e f are illustrated in Fig. 10.8. If ω D = 0 then the shapes of the output signals (10.12) and (10.13) are similar to that shown in Fig. 10.8a. In other cases, i.e. |ω D | > 0, the envelopes of (10.12) and (10.13) are additionally modulated and change as shown in Fig. 10.8b, c, respectively. The waveforms similar to that shown in Fig. 10.8a, are also observed when ( ) ω D t + TP(ϑ) R = ω D t + m(2π )

(10.14)

where m corresponds to a natural number 1, 2, 3, . . . Due to (10.1), formula (10.14) can be expressed as ω D TP(ϑ) R = 2ω0

ϑr (ϑ) T = m · 2π c PR

(10.15)

Velocities ϑr(m) ≡ ϑr which satisfy the condition (10.15) are equal to ϑr(m) = m ·

cFP(ϑ) R 2 f0

(10.16)

Fig. 10.8 Sample output signals waveforms for two different Doppler pulsations and a solid constant phase shift; b output signal of the upper channel of the detector shown in Fig. 10.7a; c output signal of the lower channel of the detector shown in Fig. 10.7a. For ω D = 00 both output signals are the same and similar to that shown in Fig. 10.8a

184

10 Basic Methods for Eliminating Spurious Signals

(ϑ)

(ϑ)

Fig. 10.9 Graphical illustration of the relationship between FP R = 1/TP R and f D for which there are first blind speed and second blind speed effects; a and b effect of the first blind speed, c and d effect of the second blind speed (ϑ) where FP(ϑ) R = 1/T P R is the frequency repetition pulse, changed due to Doppler effect, in the received signal. The both frequencies FP(ϑ) R and FP R are related to formula (10.3), so the velocities (10.16) are dependent on FP R as follows

ϑr(m) = m

cFP R 2 f 0 (1 − m FP R / f 0 )

(10.17)

In most radar devices FP R / f 0 0. The described differences are used in order to eliminate signals reflected from stationary or slowly moving objects. The examples of such objects may include slowly moving storm clouds, air turbulence (also known as the vortex effect), or various stationary terrain obstacles. A filtering circuit with a functional diagram shown in Fig. 10.10a can be used in order to eliminate reflections from the enumerated objects (referred to as a permanent echoes). In English-language based literature, this circuit referred to as the Moving Target Indicator (MTI) filter [4, 40, 51]. This single delay line canceller can be represented as 2-pulse Finite Impulse Response (FIR) filter shown in Fig. 10.10b.

10.2 Moving Objects’ Reflections Elimination Methods

185

Fig. 10.10 The single delay line canceller and its FIR filter counterpart; a functional diagram of the single delay line canceller, b functional diagram of the 2-pulse FIR filter

The acronyms AMTI (Airborne Moving Target Indicator) and GMTI (Ground Moving Target Indicator) refer to different versions of such filter, which are installed both in airborne (missile) and ground radars, respectively [4, 32]. This basic, highpass linear filter consists of a delay line with a delay tdel = TP R and an element which subtracts the signals u in (t) and u in (t − TP R ), that are the output signals of the used phase detector, see Fig. 10.7. At the output of this simple canceller circuit, a difference signal u out (t) = u in (t) − u in (t − TP R ) is obtained and fed to the user indicator (radar display). In modern radars, this signal is sent to a system that automatically makes a decision whether or not to detect an object. For this purpose, detection algorithms operating according to the criterion of a constant false alarm rate (CFAR) are most often used. If the neighboring (consecutive) signal pulses u in (t) and u in (t −TP R ) are identical in terms of amplitude, as in Fig. 10.8a, then u out (t) ≈ 0. Thus, the echo signal reflected from the stationary object is eliminated. On the contrary, in the case of reflection from a moving object, the successive signal pulses u in (t) and u in (t − TP R ) are different in terms of an amplitude, since these signals are dependent on a modulating factor cos(ω D t + ϕr e f ), where ω D = ω0 (2ϑr /c) /= 0. Consequently, due to the lack of full successive pulses compensation, the difference signal u out (t) > 0 is transmitted to the radar indicator or to the CFAR decision unit mentioned earlier. The selective properties of the filters shown in Fig. 10.10 are best described by the transmittance, ) ( ωTP R , H (1) (ω) = 1 − e− jωTP R = je− j ωTP R /2 · 2 sin 2 | ( )| | | (1) | | | H (ω)| = |2 sin ωTP R | (10.18) | | 2 | | The waveforms of | H (1) ( f )| is plotted in Fig. 10.11. Figure 10.12 shows the double delay line canceller and its FIR filter counterpart which have been widely used in practice for many decades [2, 4, 5, 11, 13].

186

10 Basic Methods for Eliminating Spurious Signals

Fig. | (1)10.11| The waveform of | H ( f )| described by formula (10.18)

Fig. 10.12 The double delay line canceller and its FIR filter counterpart; a functional diagram of the double delay line canceller, b functional diagram of the 3-pulse FIR filter

The frequency response (transmittance) H (2) (ω) of the double delay line canceller is ) ( ) ( − jωTP R 2 − jωTP R 2 ωT P R , H (ω) = 1 − e = −e · 4 sin 2 ( ) | (2) | | H (ω)| = 4 sin2 ωTP R 2 (2)

(10.19)

The selective properties of them are by the plot shown |in Fig. 10.13. | | | well illustrated Considering the transmittance | H (1) ( f )| = 2|sin(π TP R f )| and | H (2) ( f )| = 4 sin2 (π TP R f ), one can conclude that the double delay line canceller (3-pulse filter) provides a better filtration of echoes from slowly moving | that correspond to | objects | H (2) ( f )|, in comparison to Doppler frequencies close to zero. The transmittance | | the transmittance | H (1) ( f )|, is marked by a significant attenuation (a lower transmission coefficient) in the vicinity of the frequencies 0, 1/TP R , 2/TP R , 3/TP R . The spectral lines of the output signal (10.13) are focused around these frequencies. This conclusion is graphically demonstrated in Figs. 10.14 and 10.15.

10.2 Moving Objects’ Reflections Elimination Methods

187

Fig. | (2)10.13| The waveform of | H ( f )| described by formula (10.19)

Fig. 10.14 Graphical illustration of the process of filtering signal components in the frequency domain when ω D = 0

Figure 10.14 assumes that the function describing the probe signal amplitude, is a periodic function (N → ∞) with the repetition frequency FP R = 1/TP R . The spectrum of this modulating signal consists of spectral lines (harmonics) spaced along axis with an interval FP R . In the case of the signal . Nthe frequency u out (t) = k=1 kd αUm u (ϑ) (t − kTP(ϑ) R ) cos(ω D t + 0), whose modulating function also corresponds to a periodic function, each spectral component of the modulating function is multiplied by cos(ω D t + ϕr e f ). Thus, each spectral line f k (with the frequency k/TP(ϑ) R , k = 0, 1, 2, 3, . . .) of the modulating function is replaced by left and right spectral lines with frequencies (ϑ) k/TP(ϑ) R − f D and k/T P R + f D respectively, see Fig. 10.15. As the Doppler frequency

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10 Basic Methods for Eliminating Spurious Signals

Fig. 10.15 Graphical illustration of the process of filtering signal components in the frequency domain when ω D /= 0

f D = ω D /(2π ) increases, the left-sided spectral lines shift towards lower frequencies, i.e. opposite to the right-sided spectral lines. The right-sided spectral line of the k-th component and the left-sided spectral line of the (k + 1)-th component, coincide at the frequency f D = FP(ϑ) R /2. Consequently, amplitudes of two consecutive pulses, modulated by the Doppler effect, take oppositive extreme values as shown in Fig. 10.16. Fig. 10.16 The oppositive extreme values of amplitudes of two consecutive pulses, modulated by the Doppler (ϑ) effect, when f D = FP R /2

10.2 Moving Objects’ Reflections Elimination Methods

189

Subtracting these pulses, in the MTI filter system, a signal, which is the sum of their absolute values, appears on its output. If f D = FP(ϑ) R , then the amplitude modulation of the pulses disappears, see Fig. 10.8, and the observed moving object may be mistakenly perceived as a permanent or very slowly moving object. Such unacceptable situation is related to the radial velocity of the object which is equal to the so-called first blind speed ϑ = ϑr(1) = cFP(ϑ) R /(2 f 0 ) = c. f D /(2 f 0 ) The following conclusion can be derived that the carrier frequency of the probe signal and the pulse repetition frequency (within the packet) should be selected in a such way that the first blind speed is greater than the maximum speed of the potential (tracked) object. Table 10.2 shows the examples of three consecutive blind speeds which were calculated for radars operating in different frequency ranges, i.e. L, S and X. Table 10.2 shows that radars with a lower carrier frequency of the probe signal and radars with higher pulse repetition frequencies (within the packet), are marked by higher values of blind speeds. However, limiting the carrier frequency results in an increase of the main lobe’s angular width of the antenna main directivity (considering its limited geometric dimensions) and also deteriorates indirectly the angular resolution. Likewise, increasing the pulse repetition frequency (within a packet) limits the range gate and may lead to ambiguity in determining the distance to an object. For these reasons, it is crucial to eliminate the negative influence of blind frequencies. A simple solution to prevent such situation includes appropriate, cyclical change of the repetition frequency in successive sequences of radio pulses, Fig. 10.17. Table 10.2 Examples of three consecutive blind speeds which were calculated for radars operating in different frequency ranges FP R = 200 Hz (1) ϑr (km/h) (2) ϑr (km/h) ϑr(3) (km/h)

f 0 = 0.175 GHz

f 0 = 1.250 GHz

f 0 = 3.350 GHz

f 0 = 10.000 GHz

617.143

86.400

32.239

10.800

1234.285

172.800

64.447

21.600

1851.429

259.200

96.717

32.400

Fig. 10.17 An example of successive sequences of radio pulses with a cyclic change of a repetition frequency

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10 Basic Methods for Eliminating Spurious Signals

A number of broadly available publications show that a satisfactory elimination of the initial blind speeds influence is obtained for TP R2 /TP R1 = 5/4 [32, 56]. Theoretically, the cyclical carrier frequency shift in individual sequences of pulses is possible, but this implies a significant increase of device complexity and the need for ensuring synchronization, of both the transmitting and receiving side. Furthermore, the shift of the carrier frequency is used for other purposes, including protecting stations from intentional (active) interference. To describe the performance (quality) of the MTI filter two quantities are usually used, namely the Clutter Attenuation (CA) and Improvement Factor (IF). The Clutter Attenuation CA = Cin /Cout

(10.20)

of the MTI filter is defined as the ratio between the MTI filter input clutter power Cin to the output clutter power Cout . The MTI Improvement Factor is a performance measure for the clutter attenuation. It is defined as “the signal-to-clutter power ratio at the output of the MTI filter to the signal-to-clutter power ratio at the input, averaged uniformly over all target velocities of interest” [6, 7]. It is related to CA as follows, IF =

Sout CA Sin

(10.21)

where Sout /Sin is the average power gain of the MTI filter. For example the average power gain of the MTI filter composed of N identical delay lines is given by, 1 Sout /Sin = FP R

F.P R /2

| | (1) | H ( f )|2N d f =

−FP R /2

1 FP R

F.P R /2

−FP R /2

| ( )|2N | | |2 sin π f | (10.22) | | FP R

According to (10.22) the power gain Sout /Sin of the single canceller, see Fig. 10.10, is (Sout /Sin )

(1)

1 = FP R 2 = FP R

(

F.P R /2

4 sin −FP R /2 F.P R /2

−FP R /2

2

) π f df FP R

[ ( )] 2π 1 − cos f df = 2 FP R

Similarly for double delay line canceller, see Fig. 10.12, the average power gain takes a value (Sout /Sin )

(2)

1 = FP R

(

F.P R /2

16 sin4 −FP R /2

) π f df = 6 FP R

10.2 Moving Objects’ Reflections Elimination Methods

191

A functional diagram of a coherent-pulse radar, together with the previously discussed MTI filter, is shown in Fig. 10.18. Concluding the discussion on the operation of a coherent-pulse radar station with an autonomous reference signal u h (t) = Uh cos(ω0 t + ϕ0 ), one should emphasize that this type of radar station is considered to be one of the most effective radar devices. The foregoing considerations have assumed that the signal u h (t) = Uh cos(ω0 t + ϕ0 ) is generated by a stable generator located inside a radar station. In the case of the radar station located on an airplane or a ship, such a solution is not optimal due to the fact that objects, which are considered stationary with respect to the ground, are perceived as moving objects with respect to the radar station. As one may easily notice, the object’s radial velocities will differ, depending on their location with respect to the radar station. Thus, the echoes reflected from these objects will fluctuate in the same way as the echoes of signals reflected from objects moving with respect to the ground’s surface. In order to distinguish these echoes, it is crucial to use a signal reflected from stationary objects with respect to the ground’s surface, as a reference signal. Considering such solutions, the Doppler effect, resulting from a station movement, is compensated automatically. The above considerations about analog MTI filters are mainly intended to explain their principle of operation. Currently, these types of filters, as well as the phase detector preceding these filters, are mainly implemented in digital versions. For this, the signals (10.7), (10.8) and (10.9) are respectively sampled and converted into complex digital signals. The digital signal representing the phase waveform is then

Fig. 10.18 The block diagram of a radar with MTI filter

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10 Basic Methods for Eliminating Spurious Signals

processed with an appropriate digital MTI filter. At the output of this filter, a digital signal is obtained, which is fed to the decision circuit operating according to the CFAR criterion.

References 1. Bakuljev PA (2004) Radiolocation systems. Publishing House “Radio Engineering”, Moscow (in Russian) 2. Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood (MA) 3. Baskakov SI (1988) Signals and electronic circuits, 2nd edn. Publishing House “Higher School”, Moscow (in Russian) 4. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “’Soviet Radio”, Moscow (in Russian) 5. Grigorin-Rjabow WW (ed) (1970) Radar devices, theory and principles of construction. Publishing House “Soviet Radio”, Moscow (in Russian) 6. Mahafza BR (2013) Radar system analysis and design using MATLAB, 3rd edn. CRC Press, Boca Raton 7. Nathanson FE et al (1991) Radar design principles. Signal processing and the environment, 2nd edn. SciTech Publishing Inc., Mendham 8. Prinsen P (1973) A class of high-pass digital MTI filters with nonuniform PRF. In: Proceedings of IEEE, vol 61, no 8, Aug 1973, pp 1147–1148 9. Prinsen P (1973) Elimination of blind velocities of MTI radar by modulating the interpulse period. IEEE Trans Aerosp Electron Syst AES 9(5):714–724 10. Richards MA (2014) Fundamentals of radar signal processing, 2nd edn. McGraw-Hill Education, New York 11. Rosloniec S (2020) Fundamentals of the radiolocation and radionavigation, 2nd edn. Publishing House of the Military University of Technology, Warsaw (in Polish) 12. Schleher D (2010) MTI and pulsed Doppler radar with MATLAB. Artech House, Inc., Norwood 13. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York 14. Szirman JD (ed) (1970) Theoretical basis of radiolocation. Publishing House ”Soviet Radio”, Moscow (in Russian)

Chapter 11

Searching the Three-Dimensional Space with Radar Devices

The term “objects radar detection” corresponds to the following actions: • omnidirectional observation of the airspace, • observation of the land and sea areas using radar devices installed on a board of the radio-electronic reconnaissance aircrafts, • detection of objects located just below the ground surface, mainly various land mines, • authentication of the decision regarding the object’s detection, determining the object’s position (coordinates) and its complete identification. The next task includes tracking the object (route mapping) and, if necessary, targeting the object with counteractions, i.e. interceptors or rocket artillery.

11.1 The Three-Dimensional Space Observation Methods Firstly, the three-dimensional (3D) airspace search should be considered. For this purpose, radar devices are used, the transmitting-receiving antennas of which have directivity patterns similar to those shown in Fig. 11.1. Figures 11.1a, b show the cos ec2 characteristic elevation and azimuth sections of the non-electronically controlled transmitting antennas. The displacement of this characteristic in three-dimensional space is implemented by mechanical, omnidirectional antenna rotation in the azimuth plane, i.e. in the range 0 ≤ ϕ ≤ 360◦ . The directional gain of the antenna which has the characteristic of type cos ec2 , see Fig. 11.2, is expressed by the formula D(γ ) = k · cos ec2 (γ ) = k

1 sin2 (γ )

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_11

(11.1)

193

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11 Searching the Three-Dimensional Space with Radar Devices

Fig. 11.1 Cross sections of the directivity characteristics of typical radar antennas; a, c, d elevation cross sections, b azimuth cross section

Fig. 11.2 The elevation cross section of cosec2 directivity characteristic

The power of the signal reflected from object Ob is equal to Pref =

PT r 1 PT r 1 1 D(γ )η A |.ob |2 = 2 η A |.ob |2 k 2 2 r 4π r 4π sin (γ )

(11.2)

where PT r is the power of the signal supplied to the transmitting antenna, η A corresponds to the antenna energy efficiency coefficient and |.ob | is the object’s reflection coefficient modulus. The radius r , appearing in the formula (11.2), determines the object’s distance from the phase center of the transmitting antenna. According to Fig. 11.2, sin(γ ) = H/r , therefore the formula (11.2) can be expressed as follows Pref =

2 1 PT R PT r 1 2 r |. | kη kη A |.ob |2 2 = A ob 2 2 r 4π H 4π H

(11.3)

11.1 The Three-Dimensional Space Observation Methods

195

Fig. 11.3 The receiving system with multi-beam antenna; a block diagram of the receiving system, b elevation cross sections of the formed beams

The (11.3) shows that the power of the signal reflected from an object flying at a constant altitude H , does not depend on the elevation angle γ , but only on the altitude. In other words, an object flying at a constant altitude reflects the same amount of the energy of the incident signal. The azimuthal cross-sections of the receiving antennas directivity patterns are usually similar to their counterparts specified for transmitting antennas, see Fig. 11.1b. The receiving antennas directivity patterns differ significantly in elevation sections, the standard shapes of which are shown in Fig. 11.1c, d. Figure 11.1c depicts the characteristics of a non-electronically controlled multi-beam antenna which facilitates measuring an object’s flight altitude H . In the case of the receiving antennas in the shape of multi-element arrays, all receiving beams are simultaneously formed with special forming systems (known as beamformer), which include the known Blass and Butler matrices [6, 8]. The antenna-receiving system, shown in Fig. 11.3, transmits the signals received by the beams to the individual receivers. Assume that an object flies at a low height and the reflected signal permeates only through a beam 1 to a receiver 1. Similarly, the signal which permeates only through beam 2, will be directed to the associated receiver 2. If the reflected signal permeates through two or three neighboring (partially overlapping) beams, it will be proportionally directed to the beam-associated receivers. Considering the echo signal power level in the individual receivers and the beam angles of arrival, an approximation (estimate) of the object’s elevation angle, i.e. the angle γ , is determined, see Fig. 11.2. On the other hand, knowing the distance r to the object and the angle γ , a height estimate H = r sin(γ ) can be determined. Currently, the electronically controlled transmitting and receiving antennas with “pencil-like” directivity patterns are frequently used in modern radar devices. The main lobe, shown in Fig. 11.1(d), of the directivity pattern is displaced electronically in the elevation plane in the required (relatively narrow) sector γ min ≤ γ ≤ γmax . Such displacement is implemented by the appropriate phase angles changes of the currents supplying the horizontally

196

11 Searching the Three-Dimensional Space with Radar Devices

Fig. 11.4 Examples of scanning a narrow sector of space

arranged linear arrays, forming a flat (two-dimensional, curtain array) antenna array. The directivity pattern displacement in the azimuth plane can be implemented as follows, i.e. by the omnidirectional (mechanical) antenna rotation or electronically, in the case of scanning in a narrower sector ϕ min ≤ ϕ ≤ ϕmax . In addition to the discussed omnidirectional scanning, a limited space section is frequently scanned as well, also known as a sector scanning, which is shown in Fig. 11.4.

11.2 Observation of the Land and Sea Areas with Radar Devices Installed on Board of Aircrafts Land and sea areas are observed using methods similar to the ones presented earlier. In order to perform this type of tasks, radar devices placed on special (high) masts or spotter aircraft (e.g. E-9, AWACS systems) are used. Side Looking A special type of the land and sea areas observation corresponds to the side looking, which is shown in Fig. 11.5. The fundamental advantage of this relatively simple observation method is the “simplicity” of the equipment used and non-demanding requirements regarding the path and velocity of the plane. Most frequently, two identical transmitting and receiving antennas with a ”pencil” directivity patterns are suspended under the fuselage (on both sides). These characteristics are shifted electronically in a perpendicular plane towards the plane’s path, resulting in two oblong observation sectors with a small width δ and length l. These sectors shift simultaneously with the plane relocation. Consequently, due to such observation, two side looking paths with a width l, are formed. In order to obtain the most accurate “radar image” of the area under observation, one should ensure the best possible angular and range resolutions. The usage of the angular resolution, though, is particularly significant. The required range resolution is achieved by using sufficiently short probe pulses or longer pulses with internal linear frequency modulation that are compressed (matched filtration) on the receiving side. Obviously, the angular resolution is mainly determined by the geometrical dimensions of the antennas, which cannot be too large,

11.2 Observation of the Land and Sea Areas with Radar …

197

Fig. 11.5 Observation of the terrain using airborne radar with two antennas placed on both sides of the fuselage

related to the wavelength λ = c/ f of the probe signal. Further increase of the resolution for the airplane flight direction is possible through exploiting the airplane relocation at a sufficiently high linear velocity ϑ. Such resolution increasing method is referred to in the literature as a synthetic antenna aperture method [3, 4, 14]. A brief theory outline behind this observational method is provided in the next section. Observation of the Land Area with the Synthetic Aperture Radar (SAR) Method The significance of SAR method corresponds to the Doppler effect usage in order to increase the resolution towards the radar displacement which is located on board of an aircraft or a satellite. Assume that at a time instant t0 = 0, the airplane’s position is determined by the coordinate x = 0, and the detected stationary object Ob is placed at point x0 , see Fig. 11.6a. Furthermore, assume additionally that a harmonic signal, which is radiated by the antenna placed under the airplane, is expressed by the following formula u(t, x, H ) = Um cos(ω0 t + 0)

(11.4)

At any time instant t > t0 , the coordinate of the airplane x = ϑ(t − t0 ) = ϑ t, where ϑ corresponds to the airplane velocity, the vector of which is parallel to the horizontal plane, see Fig. 11.6b. The signal radiated by the antenna suspended under the plane, reaches a stationary object Ob (located at point x0 ), then reflects from this object and returns to the “hypothetical” receiving antenna, which is identical to the transmitting antenna. The same antenna is used for transmitting and receiving in a real SAR system that works with time division. The received echo, delayed part of (11.4) can be expressed as follows: u(t, x, H ) = k Um cos[ω0 (t − tdel ) + ϕr e f ]

(11.5)

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11 Searching the Three-Dimensional Space with Radar Devices

Fig. 11.6 Positions of the electromagnetic beam at two different times; a initial time t0 = 0, b time t > t0

where k corresponds to the attenuation rate determined by the range equation, ϕr e f is the phase of the complex reflection coefficient and tdel corresponds to the time which is equal to tdel =

2. 2 2D = H + (x0 − x)2 c c

(11.6)

The formula (11.6) is correct for 0 < x ≤ 2x0 . As a general rule |x0 − x| > tdel max = 2Rmax /c for Rmax corresponding to the maximum distance to the object, defined by the range equation. Assume that the object is located at the position 1 and the rotation time is counted as the beam axis passes through this position. In other words, the object is located on the beam axis at a given time instant and the echo signal obtained under such circumstances will have the maximum amplitude. After

11.3 Tracking and Radar Homing Methods

207

Tr ot /2, the beam is maximally inflected from the object’s angular position, thus the amplitude of the echo signal has the minimum value. If the object is located at the position 2, the echo signal will change similarly, whereby its maximum value is lower in comparison to the corresponding position 1. Undoubtedly, in the case of an object position on the beam precession axis, the received signal is marked by the minimum modulation coefficient m = |s(t)|max /|s(t)|min which is determined for t0 ≤ t ≤ t0 + Tr ot , for t0 indicating the beginning of the observation time. Figure 11.12b shows that such unmodulated signal is represented by plot 3. The maximum amplitude value of this signal is inversely proportional to the object’s distance from the antenna phase center, i.e. the cone apex. The position of the antenna rotation axis, which gives the unmodulated signal, indicates the direction to the object. Information regarding this direction is automatically transmitted to conjugated firecontrol radar battery. The described method of determining the direction to the object is commonly referred to as a conical scanning. One should also emphasize that the amplitude of the signals received at successive beam rotations (of the antenna) depends not only on the object’s position but also on the effective area fluctuation, which changes as the object moves. Therefore, the beam precession time Tr ot should be as short as possible. Currently, the reduction of the rotation time and elimination of numerous operating inconveniences are obtained by replacing the mechanical antenna rotation with an appropriate electronic control [1, 6, 8]. Thus, a quick (noninertial) change of the beam precession direction and indirect elimination of some stationary spurious responses, is possible. The idea of the described conical scanning is also used in the anti-aircraft defense missile systems. As an example, a single-beam system with a functional structure, as shown in Fig. 11.14, should be considered. This system is marked by the same electromagnetic beam which is used for object targeting and missile homing. The tracked target is irradiated by the missile control radar 1 using a precession beam, see Fig. 11.12a. The signal reflected from this object is used for automatic homing of the antenna system to this object in such a way that the beam precession axis coincides with the direction to an object. A launcher station, from which a guided missile is fired towards the object, is conjugated to a missile control radar. The missile contains a radar signal receiver 2, integrated with the

Fig. 11.14 A functional structure of the anti-aircraft defense missile systems with conical scanning

208

11 Searching the Three-Dimensional Space with Radar Devices

position stabilizer 3 and autopilot 4. The autopilot signal is fed to the position control servomechanisms which change the settings of the rear control ailerons, respectively, see Figs. 11.15 and 11.16. Consequently, due to the change of the ailerons position, the missile should correct its flight so that its center of gravity approaches the beam precession axis indicating the current target position. After the initial, autonomous flight segment, the missile enters the cone area (zone) in which the beam performs a precession. This phase, also known as the capture phase, is marked by the increase of a double generating angle of a cone in order to provide faster and more reliable control of the missile. The capture phase is explicitly shown in Fig. 11.16. Furthermore, the angle is then significantly reduced (cone narrowing), which enables more accurate homing and increases the range. Naturally, as Fig. 11.16 shows, the cone narrowing can be a multistage process. In order to explain the homing process, assume that a missile control radar emits an amplitude-modulated microwave signal by a beam precession. The beam’s main task includes transmitting information to the object regarding the position of the beam precession axis (indicating the direction to the object) and the reference plane orientation, i.e. horizontal x and vertical y. For this purpose, the amplitude of the emitted signal is modulated by an appropriate pulse signal (identification code) when the beam axis passes through

Fig. 11.15 The missile and its basic flight control blocks

Fig. 11.16 Graphical illustration of successive phases of missile flight control

11.3 Tracking and Radar Homing Methods

209

points P, G, L and D, Fig. 11.17, defining the position of two mutually perpendicular planes, i.e. horizontal x and vertical y. An example amplitude waveform of the probe signal (emitted by the radar) is shown in Fig. 11.18a. In the general case, the echo signal received by radar is additionally amplitudemodulated due to an uneven irradiation of the tracked object by the precessive beam. Due to the quadrature detection (insensitive to frequency and phase changes), the amplitude waveform, similar to the one shown in Fig. 11.18b, is obtained. If the object is located on the beam precession axis, then the modulation coefficient of the amplitude echo signal, due to the beam precession, has the minimum value and the necessity of changing the beam precession axis position with respect to the

Fig. 11.17 Definition of the object position versus two perpendicular planes x and y; a front view, b side view

Fig. 11.18 Examples of amplitude waveforms of the probe signal and echo signal; a amplitude course of the probe signal, b amplitude course of the echo signal

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11 Searching the Three-Dimensional Space with Radar Devices

object, is no longer required. On the contrary, when the beam precession axis is significantly inflected from the radius to the object, the received echo signal has a large modulation coefficient per rotation time Tr ot . Assume over that the amplitude waveform of this echo signal is similar to the one shown in Fig. 11.18b. As explicitly indicated, the object is located in the vertical plane x ≈ 0, see Fig. 11.17, and the direction to the object is inflected upwards with respect to the beam precession axis. The tracked object is then assumed to lie precisely in the plane x = 0, if the maximum amplitude signal values determined at points P and L were equal. In this situation, the position of the beam precession axis should be changed by uplifting. Consequently, the difference of the amplitude pulses ., corresponding to the upper G and lower D positions, should decrease, as well as the absolute value of the amplitude which corresponds to the G point. Analogically, the position of the beam precession axis is corrected horizontally x, using the amplitudes of the identification pulses in the right R and left L positions. The position correction is iterative until the beam precession axis is sufficiently and accurately coincided with the direction to the tracked object. Such tracking process corresponds to a relatively weak echo signal with an amplitude waveform, as shown in Fig. 11.18a respectively. The signal emitted by a missile control radar is received by the receiver of the guided missile. The antenna system of the receiver is rigidly connected to its body. The directivity pattern of the antenna system is directed towards the rear missile half-space, i.e. towards the homing radar. This pattern should be omnidirectional with respect to the oblong missile axis. Once the quadrature detection of the received (relatively strong) signal is performed, the amplitude waveform is obtained. This waveform, considering the missile position with respect to the beam precession axis, is similar to the ones shown in Fig. 11.18a or b. In other words, the amplitude waveform of the received signal contains information regarding the missile position with respect to the beam precession axis, precisely with respect to the coordinate system (x y), the origin of which is the point on the precession axis. The stabilizer system, conjugated with a mechanical gyroscope, is a significant component of the missile component. This gyroscope shows the vertical direction y, co-directional with the gravity direction (Fig. 11.19). The spherical rotor of the gyroscope is accelerated to a high angular velocity (about 30,000 rotations per minute) at starting position. The asynchronous electric motors are most frequently used for this purpose. During the flight, the rotation axis of the rotor always indicates the previously selected vertical direction y, regardless of the position of the missile stage. Consequently, determining the current positions (orientations) of the missile control ailerons with respect to the reference planes x and y, is possible. The ailerons’ spatial position is determined by the amplitude signal of the homing beam. Information regarding the missile position with respect to the beam precession axis (which is obtained from the received signal) along with information about its control ailerons orientation, is used to determine the correction signals (in the autopilot system, Fig. 11.15). Due to the servomechanisms, such signals change the settings of individual control ailerons, correspondingly. As mentioned before, due to the change of the ailerons position, the missile should correct its flight in such a way that its center of gravity would approach (or coincide) with the beam precession

11.4 Autonomous Methods of Flying Objects Missile Homing

211

Fig. 11.19 A mechanical gyroscope; a basic components of the mechanical gyroscope, b general view

Fig. 11.20 Graphical illustration of the three-point guidance method

axis, showing the current position of the target. The essence of the described homing process corresponds to the fact that the missile moves along a curvilinear path to the target meeting point, whereby its center of gravity continually persists on a straight line which connects the radar with the target, as shown in Fig. 11.20. For this reason, the described homing method is frequently referred to as a threepoint method [2, 5, 12].

11.4 Autonomous Methods of Flying Objects Missile Homing The homing missile method which is implemented in order to guide a missile towards a flying object is a standard-issue example of a non-autonomous method. As a result a missile is unable to independently determine its position towards an object,

212

11 Searching the Three-Dimensional Space with Radar Devices

and receives this information indirectly by an electromagnetic beam radiated by radar. Nowadays, a number of other methods is used for this purpose, mainly the autonomous ones, which include the following methods: passive, semi-active and fully active (Fig. 11.21). In the passive homing system shown in Fig. 11.22, the missile determines its position with respect to the object considering the energy (electromagnetic, thermal or light) emitted by the object. This energy is received by a specialized receiver integrated with the appropriate antenna or optical system located in its forebody. The main goal of this system is to determine, as precisely as possible, the direction, with respect to the missile axis, of energy emission (Fig. 11.22). Obviously, the autonomy of the system operation is completely dependent on the level of energy emitted by an object, or more precisely, on the strength of the signal received by the specialized receiver. If this level is too low, the system stops working. In order to eliminate this significant defect, a more complex semi-active homing system is implemented, as shown in Fig. 11.23. In the semi-active homing system, the receiver located in the frontal part of the missile receives a “specific” echo signal, i.e. a part of the radio probe signal reflected from the tracked object. In the rear part of the missile a reference signal receiver is located, which facilitates to reconstruct the probe signal carrier. The transmitters of

Fig. 11.21 Division of guidance methods into autonomous and non-autonomous

Fig. 11.22 A passive homing system

11.4 Autonomous Methods of Flying Objects Missile Homing

213

Fig. 11.23 A semi-active homing system

the external, tracking radar station, which is placed on the land area or on the deck, are the source of the probe and reference signals. The received direction of the echo signal is determined by the specialized microwave antenna array described in the following section. At this point, a bistatic system emerges which is distinguished by Forward Scattering Radiolocation (FSR). The advantages of this specific type of radiolocation are described in Chaps. 3 and 5. An additional laser probe signal (illuminating) is frequently implemented for semiactive homing, which improves its efficiency and precision when a good visibility is provided. As in the case of the passive radar system, the autonomy of the semi-active homing is dependent on the correct operation of the tracking station and the conditions of the probe signal propagation along the station—object—missile track. An active homing system shown in Fig. 11.24, indicates both, the transmitter and the receiver of the radio probe signal, which are located inside the missile and are integrated with a common microwave antenna system, if such system operates within time distribution. Therefore, in this case, one deals with a monostatic radar system with a passive response, whose advantages undoubtedly include full autonomy. Regardless of the homing system implemented for the effective approach of the missile to the object, a constant information update regarding the mutual position of those two components is necessary, throughout the tracking period. Figure 11.25 shows the basic parameters included in this information, i.e. the distance d from the missile R to the tracked object O, the missile velocity vector ϑ R , the angle α R between the direction indicated by the missile velocity vector and the axis of the

Fig. 11.24 An active homing system

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11 Searching the Three-Dimensional Space with Radar Devices

Fig. 11.25 A Definition of distances, angles and velocities used to describe the various methods of targeting a missile at an object

assumed stationary reference system, the object velocity vector ϑ O and the angle α O between the direction indicated by the object velocity vector and the mentioned axis. The angle β determines the slope of the line R O that passes through the centers of gravity of the missile and the object, with respect to the axis of reference system. In the English-language based literature, this line is referred to as a Line of Sight (LOS). All the parameters mentioned correspond to functions of time. The derived quantities, with respect to enumerated values, include the missile closing velocity to the object ϑ R O and the angular velocity . R of the missile changing the flight direction. These velocities are expressed as follows ϑR O = −

dd(t) = ϑ O cos[α O (t) − β(t)] − ϑ R cos[α R (t) − β(t)] dt .R =

dα R (t) dt

(11.24) (11.25)

While tracking, the direction of the straight line R O changes with the angular velocity . R O which is equal to .R O =

ϑ O sin[α O (t) − β(t)] − ϑ R sin[α R (t) − β(t)] dβ(t) = dt d(t)

(11.26)

The formula (11.24) derives directly from the difference between the projections of the missile velocity ϑ R and the object ϑ O on the straight line R O, see Fig. 11.25. In order to derive the formula (11.26) assume that at the time instant ti the object and the missile are placed at points Oi and Ri , respectively, see Fig. 11.26. After a short time interval .t, i.e. at a time instant ti+1 = ti + .t, the object and the missile change their positions to O i+1 and Ri+1 , respectively. Changes of these positions are due to nonzero longitudinal and perpendicular components of the object ϑ O and missile ϑ R velocity vectors.

11.4 Autonomous Methods of Flying Objects Missile Homing

215

Fig. 11.26 Positions of a guided missile and a moving object at two different times

Due to the difference of perpendicular components, i.e. ϑ O sin[α O (t) − β(t)] − ϑ R sin[(α R (t) − β(t)] the straight line Ri+1 Oi+1 is rotated with respect to the axis Ri Oi by an angle .β = βi+1 − βi . As shown in Fig. 11.26, a minor difference of angles .β = βi+1 − βi can be expressed as .β =

ϑ O sin[α O (ti ) − β(ti )] − ϑ R sin[α R (ti ) − β(ti )] .t di+1

(11.27)

where di+1 = d(ti+1 ). Thus, in the limiting case [ .RO = lim

.β .t

] =

ϑ O sin[α O (t) − β(t)] − ϑ R sin[α R (t) − β(t)] d(t)

(11.28)

.t → 0 Q.E.D. The formula (11.28) shows that at given velocities ϑ O and ϑ R , the angular velocity . R O has minimum values at long distances d and increases as the missile approaches the object. The angular velocity . R , see Eq. (11.25), should be related to . R O in such a way that its extreme values are as low as possible in the entire homing process. Thus, the missile is subjected to the smallest overloads. In rocketry, as well as in aviation, the term overload corresponds to the ratio of the external resultant force (except of the gravity force) impacting on the missile to its mass. The overload, like the external resultant force, can be represented by a vector with the same direction but the opposite sense. Thus, the overload vector expresses the apparent weight changes of the missile components due to the inertia effect. The aerodynamic forces and the power transmission system thrust are the components which determine the overload value. Obviously, in order to ensure a stable flight of the missile, these overloads should be as low as possible. At the same time, the rocket should approach the object

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11 Searching the Three-Dimensional Space with Radar Devices

in the shortest possible time. These requirements are satisfied to varying degree by the homing methods discussed as follows. Regardless of the selected method, considering information regarding the relative position of the object and the missile as well as the directions and velocities of their movement, the paths of the object and missiles are determined, together with their most convenient collision point. The “scalers” used may be located at the command post or on the missile board. The missile path, understood as the curve with center of gravity moving along, is conventionally divided into three phases (stages), i.e. the initial, middle and final phase. As a general rule, in the initial phase, which is counted from the take-off to the separation of the take-off rocket engine, the missile moves along a given straight line increasing its velocity. In other words, missile homing starts in the longest-lasting middle phase. In a short-term final phase, the missile flies along the straight section, which is determined by the homing system of a monopulse receiver or optoelectronic receiver. Curve of Pursuit Homing Method A characteristic feature of this simplest homing method is the coincidence of the missile velocity vector ϑ R at any tracking moment with the direction of the straight line R O indicated in Fig. 11.27a with a dashed line. In this case, the following equations are satisfied α R (t) − β(t) = 0, . R (t) = . R O (t) < 0 or . R (t) = . R O (t) > 0

(11.29)

which means that the position of the vector ϑ R changes concurrently with the position change of a straight line O R. As shown in Fig. 11.25, this position is determined by the angle β.

Fig. 11.27 Pursuit homing method; a definition of the direction of the missile guidance, b positions of a missile and related to them an object (target) positions

11.4 Autonomous Methods of Flying Objects Missile Homing

217

At this stage, one should emphasize that from the practical viewpoint, the coinciding condition of the vector ϑ R with the straight line O R is not satisfied accurately, due to the object’s movement and the inertia of the missile. If the information regarding the object’s position is updated (refreshed) with time interval .ts , then the path of the guided missile is approximately a polygonal chain, similar to the one shown in Fig. 11.27b. The lengths of the line segments are approximately equal to ϑ R .ts . As the Fig. 11.27 indicates, when tracking a receding object, the slope of these segments changes particularly fast in the middle phase and stabilizes in the final phase. A similar situation occurs with an object approaching the launch pad. In both cases, which are referred to as an rear hemisphere airstrike or the front hemisphere airstrike respectively, the missile hits the rear part of the object for ϑ R > ϑ O . The discussed method is frequently referred to as “the dog pursuit curve method”, because the dog chases the hare in a similar way. Due to a relatively large missile path curvature in the middle phase, this method is not applied to missile homing towards fast-moving airborne objects. The improved version of the method, which is discussed in the following subsection, is marked by a lower track curvature and an almost rectilinear end segment. Curve of Pursuit Homing Method with a Constant Angular Lead Theoretical considerations and experimental studies show that missile homing along the curve of pursuit is very dangerous if the missile velocity exceeds the velocity of the target more than twice. Consequently, the large overloads may lead to the damage of the missile. In order to avoid such a threat, a curve of pursuit homing method with slight lead is implemented. The mentioned lead consists in the inflection of the missile velocity vector ϑ R by a slight angle equal to δ = β − α R , in the direction of the target’s movement (Fig. 11.28).

Fig. 11.28 Pursuit homing method with a constant angular lead

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11 Searching the Three-Dimensional Space with Radar Devices

Fig. 11.29 Definition of angles and velocities used in the method of parallel navigation

Parallel Navigation Method Assume that an object moves along a straight line at a known constant velocity ϑ O . In this case, a missile flying at a velocity ϑ R in the direction defined by a constant angle α R , may be used to electrocute this target, see Fig. 11.29. In this predictive method, the angle value α R , for which the missile R and the target O meet at the identical point, should be selected. Figure 11.29 shows that the missile R and the object O meet at the same point when the following formula is satisfied .R O =

dβ(t) =0 dt

(11.30)

and the straight lines Ri Oi , Ri+1 Oi+1 , Ri+2 Oi+2 , …, are not parallel to the path of the object. When Eq. (11.28) is considered in the formula (11.30) one obtains ϑ O sin(α O − β) − ϑ R sin(α R − β) =0 d(t)

(11.31)

The formula (11.31) is satisfied for [ α R = β + arcsin

ϑ0 sin(α O − β) ϑR

] (11.32)

11.4 Autonomous Methods of Flying Objects Missile Homing

219

As shown in Fig. 11.29, the missile is at a point Ri and the object is at a point Oi at time instant t i . Assume the distance di = |Ri Oi | is known (measured). In this situation, the path of the moving object from the point Oi to the point of impact with the missile can be calculated according to the following formula l O = di

sin(α R − β) sin(α O − α R )

(11.33)

At the same time t = l O /ϑ O , the path of the missile is equal to l R = di cos(α R − β) + l O cos(α O − α R )

(11.34)

which is understood as the distance from the point Ri to the point of impact with an object. The discussed method, also known as the rectilinear homing method or the constant homing method, is marked by the lowest overloads (angular velocities . R and . R O ) even for slightly maneuvering targets, such as large water crafts. Contrary to the previously discussed curve of pursuit homing methods, in the case of constant homing method, an object may be destroyed at ϑ R < ϑ O . Moreover, this homing method enables the object to be electrocuted in the shortest possible time, which is significant due to the limited energy potential of the power unit, and indirectly to the range of the missile. Example 11.1 The attack scenario, as in Fig. 11.29, should be considered, assuming that: α O = π/3 rad, β = π/6 rad, di = 10000 m, ϑ O = 200 m/s and ϑ R = 300 m/s and their calculated values α R , l O and l R are equal to: α R = 0.863435 rad, l O = 18241.911 m and l R = 27362.867 m. The paths l O and l R are overcome by the object and the missile at the same time t = 91.209 s. The formulae and the results of the exemplary calculations are correct when assuming that the object velocity ϑ O and its flight direction are known and do not change during homing. This remark also applies to the velocity ϑ R and flight direction of the missile. Unfortunately, the adopted assumption is not valid in the case of objects changing the velocity or direction of flight, i.e. maneuvering. For such disadvantageous situation, the lines Ri Oi , Ri+1 Oi+1 , Ri+2 Oi+2 , …, see Fig. 11.29, are not parallel to each other, which means that the object can overtake the missile or conversely. Thus, more complex methods, known as proportional navigation methods, are currently used to guide missiles towards maneuvering airborne objects. Proportional Navigation Method The idea of guiding the missile R towards a moving object O, see Fig. 11.25, with a method of proportional approximation is expressed by the following formula dβ(t) dα R (t) =N → . R (t) = N . R O (t) dt dt

(11.35)

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11 Searching the Three-Dimensional Space with Radar Devices

where N corresponds to the navigation constant. In the English-language based literature, this method is referred to as a Proportional Navigation, however this term do not entirely reflect nature and purpose of the method [12]. The angles α R (t) and β(t) appearing in Eq. (11.35) determine the current direction of the missile velocity vector ϑ R and the current position of the straight line R O, respectively. Integrating both sides of Eq. (11.35) on obtains the equivalent equation in the interval [t0 , t] which can be expressed as follows α R (t) = Nβ(t) − [Nβ(t0 ) − α R (t0 )]

(11.36)

For N = 1 the proportional navigation method is identical to the previously discussed homing methods along the curve of pursuit. In this specific case, the difference β(t0 ) − α R (t0 ) determines the lead angle β − α R , see Fig. 11.28. This angle may have a minimum value of 0. Such a homing scenario is illustrated by Fig. 11.27. In practice, the value of the constant N is considered within the range [2−5], which ensures an acceptable length of the pursuit distance with an acceptable transverse (normal) overload of the missile. This overload, which is understood as the modulus of the overload vector, is expressed by the following formula an(R) = ϑ R

dα R (t) dt

(11.37)

When the general formula (11.35) is considered in Eq. (11.37), one obtains an(R) = ϑ R N

dβ(t) = ϑR N .R O dt

(11.38)

The overload vector is perpendicular with respect to the velocity vector ϑ R and its sense is compatible with the sense of the centrifugal force impacting the missile, i.e. the inertial force. The homing method defined by the formulae (11.35) and (11.38) is referred to as a Pure Proportional Navigation (PPN). According to Eqs. (11.35)–(11.38), the knowledge of the input data values, which include velocities ϑ R and . R O , is required in order to guide the missile effectively. In remote (non-autonomous) homing, the input data is determined by an external radar tracking station and sent to the missile by a microwave transmission. In the case of an autonomous homing, this data is obtained by a similar seeker located at the front of the missile. Figure 11.30 shows a functional diagram of the homing system which implements the discussed method. Angle β(t) is measured by the receiver of the system shown in Fig. 11.30 and is used by the on-board computer to determine the angular velocity . R O (t). Moreover, the presented system measures an acceleration an(R) (t), which is directly proportional to the angle α R (t) rate of changes. According to (11.37), the factor of proportionality is the current value of the missile velocity ϑ R (t). At the initial stage of the rocketry development, various types of mechanical and electrical acceleration measuring instruments were implemented to measure an(R) (t). Currently, such measurement is

11.4 Autonomous Methods of Flying Objects Missile Homing

221

Fig. 11.30 Block diagram of a homing system operating according to the method of Pure Proportional Navigation

performed by using special accelerometers produced in the Micro Electro Mechanical Systems (MEMS) technology. As a general rule, they are integrated with additional electronic devices, such as amplifiers, A/D converters and appropriate interfaces, which support their operation. Considering the known values . R O (t), an(R) (t) and ϑ R (t) the following deviation function is calculated .N (t) = N (t) − Nnom =

αn(R) (t) − N nom ϑ R (t) . R O (t)

(11.39)

for which N nom corresponds to the given and nominal value of the navigation constant. The function (11.39) is the autopilot input signal, which “develops” the appropriate rudder position correction. In other words, the autopilot should set these rudders in such positions for which the deviation function (11.39) decreases in absolute value and in the final phase of steering has the value close zero. This homing process description implies that the missile forward speed ϑ R (t) is known. However, the direct measurement of this velocity by a measuring instrument placed in the missile constitutes a complex technical problem. The “differential pressure gauge with a Pitot tube” airspeed indicators used in aviation, due to the pressure that quickly decreases with the altitude increase and the limitations related to the sound barrier, are inappropriate for correct measurement. In this situation, there is an indirect method of determining ϑ R (t) by numerical integration of the measured accelerations or by using the commonly referred templates, i.e. experimentally determined data [ϑ R (t0 ), ϑ R (t1 ), ϑ R (t2 ), . . . , ϑ R (t N )], saved in the computer’s memory. These data yields a set of interpolation nodes. At the selected time instant t > t0 , which is calculated from the launch moment t0 , the missile velocity is assumed to be equal to ϑ R(int) (t) of the implemented interpolating function, for example, a linear combination B—the third degree spline functions.

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11 Searching the Three-Dimensional Space with Radar Devices

The difficulties in determination of ϑ R (t) prompted development of a modified version of the discussed homing method, which uses the closing velocity ϑ R O (t), expressed by the formula (11.24), instead of the ϑ R (t). In case of active homing, see Fig. 11.24, the velocity ϑ R O (t) can be easily calculated by using the measured Doppler frequency shift f D (t) of the received echo signal. Assuming that ϑ R O (t) is the radial velocity, see Fig. 1.9, one can calculate it according to the following formula ϑ R O (t) =

f D (t) c 2 f0

(11.40)

for f 0 corresponding the probe signal frequency and c ≈ 3 × 108 m/s. Such modified homing method is described by the following equation an(R) = ϑ R O N '

dβ(t) = ϑR O N ' .R O dt

(11.41)

for which N ' is referred to as an effective navigation constant. The method discussed above is known as True Proportional Navigation (TPN) [12]. Its representative overload vector is perpendicular to the vector ϑ R O (t) and LOS, missile–object.

References 1. Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood 2. Blakelock JH (1991) Automatic control of aircrafts and missiles, 2nd edn. Wiley, New York 3. Curlander JC, McDounough RN (1991) Synthetic aperture radar, systems and signal processing. Wiley, New York 4. Harger RO (1970) Synthetic aperture radar systems: theory and design. Academic, New York 5. Jackson PB (2010) Overview of missile flight control systems. John Hopkins Appl Tech Digest 29(1):9–24 6. Johnson RC (ed) (1993) Antenna engineering handbook. 3rd edn. McGraw-Hill, New York 7. Levanon N (1988) Radar principles. Wiley, New York 8. Lipsky SE (1987) Microwave passive direction finding. Wiley, New York 9. Mor I, Seabridge A (2006) Military avionics systems. Wiley, New York 10. Peebels PZ Jr (1998) Radar principles. Wiley, New York 11. Rosloniec S (2020) Fundamentals of the radiolocaction and radionavigation (in Polish), 2nd edn. Publishing House of the Military University of Technology, Warsaw 12. Siouris GM (2004) Missile quidance and control systems. Springer, New York 13. Skolnik MI (1978) Radar handbook. McGraw-Hill, New York 14. Skolnik ML (2001) Introduction to radar systems, 3rd edn. McGraw-Hill, New York

Chapter 12

Methods of Determining the Angular Coordinates of an Object by Monopulse Radar Devices

In radiolocation and radionavigation, the determination of the angular coordinates of an object is referred to as a direction finding and direction-finders correspond to the devices implemented in this process. Direction-finders used in the initial period of radiolocation development had one receiving channel. Thus, determination of the angular coordinate (in one plane) could be performed by a sequential lobe comparison method or by a conical scanning method [9, 15]. Both methods are representatives of the sequential lobe comparison methods, see Fig. 12.1. The second group of simultaneous lobe comparison methods includes various types of monopulse methods. The full information regarding the angular coordinates of an object can be obtained considering one radio pulse reflected from this object. Consequently, the fluctuations of the echo pulse amplitude (which derives mainly from the change of the object’s effective reflective surface) do not significantly influence the accuracy of the measurement. According to [4, 8, 14], the monopulse method (also known as simultaneous lobe comparison) is marked by the simultaneous reception of the echo signal through several (frequently four) independent antennas and then the respective parameters of the received signals are compared. As a general rule, signals from one pair of antennas are used to determine the elevation angle γ , Fig. 12.2, and signals from another, similar pair of antennas, determine the azimuth angle ϕ.

12.1 Amplitude and Phase Methods of the Monopulse Radiolocation Depending on the angular coordinates determining method, considering signals received simultaneously, two basic groups of monopulse methods are distinguished, i.e. amplitude methods and phase methods. In monopulse radars, which operate according to the amplitude method, in order to determine one angular coordinate, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_12

223

224

12 Methods of Determining the Angular Coordinates of an Object …

Fig. 12.1 Pelengation methods used in the radar monopulse systems

Fig. 12.2 The local coordinate system defined in 3D space

signals received by the antenna system, which shapes two mirror-symmetrical directivity patterns deflected from the normal to the base (aperture) of the antenna by a small angle ±θ0 , are used. Figure 12.3 shows such directivity patterns (beams) which diverge in the elevation plane. A normal is understood as the direction which is perpendicular to the base or to the aperture of the antenna. As a general rule, one assumes that the normal direction corresponds simultaneously to the direction of an equal signal. This implies that the signal reflected from an object Ob located in the normal direction (.θ = 0) induces in the antenna system ports (related to the beams 1 and 2) voltages u A1 (t, θ ) and

12.1 Amplitude and Phase Methods of the Monopulse Radiolocation

225

Fig. 12.3 Directivity patterns (beams) of a monopulse system using the amplitude method of pelengation

u A2 (t, θ ) which are equal in terms of an amplitude and phase. Assume that the signal is reflected from an object located on a line deflected from the normal direction n by an angle .θ , Fig. 12.3. In this situation, the signal received by the first beam 1 is weaker in comparison to the signal received by the second beam 2. The amplitude difference of the received signals corresponds to a measure of the angle .θ . The sign of this difference indicates the direction of an object shift with respect to the equal signal direction, in an analyzed plane. The azimuth coordinate of an object ϕ is analogically determined, exploiting a second, similar antenna with two beams diverged in the azimuth plane, symmetrically around the normal n, as shown in Fig. 12.3. In monopulse devices using the phase direction finding, the angular coordinate of an object in a given plane is determined considering the signals’ phase difference which are received by two identical antennas, whose phase centers are distanced from each other by a section d, which is referred to as the base, Fig. 12.4. Assume that the directivity patterns of both considered antennas are identical in the adopted coordinate system. The following assumption also applies to the directions of the maximum power gain of both antennas. These directions should be parallel to each other and parallel to the normal n as well. This assumption implies that the signals coming to both antennas from the far-zone source, will be almost identical for an amplitude and different for phase. Let’s consider a specific case and assume that the signal reflected from the object O is received from the direction denoted in Fig. 12.4 by an angle θ . According to Fig. 12.4b, the distance between the antenna A1 and the object is equal to r1 = r0 + (d/2) sin(θ ). Likewise, the distance from the second antenna A2 to the object O is equal to r2 = r0 − (d/2) sin(θ ). The measured phase difference .ψ of the signals received by the antennas A1 and A2 is directly proportional to the distance difference .r = r1 − r2 and is equal to

226

12 Methods of Determining the Angular Coordinates of an Object …

Fig. 12.4 Phase method of pelengation. a Directivity patterns (beams) of an antenna system, b reception of the plane electromagnetic wave by individual antennas

.ψ = β.r =

2π d sin(θ ) λ

(12.1)

for β = 2π/λ and for λ corresponding to the wavelength. When changing the angle θ from 0 to π/2, rad, the phase difference .ψ changes from 0 to 2π d/λ, rad. Assuming that the angle discriminator implemented in the monopulse device operates properly for −π/2 ≤ .ψ ≤ π/2, rad, one can determine the range of the angular direction finding −θmax ≤ θ ≤ θmax for which the angle θ is defined uniquely. Formula (12.1) shows that ) ( ) ( λ λ = arcsin θmax = arcsin π 4π d 4d

(12.2)

Formula (12.1) satisfies the condition for the direction finding of the directivity pattern in the range −θmax ≤ θ ≤ θmax [8, 9, 14]. The slope of (12.1) is the

12.1 Amplitude and Phase Methods of the Monopulse Radiolocation

227

characteristic: d.ψ(θ ) d = 2π cos(θ ) dθ λ

(12.3)

corresponds to a sensitivity measure (accuracy) of the presented method. According to Formula (12.3), this sensitivity increases with the increase of the normalized base length d/λ. However, the increase of the base length results in narrowing of the explicit measurement range [−θmax , θmax ]. In order to ensure the uniqueness and the highest possible sensitivity of the measurement simultaneously, the three-antenna systems are built, i.e. with a sufficiently small d1 and sufficiently large d2 bases arranged along the same straight line. In order to determine two mutually orthogonal angular coordinates, a system consisting of four identical antennas should be implemented. The phase centers of these antennas are regularly arranged similarly, as shown in Fig. 12.5. The distance of each antenna from the origin of adopted coordinates, is equal to d/2. Assume that the direction from which the signal is received, is determined by the azimuth angle ϕ and elevation angle γ . The phase shifts of the signals received from the far zone source by the individual pairs of antennas, are determined considering the difference of path from the wave phase plane (perpendicular to the propagation direction) to these antennas. According to Fig. 12.5 and Eq. (12.1), the following formulae may be determined .ψ1−2 =

2π d 2π d cos(ϕ) cos(γ ), .ψ3−4 = sin(ϕ) cos(γ ) λ λ

(12.4)

where d is a distance between the antennas A1 and A2 , as well as A3 and A4 [4, 19]. Using the measured phase shifts .ψ1−2 and .ψ3−4 , the following angles are calculated ] ) [ ( / λ .ψ3−4 (12.5) , γ = arccos ϕ = arctg (.ψ1−2 )2 + (.ψ3−4 )2 .ψ1−2 2π d

Fig. 12.5 Definition of the azimuth angle ϕ and elevation angle γ in the assumed local coordinate system (x, y, z)

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12 Methods of Determining the Angular Coordinates of an Object …

The angles ϕ and γ (12.5) correspond to the angular coordinates of the signal source (reflecting object) in the azimuthal and elevation planes, respectively. In addition to the described amplitude and phase methods, the modern monopulse systems use mixed amplitude-phase methods that combine their best properties [8, 9, 18].

12.2 The Discriminators of the Monopulse Radar Devices The foregoing considerations show that the angular coordinates of the detected object are determined following the signals comparison received simultaneously by at least two independent receiving paths of the monopulse device. According to this principle, the signal (electric voltage) obtained at the output of the suitable discriminator, should be independent of the amplitudes’ absolute values of these signals. Moreover, the signal size should be proportional to the angle which defines the direction in the given azimuth or elevation plane from which the echo signal is received. Thus, the direction finding characteristic of a monopulse device should indicate the value and sign of this angle, i.e. being an odd function of the angle deducted from the plane of the equal signal. Accordingly, the received signals should be processed in such a way as to obtain a signal satisfying the requirements for the direction finding characteristic. This problem can be solved by implementing various types of discriminators, a brief description of which is presented in [4, 7, 9, 14]. Therefore, only the most widely used summation-differential discriminators, operating according to the amplitude and phase methods, are discussed in the following chapter. Summation-Differential Amplitude Discriminators In the receiving path of a monopulse radar station operating according to the amplitude method (see Fig. 12.1), a two-beam antenna system is distinguished as shown in Fig. 12.3, as well as the related summation-difference amplitude discriminator. Assume that the signal induced in the port 1 of the antenna system is described by u A1 (t, θ0 , .θ ) = U A · F(θ0 + .θ ) cos(ωt + ψ0 )

(12.6)

for U A corresponding to an amplitude and ψ0 the initial phase. Likewise, the signal induced in the port 2 of the considered antenna system has the following form u A2 (t, θ0 , .θ ) = U A F(θ0 − .θ ) cos(ωt + ψ0 )

(12.7)

Signals (12.6) and (12.7) are fed to the summation-differential element (. − .) which is the first component of the discriminator with the functional diagram as shown in Fig. 12.6. In the microwave range, the element function (. − .) can be satisfied by a 3 dB annular directional coupler with a length 3λ/2 or a double waveguide tee, often referred to as the magic T. According to the principle of operation of the element (. − .), the following signals appear at its outputs

12.2 The Discriminators of the Monopulse Radar Devices

229

Fig. 12.6 The block diagram of the summation-differential amplitude discriminator

UA u . (t, θ0 , .θ ) = √ [F(θ0 + .θ ) + F(θ0 − .θ )] cos(ωt + ψ0 − π/2) 2 UA u . (t, θ0 , .θ ) = √ [F(θ0 − .θ ) − F(θ0 + .θ )] cos(ωt + ψ0 − π/2) 2

(12.8)

The signals (12.8) are fed to the mixers of the respective channels (summation and differential) where are mixed with the heterodyne signal u h (t) = Uh cos(ωh t). Consequently, the signals with differential frequency components are obtained. These components are extracted by amplification in the selective amplifiers of the intermediate frequency. These amplifiers perform an additional function of bandpass filters as well. Eventually, the following signals are obtained at the outputs of these amplifiers: UA ( p) u . (t, θ0 , .θ ) = √ Uh k1 [F(θ0 + .θ ) + F(θ0 − .θ )] cos(ω p t + ψ0 2 2 − π/2 − .ψ1 ) UA ( p) u . (t, θ0 , .θ ) = √ Uh k2 [F(θ0 − .θ ) − F(θ0 + .θ )] cos(ω p t + ψ0 2 2 − π/2 − .ψ2 ) (12.9)

230

12 Methods of Determining the Angular Coordinates of an Object …

The values ω p = |ω − ωh |, k1 and k2 correspond to the amplitude transmission coefficients in the mixers and intermediate frequency amplifiers of the summation and difference channels, respectively. Likewise, .ψ1 and .ψ2 are the phase delays caused by the cascade combined mixers and the intermediate frequency amplifiers of the channels. In order to simplify the notation, the signals (12.9) can be expressed in the following, simplified form ( p)

( p)

( p)

( p)

u . (t, θ0 , .θ ) = k1 [F(θ0 + .θ ) + F(θ0 − .θ )] cos(ω p t + ψ1 ) u . (t, θ0 , .θ ) = k2 [F(θ0 − .θ ) − F(θ0 + .θ )] cos(ω p t + ψ2 )

(12.10)

√ √ ( p)) ( p) for: k1 = U A Uh k1 /(2 2), k2 = U A Uh k2 /(2 2), ψ1 = ψ0 − π/2 − .ψ1 and ψ2 = ψ0 − π/2 −.ψ2 . The signals (12.10) are fed to the inputs of the phase detector which is, in effect, a function multiplier. Due to the multiplication, a signal containing a low frequency component, which is conveyed through a low-pass filter (LPF), is obtained. Eventually, at the output of the LPF the following voltage appears u out (θ0 , .θ ) =

1 ( p) ( p) k k k P D [F 2 (θ0 + .θ ) − F 2 (θ0 − .θ )] cos(.ψ2 − .ψ1 ) 2 1 2 (12.11)

The value k P D corresponds to the total amplitude transmission coefficient of the phase detector and the LPF. Formula (12.11) can be expressed in an expanded form with respect to .θ by implementing the approximations F(θ0 ± .θ ) ≈ 0) , resulting from Taylor series expansion around a point θ0 . Once F(θ0 ) ± .θ d F(θ dθ this operation is completed and the components of the second and higher orders (with respect to .θ ) are omitted, one obtains ( p) ( p)

u out (θ0 , .θ ) ≈ 2k1 k2 k P D cos(.ψ2 − .ψ1 )F(θ0 )

d F(θ0 ) .θ dθ

(12.12)

Considering Formula (12.12), which is an odd function with respect to .θ , the following conclusions can be formulated. 1. For the angle .θ = 0, the output voltage (12.12) has a zero value, regardless of the amplitude and phase parameters of the discriminator channels characteristics. ( p) ( p) 2. Changes in the amplitude coefficients k1 , k2 , k P D and phase delays .ψ1 and .ψ2 change only the slope of the direction finding characteristic u out (θ0 , .θ ), Fig. 12.7. 3. The influence of changes in the amplitudes of signals provided by the antenna system, i.e. U A1 and U A2 , on the stability and accuracy in determining the deflection angle .θ , is minimal. Such changes can be easily compensated by the

12.2 The Discriminators of the Monopulse Radar Devices

231

Fig. 12.7 The direction finding characteristics of the discriminator shown in Fig. 12.6

implementation of an automatic gain control (AGC) of intermediate frequency amplifiers in the loop discriminator. Due the above properties the discriminator under discussion was broadly used in practice. A Summation-Differential Phase Discriminator The functional diagram of the considered summation-differential phase discriminator is shown in Fig. 12.8. Fig. 12.8 The block diagram of the summation-differential phase discriminator

232

12 Methods of Determining the Angular Coordinates of an Object …

Assume that at the outputs of the identical antennas A1 and A2 , which are arranged as shown in Fig. 12.4, the following signals occur u A1 (t, θ ) = U A F(θ ) cos(ωt − .ψ) u A2 (t, θ ) = U A F(θ ) cos(ωt + 0)

(12.13)

Subject to (12.13), .ψ = 2π d sin(θ )/λ corresponds to a wave phase shift due to the difference in the radii length r1 and r2 , as shown in Fig. 12.4. Thus, the following voltages occur at the summation and differential outputs of the applied element (. − .) UA u . (t, θ ) = √ F(θ )[cos(ωt − .ψ − π/2) + cos(ωt − π/2)] 2 UA u . (t, θ ) = √ F(θ )[cos(ωt − π/2) − cos(ωt − .ψ − π/2)] 2

(12.14)

The signals (12.4) are fed to the respective mixers activated by a heterodyne signal u h (t) = Uh cos(ωh t). Due to the mixing process, signals with differential (intermediate) pulsation components ω p = |ω − ωh | and summation pulsation components ωs = ω + ωh , are obtained, which are heavily attenuated by the intermediate frequency amplifiers. Obviously, the pulsation components ω p are selectively amplified by these amplifiers. According to Fig. 12.8, the differential channel contains a phase-shifter which delays the signal by π/2, rad. The summation and differential signals processed accordingly, are described by UA ( p) u . (t, θ ) = √ Uh k1 F(θ )[cos(ω p t − .ψ − π/2 − .ψ1 ) 2 2 + cos(ω p t − π/2 − .ψ1 )] UA ( p) u . (t, θ ) = √ Uh k2 F(θ )[cos(ω p t − π − .ψ2 ) − cos(ω p t − .ψ − π − .ψ2 )] 2 2 UA = √ Uh k2 F(θ )[cos(ω p t − .ψ − .ψ2 ) − cos(ω p t − .ψ2 )] 2 2 (12.15) The values k1 and k2 take into the consideration the influence of mixers and amplifiers on the amplitudes of the processed signals, in summation and differential channels, respectively. Likewise, .ψ1 and .ψ2 correspond to phase delays which are contributed by the mentioned mixers and amplifiers. Subsequently, the signals (12.15) are multiplied and filtered in a phase detector system. Consequently, one obtains a signal containing a following component u out (θ ) =

1 UA √ Uh k1 k2 k P D F 2 (θ ) cos(.ψ2 − .ψ1 ) sin[.ψ)] 4 2

12.3 Examples of Structural Solutions of the Monopulse Radar Station

=

] [ 1 UA 2π d sin(θ ) √ Uh k1 k2 k P D F 2 (θ ) cos(.ψ2 − .ψ1 ) sin 4 2 λ

233

(12.16)

which, when amplified, is used as a direction finding signal. Considering formula (12.16), the following conclusions can be formulated. 1. The output voltage u out (θ ) is equal to 0 for the deflection angle θ = 0, see Fig. 12.4. In other words, the zeros position of the direction finding characteristic (12.16) is independent of the amplitude and phase differences of the discriminator channels’ characteristics. Such significant property differentiates this discriminator from a structurally simpler differential phase discriminator described in [8]. 2. Small differences in phase delays .ψ2 and .ψ1 have an insignificant influence on the voltage value u out (θ ), and thereby on the accuracy of the deflection angle θ measurement. This is due to the fact that for ψ2 ≈ ψ1 , occurring in formula (12.16) the term cos(.ψ2 − .ψ1 ), decreases slightly below the maximum value which is equal to 1. 3. Unfortunately, the summation-differential phase discriminator does not eliminate the ambiguity effect, see Eq. (12.2). Furthermore in this case, such failure is most frequently eliminated by using directional antennas with relatively narrow beams (main lobes) of directivity patterns F1 (θ ) ≈ F2 (θ ) ≡ F(θ ), Fig. 12.4, and possibly low level of side lobes. Obviously, the range of an unambiguous measurement −θmax ≤ θ ≤ θmax should completely contain the main lobes, see Fig. 12.2. Summarizing, one should clearly emphasize that the discussed amplitude method is marked by one negative property, i.e. information regarding the angle θ , which is contained in the differences of the received signals phase angles, is irretrievably lost. When determining the angle θ by the phase method, likewise, information about the angle θ contained in the amplitudes of these signals, is lost as well. Thus, the simultaneous application of both mentioned methods appears to be an optimal approach, assuming that both use signals obtained from the same, non-reconfigurable antenna array.

12.3 Examples of Structural Solutions of the Monopulse Radar Station A monopulse radar station, whose aim is direction finding of an object (target) in two mutually perpendicular planes, can be built by a relevant combination of two receiving paths (antenna—discriminator) and a transmitting path including a transmitter and an additional antenna. The main task of this additional antenna is to shape a “pencil-like” directivity pattern, the axis of which coincides with the normal to the antenna aperture. Unfortunately, such monopulse radar is highly suboptimal in terms of construction for several reasons, the most important of which are listed below.

234

12 Methods of Determining the Angular Coordinates of an Object …

1. As the term suggests, a monopulse radar uses a probe signal as radio pulses, i.e. operates within time distribution. Thus, the identical switched antenna array may be implemented for both transmission and reception. This system should structure the relevant directivity pattern which is required in the transmission process, as well as a four-beam receiving characteristic. 2. In order to provide the required time coincidence between the probe pulses and the received echo signals, the operation of the transmitting and receiving paths must be synchronized by time markers produced by one standard generator common to the entire radar. 3. Heterodyne is a common component for all receiving paths, which facilitates obtaining the identical intermediate frequency in the summation and differential paths. As a result, the adverse phase shifts of the signals processed in the mixer systems are minimized. The second, common component is the automatic gain control (AGC) loop, which regulates the gain of intermediate frequency amplifiers occurring in the summation and differential paths. In order to provide a solid base for further considerations, taking the mentioned comments into account, assume that a monopulse radar, operating according to the amplitude method, is the subject of the analysis. This radar determines two angular coordinates of the detected object, which are defined in two mutually perpendicular planes, see Fig. 12.2. The spatial distribution of the four receiving antenna beams of this radar is shown in Fig. 12.9. Figure 12.9 does not cover the “pencil” summation (transmitting) characteristic, the maximum radiation direction of which coincides with the axis z. With no restrictions to generality of considerations, one can assume that these four beams are formed by an array consisting of four, structurally simple, microwave antennas, for example pyramidal horns [6, 12]. The aperture distribution of these antennas in a two-dimensional plane (x, y) is shown in Fig. 12.10. The distribution shown in Fig. 12.10, referred to as Square-Array Geometry (SAG), can be assumed as a result of the Diamond distribution, see Fig. 12.5, following the rotation around the axis z by an angle π/4 rad [5, 6]. Fig. 12.9 Graphical illustration of four receiving antenna beams used in a monopulse radar

12.3 Examples of Structural Solutions of the Monopulse Radar Station

235

Fig. 12.10 A distribution of four receiving antennas (beams) in a two-dimensional plane (x, y)

Assume that the received electromagnetic wave (TEM) gives the following voltages at the terminals of the individual antennas components: u 1 = U A1 cos(ωt + ϕ1 ), u 2 = U A2 cos(ωt + ϕ2 ), u 3 = U A3 cos(ωt + ϕ3 ) and u 4 = U A4 cos(ωt + ϕ4 ). The amplitudes and phases of these voltages contain information regarding the angular coordinates of an object which are determined with respect to the reference system (x, y, z), related to the antenna array aperture, as shown in Fig. 12.10. The mentioned high frequency voltages are fed to the corresponding ports of the analog (microwave) comparator. A typical comparator constructed in the asymmetric stripline technique, consists of four identical 3 dB directional ring couplers with a length of 3λ/2 connected mutually, as shown in Fig. 12.11. This application satisfies the role of summation-differential elements (. − .). In the microwave range a similar role can also be satisfied by a double waveguide tee (T). The sums and differences of voltages, marked in Fig. 12.11, can be expressed as follows 1 u '1 + u '2 = √ [U A1 cos(ωt + ϕ1 − π/2) + U A2 cos(ωt + ϕ2 − π/2)] 2

(12.17)

1 u '3 + u '4 = √ [U A3 cos(ωt + ϕ3 − π/2) + U A4 cos(ωt + ϕ4 − π/2)] 2

(12.18)

1 u '2 − u '1 = √ [U A2 cos(ωt + ϕ2 − π/2) − U A1 cos(ωt + ϕ1 − π/2)] 2

(12.19)

1 u '3 − u '4 = √ [U A3 cos(ωt + ϕ3 − π/2) − U A4 cos(ωt + ϕ4 − π/2)] 2

(12.20)

The following summation signal is obtained in the middle output of the comparator shown in Fig. 12.11

236

12 Methods of Determining the Angular Coordinates of an Object …

Fig. 12.11 The block diagram of the signals comparator

u. =

1 '' 1 (u 1 + u ''2 + u ''3 + u ''4 ) = − (u 1 + u 2 + u 3 + u 4 ) 2 2

(12.21)

for: u ''1 = (U A1 /2) cos(ωt + ϕ1 − π ), u ''2 = (U A2 /2) cos(ωt + ϕ2 − π ), u ''3 = (U A3 /2) cos(ωt + ϕ3 − π ) and u ''4 = (U A4 /2) cos(ωt + ϕ4 − π ). At the upper and lower outputs, one obtains respectively .u y =

] 1 [ '' 1 (u 1 + u ''2 ) − (u ''3 + u ''4 ) = − [(u 1 + u 2 ) − (u 3 + u 4 )] 2 2

(12.22)

.u x =

] 1 1 [ '' (u 3 − u ''4 ) − (u ''2 − u ''1 ) = − [(u 1 + u 3 ) − (u 2 + u 4 )] 2 2

(12.23)

The summation (12.21) and differential (12.22)–(12.23) signals are fed to the respective ports of the duplexer D (transmitting–receiving) connecting alternately the transmitter and receiver. The functional diagram of the attached, three-channel receiver is shown in Fig. 12.12. The comparator discussed earlier, see Fig. 12.11, is a microwave analog circuit, likewise the antenna array commonly integrated. Thus, such circuit is frequently considered as an integral part of the broadly understood antenna system. One should also remind that the comparator operates properly when the respective impedance adjustment of all its ports is provided. Therefore, the differential ports .u x and .u y are burdened with artificial impedances Z 0 when transmitting, see Fig. 12.12. According to the diagram shown in Fig. 12.12, the signals (12.21)–(12.23) are shifted into the intermediate frequency range and amplified respectively. Processed this way summation signal, is used in three ways. Firstly, as the reference signal for phase detectors at the outputs of which signals Az and El are obtained. Such signals contain information regarding the angular coordinates of the tracked object, which are determined in the azimuth and elevation planes. Secondly, the summation signal

12.3 Examples of Structural Solutions of the Monopulse Radar Station

237

Fig. 12.12 The block diagram of the attached, three-channel receiver

after amplitude detection and amplification is used to measure the delay time of an echo signal with respect to the probe signal, i.e. to measure the distance. The third property of the summation signal, is to control an automatic gain control loop including three signal amplifiers of intermediate frequency. The discussed monopulse radar together and the tracked object form an active radiolocation system with a passive response. The range of this system is directly proportional to the fourth root of the radar’s energy potential and the effective area of the object’s reflection. Consequently, at the bound of the range, the received summation signal is weak as are the obtained differential signals. In other words, the quotient of a small differential signal to a weak summation signal, which is determined by the electronic system, may be significantly burdened with an error. Therefore, in order to ensure a sufficient discrimination of summation and differential signals, an automatic gain control of the signals’ intermediate frequency amplifiers is implemented, exploiting the previously mentioned loop, see Fig. 12.12. Differential signals Az and El are used to control the servomechanism of the antenna system. The aim of this servomechanism is to orient the antenna system in such a way that the signals Az and El have extremely low values with a sufficiently strong summation signal. In such case, the normal to the antenna array aperture indicates the direction from which the signal reflected from the tracked object is received and so is the direction to the object. The fundamental purpose of the described monopulse radar structural solution was specifying its basic functional blocks and discussing their role in creating signals that

238

12 Methods of Determining the Angular Coordinates of an Object …

contain information regarding the position of the object with respect to the aperture of the antenna system. However, one should emphasize that the presented solution (consisting of analog components) reflects the technology of the seventies. Currently, due to a large progress of Monolithic Microwave Integrated Circuits (MMICs) technology, multiprocessor computers and digital signal processing methods development, the configuration of such radars has changed significantly. The most important properties and functional components are discussed next. 1. In the modern radar structural solution, the antenna system consists of Transmit/Receive (T/R) moduli integrated with half-wave, spiral, patch, slot or horn antennas [11, 14, 15]. Therefore, in this case, one deals with a spatially scattered transmitter and receiver. The functional diagram of a typical T/R modulus with a ferrite circulator acting as a duplexer (antenna switch) is shown in Fig. 12.13. For more modern and miniaturized structures, the ferrite circulator is replaced with a monolithic Single Pole Double Throw (SPDT). The basic microwave components of the T/R module shown in Fig. 12.13 consist of: CA—control amplifier, PA—power amplifier, SRA—step-regulated amplifier, LNA—low-noise amplifier, L—limiter and APS—adjustable (multibit) phase shifter. As a general rule, a multi-bit attenuator, which additionally serves as a separator, is placed between the two levels of the SRA amplifier. Figure 12.13 does not include the power supply systems and the SRA control signals, the adjustable phase shifter and three switches T–R which are shown in Fig. 12.13 as synchronously switched keys. The systems which compensate

Fig. 12.13 The block diagram of a typical T/R modulus with a ferrite circulator acting as a duplexer (antenna switch)

12.3 Examples of Structural Solutions of the Monopulse Radar Station

239

the transmittance shift of the transmitting and receiving paths evoked by various factors, mainly thermal, were not considered either. 2. Most frequently the antenna array formed from horn antennas, structures the required transmitting directivity pattern. Such implementation is possible by stimulating the individual tubes with equal currents of the identical frequency and respectively selected amplitudes and phases. For this purpose, a common harmonic carrier signal u n (t) with a carrier frequency f 0 is fed to all T/R moduli. The second fed common signal corresponds to the signal that controls all T–R switches (transmit–receive) and is frequently referred to as a keying signal u k (t), the typical shape of which is shown in Fig. 12.14. A pulse of duration ti and amplitude Uk0 polarizes the T–R switches in order to set them to the T position. Consequently, the duration ti of the keying signal pulse is approximately equal to the duration of the probe pulse. This also applies to the repetition frequency FP R of the keying and probe signals. In a general case, the information regarding the required amplitude and phase of the signal emitted by a particular T/R module is sent to the SRA and APS of that module by two additional channels. If the radiating elements (horns) are activated with currents of equal amplitudes and identical phases, such activation is referred to as uniform—cophasal. With such activation, the direction of the maximum transmitting directivity pattern radiation coincides with the normal to the antenna aperture, and the aperture utilization ratio has a maximum value. 3. A similar, scattered structure has the receiver which is the result of setting all the T–R switches of the T/R moduli in the position R, see Fig. 12.13. Likewise, the settings of the SRA, the APS, and all modules T /R should be equal. In such case, the microwave signals u 1 = U A1 cos(ωt + ϕ1 ), u 2 = U A2 cos(ωt + ϕ2 ), u 3 = U A3 cos(ωt +ϕ3 ) and u 4 = U A4 cos(ωt +ϕ4 ) obtained at the antenna outputs, are full equivalents of the signals fed to the input ports of the comparator shown in Fig. 12.11. Figure 12.16 shows the radar structural solution in which signals are fed to the converters (u → I, Q) ports with a functional diagram. The principle of operation of such converters, referred to as Digital Down Converter (DDC), Fig. 12.15, is similar to the operation of the converter described in Sect. 9.3. In the analog segment of the illustrated converter, including the mixer M, the band-pass filter B P F( f p ) and the sampling circuit SC, the shifting of the processed signal u(t) = U (t) cos(ω0 t − ϕu ) into the frequency f p = |ω0 − ωh1 |/(2π ) Fig. 12.14 A time domain waveform of a keying signal

240

12 Methods of Determining the Angular Coordinates of an Object …

Fig. 12.15 The block diagram of the analog–digital converter {u(t) → I [m], Q[m]}

Fig. 12.16 The block diagram of a monopulse radar with T/R moduli and analog–digital converters

12.3 Examples of Structural Solutions of the Monopulse Radar Station

241

and sampling with a relevant time interval Ts , are implemented. The necessity of shifting the processed signal u(t) into the intermediate frequency range, in the order of MHz, is mainly due to the insufficient operation speed of the available sampling circuits SC and multi-bit A/D converters. The sample sequence g[m] ≡ g[tm = mTs ] = U (mTs ) cos(ω p mTs − ϕu ) is fed to an A/D converter, for m = 1, 2, 3, . . ., which corresponds to a sample number and the related time interval tm = mTs . In the A/D converter, the individual analog samples g[m] are quantized and replaced by the representing digital signals, i.e. numbers. Such sample values are then multiplied by the digitally recorded signals of the second heterodyne, ) (Q) i.e. u (I h2 (t) = cos(ω p mTs ) and u h2 (t) = sin(ω p mTs ). These signals are generated by the Numerically Controlled Oscillator (NCO). Due to the multiplication one obtains ) I ' [m] = g[m] · u (I h2 (t) = U (mTs ) cos(ω p mTs − ϕu ) · cos(ω p mTs ) 1 1 = U (mTs ) cos(2ω p mTs − ϕu ) + U (mTs ) cos(−ϕu ) 2 2

(12.24)

Q ' [m] = g[m] · u (Q) h2 (t) = U (mTs ) cos(ω p mTs − ϕu ) · sin(ω p mTs ) 1 1 = U (mTs ) sin(2ω p mTs − ϕu ) + U (mTs ) sin(ϕu ) (12.25) 2 2 When the high-frequency components are filtered out from the signals (12.24) and (12.25), two sequences of numbers are obtained, using digital low-pass filters L P F(C) : I [m] = 0.5U (mTs ) cos(ϕu ) and Q[m] = 0.5U (mTs ) sin(ϕu ). It is not difficult to determine that the numbers I [m] and Q[m] contain complete information regarding the amplitude and phase of the processed signal u(t = mTs ). The functional diagram of the modernized version of the discussed radar, for which four converters were implemented, is shown in Fig. 12.16. The digital signals (I1 , Q 1 ), (I2 , Q 2 ), (I3 , Q 3 ) and (I4 , Q 4 ) which are obtained at the outputs of these converters, contain complete information regarding the amplitudes and phases of the analog signals u 1 , u 2 , u 3 and u 4 , on the basis of which they were created. According to Fig. 12.16, these signals are fed to the processor, which processes them according to a program reflecting the operation of an analog comparator and a three-channel receiver, see Figs. 12.11 and 12.12. The digital signals representing both summation and two differential signals, which contain information regarding the object’s location with respect to the antenna aperture, are obtained at three outputs of the processor. Considering these signals, the time courses, i.e. Az(t) and El(t), are reconstructed by A/D converters. Likewise the analog version of the discussed radar, the differential signals Az and El are used in order to control the antenna system servomechanism, which orientates the antenna system in such a way that these signals have extremely low values with a sufficiently strong summation signal. In this situation, the normal to the antenna array aperture will indicate the receiving direction of the signal reflected from the tracked object, and thus the direction to the object is received as well. For both construction versions of the discussed radar, four horn antennas are integrated with each other. These horns, which are

242

12 Methods of Determining the Angular Coordinates of an Object …

correctly powered, structure a single-beam transmitting characteristic and facilitate obtaining four independent receiving beams, see Figs. 12.9 and 12.10. For this use, the function of such horns corresponds to a beamforming system. In the case of implementing other elements, such as microstrip patches distributed in the nodes of a rectangular or hexagonal mesh, i.e. a multi-element flat array, implementing an additional, special beamforming system, is necessary. In recent years, such situation was solved by building a complex microstrip or waveguide systems, which include such classic representatives as Butler and Blass matrices [6, 12]. In modern radars with T/R moduli and converters (u → I, Q), the discussed beams are formed differently, which in the English-language based literature is referred to as a Digital Beam Forming (DBF). In order to illustrate the significance of this directivity pattern digital forming method of a radar antenna, one should demonstrate an example of a linear antenna array with a binary structure, similar to the one shown in Fig. 12.17. This linear and regular array consists of 8 identical straight antennas (hereinafter referred to as elements) and a 9-port analog circuit that serves as a signal combiner or divider. One assumes that this system introduces identical time delays (as well as phase delays) for all signals sent from the individual elements to ports .. On the contrary, the energy of the signals received by the individual elements is transmitted unequally to ports .. As a general rule, the power of these signals decreases together when increasing distance to the geometric center of the antenna array. Consequently, it is possible to lower the side lobes considering with respect to the main lobe. This problem is discussed in detail in Examples 5.3 and 5.4 of the handbook [12]. Example 5.3 shows an 8-element array, assuming that d = 0.7λ and the following amplitudes distribution of equivalent currents is equal to I4 = I5 = 1, I3 = I6 = 0.759445, I2 = I5 = 0.417904, I1 = I8 = 0.146097, which characterize the contribution of signals from individual elements to the output signal. The normalized group directivity pattern F(θ ), designated for this antenna array, is shown in Fig. 12.18. Subject to (12.26), ψ corresponds to the phase shift which is contributed by the combiner/divider. The coefficients (12.26) also determine the signal transmission from the plane of the identical phase (wave front) to the port .. In the general case, i.e. for θ /= 0 rad, the complex amplitudes of the equivalent signals, induced in the individual elements, will have different phase angles. According to Fig. 12.19 the phase angle of the signal amplitude related to the k-element will differ from the phase angle of the complex amplitude of the signal representing 1-element by Fig. 12.17 A linear antenna array with an analog divider/combiner of microwave signal

12.3 Examples of Structural Solutions of the Monopulse Radar Station

243

Fig. 12.18 The normalized group directivity pattern F(θ )

Fig. 12.19 Reception of a flat electromagnetic wave by individual elements of a linear antenna array

.ϕk = β(k − 1)d sin(θ ), for β = 2π/λ, k = 1, 2, 3, . . . , 8. tk (θ = 0) = |tk | exp(− j ψ) for k = 1, 2, 3, . . . , 8

(12.26)

The difference .ϕk will change abruptly by the increase .ϕ = βd sin(θ ) for the subsequent index values k. Consequently, the following vector is defined [ ] [V] ≡ 1, exp(− j.ϕ), exp(− j2.ϕ), . . . , exp[− j (N − 1).ϕ]

(12.27)

This vector is referred to as Array Propagation Vector (APV). For θ = 0 rad, the values of all vector components are equal to 1. Implementing (12.26) and (12.27), the signal transmission coefficients between the wave front, marked in Fig. 12.19, and the port, . can be expressed as follows tk (θ ) = tk (θ = 0) · exp[− j (k − 1).ϕ] for k = 1, 2, 3, . . . , 8 and ψ = 0 (12.28) Adopting a constant phase delay ψ = 0 does not restrict the generality of considerations.

244

12 Methods of Determining the Angular Coordinates of an Object …

In order to satisfy the cophasal condition of all signals summed in the ports ., for θ /= 0 rad, the adder system should be modified by introducing the adjustable phase shifters to compensate the shifts .ϕk = −(k −1).ϕ, Fig. 12.20. In the time domain, the signals cophasal condition is expressed by the equality of the passage times of these signals from the wave front through individual elements 1, 2, 3, . . . , 8, to the port .. The foregoing considerations show that due to selecting the relevant phase shifts implemented by the phase shifters, the angular position of the directivity pattern main lobe is changed, which constitutes the theoretical basis for the phase scanning [3, 26, 46]. For an antenna with digital forming of the directivity patterns, see Fig. 12.21, operations similar to those implemented in the analog combiner/divider with phase shifters, are performed mathematically. Assuming that all T/R moduli contribute identical phase shifts and minimum attenuations, these operations result in multiplying each signal sk [m] = Ik [m] + j Q k [m] by the factor wk = |tk | exp[ jβ(k − 1)d sin(.θ )] for k = 1, 2, 3, . . . , 8

(12.29)

Considering (12.29), .θ corresponds to the given (discrete) value of the main lobe deflection angle of the formed directivity pattern array. The discussed angle is defined with respect to the normal, i.e. likewise the angle θ , see Fig. 12.19. The moduli |tk | of the weight coefficients (12.29) have a decisive influence on the directivity pattern shape expressed among others by the angular width of the main lobe and the level of the side lobes. Their nominal values, which were determined for a given (specific) characteristic, are downloaded by the control program from the memory of the Central Processing Unit computer (CPU). In contrast, the phase angles of these coefficients determine the angular shift .θ of the formed characteristic main lobe with respect to the normal n to the aperture array. Consequently, due to the multiplication of the coefficients (12.29) by the signals sk [m], weighted signals yk [m] = wk sk [m] are obtained, with the sum corresponding to the output signal y. [m]. Due to the fact that the signals sk [m] and weighting factors wk are stored in digital form, the directivity pattern array forming is possible by performing the Fig. 12.20 A linear antenna array with an analog divider/combiner of microwave signal and adjustable phase shifters

12.3 Examples of Structural Solutions of the Monopulse Radar Station

245

Fig. 12.21 The block diagram of an antenna with digital forming of the directivity patterns

discussed earlier mathematical operations. Such operations are performed numerically according to a dedicated computer program, shown in Fig. 12.21 as a Processor (CPU)/Adaptive algorithm. The implied task of digital forming of the linear antenna array directivity pattern can be implemented according to the algorithm described as follows Example 12.1 One of the basic tasks of the central unit (CPU), is to determine such values of weighting factors wk for which the formed characteristic F(θ ) will have the required shape, and its main lobe will be deflected from the normal to the aperture array by a given angle .θ . This problem can be easily solved by implementing distributions of equivalent current amplitudes, representing signals received by the individual array elements. Numerous examples of such distributions, including Dolph– Chebyshev, Taylor, Gauss, cos2 , cos2 on a pedestal, etc., are available in the literature devoted to the antenna technique. As an example, an array consisting of N = 16 isotropic elements distributed along a straight line with equal spacing d = 0.7λ, should be considered. The amplitudes distribution of the equivalent currents Ik , for

246

12 Methods of Determining the Angular Coordinates of an Object …

k = 1, 2, 3, . . . , 16, which is determined for this array by the Dolph-Chebyshev method, is incorporated in the second column of Table 12.1 [12, 15]. The signal power Pk transmitted by the k − th element to the summation port ., is directly proportional to |Ik |2 and can be expressed by the dependence Pk = χ Z 0 |Ik |2 . In this example Z 0 corresponds to the characteristic impedance of the waveguide and χ indicates the coefficient of the signal shape. On the other hand, the power P. of the composite signal in the summation port . is the sum of signals power transmitted Table 12.1 The parameters of the linear array antenna determined in Example 12.1 N = 16, d = 0.7λ, .θ1 = −0.119330 rad, .θ2 = 0.323050 rad k

Ik

|tk |

w k (.θ1 ), .ψ = −30◦

w k (.θ2 ), .ψ = 80◦

1

0.113761

0.043734441

0.043734 + j 0.000000

0.043734 + j 0.000000

2

0.196367

0.075496075

0.065377 − j 0.037745

0.013108 + j 0.074344

3

0.331950

0.127615334

0.063807 − j 0.011052

−0.119919 + j 0.043647

4

0.492615

0.189381618

0.000000 − j 0.189382

−0.094691 − j 0.164009

5

0.661368

0.254257261

−0.127128 − j 0.220193

0.194772 − j 0.163433

6

0.816304

0.313821083

−0.271777 − j 0.156910

0.240400 + j 0.201720

7

0.935381

0.359599202

−0.359599 − j 0.000000

−0.179799 + j 0.311422

8

1.000000

0.384441424

−0.332936 + j 0.192220

−0.361257 − j 0.131486

9

1.000000

0.384441424

−0.192220 + j 0.332936

0.066757 − j 0.378601

10

0.935381

0.359599202

0.000000 + j 0.359599

0.359599 + j 0.000000

11

0.816304

0.313821083

0.156910 + j 0.271778

0.054494 + j 0.309053

12

0.661368

0.254257261

0.220193 + j 0.127128

−0.238923 + j 0.086961

13

0.492615

0.189381618

0.189381 + j 0.000000

−0.094691 − j 0.164009

14

0.331950

0.127615334

0.110518 − j 0.063807

0.097759 − j 0.082029

15

0.196367

0.075496075

0.037746 − j 0.065377

0.057830 + j 0.048525

16

0.113761

0.043734441

0.000000 − j 0.043734

−0.021867 + j 0.037875

12.3 Examples of Structural Solutions of the Monopulse Radar Station

247

by the individual elements and is 16 .

P. =

Pk = χ Z 0

k=1

16 .

|Ik |2

(12.30)

k=1

The power ratio Pk /P. is identical to the square of the module (absolute value) of the transmission coefficient tk , which can be expressed as follows / |tk | =

|Ik | Pk =√ P. P.

(12.31)

The values of the transmission coefficients moduli, calculated according to formula (12.31), are shown in the third column of Table 12.1. According to (12.29), the determined weighting factors are equal to wk = |tk | exp[ j (k − 1).ψ)] for k = 1, 2, 3, . . . , 16

(12.32)

Subject to (12.32), .ψ = β · d sin(.θ ) corresponds to a given deflection angle. In a special case, for .θ = 0, wk = |tk | for k = 1, 2, 3, . . . , 16. The weighting factors (12.32) calculated for the given shifts .θ1 = −0.119330 rad and .θ2 = 0.323050 rad, are entered in the fourth and fifth columns of Table 12.1, respectively. The off-specification directivity pattern f (θ ) of the discussed array, is related to the coefficients (12.32) with the following formula | 16 | |. | | | f (θ ) = | wk exp[− jβd(k − 1) sin(θ )]| | |

(12.33)

k=1

Sometimes (12.33) is expressed in matrix form, viz f (θ ) = |[V] · [W]|

(12.34)

for: [W] = [w1 , w2 , . . . , w16 ]T and [ ] [V] ≡ 1, exp[− jβd sin(θ )], exp[− j2βd sin(θ )], . . . , exp[− j (16 − 1)βd sin(θ )] . For the normal direction (θ = 0), all elements of the propagation vector [V] are equal to 1. Moreover, assume that the main lobe of F(θ ) is directed along the case, θ = 0 and .θ = 0, normal, which occurs for .θ = 0. In |this particular | | |. |t | The normalized characteristics Eq. (12.33) reduces to f (θ ) ≡ f max = | 16 |. k=1 k F(θ ) = f (θ )/ f max , calculated according to the weighting factors indicated in the third, fourth and fifth columns of Table 12.1, are shown in Figs. 12.22, 12.23 and 12.24, respectively.

248

12 Methods of Determining the Angular Coordinates of an Object …

Fig. 12.22 The normalized characteristics F(θ ) = f (θ )/ f max calculated for .ψ = 0◦

Fig. 12.23 The normalized characteristic F(θ ) = f (θ )/ f max calculated for .ψ = −30◦

Fig. 12.24 The normalized characteristic F(θ ) = f (θ )/ f max calculated for .ψ = 80◦

In the characteristic F(θ ), shown in Fig. 12.24, for θ < −75◦ , a fragment referred to as a big spurious lobe or, more commonly, a diffraction lobe, occurs. This spurious lobe can be eliminated by reducing the spacing d/λ between array elements to a relevant value. In the limiting case, i.e. for any value .ϕ, the diffraction lobe does not occur, for d ≤ 0.5λ. In real arrays, the additional side lobes attenuation, including diffraction ones, is given due to the directional properties of the antennas which are used as elements. The discussed example of linear antenna array, consists of 16 identical elements. The signal received by these elements is represented by digital signals s1 [m], s2 [m],

12.3 Examples of Structural Solutions of the Monopulse Radar Station

249

s3 [m], …, s16 [m], see Fig. 12.21, which may correspond to elements of a row signal vector [S[m]]. Likewise, the weighting factors w1 , w2 , . . . , w16 , can be considered as elements of the column directional vector [W]. If the signal received by the antenna array is the sum of several signals not correlated with each other, then such signal can be described by a matrix whose rows represent individual component signals. In the case of three signals u 1 , u 2 , u 3 , with amplitudes U1 , U2 , U3 , received from different directions (e.g., θ1 = 18.5◦ , θ2 = −7◦ and θ3 = −30◦ ), the representative matrix can be expressed as follows [

] S 3 [m] ⎡ ⎤ U1 [m] U1 [m] exp(− j.ϕ1 ) U1 [m] exp(− j2.ϕ1 ) . . . U1 [m] exp(− j15.ϕ1 ) ⎢ ⎥ = ⎣ U2 [m] U2 [m] exp(− j.ϕ2 ) U2 [m] exp(−2 j.ϕ2 ) . . . U2 [m] exp(− j15.ϕ2 ) ⎦ U3 [m] U3 [m] exp(− j.ϕ3 ) U3 [m] exp(− j2.ϕ3 ) . . . U3 [m] exp(− j15.ϕ3 )

Multiplying this matrix by the directional vector ]T [ [W] = |t1 | |t2 | exp( j.ψ) |t3 | exp( j2.ψ) . . . |t16 | exp( j15.ψ) for .ψ = βd sin(.θ ), one obtains the vector [Y. [m]], which represents the output signal ⎤ 16 . |t | U exp[ j (k − 1)(−.ϕ [m] + .ψ)] k 1 ⎥ ⎢ 1 k=1 ⎥ ⎢ 16 ⎥ ⎢ [ (3) ] . ⎢ |tk | exp[ j (k − 1)(−.ϕ2 + .ψ)] ⎥ [Y. [m]] = S [m] · [W] = ⎢ U2 [m] ⎥ k=1 ⎥ ⎢ ⎦ ⎣ 16 . |tk | exp[ j (k − 1)(−.ϕ3 + .ψ)] U3 [m] ⎡

k=1

The sum of elements of this column vector represents the output signal y. [m] = U1 [m]

16 .

|tk | exp[ j (k − 1)(−.ϕ1 + .ψ)]

k=1

+ U2 [m]

16 .

|tk | exp[ j (k − 1)(−.ϕ2 + .ψ)]

k=1

+ U3 [m]

16 .

|tk | exp[ j (k − 1)(−.ϕ3 + .ψ)]

k=1

For .ψ = .ϕ1 , as well as .θ = .θ1 , the signal (12.35) is equal to: y. [m] = U1 [m]

16 . k=1

|tk |

(12.35)

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12 Methods of Determining the Angular Coordinates of an Object …

+ U2 [m]

16 .

|tk | exp[ j (k − 1)(−.ϕ2 + .ψ)]

k=1

+ U3 [m]

16 .

|tk | exp[ j (k − 1)(−.ϕ3 + .ψ)]

(12.36)

k=1

. The first component of the sum (12.36), i.e. U1 [m] 16 k=1 |tk | → U1 [m] · F(θ )max , indicates an equivalent of the signal u 1 , crossing the main lobe of the directivity pattern F(θ ) determined for .θ = θ1 , with no attenuation. On the other hand, the remaining two components correspond to signals u 2 and u 3 , which are strongly attenuated (≥ 40 dB) by the side lobes of this characteristic. Such conclusion can be easily demonstrated by implementing the geometric interpretation of the individual components (12.36). The first of them can be considered as the sum of the component vectors U1 [m]|tk |, for k = 1, 2, . . . , 16, directed in the same direction. The length of the resultant is, in this specific case, equal to the sum of the individual component vectors lengths and has the maximum value. The representing vectors U2 [m]|tk | exp[ j (k − 1)(−.ϕ2 + .ψ)] and U3 [m]|tk | exp[ j (k − 1)(−.ϕ3 + .ψ)], for k = 1, 2, . . . , 16, have different directions. Due to this reason, when the vectors are summed, the effect of full or partial compensation occurs, so that the resultant with signals u 2 and u 3 have minor lengths. Formula (12.36) shows that the degree of a mutual compensation of the component vectors depends primarily on the distribution {|tk |}, the beam deflection .θ = θ1 and the directions θ2 and θ3 from which the “spatially filtered” signals u 2 and u 3 come, respectively. The term of spatial filtering should be considered as the elimination (attenuation) of signals which are fed to the antenna from spurious directions. In addition to the discussed summation characteristic f (θ ) ≡ f . (θ ), a differential characteristic f . (θ ), with a shape similar to the one shown in Fig. 1.21, is implemented in radiolocation and in radionavigation. Such characteristic has a minimum value, theoretically equal to 0, for the direction θ , in which the related summation characteristic f . (θ ) has a maximum value. Close to this direction, the differential characteristic f . (θ ) is marked by a high steepness of the slopes, which enables a precise determination of the direction from which the signal is received. The antenna array, considered in this example, consists of an even number of elements, and the implemented distribution of coefficients |tk |, marked in the third column of Table 12.1, is mirror-symmetrical with respect to the mid-plane of the array or to its phase center. Zero deflection of the main lobe .θ = 0, provides a cophasal distribution of weighting factors, identical to the distribution {|tk |}. The normalized summation characteristic F(θ ) = f . (θ )/ f . (θ )max of such array is shown in Fig. 12.22. This characteristic corresponds to a differential characteristic F. (θ ) = f . (θ )/ f . (θ )max that can be calculated according to the following formula | 8 |. 1 | |tk | exp[− jβd(k − 1) sin(θ )] F. (θ ) = | f . (θ )max | k=1

12.3 Examples of Structural Solutions of the Monopulse Radar Station

−

16 . k=9

| | | |tk | exp[− jβd(k − 1) sin(θ )]| |

251

(12.37)

for: f . (θ )max

| 8 |. | |tk | exp[− jβd(k − 1) sin(θ )] = max| | k=1 | 16 | . | |tk | exp[− jβd(k − 1) sin(θ )]| − | k=9

For the data assumed in this example, the value f . (θ )max = 2.254927 is obtained for θ ≈ 5◦ 5' and θ ≈ −5◦ 5' . The calculation results of characteristics F(θ ) and F. (θ ), performed for −6◦ ≤ θ ≤ 6◦ respectively, according formulae (12.33) and (12.37), are partially depicted in Fig. 12.25. Presenting the results in the range shown in Fig. 12.25 is not advisable due to the fact that the determined characteristics F(θ ) and F. (θ ) are even functions. One should also emphasize that the described differential characteristic forming method is useful only for cophasal arrays, which shape the characteristic F(θ ) with the main lobe directed along the normal (perpendicular) to the aperture, see Sect. 14.3. In addition, the distribution {|tk |} must be mirror-symmetrical with respect to the phase center of an array, likewise the distribution shown in the third column of Table 12.1. In the case of shaping arrays F(θ ) with the main lobe deflected from the normal, the differential characteristic F. (θ ) is obtained by differentiation of the related summation characteristic F(θ ). Assume that θi ≡ .θi defines the direction of the main lobe F(θ ) and the angle θs ≡ .θs defines the direction of the received signal. For the direction θi , the function | F(θ|) has a maximum value, equal to 1. According to | )| takes value equal to 0 for the identical direction. Rolle’s theorem, the deriviate | d F(θ dθ | Fig. 12.25 The calculated characteristics F(θ ) and F. (θ )

252

12 Methods of Determining the Angular Coordinates of an Object …

Table 12.2 The calculation results that have been made in the close vicinity of θi N = 16, d = 0.7λ, θs = 0.323050 rad, h = 8.726646 × 10−3 rad |d | | F(θi − θs )| i θi rad F(θi − θs ) → Comments dθ y. (θi − θs ) 1

0.279252

0.826595901

2

0.287979

0.886066720

3

0.296796

0.934512049

4.856276

4

0.305432

0.970349935

3.334366

5

0.314159

0.992401521

1.706469

6

0.322886

0.999987514

0.031619

7

0.331612

0.992991656

1.629799

8

0.340339

0.971759841

3.219074

9

0.349065

0.937172324

4.682023

10

0.357795

0.890543539

11

0.366519

0.833561530

0.314159 ≤ θi∗ ≤ 0.331612

Consequently, due to differentiation F(θ ) in the close vicinity θi , a function F. (θ ), with a course similar to the one shown in Fig. 12.25, is obtained. Furthermore, the calculation results shown in Table 12.2, confirm the correctness of this conclusion. d F(θi − θs ) were The calculations of the derivative’s approximate values dθ performed considering the discrete values F(θi − θs ) → y. (θi − θs ), as shown in the third column of Table 12.2. These calculations were performed according to the central fourth order differential formula, i.e. −F(θk+2 ) + 8F(θk+1 ) − 8F(θk−1 ) + F(θk−2 ) d F(θk ) ≈ dθ 12h

(12.38)

for: F(θk+2 ) = F(θk + 2h), F(θk+1 ) = F(θk + h), F(θk−1 ) = F(θk − h), F(θk−2 ) = F(θk − 2h) and h corresponds to the arbitrary assumed step, equal to 30 min of arc [13]. On the other hand, the mentioned function values F(θi − θs ) were calculated according to formula (12.33), by implementing the weighting factors wk (12.32), which are relevant to the deflection (θi − θs ) ≡ .θ . The moduli of these coefficients correspond to the transmission coefficients |tk |, shown in the third column of Table 12.1. The results shown in Table 12.2 indicate that the determined direction θi∗ is contained within a narrow range 0.314159 ≤ θi∗ ≤ 0.331612, Fig. 12.26. ) contained in this range can be interpolated with a straight line, The course d F(θ dθ ) = 0, indicates the direction θi∗ ≈ 0.314159 + which intersected with the axis d F(θ dθ 2h · 1.706469/(1.706469 + 1.629779) = 0.323086 rad. The approximation of the reception direction determined this way, differs from the precise value θs = 0.323050 rad by δθ = θs − θi∗ ≈ −3.6 × 10−5 rad. This error corresponds to a linear shift of the order of 3.6 m for the object at a distance 100 km.

12.3 Examples of Structural Solutions of the Monopulse Radar Station

253

Fig. 12.26 Interpolation of function d F(θ )/dθ with a straight line in the vicinity of its zero value

The described computations can be performed simultaneously for several different directional vectors [W] in independent computational threads. In this example, such directional vectors correspond to the vectors with the elements included in the third, fourth and fifth columns of Table 12.1. Due to the fact that each vector [W] determines the characteristic F(θ ) with the angular position .θ of the main lobe (beam), given by this vector, see Figs. 12.22, 12.23 and 12.24, in this case a spatial filtration of signals received by an array, by several beams formed simultaneously or by an equivalent multibeam antenna, can be considered. The English-language based literature describes such method of simultaneous formation of several beams is referred to as Multi-Beam Digital Beam Forming (MB-DBF). An antenna with digital directivity pattern formation usually functions within time distribution. In other words, such antenna is used for both, transmission and reception. In both of these operation modes, numerous identical components are used, which undoubtedly correspond to T/R moduli. During transmission, the T/R moduli, see Fig. 12.13, emit the electromagnetic waves represented by equivalent signals with the same carrier frequency, different amplitudes and different phase angles. The amplitudes and phase angles of these signals must have relevant values, for which the radiated energy will be focused, through the entire antenna array, in a angularly narrow space sector, hereinafter referred to as a transmitting beam. Likewise the receiving antenna characteristic, the complex amplitudes phase angles of the aforementioned equivalent signals have a decisive impact on the position of the transmitting beam in 3D space. Information regarding the carrier frequency emitted by the antenna, is transmitted to all T/R moduli, most frequently as an analog harmonic reference signal. A numerically generated digital signal processed in a D/A converter is the prototype of this signal. This relatively modern technique is referred to as a Direct Digital Synthesis (DDS). At the output of the D/A converter, an analog signal is obtained, which is subsequently shifted into the microwave frequency range. Considering this signal, each T/R module produces and emits a “portion” of the probe signal with a precisely defined power and phase angle. The corresponding values of the amplitude and phase angle of the produced signal are determined by the T/R module, considering information contained in the digital control signals, which are transmitted to the this module. As

254

12 Methods of Determining the Angular Coordinates of an Object …

a general rule, the discussed control signals are generated according to dedicated control algorithms stored in the central control unit CPU.

References 1. Balanis CA (1997) Antenna theory, analysis and design, 2nd edn. Wiley, New York 2. Barton DK (1988) Modern radar system analysis. Artech House Inc., Norwood 3. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “Soviet Radio”, Moscow (in Russian) 4. Grigorin-Rjabow WW (ed) (1970) Radar devices, theory and principles of construction. Publishing House “Soviet Radio”, Moscow (in Russian) 5. Jackson PB (2010) Overview of missile flight control systems. John Hopkins Appl Tech Digest 29(1):9–24 6. Johnson RC (ed) (1993) Antenna engineering handbook, 3rd edn. McGraw-Hill, New York 7. Komarov WM et al (1991) Height and oblique distance radar meters with continuous frequency modulated wave. Foreign Radioelectronics (12):52–70 (in Russian) 8. Leonov AI, Fomichev KJ (1986) Monopulse radar. Artech House, Norwood 9. Levanon N (1988) Radar principles. Wiley, New York 10. Peebels PZ Jr (1998) Radar principles. Wiley, New York 11. Richards MA (2014) Fundamentals of radar signal processing, 2nd edn. McGraw-Hill Education, New York 12. Rosłoniec S (2006) Fundamentals of the antenna technique. Publishing House of the Warsaw University of Technology, Warsaw (in Polish) 13. Rosłoniec S (2008) Fundamental numerical methods for electrical engineering. Springer, Heidelberg 14. Rosloniec S (2020) Fundamentals of the radiolocation and radionavigation, 2nd edn. Publishing House of the Military University of Technology, Warsaw (in Polish) 15. Rosloniec S (2021) About digital beamforming the multi-element array antennas. Przegl˛ad Telekomunikacyjny (3):57–71 (in Polish) 16. Sammartino PF et al (2008) Moving target location with multistatic radar systems. In: Proceedings of the IEEE radar conference, May 2008, pp 1–6 17. Shen G et al (2008) Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment. In: Proceedings of the 5-th workshop on positioning, navigation and communication (WPNC’08), pp 71–78 18. Sherman SL (1984) Monopulse principles and techniques. Artech House, Norwood 19. Szirman JD (ed) Theoretical basis of radiolocation. Publishing House “Soviet Radio”, Moscow (in Russian)

Part II

Basic Principles of the Radionavigation

Chapter 13

Basic Terms of Radionavigation and Object Position Determining Methods

Broadly defined navigation corresponds to a scientific and technical discipline related to the relocation process of land, sea, airborne and space objects in (3D) space. Since this process takes place in real time, a four-dimensional space, including three geographical coordinates and their determination time, is considered. These coordinates should be explicitly defined and specified in the assumed reference system. A global geographic coordinate system indicates a prime example of such reference system, which characteristic lines are the equator and prime meridian, Fig. 13.1. The position of any point P on the surface of a sphere that approximates the Earth’s surface, is determined by latitude ϕ and longitude λ [1, 2, 10]. The fundamental task of a broadly understood navigation, is to bring a moving object to a specific point with an assumed accuracy, at the right time. This task is relatively easy to perform when the object moves on land or nearby a visible seashore. In these cases, numerous different stationary objects can be used as reference points. There is no doubt that the navigational task is significantly more difficult to solve when the object moves on the open sea area or in the airspace with a limited visibility of the Earth’s surface. In the case of objects moving in deep space, their position can be determined only with respect to the selected celestial bodies, which are conventionally considered as stationary reference points. A variety of navigational devices is used in order to determine the object’s position, which can be divided into the following groups: • mechanical and electromechanical devices (logs, gyro compasses etc.) • magnetic devices (magnetic compasses, magnetometers etc.) • optical and quantum-optical devices (sextants, optical locators, infrared directionfinder etc.) • echo and hydroacoustic sounders (sonic depth finder, underwater detecting system, acoustic direction-finder etc.) • radio devices (direction-finder, goniometric direction-finder, radio navigational receivers etc.)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_13

257

258

13 Basic Terms of Radionavigation and Object Position Determining …

Fig. 13.1 Geographic coordinates; equatorial plane (a) and meridian plane (b)

The mentioned devices can be used considering the requirements and the measurement performed in the environmental conditions (for example atmospheric, magnetic anomaly etc.). As a general rule, objects contain various navigational devices, which increase the possibility of performing a navigational task in various conditions and situations. For example, in submarines, both inertial devices (gyrocompasses, vector acceleration meters etc.) and satellite navigational system receivers, are installed. Inertial devices are used to determine the submarine reckoned position when submerging. When the submarine resurfaces, the GPS system, or its differential version DGPS, is used to measure its position. The present chapter discusses only a few of the most frequently used radio navigation systems, which are generally classified as autonomous and dependent (nonautonomous). In the case of autonomous systems, the navigational devices installed on board, enable determining object’s position with no additional external information received. Dependent systems on the other hand, obtain the navigation information from the external devices, from which it is transmitted to the object by radio. Both, autonomous and dependent radionavigation systems, are classified according to the operational range, accuracy in determining (measuring) the navigation parameters, operational speed, natural and intentional interference resistance, radiation resistance, operational reliability, geometric dimensions, weight, power supply method, etc. The system range corresponds to the greatest distance D between an object and the reference (radio navigation) point, which enables to measure the position with the required accuracy. In the case of short, medium and long-range systems, these distances are equal to D ≤ 800 km, D ≤ 3000 km and D > 3000 km, respectively [2, 8].

13.1 Basic Terms, Parameters and Navigation Methods

259

The time which is required to obtain sufficiently reliable navigation parameters, is particularly significant in aviation (landing phase) and in rocketry. In addition, the geometric dimensions and the weight of navigational devices installed in airplanes and rocket missiles must be low enough.

13.1 Basic Terms, Parameters and Navigation Methods The following subsection discusses some basic concepts of a broadly understood navigation. 1. The instantaneous speed of an object (for example ship) is expressed by the first path derivative with respect to time. Thus, the average speed is expressed by object’s path in the adopted time unit. 2. The object’s acceleration should be considered as the first speed derivative with respect to time or the second derivative of change in position with respect to time. 3. The trajectory of an object’s movement is the path, which its center of gravity follows in three-dimensional (3D) space. 4. The object’s track is the perpendicular projection of its trajectory onto the curvilinear surface of the Earth. 5. The course of an object is the angle measured in the azimuth plane between the north of the meridian passing through the object’s center of gravity and its direction of movement (velocity vector). In Fig. 13.2 this angle is marked with the symbol α. Colloquially, the course of the angle is referred to as the azimuth angle or the azimuth. 6. The reference point P O homing, refers to the angle measured in the azimuth plane between the north of the meridian passing through the object’s center of gravity and the direction to the given point. As Fig. 13.2 indicates, this angle is marked with the symbol β. 7. The position of an object (at a given time instant) is determined by the object’s perpendicular projection of the center of gravity on the Earth’s surface as well Fig. 13.2 Definition of course angle and bearing angle in 2D space

260

13 Basic Terms of Radionavigation and Object Position Determining …

as the height corresponding to the center of gravity distance from the Earth’s surface. 8. Observed position—navigational devices determine the position of the object with respect to the reference points, which are strictly defined at a given time instant. The group of reference points includes, among others, celestial bodies and artificial Earth satellites that emit appropriate radio signals. 9. Calculated position—the position of the object determined (estimated) considering the last observed position and measurements of a distance traveled (course, speed, acceleration), including external influences, such as winds or sea currents. Essentially, the position of an object on the Earth’s surface is determined with respect to an assumed coordinate system, most frequently geographic coordinates or a user-entered local system, similarly to radiolocation. The position of objects moving through the deep space, is determined with respect to stars and planets, whose positions are approximately known. The group of fundamental navigational parameters determined with navigational devices, includes distances, angles and time [2, 4, 10]. In radio navigation, the frequency of the received signal corresponds to a time-equivalent parameter. The group of the derivative parameters include respectively: distance differences or sums, angle differences or sums, time or frequency differences. The physical (geometric) significance of some of the discussed parameters, is shown in Fig. 13.3. Subject to Fig. 13.3, d1 , d2 and b indicate the distances, and b in particular corresponds to locating distance between the reference points P O1 and P O2 . Angles β1 and β2 imply bearings with respect to points P O1 and P O2 , respectively. Considering the measured navigational parameters, the coordinates of an object are estimated in the adopted reference system. For this purpose, appropriate equations are formulated, the graphical interpretation of which is represented by position lines or position surface in two-dimensional space. A position line indicates a set of points on the Earth’s surface that corresponds to the identical value of the estimated parameter. However, only one position line can be assigned to one navigational parameter value. If the estimated parameter corresponds to a distance, then the position line is a circle with an adequate radius, Fig. 13.4. Fig. 13.3 A navigation scenario in which angles β1 , β2 and the base b are known

13.1 Basic Terms, Parameters and Navigation Methods

261

Fig. 13.4 A navigation scenario in which radii r1 , r2 and base b are known

The first position line corresponding to the distance (radius) r1 , is a circle, a fragment (arc) of which is shown in Fig. 13.4. This arc intersects with the second position line, which is a circle with a radius r2 . The point Q, which is the point of intersection of two position lines, is one of the two points that define the position of the object with respect to the segment |P O1 , P O2 |. The situation changes when the navigation parameters correspond to angles measured in the same plane (e.g. azimuth), Fig. 13.5. A straight line p1 corresponds to a position line which is analogous to the bearing β1 . Correspondingly, a position line which is a straight line p2 , is assigned to bearing β2 . According to this assumption, both position lines are in the same plane and intersect with each other at the point Q, which defines the position of the observed object. The object’s position with respect to the points P O1 and P O2 can be easily estimated by implementing the trigonometric values, listed as follows. The location of the base b with respect to the north direction N is known, therefore the apex angles ψ1 and ψ2 can be explicitly determined when the bearings β1 and β2 are known. The distances from the points P O1 and P O2 to the object Q, are respectively equal to: |P O1 , Q| ≡ d1 = b

Fig. 13.5 A navigation scenario in which angles β1 , β2 and the base b are known

sin(ψ2 ) sin(ψ1 ) , |P O2 , Q| ≡ d2 = b sin(ψ1 + ψ2 ) sin(ψ1 + ψ2 )

(13.1)

262

13 Basic Terms of Radionavigation and Object Position Determining …

Example 13.1 Data: b = 120 km, ψ1 = 55◦ and ψ2 = 43◦ . Estimated distances: d1 = 82.644 km and d2 = 99.264 km. The adequate position surfaces correspond to the individual navigation parameters in the three-dimensional (3D) space. Undeniably, the set of points which are equally spaced from a fixed reference point, is referred to as a sphere. Likewise, the measured course angles (α) or bearing angles (β) correspond to oriented planes (in the assumed coordinate system) respectively, including points which enable to perform the navigation measurements. As a general rule, each navigation system (including radio navigation) enables the measurement of strictly defined parameters and determining the corresponding lines or position surfaces, the short characteristics of which is shown in the section that follows. An orthodrome is the line of the shortest distance between two points lying on a spherical surface. According to the stereometry, the orthodrome corresponds to the arc of a great circle crossing these points, Fig. 13.6. A great circle refers to a set (geometric locus) of points lying at the intersection of a sphere with any plane passing through its center. Assuming that the Earth’s surface is a spherical surface, the equator and particular meridians of the geographic coordinate system correspond to special cases of the great circle. One should also emphasize that the arc of a great circle passing through two given points A and B, which do not lie at the opposite ends of the diameter, is not only the shortest but also the only one that exists. On the other hand, in the case that points A and B are at the opposite ends of the diameter, an infinite number of great circles can pass through them. The orthodrome is also defined as the geometric locus of the points lying on the surface of the sphere. The sum of the distances from two given end points (e.g. A and B) is constant and has the smallest value. The loxodrome is the path of the object that intersects the geographic coordinate system meridians at an identical angle α, see Figs. 13.2 and 13.7. It is not difficult to determine that the obtained paths difference, determined according to the orthodrome and loxodrome, is low with the assumed low distances. This difference decreases to Fig. 13.6 Geometric interpretation of the orthodromic distance

13.1 Basic Terms, Parameters and Navigation Methods

263

Fig. 13.7 Geometric interpretation of the loxodrome line

zero alongside equator motion. In this particular case, the course of the angle is equal to α = π/2 rad. The second exceptional case relates to the object moving with a constant course α = 0 rad, i.e. alongside the meridian. For any other value of the angle α, the loxodrome is a fragment of a spiral line ending at the pole. A line of equal bearings refers to a set of points lying on a spherical surface. A bearing which is performed from any point of this surface to a stationary reference point, has an identical value β. A line of equal distances refers to a set of points lying on a spherical (or flat) surface, at an equal distance from a given point. The examples of such lines, shown in Fig. 13.8, include circles with radii d1 , d2 and d3 , and a common center T . These circles are located on the surface of the sphere, which is a geometric locus (position surface) of points equally distant from its center O. The line of equal distance sums refers to a set of points whose sum of distances from two given reference points (e.g. T1 and T2 ) has the same value. Figure 13.9 shows a diagram of this line in the plane which takes the form of the ellipse. Accordingly, the ellipse consists of points P(x, y), whose distances d1 and d2 from the focal points Fig. 13.8 Circles as position lines

264

13 Basic Terms of Radionavigation and Object Position Determining …

Fig. 13.9 Ellipses as position lines

T1 and T2 satisfy the equation |d1 + d2 | = 2a, where a > 0. In the limiting case for 2a → 2c, the ellipse transforms into a segment |T1 , T2 |. If the landmarks are on a spherical surface, lines of equal distance sums refer to spherical ellipses. Navigation systems (including radio navigation systems as well) using lines of equal distance sums (ellipses) as position lines, are commonly referred to as elliptical. The line of equal distance differences refers to a set of points whose distance difference from two given reference points (e.g. T1 and T2 ), is constant. When the distance difference is determined on the plane, than the lines of equal distance differences relate to the arms of a hyperbola, Fig. 13.10. In other words, a hyperbola consists of points P(x, y) whose distances d1 and d2 from the focal points T1 and T2 , satisfy the equation |d1 − d2 | = 2a. The bisector |T1 , T2 | which is expressed by the equation |d1 − d2 | = 0, is referred to as the null hyperbola. If the distance differences are determined on a spherical surface, then such phenomenon is referred to as a spherical hyperbola. As the literature indicates, the navigation (radio navigation) systems that use the lines of equal distance differences as position lines, are referred to as hyperbolic. Examples of hyperbolic systems that had a significant impact on radio navigation include Decca Navigator, LORAN-A and LORAN-C [3, 8, 9]. Navigation in deep outer space uses equations, the geometrical interpretation of which may be constituted by position surfaces. The simplest example Fig. 13.10 Branches of a hyperbola as position lines

13.1 Basic Terms, Parameters and Navigation Methods

265

Fig. 13.11 A conical surface with the same value of the Doppler frequency

of a position surface corresponding to the identical distance, is the already mentioned spherical surface. In order to determine the position of a spaceship with respect to given celestial bodies, at least three spherical surfaces are crucial to be determined. Due to the intersection of two spheres, a circle is obtained that intersects the third sphere at two points. A spaceship that determines its position, is located in one of these points. As a general rule, the approximate location of the ship is known, so the mentioned position ambiguity can be easily eliminated. In general, the position surface consists of points in the 3D space that correspond to the identical value of a given navigation parameter. In satellite radio navigation such a parameter refers to the frequency of the signal which is received from the rapidly moving artificial Earth satellite, Fig. 13.11. Assume that the satellite S shown in Fig. 13.11 moves in a circular orbit at a linear speed ϑ, which indicates a vector sum of the radial ϑr (located on the observation line) and ϑu components, where ϑu is perpendicular to ϑr . Moreover, assume that the satellite transmitter emits a harmonic signal of a frequency f s . This signal, received at the observer’s point O, will have a frequency equal to fr(O) ec , changed due to the Doppler effect, i.e. ( ) ϑr 1 + fr(O) = f s ec c

(13.2)

The signals received (measured) in the points of the conical surface, shown in Fig. 13.11, have the identical frequency value. The axis of the conical surface coincides with the direction of velocity ϑ and the location of a cone apex coincides with satellite S position. The discussed conical surface is frequently referred to as an equal Doppler effect frequency surface. Such surface, intersecting with the Earth’s surface, marks the line of equal Doppler effect frequency, abbreviated as the isodopp. In turn, the intersection of two isodopps (measured at different time instants) gives in the result two points, one of which determines the position of an object. The American TRANST–NNSS system, which was implemented in the sixties is an example of a satellite radio navigation system using conical positional surfaces. The principle of operation of this system will be discussed in Chap. 15.

266

13 Basic Terms of Radionavigation and Object Position Determining …

13.2 Classification Criterion of Radio Navigation Systems From the user’s point of view, the most natural division of radio navigation systems corresponds to the place and radio stations manner of distribution, which relate to reference points. If these stations are distributed over land area (or islands), the system that implements them is known as a terrestrial system with stationary reference points. If the reference points correspond to Earth’s artificial satellites emitting corresponding radio signals, then one may refer to a satellite system. In turn, the navigation parameters discussed in the previous subsection and the corresponding position lines (or position surfaces), constitute a convenient division criterion of terrestrial and satellite radio navigation systems into the following types. Ground Radio Navigation Systems 1. Azimuth systems (Sonne, WRM-5, Consolan, …) 2. Hyperbolic systems (Decca Navigator, LORAN C, Omega…) Satellite Radio Navigation Systems 3. Doppler systems (TRANSIT–NNSS, Tsikada, …) 4. Stadiometrical systems (BeiDou, Galileo, GPS, GLONASS,…) Antennas that enable determining the direction of the received signal are essential in azimuth radio navigation systems. The simplest antenna system used for this purpose, consists of a wire antenna (omnidirectional) and a frame antenna, rotating around a vertical axis. The more technologically advanced version of this system uses a stationary frame antenna, the rotary motion of which has been replaced by the rotation of two coils coupled together, referred to as goniometers, see Appendix H [3, 4, 6].

13.3 Radio Direction-Finders Radio direction-finding constitutes undoubtedly the earliest application of radio technology for navigation purposes. For this purpose, relatively simple receiving devices, which are referred to as radio direction-finders, are implemented. Traditionally, radio direction-finders are divided into terrestrial (stationary) and on-board (mobile). Terrestrial-based ones are used to solve various navigation tasks, including: lightning discharge (storms) detection, ballistic missile launch detection, localizing the unknown radio transmitters (direction finding), or determining the position of landing aircrafts (TLS system). Taking into account the latter application, the position of the aircraft is determined considering signal’s direction of the reception, emitted by the transmitter of its response system (IFF). Due to the terrestrial location, the transmitters can be constructed using multi-element antenna arrays. Thus, a relatively high

13.3 Radio Direction-Finders

267

bearing accuracy is possible to be ensured. The on-board radio direction-finders, which provide bearings to terrestrial radio stations, consist of relatively simple antenna systems and receiving machinery. In other words, their accuracy is lower in comparison to terrestrial-based radio direction-finders. Despite this fact, the aviation still uses them as supplementary (emergency) navigation devices. Additionally, one should emphasize that even significant bearing errors will not prevent the aircraft from reaching the radio station (radio beacon) with respect to which the bearing is performed, although not including the shortest route. As mentioned before, one of the main component of each (terrestrial and on-board) radio direction-finder, is its antenna system with a corresponding directivity pattern. A few simple structural solutions of such antennas are presented below. Frame and Magnetic Loop Antennas Frame and magnetic loop antennas are mainly used in portable radio receivers and receiving devices of various radio navigation and radioelectronic reconnaissance systems [4, 6]. A single frame (loop) antenna is a fundamental element of them and therefore is discussed below. Thus, a frame antenna constructed of a very thin conductor, lying in a vertical plane, with an ability to rotate around a vertical axis 00' , should be analyzed, Fig. 13.12. Assume that this antenna is located in the field of a linearly polarized wave whose electric field vector is directed along the axis 00' and is expressed by the following formula E(t) = E m sin(ωt)

(13.3)

Furthermore, assume that the geometrical dimensions h and l of the antenna are very low in comparison to the length λ of the received wave. According to these assumptions, electromotive forces (E M F) are induced in the vertical sides of the frame

Fig. 13.12 A loop antenna

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13 Basic Terms of Radionavigation and Object Position Determining …

E M Fab = h E m sin(ωt − ψ/2),

E M Fcd = h E m sin(ωt + ψ/2)

(13.4)

for ψ = (2π/λ)l cos(ϕ) ≡ βl cos(ϕ), and ϕ corresponds to the angle between the frame plane and the wave direction of incidence, see Fig. 13.12. Due to the assumption that the electric field vector is parallel to (ab) and (cd) sides, this field does not induce an electromotive force in the perpendicular (ac) and (bd) sides. The direction of the electromotive forces, induced in the vertical sides of the frame, is opposite (when summing the voltages induced in the antenna), therefore E M F = E M Fcd − E M Fab = E m h[sin(ωt + ψ/2) − sin(ωt − ψ/2)]

(13.5)

Considering the following trigonometric identity in (13.5) ( sin(α) − sin(β) = 2 sin

α−β 2

)

( cos

α+β 2

) (13.6)

the following equation is obtained ] βl cos(ϕ) cos(ωt) 2 ] [ βl cos(ϕ) sin(ωt + π/2) = 2E m h sin 2 [

E M F = 2E m h sin

(13.7)

One should emphasize that the electromotive force induced on the terminals of the frame antenna, is phase shifted by an angle π/2 rad with respect to the electric field intensity, see Eqs. (13.3) and (13.7). If the frame antenna consists of N identical frames (loops) connected in series, as indicated in Fig. 13.13, then the electromotive force induced on its terminals is ] [ βl cos(ϕ) sin(ωt + π/2) (13.8) E M F (N ) = 2E m N h sin 2 The following condition l 0, dY = (x − xY )2 + (y − yY )2 > 0 . . = (x − x W )2 + (y − yW )2 > 0, d Z = (x − x Z )2 + (y − y Z )2 > 0 (14.8)

dX = dW

When the formula (14.8) is taken into account in Eqs. (14.6) and (14.7), a system of two nonlinear equations is obtained, containing two unknowns, which are the receiver’s coordinates (x, y), i.e. . . (x − x X )2 + (y − y X )2 − (x − xY )2 + (y − yY )2 = χ1 .

(x − x W )2 + (y − yW )2 −

. (x − x Z )2 + (y − y Z )2 = χ1

(14.9) (14.10)

A sought solution is obtained when this system is solved, whereby the accuracy in determining the receiver’s position depends on the accuracy in determining the absolute terms χ 1 = c(t X − tY − tk X + tkY ) − b X + bY and χ2 = c(tW − t Z − tkW + tk Z ) − bW + b Z . Due to an inaccuracy in the measurement of reception times (t X , tY , tW , t Z ), fluctuations in code delays (tk X , tkY , tkW , tk Z ) and inaccuracy regarding bases length (b X , bY , bW , b Z ), some inaccuracies in determining the absolute terms may occur. Since the Eqs. (14.6) and (14.7) are nonlinear, solving them with analytical methods, with respect to the unknowns (x, y), may be problematic. Consequently, in a “pre-computer” period, a variety of supporting tools were implemented in order to obtain an approximate (sufficiently accurate) solution, in an uncomplicated way. For instance, the coverage maps, developed for a specific chain, were used, containing marked points M, X, Y, W and Z of radio station distribution. For an identical chain, transparent masks with hyperbolic lines corresponding to the discrete parameters values χ1 and χ2 , were prepared. Most frequently, the hyperbola branches, determined for particular pairs of radio stations (X–Y, W–Z), were applied using contrasting colors. The scale of these masks was identical as the scale of the coverage map. When both masks were applied to the coverage map, a point of intersection of two hyperbola branches, corresponding to the determined parameters values χ1 and χ2 , was found. Currently, the system of Eqs. (14.6) and (14.7) can be solved numerically by the on-board computer built into the receiver. Example 14.1 Organizational data of the USA 9960 LORAN–C chain.

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14 Ground, Hiperbolic Radionavigation Systems

Table 14.2 Names and geographic locations of the radio stations of the USA 9960 LORAN–C chain Radio station

Latitude

Longitude

Transmitter power (kW)

M/Seneca

0.42◦ 42' 50.716'' N

076◦ 49' 33.308'' W

800

X/Mantucket

041◦ 15' 12.046'' N

069◦ 58' 38.536'' W

400

Y/Carolina Beach

034◦ 03' 46.208'' N

077◦ 54' 46.100'' W

800

W/Caribou, Canada

046◦ 48' 27.305'' N

067◦ 55' 37.159'' W

800

Z/Dana

039◦ 51' 07.658'' N

087◦ 29' 11.586'' W

400

The names and geographic locations of the radio stations which are the component of this chain, are summarized in Table 14.2. The geographic location and the coverage shape of this chain, are shown in Fig. 14.6. Numerous observations made within more than fifty years of system operation, show that the accuracy in determining position at approximately 90 m ground wave and the range not exceeding 1100 km, was ensured. This accuracy decreased to approximately 500 m, as the receiver deflected from the center of the coverage. A significant accuracy deterioration was noticed in the case of the occurrence of the wave reflected from the ionosphere, which most frequently occurred at large ranges, i.e. greater than 2000 km (Fig. 14.7). According to [7, 14], the error in determining the position considering ionospheric wave receiving signals, during the day is in the range of 3–5 NM, and at night in the range of 5–8 NM. The acronym NM stands for nautical mile and is equal to 1852 m.

Fig. 14.6 A coverage area for the USA/9960 chain of the Loran C system

14.2 Decca Navigator and Omega Interference Systems

291

Ionized ionosphere

Fig. 14.7 A multipath transmission (propagation) due to reflections from the ionosphere

14.2 Decca Navigator and Omega Interference Systems This chapter discusses the organizational structures and principle of operation of two terrestrial interference systems, i.e. Decca Navigator and Omega. These systems had a significant impact in long-range radionavigation, [1, 7, 9]. The distance difference relevant to hyperbolic interference systems, is determined considering the measured phase differences of the signals received from the mutually synchronized radio stations, for example T1 and T2 , as shown in Fig. 14.8. In the general case, this difference is expressed by the following formula

Fig. 14.8 Definitions for distances and angles for the radionavigation systems Decca Navigator and Omega

292

14 Ground, Hiperbolic Radionavigation Systems

.ϕ =

2π ' (d − d2' ) + k · (2π ) λW 1

(14.11)

for λ W corresponding to the wavelength which is measured and k is an integer specifying a number of the hyperbolic trajectory. Subject to this, an explicit emphasis is required, namely the phase difference of the received signals can only be defined in the interval [0, 2π ]. The distance difference .d H , referred to as a hyperbolic trajectory width, corresponds to a maximum phase difference equal to 2π . This width is the smallest and is equal to λ W /2 when the receiver is on the base line, ' i.e. at a point P, see Fig. 14.8. Considering any point P beyond the base line, the hyperbolic trajectory width is determined by the following formula .d H =

1 λW 2 sin(ψ/2)

(14.12)

Subject to (14.12), ψ indicates the angle at which the base is visible from a given point P ' , see Fig. 14.8. Considering the Eq. (14.12) and Fig. 14.8, one can conclude that reducing the base length may result in faster increase of the hyperbolic trajectory width, as the receiver (of the point P ' ) deflects from the base line. The following time difference corresponds to a measured phases difference (14.11) .t =

1 .ϕ .ϕ = ωW 2π f W

(14.13)

Taking into consideration (14.13), f W corresponds to a wave frequency, at which the phase difference was measured. An explicit phase difference measurement is only possible within one hyperbolic trajectory. The number k, appearing in the formula (14.11), can be determined by counting successively traveled trajectories. Considering the measured (explicit) phase difference (14.11), the time difference (14.13) and the corresponding trajectory difference are calculated. Consequently, a position line, which occurs as a hyperbola, is possible to be determined. The focal points of this hyperbola correspond to the points in which radio stations emitting mutually synchronized harmonic signals, with frequencies indicating the multiples of a fundamental frequency f 0 , are arranged. In this manner, two position lines are determined, the common point of which determines the receiver’s position. Decca Navigator System Decca Navigator system (DNS) was introduced during World War II, is an example of a radio navigation system operating in the previously described manner. The Royal Navy (UK) was an administrator of this system which was used in Europe until the spring of 2000. Similarly to LORAN–C system discussed before, Decca Navigator system consists of chains of four (occasionally three) mutually synchronized radio stations, one of which is a master unit and the remaining ones are slave units, Fig. 14.9.

14.2 Decca Navigator and Omega Interference Systems

293

Fig. 14.9 A typical spatial configuration of the Decca Navigator chain

Table 14.3 Names stations and operation frequencies of the Decca Navigator system chain f 0 = 14.1666666 kHz Frequency

M—Master

P—Purple

R—Red

G—Green

6 f0

5 f0

8 f0

9 f0

Table 14.3 shows the names of the radio stations and operation frequencies of Decca Navigator system English chain. As Table 14.3 indicates, f 0 = 14.1666666 kHz demonstrates a fundamental frequency. The value range of typical base distances, b P , b R and bG is included from 110 to 220 km. The fundamental frequencies of other Decca Navigator chains slightly differ from the mentioned value f 0 , simplifying their identification. The frequencies which are used, are contained in the range of 70–129 kHz. In order to explain the method of determining the phase difference (14.11) (considering one hyperbolic trajectory) and to discuss the principle of operation of a single chain, assume that the master station M, emits the wave e M (t) = E M cos(mω0 t + 0)

(14.14)

for which m = 6 is the multiplier of the fundamental frequency f 0 . This wave reaches each of the slave stations, including the radio station R. Thus, the signal reaching the radio station receiver R can be expressed as follows ( )] [ bR e'R (t) = E 0' cos m · ω0 t − c

(14.15)

294

14 Ground, Hiperbolic Radionavigation Systems

for c ≈ 2.998·109 m/s and b R corresponding to the base length between radio stations M and R, see Fig. 14.9. In the radio station R, the signal (14.15) is multiplied n/m = 8/6 times and emitted after amplification. The signal processed correspondingly, can be expressed as ( )] [ bR e R (t) = E R cos n · ω0 t − c

(14.16)

The signals (14.14) and (14.16) reach the user’s receiver at the point Q, with delays proportional to the travelled distance, i.e. d M and d R , respectively. In other words, the following signals reach the input of the receiver ( )] [ dM u M (t) = U M cos m · ω0 t − c ( )] [ bR dR − u R (t) = U R cos n · ω0 t − c c

(14.17) (14.18)

The signals (14.17) and (14.18) are amplified and multiplied Wmn /m and Wmn /n times in the receiver, respectively, for which Wmn = 24 is the least common multiple of numbers m = 6 and n = 8. Consequently, signals with identical pulsation ω = Wmn ω0 and phase difference, are obtained. ( ( ) ) Wmn ω0 bR dM dR Wmn ω0 .ϕ = Wmn ω0 t − − − Wmn ω0 t − = (d M − d R ) − bR c c c c c

(14.19) When introducing the indication λW = λ0 / Wmn = c/( f 0 · Wmn ), the formula (14.19) can be expressed in the form of a position line equation dM − dR =

.ϕ λW + 2π b R = χM R 2π

(14.20)

which indicates a hyperbola with foci at points M and R, Fig. 14.9. Analogically, the parameter χ is determined, see Eq. (14.20), of the second position line (of hyperbola) that intersects the first position line at point Q, determining the receiver’s position. Similarly to LORAN–C system, the navigator determined this position with specialized chain maps (specifying the locations of radio stations) and transparent masks with identical scale containing hyperbolic lines of various colors. Each pair of stations (master and slave stations) has a corresponding position lines marked with a color assigned to the slave station. According to this principle, the position lines (hyperbolas) for the M (master station) and G (green) radio stations are marked in green. The position determining error by using Decca Navigator system ranged from several dozen meters, in the case of the receiver located close to the base line, and was

14.2 Decca Navigator and Omega Interference Systems

295

increasing to approximately 3 NM for the receiver located on the edge of the coverage (range) at winter night. The range of a single chain is limited by the hyperbolic trajectory identification and depending on the part of day, ranges from 300 to 400 NM, [9, 11]. Omega System The Omega system, the main administrator of which was the US Navy, achieved operational status in 1971. Corresponding to a hyperbolic, a significantly long-range, interference radio navigation system, covered the entire globe [2, 8, 13]. Initially, the first American-European element of this system was implemented in order to navigate strategic bombers in the North Pole area. Later, Omega system was also used in maritime navigation. Once the GPS satellite system (which is more accurate) was deployed in the nineties of the previous century, the importance of Omega system rapidly decreased in importance and was eventually withdrawn in 1997. This system consisted of 8 phase-synchronized radio stations, the names and geographical coordinates of which are shown in Table 14.4. The specified radio stations (equipped with cesium frequency standards) emitted coherent radio pulses of different durations ti , see Table 14.5, and frequencies: f 1 = 10.2 kHz, f 2 = 11.33(3) kHz and f 3 = 4 f 1 /3 = 13.6 kHz marked by deviations δ f ≈ 10−13 . The distribution of these radio stations on both sides of the equator is shown in Fig. 14.10. The measurement of the received signals phase difference was performed for the frequency f 1 and the signals with frequencies f 2 and f 3 were used primarily to identify the hyperbolic trajectories with the scaling method, [1, 3, 7]. The emission of each subsequent radio station started after the blank interval .t = 0.2 s, which was counted from the emission end of the previous radio station. As Table 14.5 indicates, a cyclic emission of signals by particular radio stations was possible in accordance to a given schedule (format). Knowledge of the radio emission format (with the total duration Tc = 0.9+0.2 + 1.0 + 0.2 + ... + 0.2 + 1.0 + 0.2 = 10 s), see Table 14.5, enabled identifying the radio Table 14.4 Names and geographical coordinates of 8 phase-synchronized radio stations of the Omega system Radio station

Latitude

Longitude

Transmitter power (kW)

A—Norway

066◦ 25' 13'' N

013◦ 08' 13'' E

10

B—Libera

006◦ 18' 20'' N

010◦ 39' 52'' W

10

C—USA (Hawaii)

021◦ 24' 17'' N

157◦ 49' 51 W

10

D—USA (Dakota)

046◦ 21' 57'' N

098◦ 20' 8'' W

10

E—France (La Reunion)

020◦ 58' 27'' S

055◦ 17' 24'' E

10

F—Argentina

043◦ 03' 13'' S

065◦ 11' 27'' W

10

G—Australia

038◦ 28' 52'' S

146◦ 56' 7'' E

1

H—Japan

034◦ 36' 56'' N

129◦ 27' 13'' E

10

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14 Ground, Hiperbolic Radionavigation Systems

Table 14.5 The format that determines the emission of signals, in the time-domain, by individual radio stations of the Omega system ti , s 0.9 A

[ f1 ]

B

.t 1.0 f3 [ f1 ]

C D

.t 1.1

.t 1.2

f3

f2

[ f1 ]

f3

.t 1.2

f3

f2

H

f3

.t

f2 f3 [ f1 ]

G

.t 1.0

f2 [ f1 ]

F

.t 0.9

f2

[ f1 ]

E

.t 1.1

f2

f2 f3

f2

[ f1 ]

f3 [ f1 ]

Fig. 14.10 Geographical positions of radio stations of the Omega system

station considering a comparison of the emission beginning with the Universal Time Coordinated (also known as Zulu time) which was synchronized with the system. In order to provide a full synchronization of the system operation, each radio station contained a receiver recording the time instants of receiving signals from other radio stations. The signal arriving from a specific radio station could be received at a different time than the theoretically determined time instant. Information regarding thus established time difference was transmitted by radio to other radio stations, allowing for the entire system synchronization. In the Omega system, operating with time distribution of the received signals, any radio station could be selected by the user as the master station.

References

297

As mentioned before, the phase difference measurement of the signals received from two different radio stations at two different time instants, was performed on the fundamental frequency f 1 = 10.2 kHz. Such measurement was possible to be performed only when the receiver enabled ‘memorization’ of the signal phase which was received earlier from the radio station selected as the master one. The contemporary technology solved this task indirectly, i.e. by comparing the phases of both received signals with the phase of the harmonic signal of a frequency equal to 10.2 kHz, generated by the local highly stable generator. The phase fluctuations of this reference signal were significantly smaller in the time interval Tc = 10 s than 2π/100 rad [2, 5]. Likewise Decca Navigator system, the equation of the first position line was determined considering the measured phase difference. Then, taking the measured phase difference received from the second pair of radio station into consideration, the equation of the second position line that intersected the first position line at the receiver position point, was similarly determined. Theoretically, an error in determining the position with the Omega system was equal to 1.5 NM in daytime and 2 NM at nighttime. An experiment indicated that this error was significantly higher and was increasing even up to 20 NM in certain regions of the globe [3, 14]. Due to this reason, in the last decade of the previous century, a differential version of this system, known as”Differential Omega”, was implemented. In this system the corrections determined for a given region were transmitted by special radio beacons, operating in the ranges 285–415 kHz. Consequently, the position determining error approximately decreased to 0.2 NM, at the distance of the receiver from the radio beacon not greater than 110 NM, and did not exceed 0.6 NM at the edge of the radio beacon range, equaling approximately 500 NM.

References 1. Bakuljiev PA, Sosnovskij AA (2005) Radionawigation systems (in Russian). Publishing house “Radiotekhnika”, Moscow 2. Creamer PM, Gupta RR, Morris PB (1994) Omega navigation system course book, volumes I and II. U.S. Coast Guard, Omega Navigation System Center, Alexandria (VA) 3. Forssell B (2008) Radionavigation systems. Artech House, Inc., Boston/London 4. Frank RL (1983) Current developments in Loran–C. In: Proceedings of the IEEE, vol 71, no 10, Oct 1983 5. Gupta RR, Morris PB (1986) Omega navigation signal characteristics. Navigation (USA) 33(3) 6. Jansky, Bailey (1962) The Loran–C system of navigation. Report prepared for U.S. Coast Guard, Washington D.C 7. Olenyuk PV, Goloushkin GV (eds) (1985) Radionavigation systems of superlong-wave. range (in Russian). Publishing house “Radio and svyaz”, Moscow 8. Pierce JA (1989) Invention of Omega. Navigation (USA) 36(2) 9. Powell C (1982) Performance of the Decca Navigator on land. In: Proceedings of IEE, vol 129, Pt. F, no 4 10. Specification of the Loran–C transmitted signal. U.S. Coast Guard, Omega Navigation System Center, Alexandria (VA) (1994)

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11. The Decca Navigator. Principles and performance of the system. Decca Navigation Cor., Aug (1979) 12. The Loran–C system of navigation. U.S. Coast Guard, Omega Navigation System Center, Alexandria (VA) (1994) 13. The Omega system of global navigation. In: Coast guard engineers digest, no 152, (1966) pp 26–33 14. Wilkes O, Gleditsch N (1987) Loran–C and Omega. A study of millitary importance of radionavigation aids. Norvegian University Press, Oslo

Chapter 15

Satellite, Doppler Radionavigation Systems

The second half of the previous century brought one of the most significant achievements in radioelectronics, namely, the development of the first and second generation of satellite radio navigation systems. The most recognized, first-generation systems include TRANSIT-NNSS (USA), Cyclone (USSR) and Tsikada (USSR). In the last two decades of the twentieth century, these systems were replaced by more accurate second-generation (stadiometric) systems such as GPS-NAVSTAR (Global Positioning System-Navigation Signal Timing and Ranging) and deriving from Russia, GLONASS (Globalnaja Navigacjnaja Sputnikowaja Sistiema), developed respectively in the USA and the USSR, [1–5, 9, 11]. A characteristic of the mentioned first-generation systems included the use of information contained in the Doppler frequency shift of signals emitted by artificial Earth satellites, circling in low, almost circular orbits. This possibility was confirmed experimentally for the first time in October 1957, i.e. after the USSR launched the first artificial Earth satellite, called Sputnik 1. The overall shape of this satellite with two pairs of telescopic antennas with lengths 2.4 m, is shown in Fig. 15.1. This spherical object, with 59 cm diameter and 83.6 kg weight, was circling the Earth around on an elliptical orbit, with apogee and perigee equal to 938 km and 214 km, respectively. An orbital velocity of this satellite was approximately equal to 8.055 km/s and the duration of one Earth orbiting estimated approximately 96 min and 12 s. After 22 days on the orbit, the alternate radio pulses were sent towards Earth at carrier frequencies 20.005 MHz and 40.002 MHz. In order to accomplish this task, two transmitters of power 1 W were used. These signals, with a changed frequency (due to the Doppler effect), were (due to the Doppler effect), were received all over the globe by both, professionals and amateur radio operators [1, 4].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_15

299

300

15 Satellite, Doppler Radionavigation Systems

Fig. 15.1 An image of the Sputnik satellite launched by the USSR

15.1 A Principle of Operation In order to gain understanding of the principle of operation of Doppler systems, a signal received from a satellite orbiting the Earth around a circular orbit at a linear velocity, should be analyzed, Fig. 15.2. Assume that a satellite transmitter sends a harmonic signal of a frequency f S . This signal, due to the Doppler effect (evoked by a satellite movement), is received at a point P as a variable frequency signal [ ] ϑ r (t) f P (t) = f S 1 + c

(15.1)

for c ≈ 2.998·108 m/s corresponding to the wave propagation velocity in free space, and ϑ r (t) indicates the radial component (towards the observer) of the satellite ϑ velocity. A typical waveform of frequency f P (t), as a function of an angle α (in the visibility range −90◦ < α < 90◦ ), is shown in Fig. 15.3. Fig. 15.2 Definition of a radial velocity ϑr

15.1 A Principle of Operation

301

Fig. 15.3 Doppler shift of the frequency of the satellite signal received at point P

The frequency increment . f (t) ≡ f D (t) = f P (t) − f S = f S ϑr (t)/c (Doppler frequency shift) together with the satellite coordinates (x S , y S , z S ) provide sufficient data, required to determine the first position line which contains the receiver (point P). In order to examine this problem in more detail, assume that the coordinates (x, y, z) of a point P, defined in the same three-dimensional coordinate system as the satellite coordinates (x S , y S , z S ), are unknown. Moreover, the adopted Cartesian coordinate system orbits together with the Earth. To simplify the analysis, also assume that the receiver’s location is not changed when determining its position. The distance is equal to: d S (t) ≡ |P, S| =

.

[x − x S (t)]2 + [y − yS (t)]2 + [z − z S (t)]2

(15.2)

The radial velocity ϑr (t) which occurs in formula (15.1), is equal to the first derivative of (15.2) with respect to time, which is expressed by the formula −ϑr (t) =

[x − x S (t)] d xdtS (t) + [y − yS (t)] dydtS (t) + [z − z S (t)] dzdtS (t) d . [d S (t)] = dt [x − x S (t)]2 + [y − yS (t)]2 + [z − z S (t)]2 (15.3)

The minus (−), occurring in front of the radial velocity indication, considers the fact that due to the distance d S (t) increase (the deflection of a satellite from the user’s receiver), a frequency of the received signal decreases. The left side of expression (15.3) corresponds to a quantity calculated according to the formula −ϑr (t) = −c · f D (t)/ f S , considering the measured Doppler frequency f D (t) = f P (t) − f S . . Thus, taking (15.3) into account, the following equation is obtained [x − x S (t)] d xdtS (t) + [y − yS (t)] dydtS (t) + [z − z S (t)] dzdtS (t) c · f D (t) . = 0 (15.4) + 2 2 2 fS [x − x S (t)] + [y − yS (t)] + [z − z S (t)]

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15 Satellite, Doppler Radionavigation Systems

If the receiver movement is taken into consideration (with respect to the Earth’s surface), the equation of the equal Doppler effect frequency surface is expressed as follows ] ] ] [ [ [ [x − x S (t)] d xdtS (t) + [y − yS (t)] dydtS (t) + [z − z S (t)] dzdtS (t) . + [x − x S (t)]2 + [y − yS (t)]2 + [z − z S (t)]2 ] ] ] [ [ [ dy(t) dz(t) [x − x S (t)] d x(t) + [y − y + [z − z (t)] (t)] S S dt dt dt c · f D (t) . + + =0 fS [x − x S (t)]2 + [y − yS (t)]2 + [z − z S (t)]2 (15.5) Despite the fact that the displacement velocity of the receiver is disproportionately low in comparison to the satellite linear velocity, undoubtedly, its omission reduces the accuracy of determining the position. Equations (15.4) and (15.5) satisfy the coordinates (x, y, z) of the points that form the conical surface, as shown in Fig. 15.4. The cone apex of this surface is located at point S (precisely in the phase center of the satellite’s transmitting antenna) with coordinates (x S , y S , z S ), which correspond to a function of time. The frequency of the received satellite signal has identical value at each point of this conical surface, thus this position surface is frequently referred to as an equal Doppler effect frequency surface. The intersection of this surface with a curvilinear surface of the Earth, marks a position line which in the literature is referred to as isodopp, [2, 4, 7]. In order to determine a position of the receiver in 3D space, three position lines (isodoppes) must be established. In other words, considering the measured Doppler frequencies f D (t1 ), f D (t2 ) and f D (t3 ), for t1 < t 2 < t3 , the corresponding radial velocities ϑr (t1 ) = c · f D (t1 )/ f S , ϑr (t2 ) = c · f D (t2 )/ f S and ϑr (t3 ) = c · f D (t3 )/ f S are estimated. Then, three equations, similar to (15.3) or (15.4), are formulated. The satellite coordinates [x S (t), y S (t), z S (t)], occurring in these equations, are transferred to the user’s receiver in the so-called navigation telegram. Considering the obtained information and known system parameters, the Fig. 15.4 A conical surface with the same Doppler frequency

15.2 The Navy Navigation System—TRANSIT

303

derivatives d x S (t)/dt, dyS (t)/dt and dz S (t)/dt are calculated, and are essential for estimating the radial velocity ϑ r (t) components. The launch of the first artificial Earth satellite was a significant event of both, a technical and political nature. Undoubtedly, this happening accelerated work on the artificial Earth satellites implementation as reference points in newly developed radio navigation systems. The first satellite radio navigation system developed in the USSR for the army (particularly for the navy) was a Doppler navigation system, referred to as Cyclone (also known as Tsiklon). As far back as 1962, a draft of an experimental navigation satellite, with the same code name, was ready. The first satellite for this system was launched in May 1967. In 1969, the accuracy in determining the position of a stationary craft of the order of 100 m, was accomplished. Due to a high unreliability of electronic devices used at that time, including computational ones, the operational testing phase of this system began in 1974 though. The last satellite of this system was launched into orbit on February 27, 1978. The Tsikada navigation system was developed as an improved version of the Cyclone Doppler system. Its military version Tsikada-M, is also known as Parus. In 1979, the merchant navy and fishing vessels of the USSR were allowed to use the navigation service. However, not until 1987, the merchant and fishing vessels could use this system around the globe. Due to this political decision and modernization process, a civilian version was created, which was using only four satellites. However, the program was ultimately terminated in 2003.

15.2 The Navy Navigation System—TRANSIT The first American experience in the field of satellite navigation (late 1950s and early 1960s) included tracking the satellites launched by the USSR. In order to determine their position, launched in 1958 a passive, multistatic MINITRACK system was implemented, using radio signals emitted by detected satellites. In 1961 the name of the system was changed to Naval Space Surveillance System (NAVSPASUR) which is respected up to this day. Moreover, NAVSPASUR is considered to be one of the fundamental space control systems over the United States of America. The TRANSIT system was developed in early seventies of the previous century, at John Hopkins University for the US Navy. Initially, i.e. from 1964, the main purpose of this system was to navigate submarines carrying Polaris ballistic missiles, [4, 9, 10]. Later, it was facilitated to sea navigation and eventually in July 1967 to other civilian users, inter alia, for hydrographic and geodetic purposes. In the end, the system was terminated in 1996 and its satellites were mainly used to monitor the ionosphere. The TRANSIT system, likewise in other satellite navigation systems, distinguishes three fundamental segments, namely space, control and measurement terrestrial, and user segment.

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15 Satellite, Doppler Radionavigation Systems

Space Segment The space segment of this system consisted of 6 active satellites located in 6 low circular orbits approximately 600 NM (1075–1112 km) high above the Earth’s surface. The first satellites of this system (e.g. TRANSIT type) had the shape of approximately 50 cm diameter sphere, Fig. 15.5a. Each of the 61 kg satellites contained solar panels supplying energy of 30 W to the battery unit. The satellite antenna system consisted of logarithmic spirals distributed over both hemispheres. Since 1980s, these satellites have been replaced by NOVA satellites, which could correct their location in orbit due to engines built-in ramjets. A proper satellite orientation with respect to the Earth’s surface (including antennas and solar panels distribution) was obtained due to the Earth’s field of gravitation. The gradient of this field determined the 30 m boom position with a half-kilogram weight at its end, Fig. 15.5b. The English-language literature refers this satellite position stabilizing method to as a gravity gradient stabilization, [10]. The orbits of the satellites were arranged likewise the meridians of the geographic coordinate system, see Fig. 15.6. The angle of arrival of these orbits’ planes was equal to 30◦ = 180◦ /6, and the angle shift of the satellites, located in adjacent orbits, was approximately equal to 60◦ . Each of the satellites at a linear velocity of approximately ϑ = 7.3 km/s, was orbiting the Earth in 107 min. Simultaneously, the Earth was rotating about its axis by an angle close to 27◦ . As the analysis covered in Appendix J confirms, the time interval, in which the satellite is located above the horizon, was equal to approximately 16 min. The transmitting devices of each satellite emitted two phase-modulated radio signals with carrier frequencies f 1 = 4.9996 · 80 = 399.968 MHz, f 2 = 4.9996 · 30 = 149.988 MHz and power 1.5–3 W, towards the Earth. A relative, short-term carrier frequencies stabilization of these coherent signals was equal to 10−11 , likewise the stabilization of a reference frequency signal f 0 = 4.9996 MHz. A simplified functional diagram of the satellite’s transceiver is shown in Fig. 15.7. Fig. 15.5 TRANSIT satellites; a Transit satellite b Nova satellite

15.2 The Navy Navigation System—TRANSIT

Fig. 15.6 A graphical illustration of satellite orbits of the TRANSIT system

Fig. 15.7 The simplified functional diagram of a satellite’s transceiver

305

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15 Satellite, Doppler Radionavigation Systems

One of the reasons for using two navigation signals with significantly different frequencies ( f 1 / f 2 = 8/3), was compensating the influence of electromagnetic wave refraction, occurring in the ionosphere [2, 6]. As practice confirmed, most users did not use this possibility, limiting themselves to receiving only one signal with a carrier frequency f 1 . The second significant reason was the reliability increase of the navigation signal reception in the environment of various interferences. The receiver of this device received signals of corrective earth stations containing, among others, the on-board clock corrections and information regarding the current position of the satellite, as well as forecast changes for the next 16 h. More information regarding this subject is provided at the end of this chapter when discussing the purpose of the measurement and control terrestrial segment. Both navigation signals, generated by the transmitter, were phase modulated (three-state phase modulation: −60◦ , 0◦ , 60◦ ) with a digital signal containing information about the orbit parameters (operational ephemeris), data enabling determination of a satellite position and its orbital velocity with respect to the Earth (accurate ephemeris) and time markers which synchronize the operation of the entire system, [7, 9]. The mentioned navigation telegram was transmitted in two-minute trains of 6103 binary bits. The “0” and “1” bits, each one with duration equal to 8 · 2.46 ms = 19.68 ms, are assigned to eight phase states, i.e. (−60◦ , 60◦ , 0◦ , 0◦ , 60◦ , −60◦ , 0◦ , 0◦ ) and (60◦ , −60◦ , 0◦ , 0◦ , −60◦ , 60◦ , 0◦ , 00 ), illustrated in Fig. 15.8. The average value of the bit phase “0” and “1” is equal to 0◦ , therefore, random changes in the carrier frequency of the navigation signals have a minimal impact on the information contained therein. Moreover, the phase step modulated signal does not change the bit-period average of the navigation signals’ carrier frequency value. The navigation telegram was transmitted as a matrix consisting of 26 lines and 6 columns, with 39-bit words as elements. These lines were transmitted sequentially one after the other, as shown in Fig. 15.9 As discussed before, the total duration of the navigation telegram was equal to TM = 6103 · 19.68 ms = 120107.04 ms, i.e. approximately in 2 min. Such transmission protocol contained 156 words and 19 final bits, Fig. 15.9. The navigation telegram began and ended at time instants that were specifying even minutes, hereinafter referred to as a markers ti and ti+1 . The last 25 bits formed a word

Fig. 15.8 Signals with the phase modulation, bit (1) and bit (0), used to transfer data from the satellites

15.2 The Navy Navigation System—TRANSIT

307

Fig. 15.9 The data format

(0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0) which was identifying the time marker and synchronizing the beginning of the next two-minute navigation telegram. The orbit parameters, listed in Fig. 15.9, were contained in the initial 22 words of the 6th column, while the words in lines from 9 to 22 of this column were changed, if new information was entered into the memory of the transceiver shown in Fig. 15.7, [2, 10]. The information regarding navigation transmitted by the satellite and the Doppler frequency measured by the user’s receiver, allows determining the receiver’s position with respect to the geocentric coordinate system. For this purpose, the TRANSIT system uses different description of the Doppler frequency dependence from the

308

15 Satellite, Doppler Radionavigation Systems (S)

Fig. 15.10 The satellite t1 and terrestrial t (E) time scales shifted by an interval δt (E)

receiver’s position (with respect to the satellite) than the description expressed by Eqs. (15.4) or (15.5). However, the significance of this description corresponds to determining (counting) the, so-called, integral Doppler value, the determining method of which is discussed as follows. In order to facilitate understanding, assume that the clocks of a satellite and a user’s stationary receiver, measure equal time units, whereby the terrestrial time scale t (E) can be shifted with respect to the satellite time scale t (S) by a fixed interval δt (E) , Fig. 15.10. Assume additionally that at time instants t1(S) and t2(S) the satellite is located at points S1 ≡ (x1(S) , y1(S) , z 1(S) ) and S2 ≡ (x2(S) , y2(S) , z 2(S) ), respectively. The satellite flight from point S1 to point S2 , in time of t1(S) − t2(S) , is shown in Fig. 15.11. At the same time instant, i.e. t1(S) −t2(S) , the satellite sends a navigation signal which reaches the receiver located at point P ≡ (x, y, z) with a certain delay and is received by the user’s receiver at the time interval t1(E) − t2(E) , for t1(E) = t1(S) + δt (E) + R1 /c, t2(E) = t2(S) + δt (E) + R2 /c and c ≈ 2.998 · 108 m/s. The carrier frequency of the signal received from a satellite (which is phase-modulated), changes when measuring t1(E) − t2(E) , due to the satellite movement (Doppler effect). Thus, this satellite must be recreated in the receiver system. The implementation of this task is described in Appendices I and J. A signal u P (t (E) ) ≡ u P (t) with a carrier frequency f P (t (E) ) ≡ f P (t) = f S [1 + ϑr (t)/c] for f S ≡ f 1 = 399.968 MHz or f S ≡ f 2 = 149.988 MHz, see Fig. 15.2 and Fig. 15.7., is obtained due to recreation of a satellite at the reception point P. To ensure the uniqueness of further considerations, assume that f S = f 1 . Fig. 15.11 A flight of satellite from point S1 to (S) (S) point S2 , in time of t1 − t2

15.2 The Navy Navigation System—TRANSIT

309

Fig. 15.12 A typical waveform of the Doppler frequency signal u D (t (E) )

When the signal u P (t) is multiplied by the harmonic reference signal (the receiver’s first heterodyne) of a frequency f h1 = 400 MHz and a low-pass filter, the signal u D (t) with the differential frequency, is obtained f D (t) = f h1 − f P (t)

(15.6)

From a mathematical viewpoint, frequency (15.6) is a plus in the range of radio visibility because the maximum Doppler frequency shift of the received signal is of the order 8.3 kHz, see Appendix J. The sum of this shift and the carrier frequency f 1 = 399.968 MHz is always less than f h1 = 400 MHz. Consequently, there are no saddle points in the signal waveform u D (t (E) ), similar to those shown in Fig. 1.18. A typical waveform u D (t (E) ) in the range .t (E) = t2(E) −t1(E) is shown in Fig. 15.12. Considering signal shown in Fig. 15.12, the integral Doppler value is obtained (E)

(E)

.t2 N=

[ f h1 − f P (t)]dt = f h1 t1(E)

(

(E)

.t2

t1(E)

)

.t2

1 · dt −

f P (t)dt =

t1(E)

(15.7)

(E)

= f h1 t2(S) − t1(S) + f h1 (R2 /c − R1 /c) −

.t2

f P (t)dt

t1(E)

The Doppler value (15.7) is equal to the number of signal cycles u D (t (E) ) in the time slot .t (E) = t2(E) ÷ t1(E) , [2, 7, 9]. This number is determined by counting the positive zero crossings u D (t (E) ), which are marked with circles in Fig. 15.12. The (E) t. t.2(S) 2 f P (t)dt occurring in (15.7) may be replaced by f S dt = f S (t2(S) − t1(s) ), integral t1(E)

t1(S)

for f S ≡ f 1 corresponding to a constant carrier frequency of a navigation signal, emitted through the satellite’s antenna. The acceptability of such “substitution” can be demonstrated as follows. In time instant t2(S) − t1(S) , a satellite transmits a navigation

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15 Satellite, Doppler Radionavigation Systems

signal containing (t2(S) − t1(S) )/TS cycles of an identical duration TS = 1/ f S . The same number of navigation signal cycles reaches the antenna’s receiver only in an altered time, in the range from t1(E) to t2(E) . Naturally, the reason for the time reception shift is the shift in the duration of particular cycles, which is demonstrated by the Doppler effect expressed in time domain. Due to the satellite deflection from the receiver, the duration of particular cycles is extended, which in the frequency domain is expressed by its reduction. As the satellite approaches the receiver, the duration of particular cycles is reduced, which is equivalent to increasing the frequency of the signal received from the satellite. Taking all these conclusions into considerations, the following formula can be obtained ( ) ( ) N = f h1 t2(S) − t1(S) + f h1 (R2 /c − R1 /c) − f S t2(S) − t1(S)

(15.8)

When (15.8) is transformed, one obtains the following equation R 2 − R1 =

( ) ] c [ N + t2(S) − t1(S) ( f S − f h1 ) f h1

(15.9)

In the geocentric coordinate system (x, y, z) /

[x − x1(S) ]2 + [y − y1(S) ]2 + [z − z 1(S) ]2 / R2 = [x − x2(S) ]2 + [y − y2(S) ]2 + [z − z 2(S) ]2 R1 =

(15.10)

When (15.10) is considered in the Eq. (15.9), one obtains / (S) (S) (S) [x − x2 ]2 + [y − y2 ]2 + [z − z 2 ]2 / ] c [ (S) (S) (S) (S) (S) − [x − x1 ]2 + [y − y1 ]2 + [z − z 1 ]2 = N + (t2 − t1 )( f S − f h1 ) f h1

(15.11)

The receiver’s coordinates (x, y, z) correspond to unknown of the Eqs. (15.11), since N is the measured integral Doppler value, and (x1(S) , y1(S) , z 1(S) ), (x2(S) , y2(S) , z 2(S) ) are the satellite coordinates at time instants t1(S) and t2(S) , differing by two minutes, respectively. These coordinates were estimated considering ephemeris data (both, approximate and accurate) transmitted in the navigation telegram, [2, 7]. As mentioned before, f h1 = 400 MHz is the carrier frequency of the reference signal generated in the receiver. Likewise, f S ≡ f 1 = 399.968 MHz is the carrier frequency of the signal emitted by the satellite transmitter, Fig. 15.7. In order to determine three unknowns, which are the coordinates (x, y, z) of the receiver, the system of three independent equations, similar to (15.11), needs to be solved. In the case of a craft located on the surface of a specific sea area, the coordinate “z” is known. Consequently, the formulated task should be solved as a system of two independent

15.2 The Navy Navigation System—TRANSIT

311

Fig. 15.13 The terrestrial and space functional segments of the TRANSIT system

equations, each of which describes a position line as a hyperbola branch. The user’s receiver was located at one of the intersection points of the two defined position lines. Control and measurement terrestrial segment The control and measurement terrestrial segment of TRANSIT navigation system consisted of 4 control stations located in Point Mugu (CA), Prospect Harbor (ME), Rosemont (MN) and Wahiawa (HI); control center, computer center, maritime observation center and timed signal service stations. The control center, computer center and correction stations were located at the system headquarters (placed in Point Mugu), where one out of 4 mentioned control stations, was located as well. The organizational structure of the discussed segment is shown in Fig. 15.13. The main task of the control stations was to measure the Doppler frequency (as a time function) of the signals emitted by each of the 6 satellites. The control center sends the measured frequencies to the computer center, in which the satellite orbits were determined, their changes for the proximate 16 h were forecasted and, most importantly, data on the current position of the satellite in orbit, were corrected. The calculation results were sent to corrective stations, which in turn sent their proper version to particular satellites. The satellite position was possible to be accurately determined at any time instant due to the satellite location data together with simultaneously sent corrections for the on-board clocks. Each satellite received updated data regarding its position and orbit every 12 h.

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15 Satellite, Doppler Radionavigation Systems

Fig. 15.14 Spatial positions of the Transit satelline and related to them position lines, a three positions of the TRANSIT satellite estimated in time intervals (tk−1 − tk ) and (tk − tk+1 ), b the intersection point of the determined position lines, N k and N k+1 , defining the receiver’s position

The user segment Due to a small number of satellites and the arrangement of their orbits’, each receiver of the TRANSIT system received the navigation signal predominantly from one satellite during its radio visibility, with a duration not exceeding 15–17 min. Due to the fact that the satellite navigation information was transmitted to the receiver within 2 min, the user could make at most 8 measurements and determine the corresponding position lines. Inherently, TRANSIT system was intended for the navigation of watercrafts. Consequently, solving the navigation issue was facilitated to some extent, since this task could be solved in two-dimensional space. In this case, the solution of the navigation task yielded in determining at least two position lines considering two measured (counted) Doppler values, see formula (15.8), and data obtained from navigation telegrams. These lines were determined considering two Doppler values, for instance N k and N k+1 , estimated in time intervals (tk−1 − tk ) and (tk − tk+1 ), shifted to each other by a multiple of 2 min. One of the intersection points of the determined position lines, defined the receiver’s position, as shown in Fig. 15.14b. In 1970s which indicates the initial period of TRANSIT system operation, the significantly large geometric dimensions and weight of digital computers inhibited their installation on submarine’s boards, not to mention aircrafts. Therefore, an external (ground-based) computer center was responsible for solving the proper system of equations. In order to accomplish this task, the necessary parameters were transmitted via radio to the computer center, see Eq. (15.11). For the security reasons, all transmitted data was encoded. As a general rule, after less than 15 min, an encoded solution was obtained via the same radio. Each satellite of TRANSIT system transmitted navigation information using two signals of carrier frequencies f 1 ≈ 150 MHz and i f 2 ≈ 400 MHz, see Fig. 15.7. Therefore, electronics companies produced both single-channel and two-channel receivers. A general view of the AN/WRN-5 dual-channel navigation receiver used on American nuclear powered submarines is shown in Fig. 15.15, [10].

References

313

Fig. 15.15 A general view of AN/WRN–5 receiver used in military applications of TRANST

However, due to the Doppler frequency shift being directly proportional to the carrier frequency of the emitted signal, the navigation information was most frequently transmitted in 400 MHz band. Additionally, the 150 MHz carrier frequency signal was used to transmit information regarding the ionosphere condition and the associated propagation correction. Over the years, the operation of TRANSIT system confirmed that a global coverage (all over the globe), regardless of weather conditions, was provided. The stationary object’s position determining error was lesser than 0.1 NM. In the case of moving objects (for example, large and difficult to stop vessels), this error increased to approximately 0.3 NM, which is completely acceptable in sea navigation. The fundamental disadvantage of TRANSIT system was the inability to make the measurement at any time instant. Due to the small number of satellites, the intervals between the radio visibility periods of the satellites above the user’s horizon, were up to 100 min. In other words, the measurement could only be made when the satellite was above the horizon, see Fig. 15.2. Such conductive situation occurred only 13 times a day (more precisely 24 · 60/107 ≈ 13.45) and at least three measurements had to be made within 16 min.

References 1. Curov EP (1977) Satellite radionavigation systems (in Russian). Publishing house “Soviet radio”, Moskwa 2. Forssell B (2008) Radionavigation systems. Artech House, Inc., Boston/London 3. Global Navigation Satellite System GLONASS. http://russianspacesystems.ru/wp-content/upl oads/2016/08/ICD_GLONASS_eng_v5.1.pdf 4. Guier WH, Weiffenbach GC (1960) A satellite Doppler navigation system. Proc. IRE 48(4) 5. Hoffmann-Wellenhof B et al (1994) Global positioning system, theory and practice, third edition. Springer, New York 6. Holejko K (1987) Precise electronic measurements of distances and angles; second edition (in Polish). WNT, Warsaw

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7. Lamparski J (1980) Mathematical foundations of the TRANSIT radionavigation system (in Polish). Adv Astronaut 12(1):79–108 8. Langley RB (2017) Innovation: GLONASS—past, present and future. GPS World, Nov 1. http://gpsworld.com/innovation-glonass-past-present-and-future/ 9. Stansell TA (1971) Transit, The navy navigation satellite system. Navigation 18(1). Spring 10. Stansell TA (1983) The Transit satellite navigation system. Report R-5933A of Magnavox Industrial Company 11. Teunissen PJ, Montebruck O (eds) (2017) Springer handbook of global navigation satellite systems. Springer International Publishing AG, Switzerland

Chapter 16

Satellite Navigation Systems

Satellite radionavigation systems are most frequently associated with the term GPS, which is an abbreviated name of the commonly used GPS—NAVSTAR system (Global Positioning System—Navigation Signal Timing and Ranging) [1, 10, 17, 43]. Despite an unlimited access to its Standard Positioning Service (SPS), knowledge of the GPS organizational structure, principles of operation and mathematical methods of processing the measured pseudo-ranges is not common. Moreover, information on this subject is rather limited and difficult to understand, considering numerous sources, based mainly on the Internet content. Therefore, the main, didactic aim of this chapter is to present the mentioned issues as clearly as possible. A standard navigation scenario was considered as an example, in which the receiver’s position is determined considering signals (data) received from four satellites. Furthermore, the analytical method algorithm of solving the system of four nonlinear equations, characteristic for GPS, is also discussed.

16.1 GPS—NAVSTAR System The division of the GPS system into three subsystems, namely the Space Segment (SS), Control Segment (CS) and the User Segment (US), does not require a more detailed demonstration. Such situation is a result of a separate organizational structure, different purpose and different principles of operation of these subsystems. Therefore, this division has been included in the present chapter, which discusses the organizational structure and principle of operation of the entire GPS system. Space Segment The space segment of the GPS system nominally consists of 24 active satellites, orbiting in 6 almost circular orbits of radii equal to Ro ≈ 26561.75 km. These radii are approximately 4.165 times longer than the Earth’s radius equal to R E = 6378.137 km. For this reason, the orbits used are referred to as high Earth orbits. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_16

315

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16 Satellite Navigation Systems

All orbits, marked with letters A, B, C, D, E and F, see Fig. 16.1a, are inclined with respect to the plane (x0y) of the celestial equator at an angle 55°, while the angle ◦ shift between two consecutive orbits at the equator is equal to 60 . On each of the 6 orbits, 4 satellites orbit at a linear velocity ϑ ≈ 3870 m/s, shifted with respect to each other by an angle 90°. A general image of different satellite generations is shown in Fig. 16.2. The orbital period of each satellite circling around the Earth is 11 h 57 min 58.3 s [10, 17, 30]. The angle shift of the satellites with respect to the satellites located on the nearest, neighboring orbit, equals approximately 40°. The space position of the GPS orbits in relation to the orbits of other similar systems is illustrated in Fig. 16.1b. Due to such constellation, at any time from any point on, or above the surface of the Earth, at least 4 satellites are visible, while their elevation angles above the horizontal plane are not lesser than 5°. The previously mentioned number of 24 active satellites is a nominal value which may be changed as the system is modernized and its functions expand. Therefore, as already assumed at the project stage, the number of satellites can be increased to 32 in order to ensure a greater accessibility in some polar regions and to increase its reliability. The reliability is also enhanced

Fig. 16.1 The space segment of GPS; a 6 almost circular orbits, b orbits of GPS versus BeiDou, GALILEO and GLONASS

Fig. 16.2 A general image of satellites of different generations

16.1 GPS—NAVSTAR System

317

by anti-radiation shields that protect satellites against ionizing cosmic radiation and possible strong laser radiation. Moreover, GPS satellites account for a platform for other special subsystems, including the Integrated Operational Nuclear Detonation Detection System (IONDDS) subsystem, which is used to detect nuclear explosions [10, 37, 43]. For this purpose, the L 3 signal with a carrier frequency f 3 = 135 · 10.23 MHz = 1381.05 MHz, is implemented. Another example is the subsystem informing about transports threats (mainly concerning by air), using the “saving lives” signal with a carrier frequency f 5 = 115 · 10.23 MHz = 1176.45 MHz [10, 13]. From the user’s viewpoint, GPS is a passive system that enables only the reception of signals emitted by particular satellites. Each of the satellites sends two microwave navigation signals towards the Earth, i.e. L 1 and L 2 , which are expressed by the following equations: L 1 : u 1 (t) = A P · Pi (t) · Di (t) cos(2π f 1 t) + A G · G i (t) · Di (t) sin(2π f 1 t) L 2 : u 2 (t) = B P · Pi (t) · Di (t) cos(2π f 2 t) (16.1) for: A P , A G i B P —the amplitudes of carrier waves f 1 = 154 · 10.23 MHz = 1575.42 MHz, f 2 = 120 · 10.23 MHz = 1227.60 MHz Di (t)—binary (−1, 1) code containing the navigation information, G i (t), Pi (t)—binary (−1, 1) pseudo-random modulating signals commonly referred to as access codes for unauthorized and authorized users. The signal strengths (16.1) of the first generation satellites emitted by the transmitters were minor and equaled approximately to 6 W. The code G i (t) is a binary code which enables to identify the ith satellite and a rough measurement of the pseudo-range between the identified satellite and the user receiver. In the English-language literature this code is referred to as a Coarse Acquisition (C/ A). Codes G i (t), for i = 1–32, which are different for different satellites, were selected from the group of orthogonal (statistically independent) Gold codes [1, 9, 10]. The clock length of each G i (t) code is equal to .TG = 10/10.23 µs. Each G i (t) code consists of 1023 clocks, thus its total length (the repetition period) is equal to TG = 1023.TG = 1 ms. The GPS system contains 37 codes, 32 codes are assigned to the space segment and the remaining 5 codes are a backup for ground radio transmitters transmitting GPS signals. The main task of these transmitters is to improve the system accessibility in selected (limited) areas, for example during armed conflicts. As mentioned previously, the code G i (t) is explicitly assigned to the ith satellite, therefore identifies this satellite. Secondly, taking the information contained in the ith navigation message transmitted by the code Di (t) into consideration, the code G i (t) enables the SPS service to be used by an unlimited number of unauthorized users (mainly civils). The aim of the Precise Code Pi (t), which is binary (−1, 1) and pseudo-random as well, is to measure pseudo-range between the user receiver and the ith satellite,

318

16 Satellite Navigation Systems

for i = 1 ÷ 32, more accurately. The codes clock length Pi (t) is ten times shorter than the codes clock G i (t) and is equal to .TP = 1/10.23 µs. Each code Pi (t) consists of N = 6.187104003 × 1012 clocks, so the overall length of each code Pi (t) is equal to TP = N .TP = 604800 s, which corresponds to 7 days. According to [1, 17], the codes Pi (t), for i = 1–38, are different, 7-days fragments of the overall code P(t), with a period equal to 266 days, 9 h, 45 min and 55.4 s. The 32 fragments of the code P(t) have been reserved for the space segment of the GPS system, which means that the remaining 6 fragments can be used for other, special purposes. By design, the code P(t) is intended for military purposes. Therefore, in the beginning of February 1994, it was encrypted by adding a confidential code W (t). As a result, the encrypted code Y (t) was developed, which was available on Precise Positioning Service (PPS) for authorized users only. The discussed pseudo-random (PRN) access codes, i.e. G i (t) and Pi (t), are timesynchronized (.TG = 10.TP ), which facilitates their reset at the beginning of each satellite week, namely at midnight when changing from Saturday to Sunday. Equally significant is the fact that codes G i (t) and Pi (t) do not interfere with each other, so they can be transmitted simultaneously using the same carrier frequency f 1 = 1575.42 MHz. One should also clarify that the term “pseudo-random”, used in this chapter, does not indicate codes of a random nature. On the contrary, these codes are generated according to a strictly defined mathematical algorithm, the description of which is discussed in more advanced literature, e.g. [9, 10, 17]. Thus, the pseudorandom codes are only apparently similar to Random Codes. The binary (−1, 1) code Di (t) occurring in signals L 1 and L 2 transfers the navigation information from the ith satellite to the user receiver. This code, transmitted at a speed of 50 bits per second, consists of 25 pulse sequences (frames) of 1500 bits length each. The transmission duration of the complete navigation information contained in the 37,500-bit code Di (t), is equal to 750 s. Each ith satellite transmits its own navigation information (navigation message), containing primarily: • ephemeris data which enable to estimate the code’s coordinates (xi , yi , z i ) at instant Tn(i) of sending the access signal (according to a watch time of ith satellite), • time Tn(i) of transferring the access signal, • a correction δTs(i) which specifies a watch time difference of ith satellite with respect to the GPS space segment, • a propagation correction considering changes in the speed of electromagnetic wave (TEM) propagation, primarily in the dynamically evolving ionosphere. These data is updated every four hours. The navigation message also sends information regarding satellite health status and their approximate locations in the celestial sphere, also referred to as almanac. In the GPS system, the Nonstandard Codes C/A (N SC) and P(N SC) may be used also, and transmitted variably in place of codes G i (t) and Pi (t). As a general rule, these codes are used in the case of any interferences in the generation process of both system codes. Many years of experience gained during the operation of the GPS system confirms the correctness of Albert Einstein’s theory of relativity which indicates the necessity

16.1 GPS—NAVSTAR System

319

Table 16.1 The main components and their locations of the GPS ground segment of control Location

Equipment

Latitude

Longitude

Colorado Springs

MCS, MS

39°, N

105°, W

Cape Canaveral

MS, SGLS

28°, N

81°, W

Ascension Island

MS, SGLS

8°, S

14°, W

Diego Garcia

MS, SGLS

7°, S

72°, E

Kwajalein

MS, SGLS

9°, N

168°, E

Hawaii

MS

21°, N

1585°, W

of taking into consideration the relativistic effects related to the extension of a clock signal on the satellite and the change of the interaction with the Earth’s gravitational field. In the GPS system, the impact of these effects is compensated by reducing the frequency of the satellite pilot oscillator by 0.00455 Hz, i.e. to the value equal to 10.22999999545 MHz, instead of 10.23 MHz [17]. Control Segment The control segment of the GPS system consists of the Master Control Station (MCS), which is a command center, four SGLS radio links and six monitor stations (MS) that track the flight paths of the satellites. These stations are located in Colorado Springs, Hawaii, Kwajalein Atoll, Diego Garcia Island and Ascension Island [1, 10, 39]. The MCS command center is located at Schriever Air Force Base, near of Colorado Springs. The coordinates of the antennas phase centers of the monitor stations, see Table 16.1, are precisely specified in the WGS-84 (World Geodetic System), since accounting for a reference point for the determined coordinates (xi , yi , z i ) of the monitored (tracked) satellites. An equally important task of monitor stations is the clocks synchronization of all satellites by emission of proper time markers produced in their immensely accurate and mutually synchronized clocks. Reflector antennas of MS stations and SGLS radio links (with a power gain G > 40 dB) are shielded with dielectric domes in order to protect them against icing, heavy winds and pollution. The displacement of directivity pattern of antenna this type, is most frequently implemented by its rotation in the horizontal plane and by a proper tilt in the elevation plane [25, 26]. Information regarding location of each of the satellites is updated once a day by the MCS and transmitted over SGLS radio links, operating in the range of 3 GHz. The satellite receives a signal through the corrugated conical horn antenna, which is insensitive to polarization changes of the incoming wave, see Fig. 16.2. Users Receivers The user segment of the GPS system includes an unlimited number of multi-channel (4 ≤ n) users’ receivers, which are usually divided into single-band and dual-band receivers. Obviously, single-band ones receive only the L 1 signal with the effective pulse bandwidth equal to 2.046 MHz. As a general rule, these receivers do not consist

320

16 Satellite Navigation Systems

of codes Pi (t) or Yi (t) decoders, and are intended for unauthorized users. The publicly available literature shows that these receivers provide the accuracy of determining the position of the order of 100 m in the horizontal plane and 156 m in the elevation plane. This accuracy appears to be sufficient for the majority of civilian users. In the case of dual-band receivers that decode the Pi (t) or Yi (t) code, a significantly increased accuracy in determining the position of the order of 16 m in the horizontal plane and 23 m in the elevation plane respectively, is obtained [1, 17]. Undoubtedly, the length of a clock code Pi (t) has a decisive influence on the accuracy improvement, which is ten times shorter than clock .TG of code G i (t). In order to clarify the physical nature of this influence, the principle of pseudo-range measurement, using of both codes, i.e. G i (t) and Pi (t), should be discussed. Thus, assume that the receiver receives both L 1 signals, from which the code G i (t) ≡ C/ A is derived. At the same time, replicas of the codes G i (t) are generated successively in the user receiver. Each code replica is gradually delayed by clock .TG and thus compared with the code G i (t) ≡ C/A of the received signal. If synchronism (the exact time-domain overlap) of the received code and the compared replica is not obtained over the entire range of time shifts (0 ≤ k · .TG ≤ TG ), a replica of the successive code G i (t) ≡ C/ A is generated and synchronization is repeated from the beginning. In the case of the received code’s synchronism and shifted replica (Fig. 16.3), the autocorrelation function has the maximum value. Thereby, the identification information is obtained which uniquely indicates a satellite from which the signal is received. The replica delay τ , which is synchronized, is considered as the signal reception time t O(i ) according to the receiver time scale t. Thus, the function Rk (t − τ ) should have only one “explicit” extremum with the greatest possible value. This condition is satisfied by the function Rk (t − τ ) with the course as in Fig. 16.4.

Fig. 16.3 Comparing the G i (t) code with its replica generated by a GPS receiver

16.1 GPS—NAVSTAR System

321

Fig. 16.4 A typical plot of correlation function Rk (t − τ )/Rkmax calculated for the G i (t) code and its replica

When the satellite is identified, the navigation message Di (t) is decoded containing information regarding time TN(i ) of the access signal transmission (according to the ith satellite time T (i ) ) and the shift δTS(i ) of the ith satellite time scale with respect to the GPS system time T (G P S) (Fig. 16.5).

Fig. 16.5 The time scales of user receiver t, GPS system T (G P S) and ith satellite T (i)

322

16 Satellite Navigation Systems

Considering TN(i) , δTS(i ) and the reception time t O(i) measured by the receiver, a pseudo-range di∗ = c [t O(i) − (TN(i ) + δTS(i) )] is estimated, for c = 299792458 m/s corresponding to the nominal velocity of the electromagnetic wave propagation in the atmosphere. This velocity may be adjusted respectively to the ionospheric correction transmitted in the navigation message. Currently, an additional band L 4 with a mid-band frequency f 4 = 1379.913 MHz has been separated in order to transmit additional information regarding the ionospheric (propagation) correction [39, 43]. If the user receiver is equipped with a precise code Pi (t) decoder, then once a satellite is identified and a rough measurement is determined (with code G i (t) ≡ C/A), the pseudo-range determination is repeated with a precise code Pi (t). The pseudo-ranges which are determined to four different satellites facilitate formulating a system of equations with unknown coordinates (x, y, z) of the receiver and the time shift δt O . The task of formulating and solving this system of equations is discussed in the next section.

16.1.1 A System of Four Stadiometric Equations Further considerations assume that the coordinates (xi , yi , z i ) of the satellites Si are known at time instants TN(i ) , which are determined according to the indications of satellite clocks, where i = 0, 1, 2, 3. Signals emitted by these satellites are received at time instants t O(i) , which are estimated by the receiver’s time t. The receiver time is shifted with respect to the GPS satellite time by an interval δt O , see Fig. 16.5. Assume that the receiver is located at point P with coordinates (x, y, z), as shown in Fig. 16.6. The indicated coordinates of the ith satellite are equal to (xi , yi , z i ). Figure 16.5 demonstrates that the transmission TN(i ) and reception t O(i ) times of the satellite signal, recorded as the satellite T (G P S) time scale, are equal to TN(i, G P S) = TN(i ) + δTS(i) and TO(i, G P S) = t O(i) + δt O , respectively. Therefore, .T (i, G P S) = TO(i, G P S) − TN(i, G P S) = t O(i ) − (TN(i) + δTS(i) ) + δt O

(16.2)

The component (TN(i ) + δTS(i ) ) in formula (16.2) is a known value because the time instant TN(i ) and correction δTS(i) are transmitted in the ith satellite’s navigation message. As shown in Fig. 16.6, the distances between point P ≡ (x, y, z) and particular satellites S i , are equal to di =

.

(x − xi )2 + (y − yi )2 + (z − z i )2 = c.T (i, G P S)

(16.3)

for i = 0, 1, 2, 3. Subject to (16.3), a following system of four independent equations is possible to be formulated.

16.1 GPS—NAVSTAR System

323

Fig. 16.6 A distance to the ith satellite

. (x − x0 )2 + (y − y0 )2 + (z − z 0 )2 − cδt O = c.T (0, G P S) − cδt O = c [t O(0) − (TN(0) + δTS(0) )] . (x − x1 )2 + (y − y1 )2 + (z − z 1 )2 − cδt O = c.T (1, G P S) − cδt O = c [t O(1) − (TN(1) + δTS(1) )] . (x − x2 )2 + (y − y2 )2 + (z − z 2 )2 − cδt O = c.T (2, G P S) − cδt O = c [t O(2) − (TN(2) + δTS(2) )] . (x − x3 )2 + (y − y3 )2 + (z − z 3 )2 − cδt O = c.T (3, G P S) − cδt O = c [t O(3) − (TN(3) + δTS(3) )]

(16.4)

The right hand side terms of these equations represent the estimated pseudo-ranges di∗ = c [t O(i) − (TN(i) +δTS(i ) )], for i = 0, 1, 2, 3. Three coordinates (x, y, z) of point P and shift δt O indicate the system (16.4) unknowns. This non-linear system can be solved with standard mathematical methods such as the least squares, Newton– Raphson iterative method or gradient optimization methods [3, 6, 20, 21, 25]. The nature of these iterative methods is also discussed in Chap. 2. However, due to the specificity of the system (16.4), a sought solution (x, y, z, δt O ) can be derived by a simple analytical method described in the example that follows. Considering (16.3), the differences of the squares distance di2 and d02 are formulated, i.e.

324

16 Satellite Navigation Systems

di2 − d02 = (x − xi )2 + (y − yi )2 + (z − z i )2 − (x − x0 )2 − (y − y0 )2 − (z − z 0 )2 = x2(x0 − xi ) + y2(y0 − yi ) + z2(z 0 − z i ) − (x02 + y02 + z 02 ) + (xi2 + yi2 + z i2 ) = (d i − d0 )(d i + d0 ) = .di (di + d0 ) = .di (.di + 2d0 )

(16.5)

for .di = di − d0 and i = 1, 2, 3. The system of three Eqs. (16.5) can be expressed in the following matrix form [A] [X ] = [B]

(16.6)

for: ⎤ ⎡ ⎤ x x0 − x1 y0 − y1 z 0 − z 1 [X ] = ⎣ y ⎦ [A] =⎣ x0 − x2 y0 − y 2 z 0 − z 2 ⎦, x0 − x3 y0 − y3 z 0 − z 3 z ⎤ ⎤ ⎡ ⎡ 2 2 2 2 2 d0 .d 1 + k 1 2d .d + .d1 + (x0 + y0 + z 0 ) − (x1 + y12 + z 12 ) 1⎣ 0 1 [B] = 2d0 .d2 + .d22 + (x02 + y02 + z 02 ) − (x22 + y22 + z 22 ) ⎦ = ⎣ d0 .d 2 + k 2 ⎦ 2 d0 .d 3 + k 3 2d0 .d3 + .d32 + (x02 + y02 + z 02 ) − (x32 + y32 + z 32 ) ] 1[ 2 ki = .di + (x02 + y02 + z 02 ) − (xi2 + yi2 + z i2 ) where i = 1, 2, 3 2 ⎡

The differences .di = di∗ − d0∗ are estimated considering the measured pseudoranges di∗ = c [t O(i ) − (TN(i) + δTS(i) )], for i = 0,1, 2, 3. The system of Eq. (16.6) can be solved by the inverse matrix method with respect to the coefficients matrix [A]. Assume that the inverse matrix with respect to matrix [A] is known, and is indicated as follows ⎡ ⎤ a11 a12 a13 (16.7) [A] −1 = ⎣ a21 a22 a23 ⎦ a31 a32 a33 The formulae which enable estimating the elements ai j of this matrix, considering the elements of a known matrix [A], are given in Appendix B. Due to a premultiplication of both sides of the Eq. (16.6) by the matrix (16.7), one obtains [A] −1 [A] [X ] = [X ] = [A] −1 [B]

(16.8)

In order to provide a clarity of further considerations, Eq. (16.8) should be expressed in the following expanded form

16.1 GPS—NAVSTAR System

325

⎡ ⎤ ⎡ ⎤⎡ ⎤ x d0 .d 1 + k 1 a11 a12 a13 ⎣ y ⎦ = ⎣ a21 a22 a23 ⎦ ⎣ d0 .d 2 + k 2 ⎦ a31 a32 a33 d0 .d 3 + k 3 z

(16.9)

Subject to (16.9), one can conclude as follows x = a 11 (d0 .d1 + k1 ) + a 12 (d0 .d2 + k2 ) + a 13 (d0 .d3 + k3 ) = d0 n 1 + m 1 y = a 21 (d0 .d1 + k1 ) + a 22 (d0 .d2 + k2 ) + a 23 (d0 .d3 + k3 ) = d0 n 2 + m 2 z = a 31 (d0 .d1 + k1 ) + a 32 (d0 .d2 + k2 ) + a 33 (d0 .d3 + k3 ) = d0 n 3 + m 3 (16.10) for: mi =

3 . j=1

ai j k j ,

ni =

3 .

ai j .d j

where i = 1, 2, 3

j=1

After considering (16.10) in the equation d02 = (x − x0 )2 + (y − y0 )2 + (z − z 0 )2 , the following formula is obtained d02 = [(m 1 − x0 ) + d0 n 1 ]2 + [(m 2 − y0 ) + d0 n 2 ] 2 + [(m 3 − z 0 ) + d0 n 3 ] 2 (16.11) Equation (16.11), after elementary multiplications and components grouping, can be expressed as a standard quadratic equation αd02 + βd0 + γ = 0

(16.12)

for: α = n 21 + n 22 + n 23 − 1, β = 2[(m 1 − x0 )n 1 + (m 2 − y0 )n 2 + (m 3 − z 0 )n 3 ] γ = (m 1 − x0 )2 + (m 2 − y0 )2 + (m 3 − z 0 )2 . Solving this equation, the root d0 ≡ d0(−) = (−β − β 2 − 4αγ )/(2α) is determined, and then the coordinates (x, y, z) of the point P are estimated according to the formulae (16.10). The discussed algorithm has been implemented in the GPSXYZ computer program. It is not difficult to demonstrate that the Eq. (16.12), formulated for a real naviga. (−) 2 − 4αγ )/(2α) and β d = (−β − tion scenario, should contain two real roots 0 . d0(+) = (−β + β 2 − 4αγ ) /(2α), which result in the following two solutions, i.e. (x (−) , y (−) , z (−) ) and ( x (+) , y (+) , z (+) ). The solution (x, y, z), which is proper for the considered navigation scenario, can be implemented according to the criterion of proximity with respect to the Earth midpoint, see Fig. 16.6. According to this criterion, the coordinates of the receiver’s position, located on or close to the Earth’s surface, should satisfy the following condition

326

16 Satellite Navigation Systems

. (x − 0)2 + (y − 0)2 + (z − 0)2 ≈ R E = 6378137 m

(16.13)

In this situation, one of the solutions (x (−) , y (−) , z (−) ) or ( x (+) , y (+) , z (+) ) which does not satisfy the condition (16.13), is considered as incorrect and should be omitted.

16.1.2 Some Results of Simulation Calculations Example 16.1 In order to confirm the correctness of the mathematical formulae derived in the previous paragraph, a series of simulation calculations were performed using real positions of four satellites, which were published in [3]. The coordinates (xi , yi , z i ) of these satellites are given in the second, third and fourth columns of Table 16.2, respectively. On the other hand, the fifth column shows the delay times corresponding to the measured pseudo-ranges di∗ = c · [t O(i) − (TN(i) + δTS(i ) )]. Considering data in Table 16.2, the coordinates x = 1, 111, 589.20 m, y = − 4, 348, 254.52 m, z = 4, 527, 349.91 m of point P, e.g. the receiver position, were derived as a result of solving the system of Eq. (16.3). The calculations were The spherical coordinates of this performed assuming that c = 299, 792, 458 m/s.. position, defined in Fig. 16.7, are equal to: r = x.2 + y 2 + z 2 = 6, 374, 943 m, ◦ ◦ ϕ = arctg(y/x) = 75 39' 36' W and γ = arctg(z/ x 2 + y 2 ) = 45 14' 58'' N. In order to estimate the fourth unknown of the system (16.3), which corresponds to interval δt O = −0.23 s, the indirectly determined distance d0 = 24, 310, 666.56 m and pseudo-delay d0∗ /c = 0.081091654 s were implemented. The obtained results are fully consistent with their counterparts published in [3]. A task of demonstrating the correctness of the following results is not demanding, as the satellite coordinates, receiver position (x, y, z), interval δt O and delays t O(i) − (TN(i) + δTS(i) ) corresponding to particular pseudo-ranges, are known. The solution x = −2, 892, 120.41 m, y = 7, 568, 775.00 m, z = −7, 209, 499.19 m, which does not satisfy Eq. (16.13), corresponds to the root d0(+) of the Eq. (16.12). This solution indicates that a receiver Table 16.2 The coordinates (xi , yi , z i ) of satellites and delay times corresponding to the measured (i )

(i )

(i )

pseudo-ranges di∗ = c · [t O − (TN + δTS )] i

xi , m

(i)

(i )

(i )

yi , m

zi , m

t O − (TN + δTS ), s

0

14,832,308.66

− 20,466,715.89

−7,428,634.75

0.311091641

1

−15,799,854.05

−13,301,129.17

17,133,838.24

0.306434539

2

1,984,818.91

−11,867,672.96

23,716,920.13

0.298809965

3

−12,480,273.19

−23,382,560.53

3,278,472.68

0.308128304

16.1 GPS—NAVSTAR System

327

Fig. 16.7 A coordinate system used

would have . to be spaced from the Earth’s surface by approximately 4476489 m, because x 2 + y 2 + z 2 = 10, 845, 626 m. Example 16.2 Considering the brought computing example, the delays t O(i ) − (TN(i) + δTS(i ) ) are determined with an accuracy up to 10−9 s, i.e. much greater than the accuracy provided by the SPS and PPS services of the GPS system. As speculated, the discussed services enable to measure the time with an accuracy equal to 0.34×10−6 s and 0.1 × 10−6 s, respectively. This significant fact was taken into consideration in the present computing example, in which the position of an aircraft, flying at high altitude, was determined (Fig. 16.8). Assume that a navigator has a hypothetical receiver, capable of measuring time delays with an accuracy up to 10−12 s. Assume additionally that this receiver has received signals from four satellites, the coordinates (xi , yi , z i ) of which are shown in the second, third and fourth columns of Table 16.3.

Fig. 16.8 Determination of position in 3D space by ranging to four satellites

328

16 Satellite Navigation Systems (i )

(i )

(i)

Table 16.3 The coordinates (xi , yi , z i ) of satellites and times t O − (TN + δTS ) determined with two different precission i

xi , m

yi , m

zi , m

(i )

(i)

t O − (TN +

(i)

(i)

t O − (TN +

δTS(i) ), s

δTS(i) ), s

0

15,524,471.17

−16,649,826.22

13,512,272.38

0.070758991542

0.0707590

1

−2,304,058.53

−23,287,906.46

11,917,038.11

0.066951697630

0.0669517

2

16,680,243.36

−3,069,625.56

20,378,551.05

0.078516024575

0.0785160

3

−14,799,931.39

−21,425,358.24

6,069,947.22

0.074922578606

0.0749226

However, the fifth column of Table 16.3 shows time delays t O(i ) − (TN(i) + δTS(i) ) s, which are determined considering these signals. These data corresponds to the position: x ∗ = 731, 123.31 m, y ∗ = −5, 449, 261.13 m, z ∗ = 3, 231, 411.17 m which should be considered as sufficiently accurate for the purpose of further discussion. As mentioned previously, the PPS service ensures the accuracy of time measurement, including delays t O(i ) − (TN(i) + δTS(i) ) of the order of 0.1 × 10−6 s. Therefore, these delays are recorded to the seventh decimal place in the sixth column of Table 16.3. The position of an aircraft, estimated considering these approximate delays, is determined by the coordinates: x = 731, 118.56 m, y = −5, 449, 258.66 m and z = 3, 231, 438.35 m. Therefore, the deviations estimated for this particular case are equal to: .x = |x ∗ − x| = 4.75 m, .y = |y ∗ − y| = 2.47 m and .z = |z ∗ − z| = 27.18 m. Despite the fact that the values of these deviations are close to the permissible ones which are specified for the PPS service, the accuracy measure of the entire system is not sufficiently reliable, due to its complexity and nonstationary nature. According to the adopted assumption, the essential role of these deviations in the discussed example is the exemplification of the impact of pseudorange measurement errors on the accuracy of determining the position of the on-board receiver. As tacitly assumed for both demonstrated computing examples, the satellites coordinates are determined with high accuracy. In fact, both the satellite coordinates (xi , yi , z i ), the time corrections δTS(i) and the delays t O(i) − (TN(i) + δTS(i) ) have various errors that vary in time. Thus, the GPS can be considered as a very complex (multi-parameter) and dynamically changing measurement system, the parameters of which and subsequent indications are mutually correlated. In other words, unlikely as it may seem, the ephemeris data, time corrections δTS(i) and δt O or even the currently determined coordinates (x, y, z) differ significantly from their counterparts, which were determined in the ‘recent past’. Considering the GPS receiver, this “historical” knowledge along with the results of current measurements is subjected to further filtering, in order to eliminate obvious errors, and prediction of the most probable coordinate estimates. Sequential data processing procedure, understood this way, is also referred to as Kalman filtration [17, 39, 43]. Consequently, a receiver gives its coordinates (x, y, z), which are not dependent only on the current measurements, but are estimated (weighted) values—which are considered as the most probable from a statistical viewpoint. Such position (x, y, z) of the plane can be easily assigned to its geographical coordinates (r, ϕ, γ ), see Fig. 16.7. Thus, the flight altitude

16.2 The System GLONASS and Its Functional Segments

329

H = r − R E (ϕ, γ ) can be estimated, for R E (ϕ, γ ) corresponding to the radius length which is calculated considering an ellipsoidal model of the Earth WGS − 84, Appendix I. The beginning of this radius coincides with the point (0, 0, 0) of the adopted coordinate system (x, y, z), see Fig. 16.6. Furthermore, WGS-84 assumes that (0, 0, 0) is a midpoint of the Earth’s mass. Due to an orbital motion of the Earth in the external field of gravitation, this point can be considered as its center of gravity. In the case of an aircraft descending, a flight altitude and descent velocity are estimated simultaneously and continuously with an on-board Continuous Wave Altimeter (CWA), which is marked by lower inertia, higher reliability and provides much greater measurement accuracy. Despite GPS being a global system, many countries and international organizations have developed their own navigational systems, such as GLONASS, Galileo, Bideou, IRNSS and QZSS. Brief descriptions of these satellite systems are provided in Sects. 16.2, 16.3 and 16.4 respectively.

16.2 The System GLONASS and Its Functional Segments The GLONASS developed in Russia mainly for military purposes has a functional structure composed of three fundamental segments, similarly as GPS. Therefore its space segment, ground based segment (so-called ground control complex) and user segment are discussed below in a such order.

16.2.1 The Space Segment and Transmitted Signals The complete space segment of the GLONASS consists of 24 satellites placed in three orbital planes marked by numbers 1, 2 and 3 counting towards Earth rotation. These orbital planes have a nominal inclination of 64.8°, with respect to plane (x0y) of the celestial equator, and are separated by 120° in longitude. There are 8 satellites in each orbital plane evenly distributed, as illustrated in Fig. 16.9a. Accuracy of satellite orientation is not worse than ±1°, but after complete installation of the satellite into his orbital slot. Maximum deviation of a satellite position relative to its nominal position (orbital slot) does not exceed ±5° on the period of lifetime [13, 22, 39, 43]. The radius of the circular satellite orbits is 25510 km. Consequently altitudes of the particular satellities are equal to 19,132 km, approximately. The satellites placed on i + 1 plane (orbit) are shifted by 15° in the argument of latitude compared to the satellites in orbit i, where i = 1, 2. The orbital period is 8/17 of a sidereal day or approximately 11 h 15 min and 44 s of the Universal Time (UT). The constellation of 24 satellites guarantees that at least five satellites are seen simultaneously from 99 percent of the Earth’s surface. Moreover, the entire territory of Russia is sufficiently well illuminated (Fig. 16.10).

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Fig. 16.9 The space segment of the own navigational systems, such as GLONASS; a three orbits of satellities, b a view of the satellite

Fig. 16.10 The coverage area of GLONASS

The Satellite Signal The GLONASS satellites broadcast their signals in two subbands of the L-band, namely L1: (1598.0625–1605.375, MHz) and L2: (1242.9375–1248.625, MHz). The nominal carrier frequency for each satellite may be calculated as follows: m m f (L1) = 1602 + m · 0.5625 MHz, f (L2) = 1246 + m · 0.4375 MHz

(16.14)

where m = 0, 1, 2, . . . , 24 is the frequency channel number contained in the almanac message. There is a constant ratio between the carrier frequencies of signals transmiting m m 0 / f (1) = 7/9. The frequencies f (L1) = 16002 MHz and by the each satellite, i.e. f (2) 0 f (L2) = 1246 MHz of 0—channel are used only for the test purposes. The carrier m m and f (L2) are permanently assigned to the satellite marked with frequencies f (L1)

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331

the number m and make it possible to identify it. In other words, GLONASS uses Frequency Division Multiple Access (FDMA) technology to discriminate the signal and related to it satellite. A GLONASS receiver can separate the signal of a particular satellite from the total incoming signal of all visible satellites by assigning different frequencies to its tracking channels. For each satellite, carrier frequencies of L1 and L2 subbands are coherently derived from a common on board time/frequency standard. The GLONASS satellites are equipped with clocks (time/frequency standards) which daily instability is not worse than 5 × 10−13 and 1 × 10−13 for the GLONASS-M satellites. An accuracy of mutual synchronization of the satellite time scales is not worse then 8 ns for the GLONASS − M satellites. The GLONASS time is generated on a base of GLONASS Central Synchronizer (CS) time. Daily instability of the Central Synchronizer hydrogen clocks in not worse than 2 × 10−15 [13]. The time scales of the GLONASS satellites are periodically compared with the CS time scale. Corrections to each on board time scale relative to GLONASS time and UTC (SU) are computed and uploaded to the satellites twice a day by control segment. The error of a scale system binding of the GLONASS UTC (SU) time scale should not exceed 1 mks. The carrier signals of frequencies (16.19) are modulated by two binary codes called C/A-code, P-code and by the data message. Contrary to the GPS, the P code signal of GLONASS is not encrypted. The signals transmitted in the L1-band are modulated by both types of binary code, whereas the L2 signal only contains the P-code. The C/A-code is generated with a frequency of 0.511 MHz and is available for civil users, for the so-called Standard Precision Navigation. The P-code is modulated with a frequency of 5.11 MHz and is called High Precision Navigation Code. The P-code is not recommended for civil use without authorization of the Russian Space Forces. These P-code may be changed by the Russian Space Forces without prior announcement. Each GLONASS satellite transmits its L1 and L2 signals on slightly different frequencies. As it has been mentioned earlier the GLONASS receiver can separate the signal of a specific satellite from the total incoming signal of all visible satellites by assigning different frequencies to its tracking channels. This technique is called Frequency Division Multiple Access (FDMA). Since there is no need to distinguish between satellites by signal modulation, all GLONASS satellites use the same code for modulation. FDMA is different from the technique used for signals transmitted by GPS satellites. Here it should be pointed out that each GPS satellite modulates its carrier with a different code. The GPS receiver identifies a particular signal by “looking” at the code modulation and discarding all signals with the different code. This technique is called Code Division Multiple Access (CDMA). Therefore, there is no need to use different frequencies in GPS, and all GPS satellites use the same frequency for L1 and L2. The GLONASS broadcast ephemeris describes a position of transmitting antenna phase center of given satellite in the PZ-90.02 reference frame described in Appendix I.

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16.2.2 The Control and User Segment The satellite constellation GLONASS is served by a so-called ground control complex (GCC). It consists of the System Control Center (SCC) in the Moscow Region and several Command Tracking Stations (CTS) located in the vast territory of Russia, among others they a located in Moscow, St. Petersburg, Yenisiejsk (central Siberian), Komsomolsk (Far East), Ternopol (now Ukraine) and Neustrelitz (Germany). The geographical locations of the mentioned stations are shown with small circles in Fig. 16.10. The letter “M” in this drawing marks the Moscow area where the command center of the system is located. The CTSs track alle GLONASS satellites in view and pass the ranging data and satellite messages to the SCC. Ranges to the satellites are measured by radar with a maximum error of between two and three meters [13]. This information is processed by the SCC to determine clock corrections, navigation messages and status information for each satellite. The updated information is transmitted to the CTSs and uploaded to the satellites. The ranges observed by the CTSs are periodically calibrated using a laser device at the Quantum Optical Tracking Stations (QOTS) that are an integral part of the GCC. The GLONASS system time is generated on the basis of the Central Synchronizer. Status information of the GLONASS satellite constellation is also provided by the Coordinational Scientific Information Center (CSIC) of the Russian Ministry of Defence. The user segment consists of an unlimited number of GLONASS receivers. Various types of receivers are commercially available. The GLONASS receivers can be classified according to signal components processed in the same way as for GPS receivers. These are: (a) L1 single frequency receivers, (b) L1 and L2 dual— frequency receivers, (c) C/A-code receivers, and (d) P-code receivers. Unlike GPS antennas, the GLONASS antennas require increased bandwidth to measure different frequencies. For a combined GLONASS/GPS receiver, the antenna band must cover GLONASS and GPS frequencies.

16.3 The System GALILEO and Its Functional Segments Galileo is Europe’s global navigation satellite system, providing a highly accurate, guaranteed global positioning service under civilian control. Galileo is not too different from the modernized GPS and GLONASS. It provide autonomous navigation and positioning services, but at the same time it is interoperable with the two other global satellite navigation systems i.e. the GPS and GLONASS. However, by providing dual frequencies as standard, Galileo deliver real-time positioning accuracy down to the meter range. It will guarantee availability of the service under all, but the most extreme circumstances and inform users within seconds of a failure of any satellite. This feature makes it appropriate for applications where safety is vital. The system Galileo achieved its operational state in year 2018.

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16.3.1 The Space Segment, Transmitted Signals and Services The Galileo functional segments are almost similar to GPS ones, but with some modification. The complete space segment consists of 30 satellites (27 active and 3 spare satellite), distributed over three orbit planes. The altitude of these circular orbits is equal to 23, 222 km and is slightly larger than for GPS. Is means that the Galileo satellites are above the GPS and Glonass satelites. The so called “repeat cycle ” for the Galileo satellite orbits is 10 days. The planes of orbits are inclined at an angle of 56◦ with respect to the celestial equator (Fig. 16.11a). The satellites of Galileo (Fig. 16.11b), are spread evenly on the particular orbits and need approximatelly 14 h and 7 min to orbit the Earth. The such constellation guarantees that at any point on the earth, there will be at least 6 satellites in the view [36, 39, 43]. Galileo Signals and Services The Galileo navigation signals are transmitted in the four frequency bands: E5a, E5b, E6 and E1 indicated in Fig. 16.12. They provide a wide bandwidth for the transmission of the Galileo signals of right-hand circular polarization (RHCP). The such frequency bands have been chosen in order to minimize interference with GPS and GlONASS systems, which operate in the same portion of the RF spectrum. All Galileo satellites will share the same nominal frequency, making use of Code

Fig. 16.11 The space segment of the GALILEO; a three orbits of satellities, b a view of the satellite

Fig. 16.12 Frequency bands used in the GALILEO system

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Division Multiple access (CDMA). Galileo uses a different modulation scheme for its ten navigation signals, namely the binary offset carrier (BOC) and quadrature phase shift keying (QPSK). The Galileo has its own system time, called Galileo System Time (GST) that is a continuous atomic time scale with a nominal constant offset with respect to the International Atomic Time (TAI). GST starts at midnight between Saturday and Sunday [f][d]. GST covers 4096 weeks before it reset to zero. The week has 604800 s and it is rolled over to 0 at the midnight between Saturday and Sunday. The navigation message includes the parameters that are needed for the conversion of GST to UTC and also GPS Time. The Galileo satellite constellation offers the capability of broadcasting globally a set of six signals supporting the open, commercial, safety-of-life and public regulated services. Each navigation signal is composed of one or two ranging codes and navigation data as well as, depending on the signal, integrity, commercial and search and rescue (SAR) data. Satellite to user distance measurements based on ranging codes and data are used in the Galileo user receivers to fulfill the different Galileo services such as. 1. Open service (OS) data aree available to all users and consist mainly of the navigation and SAR data. Open service offers positioning, navigation and timing signals, which can be accessed free of charge. 2. Commercial Service (CS) data are encrypted and provided by service providers that interface with the Galileo Control Centre. Access to those commercial data is provided directly to the users by the service providers. The signal is designed to support very precise local differential applications (submeter accuracy) using the open ( encrypted) signal overlaid with the PRS signal on E6 and also support the integration of Galileo positioning applications and wireless communications networks. 3. Safety of Life Service (SoL) is mainly relate to the integrity of the Galileo system and signal quality in space. In particular, SOL relies on satellite navigation signals without the use of additional information from WAAS and EGNOS. 4. Public Service Regulated Access (PRS) data are provided on dedicated frequencies to ensure greater continuity of service, especially in terms of security. PRS is resistant to interference, jamming and other accidental or deliberate actions.

16.3.2 The Control and User Segments The Galileo ground segment is responsible for managing the constellation of navigation satellites, controlling basic functions of the navigation mission such as orbit determination of satellites, and clock synchronization, and determining and disseminating (via the MEO satellites) the integrity information, such as the warning alerts within time-to-time alarm requirements, at global level. The ground segment consists of 2 base ground control and information processing centers that are located in Oberpfaffenhofen (near Munch, Germany) and Fucino (Italy). In addition the control

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centers are linked to 5 tracking and control stations, 9 C – band uplink stations, and about 40 sensor stations. The user segment consists of different types of receivers, with various capabilities related to the specific Galileo navigation signals. Here it should be pointed out that these signals are obtained by applying two kinds of modulation, namely binary offset carrier (BOC) and quadrature phase shift keying (QPSK). The satellite signal after the reception passes through a “front end” constituted by amplifiers and filters. Next the signal is sampled and converted into phase and quadrature components. These digital components allow the tracking of the carrier signal frequency and the code phase. They are processed also in the receiver for a posterior navigation data extraction and pseudodistances computation. These pseudodistances are filtered in the navigation system and originate data: a position, velocity and time estimation desired by the user.

16.4 Other Stadiometric Satellite Navigation Systems It is not difficult to notice that the discussed satellite navigation systems have a similar functional structure. These systems are composed of three basic segments, which are the space segment, the terrestrial monitoring and control segment, and the user segment. Principles of operation of them are also similar [43, 39]. These systems differ mainly in the spatial constellation of satellites and the navigation signals used. The above remark also applies to other similar systems such as BeiDou (Compass), IRNSS and QZSS. Therefore, the following considerations are limited to discussing only their characteristic features relating to the spatial constellation of satellites and the signals used.

16.4.1 The Chinese Navigation System BeiDou The BeiDou Satellite Navigation System (BDS) was developed and built in line with a “three - stage” strategy icluding successively upgraded versions BDS-1, BDS-2 and BDS-3. The second version of BDS is also known as COMPASS. The construction of BDS-3 started in 2009 and was completed in 2020 with the launch of the last GEO satellite. Thus, from June 2020 the space segment of BDS is fully completed and can to provide navigation services. It currently consists of 5 Geostationary Earth Orbit (GEO) satellites, 27 Medium Earth Orbit (MEO) satellites and 3 satellites (IGSO) on Inclined Geosynchronous Orbit. Thus, the total number of active satellites is 35 (Fig. 16.13a). The GEO satellites are operating in orbit at an altitude of 35786 m and positioned at 58.75, 80, 110.5, 140 and 160° E, while two non-active spare satellites at 84 and 144.5° E. The 27 MEO satellites are place in three orbital planes, 9 per plane, inclined approximately 55° relative to the Equator. The heighs (altitudes) of their

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Fig. 16.13 The BeiDou Satellite Navigation System; a the orbits of satellities, b a view of the satellite

orbits are about 21528 m. The 3 IGSO satellites are placed in the orbital plane inclined approximately 55° to the Equator and the crossing longitude is 118° E [4, 5, 39]. The BeiDou satellites transmit signals on three frequency bands (B1, B2, B3), which share common features with GPS and Galileo, e.g., CDMA. The B1, B2 and B3, frequency bands are close to the GPS L1, the GPS L5 and the Galileo E6 frequency, respectively. The nominal values of carrier frequencies of signals transmited on these bands (B1, B2, B3), are equl to 1561.098 MHz, 1207.14 MHz and 1268.52 MHz, respectively. The modern rubidium clocks applied in BeiDou are competitive with older generations of GPS frequency standards and rubidium clocks instalated on the GIOVE-A/B satellites of Galileo. The ground control segment of BeiDou consists of master control station (MCS) at Beijing and many separated monitor stations (MS) located mainly on the territory of China. Unlike other systems, the ranging is carried out through the bidirectional link by measuring the time taken for the signal to reach the receiver and then return to the satellite. The user segment includes all BeiDou receivers and the appropriate supporting equipment. Each receiver can receive navigational signals from four or more satellites in view, measure their transit times, phases of RF signals and Doppler frequency shifts. Next these data are used to determine a position in three dimensional space, as well as velocity and time. At this point it should be emphasized that BeiDou differs in two key respects from other GNSS. First, BeiDou can identify the locations of the receivers on the Earth’s surface. Second, BeiDou compatible devices can send data back to the satellites in text messages of up to 1200 Chinese characters.

16.4.2 The Indian Regional Navigation Satellite System (IRNSS) India also has been aspiring to have its own satellite positioning and navigation system for defense and civilian applications. For this aim the Indian Space Research

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Organization (ISRO) developed the concept of an independent regional satellite navigation system, which provides service only to a particular region of Asia. The Indian Regional Navigation Satellite System (IRNSS) has been implemented by ISRO to provide navigation services with an accuracy better than 20 m. IRNSS service area has been specified between 40 and 140° E in longitude and ±40° in latitude [38, 39]. It provides single and dual frequency services with L-band and S-band signals. The space segment of IRNSS consists of 7 satellites of which 3 are placed in geosynchronous equatorial orbits (GEO) at the longitude of 32.5, 83, and 131° E. The remaining 4 satellites are in 2 geosynchronous (GSO) planes with an inclination of 29° to the equatorial plane and longitude crossings at 55 and 111.5° E. The phase of the orbital planes is 180°, and the relative phasing between the satellites in the orbital planes is 56°. Such space constellation of satellites ensures a continuous visibility of minimum four satellites from anywhere in India. IRNSS provides three kinds of services, i.e. the”Standard Positioning Service” (SPS), the” Precise Positioning Service” (PPS) and the” Restricted Services for Special sers”(RS). While SPS is open for all the users with binary phase shift keying (BPSK) modulated signals, the PPS and RS are only for authorized users using receivers adopted to binary offset coding (BOC) on L and S bands. The carrier frequencies of IRNSS navigation signals used to provide the above mentioned services are 1191.795 MHz and 2491.005 MHz, respectively [38].

16.4.3 The Japanese Regional Navigation Satellite System (QZSS) The second strictly regional satellite navigation system for the Eastern Asia and Oceania is the Quasi Zenith Satellite System (QZSS) developed in Japan. QZSS is not an independent satellite system and is based on GPS, so it can be seen as an enhanced GPS version in the range of the Japanese longitude. The first demonstration satellite QSZ-1 of this system was launched in 2010 by the Japan Aerospace Exploration Agency (JAXA) [16]. The first version (generation) of QZSS (until May 2018) included four active satelites. One of them is placed in geostationary orbit. The three remaining satellites are deployed on three inclined, slightly elliptical, geosynchronous orbits. This means that the time each satellite orbits the Earth is equal to one sidereal day, i.e. 23 h 56 min and 4 s. Each plane of the geosynchronous orbit is 120 0 apart from the other two (Fig. 16.14a). For clarity, Fig. 16.14a does not show the geostationary orbit and satellite related to it. The current parameters of all four orbits and the list of orbiting satellites are provided in JAXA reports. The orbits selected ensure flying of each satellite near the zenith over Japan and Oceania region. The such constellation of QZSS satellites provide a highly accurate

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Fig. 16.14 The Japanese Regional Navigation Satellite System (QZSS); a three geosynchronous orbits, b The terrestrial tracks of QZSS satellites

positioning service covering the Japan territory and on areas close to it. The terrestrial tracks of QZSS range from 120 to 155° E and within ±45° N/S. The corresponding four terrestrial trajectories are in a shape close to the number “8”, similar to that shown in Fig. 16.14b. Due to the unique features of used satellite orbits the elevation of the QZSS are above 60° and the coverage rate is up to 100% in Japan [27]. QZSS transmits 6 navigation signals, i.e. L1-C/A, L1C, L2C, L5, L1-SAIF and LEX whose parameters are summarized in Table 16.4 [8, 15, 27]. The first four in Table 16.4 are the familiar GPS signals. The other two signals are unique to QZSS. The signal L1 - SAIF (Submeter class Augmentation with Integrity Function) is broadcast at carrier frequency equal to 1575.42 MHz. It is interoperable with GPS and is intended to provide a submeter correction signal to users. Another unique signal to be broadcast by QZSS is LEX. This experimental signal is being developed to provide high accuracy positioning that is interoperable with the signal E6, used also in Galileo. QZSS will broadcast multiple frequency commercial signals and also provide a short message service (SMS) as does the Chinese BeiDou system. In summary, the primary purpose of QZSS is to increase the availability of GPS in Japan’s numerous urban canyons, where only satellites at very high elevation can be seen. A secondary function is performance enhancement, increasing the accuracy and reliability of GPS derived navigation solutions. It should be emphasized again Table 16.4 Basic parametres of 6 navigation signals used in the QZSS Signals

Center requency, MHz

Modulation method

Comments

L1 -C/A

1575.42

BPSK(1)

First civil user signal

L1C

1575.42

BOC(1,1)

More robust version of L1 -C/A

L2C

1227.60

BPSK(1)

Second civil user signal

L5

1176.45

QPSK(10)

Third civil user signal

L1 -SAIF

1575.42

BPSK(1)

Suplement signal to SBAS L1

LEX

1278.75

QPSK(5)

Experimental signal

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that satellites of QZSS transmit navigation signals compatible with the GPS L1C/A signal, as well as with the signals of the modernized GPS, i.e. L1C, L2C and L5.

16.5 About Differential Versions of the Satellite Navigation Systems In many regions of the Earth of particular importance for the widely understood transport, it is necessary to determine the position with high accuracy, much greater than the accuracy provided by the standard service (SPS) of the GPS system. In addition, a significant requirement is also the reliability of the system operation, expressed by the probability of the appearance of large errors, for example when determining pseudorange. Therefore, already in the 90 s of the previous century, the idea of supplementing the user segment with monitoring stations sending alarm signals (flags) to users informing about incorrect operation of the system was discussed. It was then realized that these stations could additionally continuously transmit information about coordinate differences defining the known (exact) position of the monitoring station and its measured counterpart. The above-mentioned information can be sent to users via terrestrial VHF radio stations or broadcasted by means of satellites located in the appropriate slots of the geostationary orbit. For this reason, ground based augmentation systems (GBAS) and sattelite based augmentation systems (SBAS) are distinguished, respectively. Of course, the principles of operation of these systems are similar and their essence is described below.

16.5.1 A Principle of Operation In a large number of the Earth’s regions of a particular importance, for a broadly understood transport, there is a necessity of determining the position with a high accuracy, much greater than the accuracy provided by the standard service (SPS) of the GPS system. Additionally, in these regions, differential versions of the GPS system with an organizational structure, as shown in Fig. 16.15a, are most frequently implemented. The elements of the space and ground control segments, shown in Fig. 16.5, are separated by a propagation area that includes the intensively changing ionosphere and a relatively thin (about 12 km) troposphere. At this point, one should emphasize that the ionosphere and the troposphere are heterogeneous media for microwave signals L 1 and L 2 , see Eq. (16.1), emitted by highly suspended satellites of the GPS system (H ≈ 20, 183 km). Taking these media into consideration, both the change of the speed of electromagnetic wave propagation and its deflection occur. The ground control segment of the GPS differential version, hereinafter referred to as Differential Global Positioning System (DGPS), consists of multiple User

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Fig. 16.15 Different version of the satellite navigation system. a A scheme of the organizational structure, b a view of the reference station

Receivers (UR) and a Ground Reference Station (GRS). Considering GRS, on the other hand, a Reference Receiver (RR), a Central Processing Unit (CPU) and a Transmitter (T) sending appropriate corrections to users’ receivers, are distinguished. An example of a constructional solution of the GRS reference station, implemented in DGPS, supporting the automation of field works, is shown in Fig. 16.15b. This photo serves an example of the clearly visible antennas, i.e. a VHF radio link antenna and a GPS patch antenna, located under the dielectric dome, which also serves as the upper cover of the container. The Reference Receiver RR, likewise the User Receivers UR, determines its position (x R , y R , z R ) and shift δt O , considering the pseudo-range d R(i) , for i = 1, 2, 3, 4, to the visible four satellites of the space segment. Then, the determined position (x R , y R , z R ) is compared to the known (accurately determined) coordinates (x R∗ , y R∗ , z ∗R ) of the GRS station, which enables deviations determination for particular coordinates, i.e. .x = x R∗ − x R , .y = y R∗ − y R , .z = z ∗R − z R

(16.15)

If the UR navigation receivers are located sufficiently close to the reference station and the additional condition described below is satisfied, one can assume that their positions are determined considering deviations which are close to (16.15). A sufficiently close vicinity of the GRS is understood as an area with a radius of approximately 200 NM (370 km). If the UR is at the edge of this area, an angle between the radii, representing the signal paths from any satellite to the RR and UR, is of the order of one degree. In other words, these radii can still be considered as parallel and time delay changes of satellite signals, which are the result of an ionospheric and a

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tropospheric refraction, as identical. On the contrary, deviations, which are the result of navigation signals multipath and receiver noise, are not equal and cannot be mutually compensated. Fortunately, the deviations significance in terms of “a budget” of the entire system, is minor. The second condition mentioned previously, relates to the reception by all DGPS receivers at the same time of navigation signals which derive from the same set of four satellites. If RR and UR receive signals from the same satellites, then the conclusion regarding the deviations equality in determining the position coordinates due to errors contributed by the space segment, is correct. These errors include determining ephemeris, errors of on-board satellite clocks and possibly, the deliberately introduced Selective Availability (SA) errors. Thus, when satisfying these both discussed conditions, the deviations (16.15) could be transmitted to all user receivers as corrections to particular coordinates. Unfortunately, the condition for all RR and UR receivers to receive the navigation signals emitted by the same set of four satellites, is demanding enough to be satisfied. Thus, the DGPS system, which directly determines the deviations (16.15) to the particular coordinates of the UR receivers position, has not found a broader application. Currently, in order to eliminate errors caused by the space segment, the ionosphere and the troposphere, the most commonly implemented are Pseudo-range Corrections (PRC) which are determined as follows. Assume that the RR receiver has determined the position (x R , y R , z R ) and correction δt O for the reference station considering the measured pseudo-ranges d R∗(k) = c[t O(k) − (TN(k) + δTS(k) )], see the system of Eqs. (16.3). Then, for the known (accurate) position (x R∗ , y R∗ , z ∗R ) of the RR receiver and the satellites coordinates [xk (t), yk (t), z k (t)], transmitted in the navigation message, the following geometric distances are estimated / R (k) (t) = [xk (t) − x R∗ ]2 + [yk (t) − y R∗ ]2 + [z k (t) − z ∗R ]2 (16.16) R to all k = 1, 2, 3, 4, 5, . . . , N , of currently visible satellites. The distances (16.16) are compared to the distances d R(k) = d R∗(k) + cδt O , determined by the RR receiver for a selected time instant t0 . As a result, the following differences are obtained (k) .d R(k) (t0 ) = R (k) R (t0 ) − d R (t0 )

(16.17)

In the following step, the differential approximations of their first derivatives are determined, i.e. d [ (k) ] .d R(k) (t0 + .t) − .d R(k) (t0 ) .d R (t0 ) ≈ dt .t

(16.18)

Differences (16.17) and related derivatives approximations (16.18) are transmitted to users immediately, e.g. with VHF radio link, which is indicated in Fig. 16.15 as “Data link”. Each of the UR receivers selects the proper data set from the transmitted packet. The following correction is estimated for t1 ≥ t0 + .t in the user receiver

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.dU(k)R (t1 ) = .d R(k) (t0 ) +

d [ (k) ] .d R (t0 ) · (t1 − t0 ), dt

(16.19)

This correction is then added to the pseudo-range dU∗(k) R (t1 ), measured by this receiver. In a final stage, the position of the UR receiver is determined considering (k) the four pseudo-ranges dU∗(k) R (t1 ) + .dU R (t1 ), corrected as described above. The results of many years of operation show that the Geometric Dilution of Precision (GDOP) area of the DGPS system is multiple times smaller than the corresponding area determined with a standard SPS service of the GPS system. The largest linear size of this reduced area is of the order of one meter. It is not difficult to imagine a similarly operating low-range DGPS, whose ground control segment is located near the runway. One of the task of such DGPS is to increase the accuracy of determining the position of aircrafts descending, see Figs.17.1 and 17.2. Currently, this type of DGPS, is referred to as a Ground Based Augmentation System (GBAS), which is installed at a large number of airports in order to duplicate or possibly replace the troublesome ILS system, described in Chap. 17. The GBAS ground control segment most frequently consists of two or three RR receivers of the GPS signal, a common processing unit CPU, and a common radio link VHF. The pseudo-range corrections processed by the CPU are transmitted to the aircraft on-board receivers along with additional data, specifying current local conditions, e.g. terrestrial or atmospheric. The implemented VHF radio link operates within a metric wave range, see Table 1.1. Consequently, the implementation of the antennas with relatively narrow directivity pattern and a low level of side lobes, is facilitated. Due to this, the signal reaching the incoming aircraft does not contain significant reflections from various terrain obstacles and technical infrastructure elements of an airport. Thus, the conditions that are created, aim for descending support which is referred to as a Localizer Performance with Vertical Guidance (LPV).

16.5.2 Examples of the Satellite Based Augmentation Systems (SBAS) Several regional DGPS systems operate in a maner similar to the one described above. Among them are WAAS (Wide Area Augmentation System), EGNOS (European Geostationary Navigation Overlay Service), GAGAN (GPS Aided Geosynchronous Augmentation Navigation) and MSAS (Multi-functional Satellite Augmentation System). They are often referred to in the literature by the common acronym SBAS (Satellite Based Augmentation Systems) due to the fact that pseudorange corrections are sent to users using microwave retransmission devices located on Earth’s geostationary satellites. A functional diagram of an augmenting system (SBAS) operating in this way is shown in Fig. 16.16. Regions of the globe “supported” by the above-mentioned, already put into service, systems are shown in Fig. 16.17.

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343

Many similar regional systems are currently under construction, the most advanced of which are those listed in Table 16.5 and discussed below. For obvious reasons, Table 16.5 does not include many systems in the testing or development phase, such as the South American System (SACCA) or its Australian and Algerian equivalents. European Geostationary Navigation Overlay Service (EGNOS) The operators of this Europe’s regional SBAS are the European Space Agency, European Commission and Eurocontrol, and European Organization for Satety of Air Navigation. EGNOS, similarly as other SBAS, improves the accuracy and reliability of GNSS information by correcting signal measurement errors and by providing information about the accuracy, integrity, and availability of its signals [14]. Its updated version known as EGNOS V2, commissioned in 2011, can be seen as a

Fig. 16.16 The block diagram of the SBAS system

Fig. 16.17 Coverage areas of SBAS systems used currently

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Table 16.5 Examples of systems augmenting the satellite navigation Name

Acronym

Regiom

Agency for Aerial Navigation Safety in Africa

ASECNA

Midle Africa

BeiDou Satellite Based Augmentation System

BDSBAS

China

European Geostationary Navigation Overlay Service

EGNOS

Europa

GPS Aided Geosynchronous Augmentation Navigation

GAGAN

Indie

Korean Augmentation Satellite System

KAAS

Korea

Multi - functional Satellite Augmentation System

MSAS

Japan

System for Differential Correction and Monitoring

SDCM

Russia

Wide Area Augmentation System

WAAS

North America

composition of four functional segments, such as: space segment, ground segment, terrestial support segment and user segment, which are presented below. Currently space segment of EGNOS is composed of 3 geostationary satellites broadcasting corrections and integrity information for GPS satellites in the L1 frequency band (1575.42 MHz). These satellites are identified by the following codes PRN123-(31° E), PRN136-(5° E) and PRN126-(63.9° E). As geostationary satellites, they orbit at speed ϑ ≈ 3.074651 km/s in a common circular orbit with a radius RG ≈ 42,164 km lying in the same plane as the equator. They make a full rotation, in line with the Earth’s rotation, during one sidereal day equal to T = 86164 s. This space segment configuration provides a high level of redundancy over the whole service area in the event of a failure in the geostationary satellite link. EGNOS operations are handled in such a way that, at any point in time, at least two GEOs broadcast an operational signal. The ground segment of EGNOS comprises a network of 40 Ranging Integrity Monitoring Stations (RIMS), 4 Mission Control Centers (MCC), 6 Navigation Land Earth Stations (NLES)—2 per each GEO satellite, and the EGNOS Wide Area Network (EWAN), which provides the communication network for all the components of the ground segment under discussion. The main function of RIMS is to collect measurements from GPS satellites and to transmit these raw data every second to the Central Processing Facilities (CPF) of each MCC. The configuration used for the initial EGNOS OS includes 40 RIMS sites located over a wide geographical area (Fig. 16.18). Mission Control Centres (MCC) receive the information from the RIMS and generate correction messages to improve satellite signal accuracy and information messages on the status of the satellites (integrity). Therefore, they are often referred to as the EGNOS brain. The NLES transmit the EGNOS message received from the central processing facility (MCC) to the GEO satellites for broadcasting to users and to ensure synchronization with the GPS signal. The geographical locations of MCC and NLES are also shown in Fig. 16.18.

16.5 About Differential Versions of the Satellite Navigation Systems

345

Fig. 16.18 EGNOS terrestrial segment components

The transmitters installed on these GEO satellites retransmit (broadcast) the corrections determined by RIMS and MCC to the relevant regions of Europe, the Atlantic and North Africa [40]. The signals emitted by these satellite transmitters are similar in structure to a standard GPS signal L1 , in L1 band, and are transmitted on a carrier of the same frequency f 1 = 1575.42 MHz. However, they are coded in such a way that they correlate with GPS beacons as little as possible. In addition to the above-mentioned stations/centers, the system has other ground support installations involved in system operations planning and performance assessment, namely the Performance Assessment and Checkout Facility (PACF) and the Application Specific Qualification Facility (ASQF) which are operated by the EGNOS Service Provider (ESSP). The EGNOS user segment is comprised of EGNOS receivers that enable their users to accurately compute their positions with integrity. To receive EGNOS signals, the end user must use an EGNOS - compatible receiver adopted to such market segments as agriculture, aviation, maritime, rail, mapping/surveying, road and location based services (LBS). Currently, the main services offered by EGNOS are Open Service (OS) and Safety of Life (SOL) available since 2009 and 2011, respectively. In addition to corrections to the pseudo-distance, sent in the Open Service (OS), a signal is sent confirming their credibility and information about the current state of the entire GPS system.

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This information is sent as part of the Safety of Life Service (SOL), using a carrier wave with a frequency f 5 = 1176.45 MHz. The most common warning is information about a decrease in the accuracy of pseudo-range measurement resulting from the failure of some GPS or GLONASS satellites, or due to severe ionospheric disturbances. This information is particularly important for pilots of aircraft approaching to landing. During this operation, the center of gravity of the airplane should be on the descent path, and any deviations, mainly elevation, in the landing zone specified in Figs. 18.1 and 18.2, should not exceed 1 m. The next generation of EGNOS, designated as EGNOS V3, will augment only GPS and Galileo constellations in the L1/E1 and L5/E5a frequency bands and will extend the service area to the entire landmasses of EU member states. The European Commission assumed that EGNOS V3 operational transition will start in 2024 and that legacy services will be continued together with two additional features. First, EGNOS V3 will augment the Galileo positioning service additionally. Second, it will provide correction data and integrity information in the L1/E1 and L5/E5a bands. Consequently, the above modernized version will be able to ensure high positioning accuracy and reliable EGNOS services throughout Europe for the post 2024 period. This is especially important for air transport. For instance, EGNOS V3 will enable the automatic landing of airplanes - under the supervision of the flight crew—at “Category I (the lowest)” aerodromes in weather conditions where it would otherwise be unsafe or impossible to do so. Wide Area Augmentation System (WAAS) WAAS was commissioned for service in July 2003 and has undergone many changes with many improvements to its service since that time [42]. Currently it provides extremely accurate navigation capability by augmenting the GPS. It was developed for civil aviation by the Federal Aviation Administration (FAA) and covers most of the U.S. National Airspace System (NAS) as well as parts of Canada and Mexico. The space segment of WAAS cretate three geostationary (GEOs) satellites. The current GEOs are the Intelsat Galaxy XV satellite at 133° W (labeled CRW and using PRN 138), the Telesat ANIK F1R satellite at 107° W (labeled CRE and using PRN 137), and the EUTELSAT 117 West B at 117° W (labeled SM9 and using PRN 131). Each GEO satellite cooperates with two independent ground uplink stations (GUSs). Consequently the total number of uplink stations, belonging to the ground segment, is equal to six. The ground segment of WAAS consists of 38 reference stations (WRS) placed in the central area of United States, in Alaska, Hawaii, Puerto Rico, Canada and Mexico [24]. The central locations of 20 WRSs are illustrated in Fig. 16.19. Additionaly, the ground segment includes three wide area master stations (WMSs) conected with (WRS) by means of a land-based communications network. Each of the three WMSs has a corrections and verification (C&V) processor that consists of two parts, i.e. a Corrections Processor (CP) and a Safety Processor (SP). The CP performs an initial screening of the data to identify and remove outliers. The resulting output is fed into filters that estimate the receiver and satellite Inter-Frequency Biases (IFBs) the WRE clock offsets, the satellite orbital locations, and the satellite clock

16.5 About Differential Versions of the Satellite Navigation Systems

347

Fig. 16.19 Terrestial and space components of WAAS

offsets. These are then passed along to the SP for evaluation. The SP is responsible for ensuring the safety of the WAAS output. It will decide what information will be sent to the user and to what level such information can be trusted. The SP performs its own independent data screening on the input WRE data. Multi-function Satellite Augmentation Service (MSAS) MSAS was put into operation in September 2007 [32]. It has been assumed in the designing that MSAS has to cover and control the airspace associated with Japan. MSAS works by processing GPS data collected by a network of reference stations to generate the SBAS message which is uploaded to the GEO satellites. The GEO satellites broadcast this information to the user receivers, which compute the aircraft positioning and inform on potential alert messages. The main functional segments of MSAS are: the MSAS space segment, MSAS ground segment and user segment [37]. The space segment consists of two geosynchronous satellites (GEOs), called Multifunctional Transport Satellites (MTSAT), with identification codes PRN 129 and PRN 137. These satellites are located on the geosynchronous orbit at 140 and 145° E. They are primarily intended to transmit correction messages generated by master control stations (MCS) for reception by the user segment, as well as, are used for meteorological purposes. MSAS ground segment consists of six ground monitoring stations (GMS) located in the Japanese Islands, as it ilustrated and Fig. 16.20. Among them are two main control stations (MCS) located in Kobe (KASC) and Hitachi–Ota (HASC), that compute precise differential corrections and integrity bounds and send them to the MTSAT for rebroadcast to the user segment. The MSAS

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16 Satellite Navigation Systems

Fig. 16.20 Components of the terrestial segment of MSAS

ground segment is also completed with two monitor and ranging stations (MRS) located in Hawaii (USA) and Canberra (Australia), whose purpose is primarily the correct orbit determination of the MTSAT satellites, and they also work as GMS stations. The user segment consists of GPS and SBAS enabled receivers that use information transmitted from each GPS satellite to determine its location and current time, and receive MSAS corrections from the space segment. GPS Aided Geosynchronous Augmentation Navigation (GAGAN) The GPS Aided Geosynchronous Augmentation Navigation in India is called GeoAided Geo-Augmented Navigation (GAGAN). Its space segment is composed by three GEO satellites GSAT-8 at 55° E, GSAT-10 at 83° E, and GSAT-15 at 93.5° E. These satellites broadcast signals using codes PRN 127, PRN 128 and PRN 132, respectively [39, 43]. Currently the ground segment of GAGAN icludes the following components (Fig. 16.21). (1) Indian Reference Stations (INRES) at 15 locations across India territory. (2) Two Indian Master Control Centers (INMCC) located at Bangalore. (3) Three Indian Land Uplink Stations (INLUS) are located at Bangalore and New Delhi. (4) A data communication subsystem composed of two optical fiber communication network and two small aperture terminal (VSAT) links. The INRES collect measurement data and message from all the GPS and GEO satellites in view and forward them in real time (every second) to INMCC for further processing. The SBAS messages generated by INMCC are sent to ILNUS, operating in C-band, that formats them for GPS compatibility and uplinks to GEO satellites.

16.5 About Differential Versions of the Satellite Navigation Systems

349

Fig. 16.21 Components of the GAGAN ground-based segment

Similarly as in other SBAS, discussed previously, the SBAS messages are broadcasted via GEO satellites to the user community. They allow SBAS receiver to remove errors in the GPS position solution, thus allowing for a significant increase location accuracy and reliability. Along with position corrections, the confidence parameters (integrity) are also computed and provided to users as messages. System for Differential Correction and Monitoring (SDCM) Russia has been developed own System for Differential Correction and Monitoring (SDCM) augmenting mainly the GLONASS. Currently it is composed of 19 operational measuring points (MPs) in Russia, one in South America (Brasilia City), and three in Antarctica. More detailed information on their distribution can be found in many publications, for instance in [22, 34]. The ground segment of SDCM includes also three microwave uplink stations. Two of them are allocated in Moscow region and one in Khabarovs. The space segment of SDCM broadcasts the navigation and integrity messages via three GEO satellites marked as: Luch - 5A at 167° E, Luch 5B at 16° W, and Luch - 5 V at 95° E. For this purpose the folowing codes PRN 125, PRN 140 and PRN 141 are used. These codes are coordinated with the operator of the GPS to insure compatibility with GPS and other GPS—like signal broadcasts. Due to such a constellation of used geostationary satellites, minimal overlap between the areas illuminated by them was achieved over the vast territory of Russia. During its design, it was assumed that it would provide navigational corrections and integrity messages for signals received from GLONASS and GPS constelations. Thus SDCM differs from other SBAS by an important feature, namely it provides corrections and

350

16 Satellite Navigation Systems

integrity messages based on both GLONASS and GPS signals, while other SBASs augmente only GPS signals. BeiDou Satellite Based Augmentation System (BDSBAS) China is developing its BeiDou Satellite Augmentation System (BDSBAS) on the assumption that navigational corrections will be delivered to users via three BeiDou GEO satellites. China successfully launched the first GEO-1 satellite (SBAS PRN 130) on November 2018 that began broadcasting its augmentation signal. Two SBAS satellites remain to be launched. In accordance with international regulations, all satellites will be located at 80, 110.5 and 140° E and using PRN codes 130, 143 and 144 [7]. The publications available before 2020 show that the system is still being expanded and has not yet reached the operational phase [31]. Although BDSBAS broadcasts an augmentation signal for relatively long time, it is still being tested and therefore cannot yet provide services for civil aviation. Consequently, the final number of reference stations and their locations have not yet been established definitelly. Before 2020 its ground segment has been composed of 24 monitoring stations only, whose locations are illustrated in Fig. 16.22. BDSBAS will improve the accuracy of the user’s position as measured by GPS and BeiDou. However, according to the authors of the review article [31], the BDSBAS is still being refined to meet the special requirements of air transport. For this key reason, it has not yet been officially put into operation. Korean Augmentation Satellite System (KASS) The Republic of South Korea has announced that the Korean Augmentation Satellite System (KASS), primarily developed for aviation applications, will be completed in around 2023 [11, 41]. The territory of the Korean Peninsula is much smaller than the territory of Russia or China, so two GEO satellites are enough to effectively and

Fig. 16.22 Monitoring stations of the terrestial segment of BDSBAS

References

351

reliably illuminate it, as well as adjacent areas. For this purpose they will be located at 113, 116° E or 128° E. The satellites will transmit integrity and correction data to SBAS receivers. In this way they will be used for ensuring the three basic services, i.e., FSS (Fixed satellite service), BSS (Broadcasting satellite service) and EESS (Earth exploration satellite service). The navigation signals will be transmitted by satellites on the L1 and L5 carriers. The transmitted waves will have a clockwise circular polarization. The ground segment of KASS will be composed of the following functional units: – 7 KASS Reference Stations (KRS), – 2 KASS Processing Stations (KPS), – 2 KASS Control Stations (KCS), and – 3 KASS Uplink Stations (KUS). All 7 KRS will be located near the South Korean coastline. The remaining units of the ground segment are located in the central part of Korea [12]. With 2 KPS, SBAS service within the service area will not be disrupted due to natural disaster and other effects. KUS enables the reception of KASS data broadcasted by GEO satellites (space segment) and transfer them to KPS via a terrestrial communication link or satellite link. The C frequency band will be used for data transmission between the space and ground segments. KRS receives GPS L1 (1575.42 MHz), L2 (1227.6 MHz) and L5 (1176.45 MHz) signals from GPS satellites for monitoring GPS signals as well as for calculating and correcting for ionospheric delays in signal propagation time. KRS has the function to collect the basic data required for ranging of the KASS satellites position to create the ranging data (positioning data equivalent to that of GPS) in addition to KUS functions. As usually, also in this case the user segment consists of KASS receivers allocated in the air, on the ocean and on land. They determine their geographical positions on the basis signals of the GPS constellations and the KASS signal. Agency for Aerial Navigation Safety in Africa and Madagascar (ASECNA) The Air Navigation Safety Agency in Africa and Madagascar (ASECNA) has investigating the possibility of building SBAS over a region of Africa including: Benin, Burkina, Cameroon, Central African Republic, Congo, Ivory Coast, Gabon, Guinea Bissau, Equatorial Guinea, Madagascar, Mali, Mauritania, Niger, Senegal, Chad, Togo and the Union of the Comoros [2]. These listed 17 states jointly manage their airspace and are interested in developing SBAS to augmenting the air transport, terminal operations and approaches in the region for which they are responsible. They conducted tests in coordination with EGNOS and investigate the adaptation of EGNOS algorithms to service delivery in Africa [23].

References 1. Ahmed E-R (2002) Introduction to GPS—the global positioning system. Artech House, Inc., Norwood (MA) 2. Agency for Aerial Navigation Safety in Africa and Madagascar. https://www.asecna.aero/index. php/en/

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3. Awange JL, Grafarend EW (2002) Algebraic solution of GPS pseudoranging equations. GPS Solutions 5(4):20–32 4. BeiDou Navigation Satellite System Open Service Performance Standard. http://www2.unb. ca/gge/Resources/beidou_open_service_performance_standard_ver1.0.pdf 5. BeiDou Navigation Satellite System, Signal in Space, Interface Control Document. http:// www2.unb.ca/gge/Resources/beidou_icd_english_ver2.0.pdf 6. Chaffe J, Abel J (1994) On the exact solutions of pesudorange equations. IEEE Trans Aerospace Electron Syst AES-30(4):1021–1030 7. China Satellite Navigation Office (2017) Update on BeiDou navigation satellite system. Presented at 12th Meeting of the International Committee on Global Navigation Satellite Systems (ICG), Kyoto, Japan, Dec 2017. http://www.unoosa.org/documents/pdf/icg/2017/05_ icg12.pdf 8. Description of systems and networks in the radionavigation—satellite service Recommendation ITU—R M.1787-3, 2018. https://www.itu.int/dms_pubrec/itu-r/rec/m/R-REC-M.1787-3-201 803-I!!PDF-E.pdf 9. Forssell B (2008) Radionavigation systems. Artech House, Inc., Boston 10. Getting IA (1993) The global positioning system. IEEE Spectrum 36–47 11. Gi WN (2020) KASS in operation. What do we expect it? Industry developments, Sept 2020. https://gnss.asia/blog/kass-in-operation-what-do-we-expect-from-it/ 12. Gi WN (2018) The development status of Korean SBAS. Working paper presented at 13 ICAO Air Navigation Conference, Montre’al, Canada, Oct 2018. https://www.icao.int/Meetings/anc onf13/Documents/WP/wp_240_en.pdf 13. Global Navigation Satellite System GLONASS. http://russianspacesystems.ru/wp-content/upl oads/2016/08/ICD_GLONASS_eng_v5.1.pdf 14. Global Navigation Satellite Systems Agency, EGNOS Safety of Life (SoL) Service Definition Document 3.1, Sept 2016. https://egnos-user-support.esspsas.eu/new_egnos_ops/sites/default/ files/library/official_docs/egnos_sol_sdd_in_force.pdf 15. Hama S et al (2010) Quasi–Zenith satellite navigation system (QZSS) project. J Nat Inst Inform Commun Technol 57(3/4):289–296 16. Hiroyuki N et al (2010) Development of the QZSS and high accuracy positioning experiment system flight model. NEC Tech J 5(3):93–97 17. Hoffmann-Wellenhof B et al (1991) Global positioning system, theory and practice, 3rd edn. Springer, New York 18. Johnson RC (ed) (1993) Antenna engineering handbook, McGraw-Hill, New York 19. Kayton M, Fried WR (1996) Avionic navigation systems, 2nd edn. Wiley, New York 20. Krause LO (1987) A direct solution to GPS-type navigation equations. IEEE Trans. Aerospace Electron Syst AES-23(2):225–232. 21. Langley RB (1991) The mathematics of GPS. GPS World 2(7):45–50 22. Langley RB (2017) Innovation: GLONASS—past, present and future. GPS World, 1 Nov 2017. http://gpsworld.com/innovation-glonass-past-present-and-future/ 23. NIGCOMSAT, ASECNA, Thales Alenia to power first African early SBAS open service. Space in Africa, September 2020. https://africanews.space/nigcomsat-asecna-thales-alenia-to-powerfirst-african-early-sbas-open-service/ 24. Lawrence D et al (2007) Wide area augmentation satellite system (WASS). In: Proceedings of the 20th international technical meeting of the Satellite Division of the Institute of Navigation, Fort Worth, TX, September 2007, pp 892–899 25. Leva J (1996) An alternative closed-form solution to the GPS pseudo-range equations. IEEE Trans Aerospace Electron Syst AES-32(4):1430–1439 26. Milligan TA (1985) Modern antenna design. McGraw-Hill, New York 27. Quasi—Zenith Satellite System (QZSS). https://qzss.go.jp/en/overview/services/sv01_what. html 28. Rosłoniec S (2006) Fundamentals of the antenna technique (in Polish). Publishing house of the Warsaw University of Technology, Warsaw 29. Rosłoniec S (2008) Fundamental numerical methods for electrical engineering. Springer, Berlin

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30. Rosloniec S (2020) Fundamentals of the radiolocaction and radionavigation, second edition (in Polish). Publishing house of the Military University of Technology, Warsaw 31. Li R et al, (2020) “Advances in BeiDou Navigation Satellite System (BDS) and satellite navigation augmentation technologies”. Springer Open J. https://satellite-navigation.springeropen. com/articles/10.1186/s43020-020-00010-2 32. Sakai T, Tashiro H (2013) MSAS status. In: Proceedings of the 26th international technical meeting of the satellite division of the institute of navigation, Nashville, TN, pp 2343–2360, Sept 2013 33. Satellite Based Augmentation Systems (SBAS). http://web.stanford.edu/group/scpnt/gpslab/ pubs/books/Chapter14SBAS.pdf 34. Sernov V. GNSS GLONASS augmentation system—SDCM, Presentations/6.pdf. https://www. unoosa.org/documents/pdf/psa/activities/2015/RussiaGNSS/ 35. Sanz Subirana J et al (2013) GNSS data processing, vol 1: fundamentals and algorithms ESA, TM-23/1, May 2013. https://gssc.esa.int/navipedia/index.php/GNSS:Tools 36. Signal-in-space interface control. Galileo_OS_SIS_ICD_v2.0.pdf. European GNSS (Galileo) open service, Issue 2.0, Jan 2021. https://www.gsc-europa.eu/sites/default/files/sites/all/files/ 37. Saito S (2019) MSAS system development—ICAO. GBAS/SBAS International Workshop, Seoul, 3–5 June 2019. https://www.icao.int/APAC/APAC-RSO/GBASSBAS%20Implementa tion%20Workshop/1-6_MSAS%20System%20Development_Rev2%20(S%20Saito).pdf 38. Swamy KTC. Global navigation satellite system and augmentation. https://www.ias.ac.in/art icle/fulltext/reso/022/12/1155-1174 39. Teunissen PJ, Montebruck O (eds) (2017) Springer handbook of global navigation satellite systems. Springer, Switzerland 40. User guide for EGNOS application developers CNES–ESA, ED 2.0, pp 1–111. 15 Dec 2011. https://sciences-techniques.cnes.fr/sites/default/files/migration/automne/standard/ 2014_05/p7853_cb8d73df13dfd104092d3f8c5cc2062bguide_egnos_2011_GB_112_P.pdf 41. Yun S (2019) KASS program status. EGNOS workshop, Rome , Italy, Sept 2019. https://egnos-user-support.essp-sas.eu/new_egnos_ops/sites/default/files/EWS19%20K ARI%20-%20KASS%20programme%20status.pdf 42. Walter T et al (2018) WAAS at 15. In: Proceedings of the international technical meeting of the Institute of Navigation, Reston, VA, Jan 2018. 43. Zogg JM. GPS, essential of satellite navigation. Compendium. https://zogg-jm.ch/Dateien/ GPS_Compendium(GPS-X-02007).pdf

Chapter 17

Aircraft Landing Aid Systems

The fundamental task of air radio navigation is to provide a safe flight of an aircraft (mainly an airplane) on a given path with the required accuracy and time. For this purpose, various navigation systems are implemented, both global (GPS, GLONASS, …) and local (DGPS, VOR/DME, single LORAN-C, System 4000) [8, 10]. Final and difficult stage of each flight corresponds to landing, which should be supported by more accurate and reliable radio navigation systems. These systems must meet the specific requirements imposed by International Civil Aviation Organization (ICAO). In other words, these systems must ensure safe landing of airplanes in various weather conditions. Depending on the proper radio navigation system equipment and terrain conditions, airports are prepared to satisfy this task to a various extent, which is formally specified by their category. According to ICAO regulations, airports are divided into three fundamental categories, from which the I category is lowest, [2, 10]. The basic requirements imposed on airports of particular categories, are shown in Table 17.1. The Runway Visual Range (RVR) parameter, shown in Table 17.1, determines the distance at which the pilot is able to clearly see the beginning (threshold) of the runway (RW/LW). The significance of some of the parameters demonstrated in this table, is shown in Fig. 17.1. The deviations .αmax and .βmax define the acceptable angular errors in determining the flight path deflection from the course plane and the descent plane, respectively. The radio aperture placed on-board the aircraft should facilitate the entry to the approach zone and subsequently to the touchdown zone within the range of a landing aid system. In the touchdown zone the following are determined: flight altitude (with an altimeter), distance to the touchdown point and angular deviations .α, .β, defined in Fig. 17.2. Subject to Fig. 17.2, the following symbols have been adopted: P—plane, RW/LW—runway/landingway, CL—course line, GP—glide path, FP—flight path, .α—course deviation (azimuth) and .β—elevation deviation. According to ICAO requirements, a width of runway of category I airport should be no lesser than 45 m

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_17

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17 Aircraft Landing Aid Systems

Table 17.1 The basic requirements imposed on airports of particular categories Category

Dmax , m

Hmax , m

Hmin , m

RVR, m

δαmax ,◦

δβmax ,◦

I

45,000

20,000

60

>800

±0.15

±0.22

±0.043

±0.12

II

45,000

20,000

30

>400

III A

45,000

20,000

30

>200

III B

45,000

20,000

30

>50

III C

45,000

20,000

–

Fig. 17.1 Definition of landing, touchdown and approach zones around a runway; a top view, b side view

Fig. 17.2 Definitions of the CL—course line, GP—glide path, FP—flight path, .α—course deviation (azimuth) and .β—elevation deviation

and a length no lesser than 2500 m. The runway dimensions of modern airports are more standardized, namely 60 × 4000 m. Radio navigation systems installed at the airports of category II and III should enable safe landing with an autopilot. This requirement can be demonstrated as follows. When the airplane exits low-hanging clouds (at the altitude of Hmin = 30 m), the pilot has only 13−15 s for levelling out the flight (positioning the wings as parallel to the runway as possible), climbing compensation and soft touchdown. The given time interval 13−15 s results from the rate of descent, which is within the range of

17.1 Instrument Landing System (ILS)

357

1.8−2.3 m/s for the majority of airplanes. In view of a careful observation of onboard navigation instruments that requires approximately 3 s, the pilot is not able to precisely perform all the mentioned operations. In order to solve the mentioned tasks effectively, a large number of radio navigation systems have been developed, the most known of which refer to ILS, MLS and TLS. Although in the era of satellite navigation these systems may be considered obsolete, they are still broadly implemented, since in several respects they surpass the Local Area Augmentation Systems (LAAS). Therefore, the principles of operation of these systems are the subject of further consideration.

17.1 Instrument Landing System (ILS) Referring to history, a first radio landing aid system corresponds to the Instrument Landing System (ILS). A first fully automatic landing with ILS was carried out on January 26, 1938 at the Pittsburgh Airport (PA) during a snow-storm. The experimental versions of the ILS, authorized by Civil Aeronautics Administration (CAA), were installed in 1941 at six American airports. In 1949, ICAO approved the ILS system as suitable for most civil airports of category I. Currently, the improved versions of the ILS system are still in service at numerous airports of categories I, II and even IIIA [1, 8, 10]. The SP-80 system is a counterpart of the ILS developed in the USSR [1, 16]. System Organizational Structure Each version of the ILS system distinguishes a ground segment, air space and a multichannel receiver placed on board the airplane. The ground segment consists of the following integrated subsystems: • Localizer array antenna (LOC), • Glidepath array antenna (GP), • Marker beacons which specify the distance from an airplane to the recommended touchdown point and cooperate with the radio system • An approach Lighting System (ALS). The contemporary versions of ILS use the Distance Measuring Equipment (DME) in order to replace marker beacons. Another important piece of equipment corresponds to a ground transmissometer, which informs about the light beam weakening by the tropospheric layer surrounding a runway. Indirectly, transmissometer facilitates determining a current visibility of a runway threshold and the lighting system installed on it. This information is transmitted to the pilot by radio from a control tower. Localizer array antennas A1 and A2 use one of the 40 narrow radio channels. Their mid- frequencies f 0l in the range 108.10÷111.95 MHz are given in the second and fifth columns of Table 17.2. The third and sixth columns of this table demonstrate the mid-channel frequencies f 0g of the corresponding radio channels, used by the glidepath array antennas A3 and A4 . These frequencies are within the range of

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17 Aircraft Landing Aid Systems

329.15−335.00 MHz. On board of the landing airplane there is a receiver that is able to receive all radio signals emitted by the ground segment. The arrangement of the discussed radio beacons around the runway is shown in Fig. 17.3. Subject to Fig. 17.3, three distance marker beacons Am , emit modulated proper radio signals in limited (fin-type) areas. Consequently, an aircraft passing through Table 17.2 The center frequencies, f 0l and f 0g , of individual ILS channels Channel

f 0l , MHz

f 0g , MHz

Channel

f 0l , MHz

f 0g , MHz

1

108.10

334.70

21

110.10

334.40

2

108.15

334.55

22

110.15

334.25

3

108.30

334.10

23

110.30

335.00

4

108.35

333.95

24

110.35

334.85

5

108.50

329.90

25

110.50

329.60

6

108.55

329.75

26

110.55

329.45

7

108.70

330.50

27

110.70

330.20

8

108.75

330.35

28

110.75

330.05

9

108.90

329.30

29

110.90

330.80

10

108.95

329.15

30

110.95

330.65

11

109.10

331.40

31

111.10

331.70

12

109.15

331.25

32

111.15

331.55

13

109.30

332.00

33

111.30

332.30

14

109.35

331.85

34

111.35

332.15

15

109.50

332.60

35

111.50

332.90

16

109.55

332.45

36

111.55

332.75

17

109.70

333.20

37

111.70

333.50

18

109.75

333.05

38

111.75

333.35

19

109.90

333.80

39

111.90

331.10

20

109.95

333.65

40

111.95

330.95

Fig. 17.3 Antennas and distance marker beacons of ILS around a runway

17.1 Instrument Landing System (ILS)

359

Table 17.3 The light and acoustic signals characterizing the individual distance markers Marker

Distance to R

FX M , Hz

Flickering light

Morse code

Outside

3.9 nm; 7200 m

400

Blue

----

Middle

3500 ft; 1100 m

1300

Amber

. -. -

Inside

100 ft; 30 m

3000

White

...

these narrow areas, is able to approximate its distance to the beginning of the runway. Typical distance values d6 , d5 and d3 are given in the second column of Table 17.3. The localizer array antennas of ILS (A1 and A2 ) transmit periodically a proper identification signal for a given airport indicated as three or four Morse code letters. This signal indicated as respectively short and long acoustic pulses of a frequency 1020 Hz, is obtained by amplitude detection of the mentioned radio identification signal. If there are several runways at a given airport, their identification signals are also transmitted. For example, the John Kennedy airport in New York is marked with the IJFK symbol, and its 04R runway with the IHIQ symbol. A first, important task of the localizer array antenna is to introduce the airplane into the closest possible vicinity of the localizer plane. Likewise, the glidepath array antennas should facilitate the pilot to bring the aircraft to the glidepath as early as possible. The intersection of localizer plane and glidepaths determine the GP line (path) on which the distance of an airplane approaching landing should be no lesser than 6 NM (11 km), counting from the beginning of the runway. The main task of the localizer array antenna is to determine the localizer plane, perpendicular to the runway and intersecting this runway along its longitudinal axis, as accurately as possible, see Fig. 17.2. The glidepath array antennas determine a plane which is perpendicular to the localizer plane and inclined with respect to the runway at a minor angle 2.0◦ ≤ β0 ≤ 4.5◦ . Most frequently, one assumes that β0 ≤ 3◦ . This plane intersects with the runway along the touchdown line approximately 300 m from its beginning. Considering the signals emitted by the localizer array antennas (A1 and A2 ), a current azimuth deviation value .α is determined in the airplane receiver, see Fig. 17.2. Likewise, considering signals emitted by the radio beacons (A3 and A4 ), a current value of the elevation deviation is determined. The pilot’s task is to change a flight path so as to approach the descent line for which .α = 0 and .β = 0. The method of determining the deviations .α and .β is the subject of further discussion in the present chapter. Marker beacons emit radio signals of a power equal to 3 W in a narrow angular spatial sectors. Due to this fact the airplanes passing through these sectors are able to approximate their distance to the beginning of the runway. The ILS system uses three marker beacons, i.e. the outside, middle and inside, emitting signals with a carrier frequency 75 MHz, modulated by an acoustic signal corresponding to dashes or dots in Morse code, see Table 17.3. The on-board receiver of an airplane flying through a narrow fin beam of a distance radio beacon, generates

360

17 Aircraft Landing Aid Systems

an interrupted acoustic signal of a proper frequency FX M and activates proper (in terms of color) light signaling. The Determining Principle of an Aircraft Position with Respect to the Course Plane The localizer array antennas (A1 and A2 ) emit the electromagnetic wave towards the incoming airplane. This wave is accumulated in two mirror-symmetrical beams FA1 (α0 +.α) and FA2 (α0 −.α), and their azimuth and elevation sections are shown in Fig. 17.4. The angular widths θ3dB of these beams, specified in the azimuth plane, should be approximately equal to 5 ± 1◦ with their line inclination angle 2α0 not exceeding 5◦ . The antenna A1 is fed by a continuous VHF signal of a carrier frequency f 0l (see Table 17.2) with an amplitude modulated harmonic signal of a frequency F1 = 90 Hz. Antenna A2 is fed in a similar manner, however frequency of the modulating signal equals F2 = 150 Hz. Thus, the signals received by the aircraft on-board receiver, the course of which is determined by the angular deviation .α, see Fig. 17.4a, are described by the following formulae: u 1 (t, .α) = Um FA1 (α0 + .α)[1 + m 1 cos(.1 t)] cos(ω0l t) = Um FA1 (α0 + .α) cos(ω0l t) + 0.5Um FA1 (α0 + .α)m 1 cos[(ω0l + .1 )t] + 0.5Um FA1 (α0 + .α)m 1 cos[(ω0l − .1 )t]

(17.1)

Fig. 17.4 Electromagnetic beams shaped by localizer array antennas A1 and A2 ; a top view, b side view

17.1 Instrument Landing System (ILS)

361

Fig. 17.5 The amplitude spectra of received signals (17.1) and (17.2)

u 2 (t, .α) = Um FA2 (α0 − .α)[1 + m 2 cos(.2 t)] cos(ω0l t) = Um FA2 (α0 − .α) cos(ω0l t) + 0.5Um FA2 (α0 − .α)m 2 cos[(ω0l + .2 )t] + 0.5Um FA2 (α0 − .α)m 2 cos[(ω0l − .2 )t]

(17.2)

for: ω0l = 2π f 0l , .1 = 2π F1 , .2 = 2π F2 , m 1 = 0.2 and m 2 = 0.2. The amplitude spectra of both received signals are illustrated in Fig. 17.5. The sum of signals (17.1) and (17.2) can be expressed as follows u . (t, .α) = Um F. (.α)[1 + M1 cos(.1 t) + M2 cos(.2 t)] cos(ω0l t)

(17.3)

for: F. (.α) = FA1 (α0 + .α) + FA2 (α0 − .α), M1 =

m 2 FA2 (α0 − .α) m 1 FA1 (α0 + .α) , M2 = , F. (.α) F. (.α)

M1 + M2 < 1

A navigation parameter which is determined considering signal (17.3), corresponds to a Difference in the Depth of Modulation (DDM). .Ma = M1 − M2 =

m 1 FA1 (α0 + .α) − m 2 FA2 (α0 − .α) F. (.α)

(17.4)

In the case m 1 ≈ m 2 = m formula (17.4) is simplified to a following form: .Ma = m

F. (.α) FA1 (α0 + .α) − FA2 (α0 − .α) =m FA1 (α0 + .α) + FA2 (α0 − .α) F. (.α)

(17.5)

The difference (17.5) corresponds to a must value, therefore is independent of the distance between the airplane and the localizer array antennas. The foregoing considerations show that the signal (17.3) is amplitude modulated both in time and space domain. This means that for FA1 (α) ≈ FA2 (α), selecting such coefficient values m 1 and m 2 , for which the position of the plane in a course plane (.α = 0) will correspond to a difference .Ma = 0, is possible. When deflecting into the right (entering a celestial sphere; .α > 0) the difference .Ma < 0. In the opposite case

362

17 Aircraft Landing Aid Systems

(entering a yellow zone; .α < 0) the difference .Ma > 0. Thus, the function .Ma (.α) satisfies the conditions which is set for direction finding characteristics and can be used explicitly to visualize the position of an aircraft with respect to the course plane and, alternatively, to control the autopilot. A voltage signal, which is linearly proportional to a difference in the depth of modulation (17.4), is obtained in an electronic circuit with a functional diagram, as shown in Fig. 17.6. In general, the narrow-band-pass filters (BPF), present in this circuit, should be precisely tuned to the frequencies of the modulation components, which vary due to the Doppler effect. This effect is expressed by the dependencies F1D = F1 (1+ϑr /c), F2D = F2 (1 + ϑr /c), for ϑr corresponding to a radial velocity of an airplane with respect to the wave source, and c = 3 × 108 m/s is the speed of light. Considering a typical case, if ϑr = 200 km/h, then a frequency F2D = F2 + .F2D = 150 + 0.00027(7) Hz. This example demonstrates that the Doppler modulation frequencies shift is insignificantly small and there is no need to retune the mentioned filters. In order to provide a range no lesser than 25 NM (≈ 46 km), the localizer array antennas (A1 and A2 ) radiate a wave of approximately 100 W and linear horizontal polarization. The main disadvantage of the discussed basic solution, is a low direction finding sensitivity, which can be improved by increasing a line inclination angle 2α0 of the antenna system beams, see Fig. 17.4a. Unfortunately, increasing this angle reduces the range of the system in the course plane (.α = 0). In such situation, a reasonable compromise between range and direction finding sensitivity, is necessary. As most frequently assumed, the beams FA1 and FA2 intersect with each other in the course plane (.α = 0) at the half power level, i.e. −3.01 dB. Another, more effective solution is the formation by the system of antennas A1 and A2 (considered as a two-element antenna array) of two independent beams, namely summation and differential. In order to clarify a significance of this more modern solution, assume that the directivity patterns (radiation) of antennas A1 and A2 , are similar to those shown in Fig. 17.4a. The signals that activate each of these antennas, contain two components, hereinafter referred to as summation and differential. According to this, a signal reaching the on-board receiver from the radio beacon A1 , i.e.

Fig. 17.6 The block diagram of an on-board apparatus of the ILS system

17.1 Instrument Landing System (ILS)

363

(.) u 1. (t, .α) = u (.) 1 (t, .α) + u 1 (t, .α)

(17.6)

corresponds to a sum of the following components: [ ] (.) (.) u (.) (t, .α) = U F (α + .α) 1 + m cos(. t) + m cos(. t) cos(ω0l t) m A1 0 1 2 1 1 2 (17.7) [ ] (.) (.) u (.) 1 (t, .α) = Um F A1 (α0 + .α) m 1 cos(.1 t) − m 2 cos(.2 t) cos(ω0l t) (17.8) A signal reaching the receiver from radio beacon A2 is similar (.) u 2. (t, .α) = u (.) 2 (t, .α) + u 2 (t, .α)

(17.9)

for: [ ] (.) (.) u (.) (t, .α) = U F (α − .α) 1 + m cos(. t) + m cos(. t) cos(ω0l t) m A2 0 1 2 2 1 2 (17.10) [ ] (.) (.) u (.) 2 (t, .α) = Um F A2 (α0 − .α) −m 1 cos(.1 t) + m 2 cos(.2 t) cos(ω0l t) (17.11) A sum of the components (17.7) and (17.10), i.e. [ ] (.) u . (t, .α) = Um F. (.α) · 1 + m (.) cos(. t) + m cos(. t) cos(ω0l t) 1 2 1 2 = Um F. (.α) cos(ω0l t) + 0.5Um F. (.α)m (.) 1 cos[(ω0l + .1 )t] + 0.5Um F. (.α)m (.) 1 cos[(ω0l − .1 )t] + 0.5Um F. (.α)m (.) 2 cos[(ω0l + .2 )t] + 0.5Um F. (.α)m (.) 2 cos[(ω0l − .2 )t]

(17.12)

forms the so-called summative directivity pattern F. (.α) ≡ FA1 (α0 + .α) + FA2 (α0 − .α) with azimuthal section, similar to one shown in Fig. 17.7. This directivity pattern is an even function and has positive values for both, negative and positive angular deviations .α. Likewise, the sum of the components (17.8) and (17.11), i.e. [ ] (.) u . (t, .α) = Um F. (.α) m (.) 1 cos(.1 t) − m 2 cos(.2 t) cos(ω0l t) = 0.5Um F. (.α)m (.) 1 cos[(ω0l + .1 )t]

364

17 Aircraft Landing Aid Systems

Fig. 17.7 The spectra of signals emitted by the particular electromagnetic beams

+ 0.5Um F. (.α)m (.) 1 cos[(ω0l − .1 )t] − 0.5Um F. (.α)m (.) 2 cos[(ω0l + .2 )t] − 0.5Um F. (.α)m (.) 2 cos[(ω0l − .2 )t]

(17.13)

forms the so-called differential directivity pattern F. (.α) ≡ FA1 (α0 + .α) − FA2 (α0 − .α). The azimuth section of this directivity pattern is also demonstrated in Fig. 17.7. Contrary to the previously defined summation characteristic F. (.α), a differential characteristic F. (.α) is an odd function. Moreover, if |FA1 (α0 + .α)| = |FA2 (α0 − .α)| is satisfied, this characteristic is equal to zero for .α = 0, i.e. in the course plane. The frequency spectra of the signal (17.12) specified for different values of the azimuth deviation .a, are shown in Fig. 17.8a. The signal spectra (17.13) defined for the same angular deviations .a, is shown in Fig. 17.8b. On the other hand, Fig. 17.8c shows the spectra of the sum of signals (17.12) and (17.13), which are simultaneously received by the airplane on-board receiver. Considering the sum of signals (17.12) and (17.13), i.e. u(t, .α) = u . (t, .α) + u . (t, .α) = Um F. (.α) cos(ω0l t) (.) + 0.5Um [F. (.α)m (.) 1 + F. (.α)m 1 ] cos[(ω0l + .1 )t] (.) + 0.5Um [F. (.α)m (.) 1 + F. (.α)m 1 ] cos[(ω0l − .1 )t] (.) + 0.5Um [F. (.α)m (.) 2 − F. (.α)m 2 ] cos[(ω0l + .2 )t] (.) + 0.5Um [F. (.α)m (.) 2 − F. (.α)m 2 ] cos[(ω0l − .2 )t]

(17.14)

a difference in the depth of modulation with the signals of acoustic frequencies F1 and F2 , is determined. For this purpose, formula (17.14) should be expressed as

17.1 Instrument Landing System (ILS)

365

Fig. 17.8 The spectra of signals; a described by (17.12), b described by (17.13), c simultaneously received by the airplane on-board receiver

follows: u(t, .α) = Um F. (.α)[1 + M1α cos(.1 t) + M2α cos(.2 t)] cos(ω0l t)

(17.15)

for: M1α =

(.) (.) F. (.α)m (.) F. (.α)m (.) 2 − F. (.α)m 2 1 + F. (.α)m 1 , M2α = F. (.α) F. (.α) (17.16) M1α + M2α < 1

In practice, it is most frequently assumed that m (.) = m (.) = m (.) = m (.) = m. 1 1 2 2 Considering this assumption, formulae (17.16) are simplified into M1α =

2m · FA1 (α0 + .α) 2m · FA2 (α0 − .α) , M2α = F. (.α) F. (.α)

(17.17)

In this case, the determined difference in the depth of modulation .Ma = M1α − M2α = 2m

F. (.α) FA1 (α0 + .α) − FA2 (α0 − .α) = 2m F. (.α) F. (.α) (17.18)

is twice as large as the corresponding difference described by formula (17.5). This means that the analyzed system is marked by a twice increased direction finding sensitivity, while the maximum range in the course plane is provided, i.e. for .α = 0.

366

17 Aircraft Landing Aid Systems

Also considering this case, the difference (17.18) is a must value, which means that its value does not depend on the distance between the localizer array antenna and the airplane. In order to specify the influence of the beams 2α0 line inclination angle, see formulae (17.6)–(17.11), on the direction finding sensitivity in the course plane, a following derivative should be derived 1 d F. (.α) F. (.α) d F. (.α) d.Mα = 2m · − 2m 2 · d.α F. (.α) d.α d.α F. (.α)

(17.19)

for: d F. (.α) d FA1 (α0 + .α) d FA2 (α0 − .α) = + d.α d(α0 + .α) d(α0 − .α)

(17.20)

d FA1 (α0 + .α) d FA2 (α0 − .α) d F. (.α) = − d.α d(α0 + .α) d(α0 − .α)

(17.21)

If the characteristics FA1 (.α) and FA2 (.α) are identical even functions, then the component (17.21) has an insignificantly small value when the aircraft is close to the course plane, i.e. for .α → 0. In this situation the derivative (17.19) can be expressed in the following simplified form [ ] 2m d F. (.α) 2m d FA1 (α0 + .α) d FA2 (α0 − .α) d.Mα ≈ · = + d.α F. (.α) d.α F. (.α) d(α0 + .α) d(α0 − .α) (17.22) The analysis of formula (17.22) shows that the direction finding sensitivity determined in the course plane is the greater the narrower the angular characteristics FA1 (.α) and FA2 (.α) are. As assumed for the majority versions of the ILS system, the corresponding summation characteristic F. (.α) ≡ FA1 (α0 + .α) + FA2 (α0 − .α) has 3 dB angular width, which does not exceed 12◦ , as shown in Fig. 17.9. Considering such narrow summation characteristics, locating the correct angular approach sector by an airplane is a demanding task. However, this is facilitated by a clearance signal emitted by an additional radio beacon in a relatively wide 70◦ angular sector. A clearance signal is emitted at a carrier frequency f 0c , which is very similar to a frequency of the localizer array antenna ( f 0l − f 0c = ±10 kHz), by two independent beams, hatched in Fig. 17.9, while the signal emitted by the beam FC1 (α) is amplitude modulated acoustic signal with a frequency F1 = 90 Hz. Likewise, the signal emitted by the beam FC2 (α) is amplitude modulated acoustic signal with a frequency F2 = 150 Hz. Considering a difference in the depth of modulation of the clearance signal’s acoustic components and the knowledge of the airport topology, a pilot obtains preliminary information regarding his position with respect to the course plane, which undoubtfully facilitates the aircraft to enter the approach sector within the ILS operating range. A similar problem was discussed

17.1 Instrument Landing System (ILS)

367

Fig. 17.9 The electromagnetic beams of the ILS system

in detail in the beginning of this chapter, see Eqs. (17.1)–(17.5). Due to a relatively large line inclination angle of the FC1 (α) and FC2 (α) characteristics, the clearance signal is weak in the vicinity of the course plane in comparison to the signal received from the localizer array antenna A1 and A2 , and thus has insignificant influence on the direction finding result. Moreover, this signal is additionally attenuated in a non-linear receiver path by a significantly stronger signal received from the already mentioned localizer array antennas. The discussed solution, which uses an additional radio beacon emitting a clearance signal, is broadly implemented in the currently operated ILS system installations. The Determining Principle of an Airplane Position in the Descent Plane In the previous subsection, a method of determining a course plane by comparing the modulation depth of the received signal’s acoustic components, was discussed. The same method is applied in order to determine a descent (elevation) plane, which determines a descent line by intersecting with the course plane. For this purpose, radio beacons A3 and A4 , shaping two relatively narrow beams with elevation sections FA3 (β) and FA4 (β), are implemented. Both glidepath array antennas, i.e. A3 and A4 , are powered with continuous UHF signals which are amplitude modulated by harmonic signals with corresponding frequencies F1 = 90 Hz and F2 = 150 Hz. The carrier frequencies of these signals, included within a range of 328.70−335.00 MHz, are assigned to particular channels which are specified in the third and sixth column of Table 17.2. The direction of the antenna A4 maximum radiation is inclined with respect to the azimuth plane (of the runway) at a slight angle β4 > β0 ≈ 3◦ . Accordingly, the direction of the antenna A3 maximum radiation, defined by the angle β3 , satisfies the condition β3 < β0 , see Fig. 17.10a. The radiation pattern of the antenna A4 , i.e. FA4 (β), intersects the characteristic FA3 (β) of the antenna A3 in the plane of an equal signal (.β = 0) which should coincide with a required descent plane. The characteristics formed by the antennas A3 and A4 should provide a stable coverage in the sector ±8◦ (with respect to the runway axis) in the azimuth plane and 0.45β0 − 1.75β0 in the elevation plane [8, 16]. The power of the electromagnetic wave with horizontal polarization radiated by these antennas is of the order of 5 W, so a range given along the direction .α = 0 should be no lesser than 10 NM (18.52 km). The radio beacon antennas A3 and A4 form

368

17 Aircraft Landing Aid Systems

a two-element array which is able to form the summation and differential radiation pattern, likewise the localizer array antennas. For this purpose, these antennas are fed with signals similar to those feeding the localizer array antennas. Assume that the signal received by the on-board receiver from the radio beacon A3 can be expressed as follows (.) u 3. (t, .β) = u (.) 3 (t, .β) + u 3 (t, .β)

(17.23)

for: (.) u (.) 3 (t, .β) = Um F A3 (β0 + .β − β3 )[1 + m 1 cos(.1 t)

+ m (.) 2 cos(.2 t)] cos(ω0g t)

(17.24)

(.) u (.) 3 (t, .β) = Um F A3 (β0 + .β − β3 )[m 1 cos(.1 t)

− m (.) 2 cos(.2 t)] cos(ω0g t)

(17.25)

A signal reaching the receiver from the radio beacon A4 looks alike (.) u 4. (t, .β) = u (.) 4 (t, .β) + u 4 (t, .β)

(17.26)

for: (.) u (.) 4 (t, .β) = Um F A4 (β4 − β0 − .β)[1 + m 1 cos(.1 t)

Fig. 17.10 Elevation cross-sections of various electromagnetic beams formed by antennas A3 and A4 and spectra of the signals emitted by them; a beams of a simplest antenna solution, b beams of the antenna solution more useful for practice

17.1 Instrument Landing System (ILS)

+ m (.) 2 cos(.2 t)] cos(ω0g t)

369

(17.27)

(.) u (.) 4 (t, .β) = Um F A4 (β4 − β0 − .β)[−m 1 cos(.1 t)

+ m (.) 2 cos(.2 t)] cos(ω0g t)

(17.28)

The sum of the components (17.24) and (17.27), i.e. [ ] (.) u . (t, .β) = Um F. (.β) · 1 + m (.) 1 cos(.1 t) + m 2 cos(.2 t) cos(ω0g t) = Um F. (.β) cos(ω0g t) + 0.5Um F. (.β)m (.) 1 cos[(ω0g + .1 )t] + 0.5Um F. (.β)m (.) 1 cos[(ω0g − .1 )t] + 0.5Um F. (.β)m (.) 2 cos[(ω0g + .2 )t] + 0.5Um F. (.β)m (.) 2 cos[(ω0g − .2 )t]

(17.29)

forms a summation directivity pattern F. (.β) ≡ FA3 (β0 + .β − β3 ) + FA4 (β4 − β0 − .β) in the space, with an elevation section similar to the one shown in Fig. 17.10b. Likewise, the sum of the components (17.25) and (17.28) i.e. [ ] (.) u . (t, .β) = Um F. (.β) m (.) cos(. t) − m cos(. t) cos(ω0g t) 1 2 1 2 = 0.5Um F. (.β)m (.) 1 cos[(ω0g + .1 )t] + 0.5Um F. (.β)m (.) 1 cos[(ω0g − .1 )t] − 0.5Um F. (.β)m (.) 2 cos[(ω0g + .2 )t] − 0.5Um F. (.β)m (.) 2 cos[(ω0g − .2 )t]

(17.30)

forms the so-called differential directivity pattern F. (.β) ≡ FA3 (β0 + .β − β3 ) − FA4 (β4 −β0 −.β). The frequency spectra of the signals (17.29) and (17.30) specified for different values of the elevation deviation .β, are shown on the left side of Fig. 17.10b. On the right side of this figure, the spectra of the signals (17.29) and (17.30) sum, received simultaneously by the on-board receiver, are demonstrated. Considering the sum of signals (17.29) and (17.30), i.e. u(t, .β) = u . (t, .β) + u . (t, .β) = Um F. (.α) cos(ω0g t) (.) + 0.5Um [F. (.β)m (.) 1 + F. (.β)m 1 ] cos[(ω0g + .1 )t] (.) + 0.5Um [F. (.β)m (.) 1 + F. (.β)m 1 ] cos[(ω0g − .1 )t] (.) + 0.5Um [F. (.β)m (.) 2 − F. (.β)m 2 ] cos[(ω0g + .2 )t] (.) + 0.5Um [F. (.β)m (.) 2 − F. (.β)m 2 ] cos[(ω0g − .2 )t]

(17.31)

370

17 Aircraft Landing Aid Systems

a difference in the depth of modulation with the signals of acoustic frequencies F1 and F2 , is determined. For this purpose, formula (17.31) should be expressed as follows [ ] u(t, .β) = Um F. (.β) 1 + M1β cos(.1 t) + M2β cos(.2 t) cos(ω0g t) (17.32) for: M1β =

(.) (.) F. (.β)m (.) F. (.β)m (.) 1 + F. (.β)m 1 2 − F. (.β)m 2 , M2β = F. (.β) F. (.β) (17.33) M1β + M2β < 1

As most frequently assumed in practice, m (.) = m (.) = m (.) = m (.) = m, thus 1 1 2 2 formulae (17.33) are simplified to the following form M1β =

2m FA4 (β4 − β0 − .β) 2m FA3 (β0 + .β − β3 ) , M2β = F. (.β) F. (.β)

The determined difference in the depth of modulation .Mβ = M1β − M2β = 2m

F. (.β) F. (.β)

(17.34)

satisfies the conditions of the direction finding characteristics [1, 7, 16]. One of the main causes of direction finding errors, both in the azimuth and elevation planes, are spurious signals which are created due to reflections from the earth’s surface and objects that ale placed on it. Limiting the reflections influence by increasing the angle β3 is impossible because β3 < β0 ≈ 3◦ . Forming a very narrow electromagnetic beam (in the elevation plane) with as possibly low level of side lobes in the UHF range as possible, and providing stationarity of this beam in a changing environment, is a demanding technical task. In order to find an effective solution to this task, recognizing the mechanism of the reflected signals influence on the resultant difference in the depth of modulation, is required. Assume that the airplane is at point P1 , being placed on the trajectory forming an angle .β with the GP glide path, see Fig. 17.11. Assume additionally that the on-board receiver receives the directly incoming signals and signals reflected from a point object Oi , placed on a line inclined with respect to a glide path GP, at an angle .βi(ob) . The difference in the depth of modulation, which is determined considering the directly incoming signals .Mβ (.β), is specified by formula (17.34). The same formula determines the difference in the depth of modulation of the signals reaching the point Oi . The signals reflected from the object Oi reach the on-board receiver along the same straight line Oi − P1 , preserving the difference in the depth of modulation. The amplitudes of these signals will be significantly reduced since only a small energy fraction of the incident signal

17.1 Instrument Landing System (ILS)

371

will propagate towards the airplane. Thus, assume that the simultaneous processes of “dissipative reflection” and omnidirectional scattering are characterized by the equivalent reflection coefficient .i . This means that the amplitudes of modulation components of signals reaching the receiver indirectly, are .i -times smaller in comparison to the corresponding amplitudes of these signals, just in front of the object Oi . All signals reaching the antenna of the on-board receiver generate an electromotive power proportional to the vector sum of their components. Consequently, the signal incoming indirectly causes a spurious change of difference in the depth of modulation by a certain value which is proportional to the reflection coefficient .i and the phase shift ϕi between the phases of the signals reaching the receiver directly and indirectly, i.e. by reflecting from the object Oi . Such reasoning enables one to express the resultant of difference in the depth of modulation with the following formula .Mβ(t) = .Mβ (.β) + .i .Mβ (.βi(ob) ) cos(ϕi )

(17.35)

Formula (17.35) can be generalized to a case in which a simultaneous reflection from many point objects occurs, both scattered and focused. Thus, considering the reflection from N objects, the following formula can be expressed .Mβ(t) = .Mβ (.β) +

N .

.i .Mβ (.βi(ob) ) cos(ϕi )

(17.36)

i=1

A complete, quantitative analysis of the impact of reflections from terrain obstacles on the resultant of difference in the depth of modulation, is not possible. Numerous computer simulations and experimental tests confirmed that due to terrain reflections, the surface of an even signal (β = β0 ) is unevenly folded with respect to the assumed

Fig. 17.11 Graphic illustration of the reflection from an to terrain obstacle; a a side view, b the waviness of a glide path due to terrain reflections

372

17 Aircraft Landing Aid Systems

Fig. 17.12 The view of typical ILS antennas; a localizer array antenna, b glide path array antenna

flat descent surface. The folded surface section in the course plane is similar to the one shown in Fig. 17.11b. The section of an even signal surface in the azimuth plane indicates a similar course as well, which is demonstrated by the mechanism of terrain obstacles reflections, identical to the one discussed. One of the most effective methods of eliminating the impact of the discussed reflections, is to prevent their occurrence. For this purpose, more complex antenna systems are built in order to compensate the electromagnetic field by the earth’s surface, i.e. for 0 ≤ β ≤ 0.3β0 ≈ 1◦ . An example of such system is the system of three identical reflector antennas, shown in Fig. 17.12b, placed at heights h1 = h, h2 = 2h and h3 = 3h above the wellconducting earth surface [1, 16]. Considering this three-element array, the highest antenna serves mainly as a compensating antenna. Due to the proper amplitude-phase power supply of these antennas, the required distribution of the electromagnetic field in the elevation plane is obtained, including its compensation close to the earth’s surface. Furthermore, a proper leveling of the extensive area around the runway, is required. The condition of this area should be systematically examined and any terrain obstacles removed. The Structural Solutions Examples of Some ILS Devices According to the previously discussed ILS system operation, its main devices include antennas determining planes of equal signals, coinciding with the course planes (.α = 0) and descent planes (β0 , .β = 0). Figure 17.12a shows a typical antenna system structural solution of a localizer array antenna, located at the end of the runway. On the other hand, Fig. 17.12b shows the example of structural solutions for the glidepath array antennas, which are placed beside the runway, most frequently 300 m from the runway threshold and approximately 100 m from its center line. For some airports, for example military ones, these distances can be significantly shorter [10, 16]. Main Weaknesses and Limitations of the ILS Application In over sixty years of ILS system operation, both aviation and radiocommunication have faced an intensive and turbulent development, the effects of which are visible

17.2 Microwave Landing System (MLS)

373

almost everywhere. Undoubtedly, it is no wonder that from the perspective of current times this system is perceived as outdated. The main disadvantages of ILS include: • • • •

relatively low frequency of operation, a small number (40) of available channels, constant descent line (path) β0 inclination, high sensitivity to signals reflected from the earth’s surface and airport infrastructure, • difficulty in providing the far radiation zone for radio beacons placed near the runway threshold, • a spatially developed large dimensions antenna system • large preparation costs and proper maintenance of the area that adjacent to the runway. In the light of the enlisted disadvantages and limitations regarding the ILS application, the discussed system appears to be a “fading star”. This conclusion is supported by the fact that currently, out of the approximately 5700 public airports in the US, only a slightly more than 1000 use its latest installations. Most frequently, ILS is duplicated or completely replaced with more modern systems described in the following subsections.

17.2 Microwave Landing System (MLS) Nowadays, different versions of the Australian Microwave Landing System (MLS) are implemented, mainly at larger airports, including military airports adapted for space shuttles landing. The first name of this system was referred to as Time Reference Scanning Beam (TRSB) which fell the initial period of its development. The operational and technical assumptions, adopted in 1978, were made considering the results of the ILS system operational researches, conducted over the course of years. Consequently, the organizational structure and principles of operation of MLS particular subsystems (devices), were impacted, which will be discussed in the section that follows. MLS Organizational Structure Likewise ILS, the MLS system consists of two segments, namely the ground one, located near the runway and the receiver with the antenna placed on board the aircraft. The basic version of the ground segment consists of the following devices (subsystems), the operation of which is strictly synchronized in the time domain. 1. A microwave station emitting an unmodulated microwave signal in the area of a narrow fin beam moving in the azimuth plane (AZ). 2. A microwave station emitting an unmodulated microwave signal in the area of a narrow fin beam moving in the elevation plane (EL).

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17 Aircraft Landing Aid Systems

Fig. 17.13 Fundamental functional devices of the MLS located around a runway

3. Distance Measuring Equipment (DME, DME/P). 4. Microwave radio beacon emitting preamble and illuminating signals. The extended version of the ground segment has additionally two microwave stations emitting electronically controlled beams in the rear area (also referred to as return azimuth and elevation area) and an alignment radio beacon. Figure 17.13 shows a typical arrangement of discussed devices around a runway RW/LW. The presented sketch does not include the illuminating radio beacon, the location of which is shown in Fig. 17.20. The indicated CU and FG components correspond to a central Control Unit and the Flare Guidance, respectively. The central unit of the system is connected with all its components by underground channels, which most frequently correspond to coaxial lines. System remote control performed from the control tower, is possible with the CU unit. Considering the indications of the MLS on-board segment devices, three basic navigation parameters are determined, namely the angular deviations .α and .β, and the distance to the Glide Path Intercept Point (GPIP). The distance measurement of an aircraft from the GPIP is achieved by DME/P, operating in the L-band. A C-band microwave receiver (integrated with a non-polarization-selective antenna system) is an on-board device particularly developed for the MLS system that receives signals from the previously mentioned microwave azimuth and elevation stations. Due to these signals, determining the angular deviations .α and .β = β − β0 , which are specified with respect to the assumed glide path GP, is possible. The glide path angular coordinates are determined by the angles .α = 0 and β0 with the angle β0 value being selected by the navigator pilot within the acceptable range for the particular type of an airplane. All the discussed components of the ground segment operate cyclically in a strictly defined time intervals, i.e. with time distribution. Consequently, using only one common frequency channel and a single-channel onboard receiver, is possible. The operation of the entire system is managed by its operating system, located in the CU central unit. The operation order of particular subsystems is specified by the protocol, the structure of which is shown in Fig. 17.14.

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375

Fig. 17.14 The time structure of the MLS protocol

The symbols used in Fig. 17.14 correspond to abbreviations of the English language names of the functions (operations) performed, i.e. • • • •

EL (scanning in the Elevation plane) AZ (scanning in the Azimuth plane) BAZ (scanning in the Back Azimuth plane) SUP (Supplement).

The proper navigation parameters are determined in a similar sequence in the on-board receiver. The tasks highlighted in Fig. 17.14, are performed cyclically in order to update the values of the determined parameters. According to the available documentation, the information refresh periodicity regarding the azimuth deviation .α is approximately equal to 13 Hz [10, 14]. Considering the effective system operation, the elevation deviation .β, which is updated with a frequency 39 Hz, is more important. Information regarding the system operation, i.e. about the performed functions (scanning in the azimuth plane, scanning in the elevation plane, scanning in the azimuth plane in the back azimuth area, etc.…) is transmitted to the on-board receiver by a radio beacon with a low directivity antenna. This means that this radio beacon emits the wave in a relatively wide angular sector coinciding excessively the scanning sector by azimuth and elevation microwave stations. Each space scanning cycle is preceded by a radio beacon signal lasting approximately 2560 μs. The main components of this system signal are: a preamble signal, a sector signal and a first test signal, referred to as “to”. The first emitted signal corresponds to a preamble signal containing: • a fragment of the carrier wave used (in order to synchronize the phase of the wave reproduced in the receiver), • a code-modulated time synchronization signal determining the beginning of a given space scanning cycle, • a code-modulated identification signal of the operation (function) that will be performed (e.g. scanning in the elevation plane). The transmitted sector signal consists of scanning sector identification pulses which determine the position of the system’s ground segment antennas and eliminate false bearings when the aircraft is out of the scanning sector. A third component of the system signal is the “to” pulse which determines the direction of the microwave

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17 Aircraft Landing Aid Systems

Fig. 17.15 Illustration of scanning the aerospace area around the airport runway in the azimuth and elevation planes

scanning beam movement in the azimuth plane. In addition, the radio beacon transmits clearance signals at the beginning and end of each scan cycle in the azimuth plane. Space Scanning in the Azimuth Plane The operation of a microwave station scanning a ground sector of space in the azimuth and elevation planes is illustrated in Figs. 17.15 and 17.16, respectively. Scanning in the azimuth plane starts from a position αmin = −40◦ and continues (a) to the position αmax = +40◦ (primary move’ at a constant angular velocity ωsc ◦ “To”). When the position +40 is obtained, a short break (t p ) in the beam movement and in the emission of electromagnetic wave occurs, followed by the “From” to the output position αmin = −40◦ . During this two-phase scanning cycle, the aircraft is illuminated twice by a narrow beam of unmodulated electromagnetic wave, which is obviously registered by the on-board receiver. The received signals are processed in the system as shown in Fig. 17.17, which aims for defining as precisely as possible the time interval ta = ta (.α) between these signals. This interval explicitly defines the angle .α ≡ α. The threshold voltage UT hr fed (α) of from the receiver to the pulse forming circuit, determines the width ti = θ0.5 /ωsc ◦ (α) ◦ the steep slope pulses formed by this circuit. For θ0.5 = 1 and ωsc = 0.02 /μs, time ti = 50 μs. In order to determine the dependence which links the angular deviation .α with the time interval ta , assume that t1 corresponds to a required time of the beam to move from the position determined by the angle .α to the extreme position for which .α = .αmax ≡ αmax . According to this assumption, the time interval ta ,

17.2 Microwave Landing System (MLS)

377

Fig. 17.16 Illustration of the MLS system operating principle; a scanning in the azimuth plane, b measurement of the time interval ta (.α)

Fig. 17.17 The block diagram of an on-board device determining the deviation .α

after which the beam returns to the position .α (during From), is equal to ta (.α) = t1 + t p + t1 = 2t1 + t p

(17.37)

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17 Aircraft Landing Aid Systems

Subject to (17.37), t p is an interval during which a stationary beam is placed at its extreme position. Assume additionally that considering the pulses received by the receiver of an aircraft, placed on the course line, a value of the interval ta (.α = 0) = t0 is known. At time instant t1 , the azimuth microwave station beam sweeps space sector with an angular width equal to (a) αmax − .α = ωsc t1 =

] 1 (a) [ ωsc ta (.α) − t p 2

(17.38)

Formula (17.38) yields ] 1 (a) [ ta (.α) − t p .α = αmax − ωsc 2

(17.39)

In order to eliminate the value αmax from formula (17.39), this value should be expressed by a given interval value ta (.α = 0) = t0 . According to the principle of operation of the discussed system αmax =

1 (a) ω (t0 − t p ) 2 sc

(17.40)

After considering formula (17.40) in Eq. (17.39), the following is obtained .α =

] 1 (a) 1 (a) [ 1 (a) ωsc (t0 − t p ) − ωsc ta (.α) − t p = ωsc [t0 − ta (.α)] 2 2 2

(17.41)

The approach sector scanning cycle completes transmission of the second test signal “from” by the radio beacon which is possible by moving a microwave beam in the azimuth plane. This cycle of approximately 16,000 μs (including the radio beacon signal) is repeated with the frequency of 13 Hz. Space Scanning in the Elevation Plane The operation of the microwave station scanning the approach sector can be analyzed likewise by moving the microwave beam in the elevation plane, Fig. 17.18. Scanning starts with the position βmin = 0.9◦ and continues at a constant angular (β) velocity ωsk towards an upper position βmax ≈ 15◦ . After reaching the upper posi(β) tion, a beam is stopped for a period of t p ≡ t p , during which no electromagnetic wave is emitted. Finally, when the pause is over, the scanning process continues at the (β) same angular velocity ωsc towards a lower position. While the beam is moving up and down, the aircraft is illuminated by a twice unmodulated electromagnetic wave, which excites two pulse signals distant from each other by a time instant tβ ≡ tβ (β) in its receiver. These pulses are subsequently processed in the circuit, as shown in Fig. 17.19, whose function is similar to the circuit shown in Fig. 17.17. Assume that the aircraft angular position in the elevation plane is determined by the angle β. Correspondingly, the angular sector 2(βmax − β) is swept by the beam in (β) a time tβ (β), including the pause time t p . The principle of operation of the discussed

17.2 Microwave Landing System (MLS)

379

Fig. 17.18 Illustration of the MLS system operating principle; a scanning in the elevation plane, b measurement of the time interval tβ (β)

Fig. 17.19 The block diagram of the on-board device determining the angle deviation .β − β0 Fig. 17.20 The azimuth sections of directivity patterns FA1 and FA2 shaped by the backlight antennas

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17 Aircraft Landing Aid Systems

components shows that tβ (β) =

2(βmax − β) (β) ωsc

+ t p(β)

(17.42)

For β = β0 , a reference time tβ (β0 ) = tβ0 =

2(βmax − β0 ) (β)

ωsc

+ t p(β) .

(17.43)

Subtracting Eq. (17.43) both sides from formula (17.42), one obtains tβ (β) − tβ0 =

2(β0 − β) (β)

ωsc

(17.44)

Ultimately, .β = (β − β0 ) =

] 1 (β) [ ωsc tβ0 − tβ (β) 2

(17.45)

A value given by the pilot (navigator) is the time interval tβ0 corresponding to the glide path inclined towards the runway at the required angle β0 . The scanning process of the approach sector by moving the microwave beam in the elevation plane is repeated with frequency 39 Hz. The Principle of Operation of the Radio Clearance Subsystem Likewise the previously discussed ILS system, the MLS system consists of a radio backlight component that facilitates the aircrafts to enter the scanning sector. Assume that the antenna system of this subsystem consists of two separate antennas A1 and A2 with directivity patterns FA1 and FA2 , the azimuth sections of which are shown in Fig. 17.20. According to Fig. 17.20, the directivity patterns FA1 (α) and FA2 (α) are arranged on the left and right sides of the scanning sector in the azimuth plane. The antenna A1 of the backlight system sends a short 50 μs pulse just before starting the scanning process and immediately after its completion, i.e. when a scanning beam is in the position −αmax . As Fig. 17.21 indicates, these pulses are marked with symbols U p1 . (α) = 0.02◦ /μs and αmax = 40◦ time of primary (To) and return (From) motions For ωsc (α) = 4000 μs. are equal to T = 2αmax /ωsc The length of the break t p is roughly equal to 0.1 T and estimates approximately 400 μs. Similar pulses are sent by the antenna A2 when the beam completes scanning with its primary motion (To) and before starting scanning with the return motion (From). When both these pulses, indicated in Fig. 17.21 with symbols U p2 , are transmitted, the beam is placed in the position αmax . Backlight pulses are received by the on-board receiver and their amplitude waveforms depend on the position of an aircraft with respect to the course plane, α = 0◦ . This dependence is demonstrated in Fig. 17.22.

17.2 Microwave Landing System (MLS)

381

Fig. 17.21 Radio pulses of the backlight emitted by antennas A1 and A2 , see Fig. 7.20

Fig. 17.22 Graphic illustration of the principle of operation of the radio backlight system

Considering a time waveform of these pulses and information on the geographical location of the scanning sector at a given airport, a proper decision regarding turning left, flying straight or turning right is made. Such decision shall be taken if the received signals differ in level by more than 15 dB [8, 16]. One should also emphasize that clearance signals do not interfere in determining the navigation parameters, because all MLS subsystems operate within the so-called time distribution. Therefore, the minimum range of the clearance system may be significantly greater than the minimum scanning range in the azimuth plane, i.e. from 20 nm. Examples of Some MLS System Devices Structural Solutions The fundamental devices of the MLS ground segment include two microwave stations with phase-controlled flat antenna arrays. A general view of the microwave station scanning a (3D) space in the azimuthal plane is shown in Fig. 17.23a. Likewise, Fig. 17.23b shows a general view of a microwave station scanning a (3D) space in the elevation plane. Both presented photographs show two pairs of lamps emitting respectively in poor visibility conditions intermittent signal lights. Basic Technical and Operational Parameters of the MLS System Frequency operating range: 5031.0−5090.7 MHz. This scope includes specified 200 transmission channels, arranged 300 kHz apart.

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Fig. 17.23 A general view of the microwave stations of MLS; a microwave station that scanning in the azimuth plane, b microwave station that scanning in the elevation plane

Angular scanning range in the azimuth plane

±40◦

Azimuth beam width in the azimuth plane

θ0.5 = 1.0◦ or 2.0◦

Angular scanning range in the elevation plane

0.9−15◦

Elevation beam width in the elevation plane

θ0.5 = 1.0◦ or 1.5◦

Scanning angular speed ωsk (in both planes)

0.01−0.02◦ /μs

Minimum power of the electromagnetic wave emitted in the beams

Pmin = 40 W

Polarization (electric field) of the emitted wave

−vertical

Azimuth coordinate (.α) refresh information frequency

13 Hz

Elevation coordinate (.β) refresh information frequency

39 Hz

Accuracy in determining the angular deviations .α and .β on glide path

≈ 0.05◦

Minimum range in the front area (20 NM)

≈ 37 km

Minimum range in the rear area (7 NM)

≈ 13 km

Power supply of the azimuth microwave station

≈ 6 kW

Power supply of the elevation microwave station

≈ 5 kW

17.3 Transponder Landing System (TLS) The Transponder Landing System (TLS) was developed in the early 1990s by the American company, Advanced Navigation and Positioning, Corp. as an alternative solution to the now obsolete ILS system. In 1992, the system’s first installations

17.3 Transponder Landing System (TLS)

383

tests were commissioned. In 1998, after six years of trial operation, TLS obtained the Federal Aviation Administration (FAA) certificate, allowing for its operation at small airports of I (lowest) category [12, 17]. The principle of operation of this system is not complicated and consists in determining the angular coordinates of the aircraft position considering a signal sent by its ATCRBS transponder, which is used by the secondary radar system. This signal is a response to a 3/A mode query sent by the ground interrogator. The receiving direction of this signal is determined with a ground-based set of two microwave interferometers cooperating with the central unit. The measurement result in the azimuth α and elevation γ = β angles which enable the calculation of angular deviations .α and .β. These deviations are respectively determined with respect to the course planes and glide paths, i.e. likewise the previously discussed ILS system. The information regarding the angular deviations .α and .β is transmitted to the on-board receiver with two radio signals emulating the relevant signals of the ILS system. As a result, these signals can be received by the ILS receiving apparatus. TLS Organizational Structure The fundamental components of the TLS ground segment consist of: • • • •

Interrogator (INT), Built in test equipment (BITE) Azimuth and Elevation interferometers (AI and EI) radio beacons operating in the VHF and UHF ranges, emitting amplitude modulated signals emulating the ILS signals, • Base Station (BS) that controls the system operation and performs the necessary navigation calculations. Likewise the ILS and MLS systems, in order to determine a distance from an aircraft to the expected glide path intercept point, the DME is used. The mentioned system devices are arranged near the runway as shown in Fig. 17.24.

Fig. 17.24 Fundamental functional devices of the TLS located around a runway

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The on-board segment of the system consists of a multichannel receiver, identical to the receiver used in the ILS system and to a standard Air Traffic Control Radar Beacon System (ATCRBS) transmitter (transponder) sending a response signal as an Instrument Flight Rules (IFR) code. Before the landing approach, a pilot receives information regarding the TLS system installed at the airport and the operating frequency at which radio signals containing navigation information will be sent to this system. The main task of the INT is to activate the on-board transponder. To this end, INT emits in the 90◦ approach sector electromagnetic wave pulses with a carrier frequency f i = 1030 MHz, low repetition frequency (≈ 10 Hz) and low power, which relate to a standard protocol. This reduces the influence of the INT on other systems such as the secondary radar, approach radar and anti-collision radar. Due to the fact that the TLS system uses a standard ATCRBS transponder, the format of signals emitted by the ground interrogator is required to be compatible with the 3/A format used in the secondary radar system, which is referred in the military version to as the Identification Friend or Foe (IFF). In addition, these signals must be emitted in a strictly defined manner in the approach plane, in order to eliminate responses of airplanes placed outside that area. The signal emitted by the on-board transponder with a carrier frequency f t = 1090 MHz is received by the antenna system consisting of a set of two microwave reflectometers. These reflectometers determine an aircraft position in three-dimensional space using phase direction finding method, with respect to the coordinate system associated with them. This position is specified by the azimuth angle α, elevation angle γ and distance. The last of the mentioned parameters is measured by the DME cooperating with the system. Considering the discussed navigation parameters, a ground distance is determined in the azimuth plane. Consequently, for the specified distance values, see Table 17.3, radio transmission of relevant acoustic signals (Morse code) emulating the marker beacons Am operation, see Fig. 17.3, is possible. A fundamental task of the base station processor is to calculate the geographic coordinates of the current aircraft position, flight altitude H and angular deviations .α and .β. These deviations are determined considering the assumed course and descent planes, the position of which, with respect to the runway RW/LW, can be programmatically changed by the system operator. As mentioned earlier, the data on the aircraft’s position with respect to the course and descent planes are transferred immediately to the relevant on-board navigation devices, which are identical to those used by the ILS system. As a result, the information regarding the angular deviation .α is transmitted to the on-board receiver with a VHF signal similar to the signal (17.3), i.e. u α (t, .α) = Um FAL (.α)[1 + m 1α cos(.1 t) + m 2α cos(.2 t)] cos(ω0l t) (17.46) for: FAL (.α)—VHF radio beacon radiation pattern

17.3 Transponder Landing System (TLS)

385

ω0l = 2π f 0l , .1 = 2π F1 , F1 = 90 Hz, .2 = 2π F2 , F2 = 150 Hz f 0l —center frequency of one of the 40 channels indicated in Table 17.2, m 1α = m 0α [1 + sin(.α)], m 2α = m 0α [1 − sin(.α)], m 0α = 0.5(m 1α + m 2α )—average value of the gap factor, should be selected so as to satisfy the condition m 1α + m 2α < 1. In the on-board receiver which is identical to the ILS system receiver, considering the signal (17.46), the normalized modulation difference in the depth of modulation is determined .m α =

m 1α − m 2α = sin(.α) m 1α + m 2α

(17.47)

Then the angular deviation .α = arcsin(.m α ) is calculated. The average value of the gap factor m 0α should be as large as possible in order to ensure sufficient accuracy in determining the difference .m α . However, one should remember that the factor m 0α should always be lower than 0.5. The information regarding angular deviation .β is transmitted likewise. For this purpose, a UHF signal, similar to signal (17.46), is transmitted to the on-board receiver i.e. [ ] u β (t, .β) = Um FAG (.β) 1 + m 1β cos(.1 t) + m 2β cos(.2 t) cos(ω0g t) (17.48) for: FAG (.β)—UHF radio beacon radiation pattern ω0g = 2π f 0g , .1 = 2π F1 , F1 = 90 Hz, .2 = 2π F2 , F2 = 150 Hz, f 0g —center frequency of one of the 40 channels indicated in the third and sixth column of Table 17.2, m 1β = m 0β [1 + sin(.β)], m 2β = m 0β [1 − sin(.β)], m 0β = 0.5(m 1β + m 2β )—average value of the gap factor, should be selected so as to satisfy the condition m 1β + m 2β < 1. Also in this case, an average value of the gap factor m 0β should be as large as possible in order to provide sufficient accuracy in determining the difference .m β . The average value of the gap factor m 0β should be lower than 0.5. Considering signal (17.48), the normalized difference in the depth of modulation is determined in the on-board receiver .m β =

m 1β − m 2β = sin(.β) m 1β + m 2β

(17.49)

Subject to (17.49), the deviation .β = arcsin(.m β ) is calculated in the on-board receiver. Measurement of the Angular Coordinate of an Aircraft’s Position in Two-Dimensional Space with an Interferometer Considering interferometers that use the phase direction finding method, the angular coordinate of an object in a selected plane is determined taking the phase difference

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17 Aircraft Landing Aid Systems

of signals received by two identical antennas into account. The phase centers of these antennas are distant by a distance d, hereinafter referred to as the base, see Fig. 12.4. Assume that the radiation patterns of both considered antennas are identical in a given coordinate system. This assumption also applies to directions of both antennas maximum radiation, which should be parallel to each other as well as to the normal n. Taking this assumption into consideration, signals received in the receiving channels, connected to each of the antennas, have the same amplitude value and different signal phase values. The estimated phase difference of the signals depends on the signal’s receiving direction. This corresponds to paths difference, the distance of which is indicated by the electromagnetic wave front (an identical phase plane) before reaching the phase center of each antenna, Fig. 12.4b. The distance r1 from point O to the phase center A1 of the first antenna differs from the distance r2 to the phase center A2 of the second antenna, by value .r which depends on the angular position of an object at point O. ( ) ( ) d d .r = r1 − r2 = r0 + sin(θ ) − r0 − sin(θ ) = d sin(θ ) 2 2

(17.50)

The distance difference covered by the electromagnetic wave front introduces a phase shift .ψ which depends on the received electromagnetic wave length, i.e. .ψ =

2π 2π .r = d sin(θ ) λ λ

(17.51)

The demonstrated formula (17.51) facilitates determining the position of an object when being placed on the measured plane. In order to provide a correct system operation, a phase discriminator operating properly in the relevant range changes of the received signal phase, is implemented. Assume that the angle of an object is within the range from −π/2 to π/2, then the phase difference .ψ of the received signals estimates in the range [−2π d/λ; 2π d/λ]. The angle discriminator, physically implemented, operates properly, i.e. giving an explicit result, if −π/2 ≤ .ψ ≤ π/2. This range corresponds to the range of the angle changes θ , which is contained within [−θmax ; θmax ]. The maximum value of the angle θ can be determined from the equation θmax = arcsin[λ/(4d)]. Formula (17.51) is an odd function with respect to θ and is equal to zero for θ = 0. The slope of this function (direction finding characteristics) is a sensitivity measure and, indirectly, the accuracy measure of the discussed method. d d[.ψ(θ )] = 2π cos(θ ) dθ λ

(17.52)

The analysis of Eq. (17.52) shows that in order to develop the method sensitivity, the base length d should be increased. Unfortunately, this results in narrowing the range [−θmax ; θmax ] of an explicit measurement. Thus, in order to provide the

17.3 Transponder Landing System (TLS)

387

Fig. 17.25 The block diagram of the three antenna interferometer

uniqueness of an angle θ measurement and the highest possible sensitivity simultaneously, the three-antenna systems are frequently constructed, i.e. with a relatively large d1 + d2 and low d2 bases, arranged along a straight line as shown in Fig. 17.25. Considering phase differences ϕ1 and ϕ3 of signals received by the antennas A1 and A3 , the coordinate θ is determined with high accuracy, however this estimation is not explicit. Such ambiguity is eliminated by using the phases ϕ2 and ϕ3 of signals received by the antennas A2 and A3 . Measurement of Two Angular Coordinates of an Aircraft’s Position in Three-Dimensional Space with a System of Two Interferometers In the case of objects located in three-dimensional (3D) space, two pairs of antennas arranged in two mutually perpendicular planes, are used to determine the angular coordinates ϕ and γ, see Fig. 12.5. The signals phase difference in the pair of antennas A1 and A2 and the signals phase difference in the pair of antennas A3 and A4 can be determined considering the path difference from the projection of the wave path onto the plane x0y to the phase centers of the relative antennas located in this plane. .ψ1−2 =

2π 2π d cos(ϕ) cos(γ ), .ψ3−4 = d sin(ϕ) cos(γ ) λ λ

(17.53)

Considering the estimated phase shifts .ψ 1-2 and .ψ 3-4 , the values of the angles ϕ and γ can be determined. ] ) [ ( λ . .ψ3−4 2 2 , γ = arccos ϕ = arctg (.ψ1−2 ) + (.ψ3−4 ) .ψ1−2 2π d

(17.54)

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17 Aircraft Landing Aid Systems

Fig. 17.26 A view of fundamental components of the TLS. a the antenna system of the interrogator, b antennas of the azimuth interferometer, c the elevation interferometer

The angles (17.54) correspond to the estimates of the specified angular coordinates, respectively in the azimuth and elevation planes [14]. Examples of Structural Solution for Some TLS System Devices As explained at the beginning of Sect. 17.3, the fundamental radio devices of the TLS ground segment include: an interrogator (INT), azimuth interferometer (AI), elevation interferometer (EI), VHF radio beacon which transmits the azimuth deviation .α information, UHF radio beacon which transmits the elevation deviation .β information, built in test equipment (BITE) and base station (BS) that controls the operation of the enumerated devices and performs the necessary navigation calculations. All the mentioned devices are related to the base station with coaxial lines laid in the ground. Figure 17.26a shows the constructional structure of the INT antenna system. According to [12], each of the two vertically arranged columns of an antenna array shown in Fig. 17.26a, contains 6 powered (controlled) radiating elements, respectively. The height of columns, including the frame these columns are placed on, does not exceed 2 m. A single wire antenna (with omnidirectional radiation pattern in the azimuth plane) placed above one of these columns, serves as a SLS, emitting a radio pulse P2 of the interrogation signal in the 3/A mode. As in the case of the secondary radar, the discussed antenna array emits a vertically polarized electromagnetic wave with a carrier frequency f = 1030 MHz, which is modulated in time and space. As mentioned previously, the SLS shapes the omnidirectional reference beam coinciding excessively all side lobes of the interrogate beam. Both radiation patterns, shaped by the discussed antenna system, have a similar, fin-type shape in the elevation plane with maximum radiation directions inclined at an angle +3◦ with respect to the horizontal plane, i.e. runway plane. The azimuth reflectometer system consists of three vertically oriented antennas, each of them, taking the form of linear array of 6 radiating elements, facilitates wave reception of a vertically polarized carrier frequency f = 1090 MHz. These antennas are arranged on a common base, as shown in Fig. 17.26b.

17.3 Transponder Landing System (TLS)

389

The height of these antennas including the frame these columns are placed on, does not exceed 2 m. One of the extremely positioned antennas of the discussed interferometer, is implemented as a reference antenna and the two remaining ones, as measurement antennas. They are frequently referred to as rough and precise measurement antennas. The signals received by the particular antennas are processed in a three-channel receiver, see Fig. 17.25. The elevation interferometer of the TLS system exploits an antenna array as a 9 m linear array consisting of four subarrays, known as reference antenna and measurement antennas, respectively. The overall structure of the discussed antenna system is shown in Fig. 17.26c. As demonstrated, the lowest is the reference antenna, subsequently a low resolution, medium resolution and high resolution antenna. The signals received by these antennas are simultaneously processed in a four-channel receiver. Serial processing in a system of a two-channel receiver preceded by an electronic commutator system connecting the selected antenna to the measurement channel, is possible as well. Both the reference antenna as well as each of the three measurement antennas consists of 10 respectively connected radiating elements. The calibration/built in test equipment station (BITE), see Fig. 17.24, is placed at a sufficient distance in front of the antennas of the azimuth and elevation interferometers. This station sends a test signal to antennas of the mentioned interferometers, at a carrier frequency f = 1090 MHz in order to determine their operation accuracy and, if necessary, to implement essential corrections. The base station managing the operation of the entire system is placed in an air-conditioned container, which also includes all the transmitting and receiving devices of the system, i.e. the interrogator transmitter, VHF and UHF radio beacon transmitters, the UHF radio link transmitter and both interferometers receivers. Next to the discussed container of the base station, there is a 9 m tower with VHF, UHF radio beacon antennas and UHF radio lines antennas placed on [12, 17, 18]. The particular components of the military version of this system, known as “Firefly”, are constructed in a similar manner as well [17, 18]. All of them can be transported simultaneously with a small trailer pulled by an off-road vehicle. As a result, the transport by airway is significantly facilitated. For instance during Balkan conflict the Firefly was transported to Dubrovnik airport by a C-130 aircraft and assembled completely by two men within four hours. The Main Disadvantages and Limitations of the TLS System Implementation At a given time instant, the system’s ground station cooperates with only one interrogator, which means that the position of only one aircraft can be determined and the system guides it to the approach sector and subsequently to the glide path, which can be modified during landing. The operation of other airplanes in the approach area is implemented with time division. Thus, the TLS system is primarily exploited at small airports with low air traffic.

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17.4 Ground and Satellite Based Augmentation Systems Used for a Precision Approach On the basis of reading the previous chapters on ILS, MLS and TLS systems, it can be concluded that the technical infrastructure of these systems is very complex and quite expensive to operate. Therefore, with the advent of satellite navigation systems, the idea was born to adapt them to support the precise approach and landing of aircraft. However, the main problem was that since GPS was not designed to be a safety of life system, it can sometimes provide misleading information. In order to avoid such a dangerous situation, it was proposed to supplement the system with a network of monitoring stations that would send flags to the user in the event of a complete system failure or incorrect operation. It was then realized that this network could also correct errors, leading to better accuracy and availability. Finally, the idea of broadcasting the navigation data and flags from a geostationary satellite was proposed. All three of these ideas are the key components of SBAS being dedicated for aviation needs. The communication network of SBAS carries the navigation data to and from the master station. It needs to be redundant and reliable because the information is time critical, so it cannot get lost or delayed. Consequently, the communication network must fulfil rigorous requirements in terms of latency (no more than 50 ms in the case of WAAS) and reliability. WAAS requires that more than 99.9% of messages reach their intended destination on time along each channel and that the two parallel channels achieve at least 99.999% reliability. In other words the SBAS was especially designed to replace a large number of distributed terrestrial navigational aids with a single integrated system. A list of eliminated in this manner radio devices is rather long and includes: Non-Directional Beacons (NDBs), Distance Measuring Equipment (DME), Tactical Air Navigation systems (TACANs), VHF Omnidirectional Range systems (VORs), and Category I Instrument Landing Systems (ILSs) [3, 5]. Generally an SBAS is capable of providing guidance for all phases of flight including takeoff, ascent, enroute, terminal area, and approach. An SBAS has a strict upper limit on the length of time that erroneous information could be presented to the pilot. The Time-To-Alert (TTA) for an SBAS is six seconds in order to support operations where the aircraft is near to the ground. Each SBAS also evaluates the effects of the ionosphere on the ranging signals. Differential corrections and confidence bounds are produced to improve the nominal positioning accuracy and alert the user when the ionosphere may be creating unacceptably large errors. SBAS has been used for many years to guide aircraft both at altitude and to within 200 ft of the ground. From the observations it was found that SBAS allows to determine the user’s position with an accuracy of about 2 m in the horizontal as well as vertical planes [2, 9]. Currently, only approaches of category 1 using SBAS have regularity of ICAO. Precisely, these regulations apply mainly to the Wide Area Augmentation System (WAAS) covering North America and the European Geostationary Navigation Overlay Service (EGNOS) covering Europe. The principle of operation of SBAS

17.4 Ground and Satellite Based Augmentation Systems Used …

391

Fig. 17.27 Local area augmentation system supporting landing of aircrafts; a a scheme of the organization structure, b a view of the reference station integrated with the VHF radio link

is described in Sect. 16.5.1. Therefore, only the functional structure and the principle of operation of its simplified version, known as the Ground Based Augmentation System (GBAS), are presented below. At this point, it’s worth noting that the US Federal Aviation Administration (FAA) initially called GBAS as LAAS—Local Area Augmentation System. The application of GBAS to precision approach is described as the GBAS Landing System or shortly as GLS. A GBAS provides differential corrections and integrity monitoring of Global Navigation Satellite Systems (GNSS) data using as input data either three or four GNSS satellite signals received by reference stations. The differential correction message computed from this data is then continually broadcast omni-directionally (twice every second) by a ground transmitter using a VHF frequency broadcast (VDB: 108.025−117.957 MHz) which is effective within an approximate 23 nm radius of the host airport, Fig. 17.27. As a rule, the leading reference station (master) is integrated with the information processing center (GF), as it is illustrated in Fig. 17.27a [4]. The adjacent drawing, i.e. Fig. 17.27b, shows a design solution of one of the reference stations integrated with the VHF radio link. GBAS is mainly used to facilitate GNSS-based precision approaches, which are more flexible in designing than is possible with ILS. While the main purpose of GBAS is to ensure signal integrity, it also increases signal accuracy, with position errors shown below one meter in both the horizontal and vertical planes. One GBAS technical infrastructure at the airport supports the approach and landing of aircraft on multiple runways, as well as departures from multiple runways and ground movement for all GBAS equipped aircraft. Overall, GBAS appears to be a good alternative both technically and economically. Only one radio station can serve an airport with several runways. But this system also has a major drawback, namely it is more prone to RF interference than ILS. Moreover, the navigation signals received from GNSS may also be disturbed by local jamming that is not controlled by the airport operator. One parasitic emitter can disable the entire infrastructure. Due to these drawbacks, GBAS

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17 Aircraft Landing Aid Systems

is unlikely to completely replace ILS or MLS in the near future at category II or III aerodromes. In other words, research leading to the elimination of these and similar significant drawbacks should be carried out due to the benefits of GBAS [4].

References 1. Bakuljiev PA, Sosnovskij AA (2005) Radionavigation systems. Publishing House “Radiotekhnika”, Moscow (in Russian) 2. De Blas FJ, Sanchez MA (2010) The EGNOS services provision within the single European sky—the start of the safety-of-life service. In: Proceedings of the 23rd international technical meeting of the Satellite Division of the Institute of Navigation, Portland, OR, Sept 2010, pp 1984–1993 3. Ground Based Augmentation System (GBAS), German Aerospace Center (DLR), GBAS_EN_2019. https://www.dlr.de/fl/en/Portaldata/14/Resources/dokumente/veroeffen tlichungen/GBAS_EN_web.pdf 4. Heinke T, Feuerle T. Flight inspecting ground based augmentation systems (GBAS). http:// www.icasc.co/sites/faa/uploads/documents/resources/ 5. Holland F et al (1973) Structure of the airspace. IEEE Trans Commun 21(5):382–398 6. International standards and recommended practices and procedures for air navigation services: aeronautical telecommunications, Annex 10 to the convention on international civil aviation, 5th edn., vol 1. Part 1—equipment and systems (1996). International Civil Aviation Organization 7. Johnson RC (ed) (1993) Antenna engineering handbook, 3rd edn. McGraw-Hill, Inc., New York 8. Kayton M, Fried WR (1996) Avionic navigation systems, 2nd edn. Wiley, New York 9. Lawrence D et al (2007) Wide Area Augmentation System (WAAS)—program status. In: Proceedings of the 20th international technical meeting of the Satellite Division of the Institute of Navigation, Fort Worth, TX, Sept 2007, pp 892–899 10. Moir I, Seabridge A (2006) Military avionics systems. Wiley, New York 11. Parkinson WB, Spilker JJ (1996) Global positioning system, theory and applications. American Institute of Aeronautics and Astronautics, Washington 12. Picou G. Transponder landing system—a new approach using familiar radios. www.gaavon ics.com/tls.htm 13. Rosłoniec S (2006) Fundamentals of the antenna technique. Publishing House of the Warsaw University of Technology, Warsaw (in Polish) 14. Rosloniec S (2020) Fundamentals of the radiolocation and radionavigation, 2nd edn. Publishing House of the Military University of Technology, Warsaw (in Polish) 15. Sauta OI et al (2019) Principles of radio navigation for ground and ship-based aircrafts. Springer, Heidelberg 16. Sosnovskij AA (ed) (1990) Aviation radionavigation—handbook. Publishing House “Transport”, Moscow 17. TLS—Transponder landing system. Advanced Navigation and Positioning Corporation. http:// www.anpc.com/wp-content/uploads/2012/01/TLS_ANPC_English.pdf 18. Winner K. Application of the transponder landing system to achieve airport accessibility. Advanced Navigation and Positioning Corporation. https://www.scribd.com/document/862 13795/Application-of-the-Transponder-Landing-System-to-Achieve-Airport-Accessibility

Chapter 18

Radio Beacons and Distance Measuring Equipment Supporting Flight and Landing of the Aircrafts

The present chapter discusses the topic of VHF Omni Directional Range (VOR, D-VOR). The first discussed radio beacon refers to a phase radio beacon, operating according to a phase method. Next the principle of operation of radio beacon using the Doppler effect is discussed, which is known as Doppler VOR beacon. The last and the most precise radio beacon is a pulse radio beacon, i.e. operating according to the amplitude-pulse method. As a general rule, the mentioned radio beacons are integrated with a distance measuring equipment, forming the angulardistance VOR/DME or D-VOR/DME sets. Each of these sets uses radio waves with carrier frequencies in the band 108–117.9 MHz, whereby a part of this band, namely 108–112 MHz, is also used by the ILS localizer array antenna, see Table 17.2. Thus, considering the following sub-band, the frequency channels of the VOR/DME set are arranged every 200 kHz. As with the previously discussed ILS, MLS and TLS systems, VOR/DME set cyclically sends its identification signal in the form of three letters of the Morse alphabet. Considering the location, the power of the electromagnetic wave, radiated by VOR, ranges from 100 to 200 W. Furthermore, the radio beacons used are marked with power equal to 50 W and with symbol indication T-VOR. As a general rule, these radio beacons are installed near the runway. The signals emitted by VOR/DME set carry information regarding the angle (direction) α, in which the wave is currently radiated, and about the time interval ∆t. Considering ∆t, a slant range R ≡ |P V | between the radio beacon V and the airplane P is determined, Fig. 18.1. Consequently, determining the current position of an aircraft with respect to the known geographic position of VOR/DME set, is facilitated. The reference system of a passing aircraft is always considered as the surface of the earth, together with the superimposed geographic coordinate system. According to this convention, | | the position of an aircraft is explicitly determined by the flight altitude H ≡ | P P ' | and the geographical coordinates of the point P ' , in which the perpendicular P−P ' (passing through the aircraft’s center of gravity) intersects the earth’s surface. As assumed, the planes S2 and S3 , shown in Fig. 18.1, intersect along the axis z and are perpendicular to the horizontal plane S1 . On the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6_18

393

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18 Radio Beacons and Distance Measuring Equipment Supporting …

Fig. 18.1 Graphical illustration of a slant (oblique) (range R ≡ |P V | between the radio beacon V and the aircraft P

Fig. 18.2 The radionavigation scenario with two radio beacons V1 and V2 . a Definition of oblique ranges and their of perpendicular projections, b definition of angles α1 , α2 , ϕ1 , ϕ2 and ψ

other hand, the plane S1 is perpendicular to the axis z which coincides with the direction of gravity. These assumptions are fully justified considering the negligibly low curvature of the Earth, which is usually satisfied when using short-range radio navigation systems. The azimuth angle α indicates a navigation parameter determined on board the aircraft and is defined in the horizontal plane | S1| between √ the direction of true North N and a straight line with a segment D = | P ' V | = R 2 − H 2 placed on it, for H corresponding to the flight altitude measured by the on-board altimeter. Considering specified time interval ∆t = R/c, a ground distance D is calculated simultaneously. The point P ' and height H , explicitly determine the position of an aircraft with respect to the known, geographical location of the VOR/DME set. Such location can be also determined if two independent bearings, performed with respect to two nearest radio beacons with known geographic positions, are considered, see Fig. 18.2.

18.1 Phase Radio Beacon

395

In this case, the known values correspond to the base length d0 and its slope ψ with respect to the meridian line, coinciding with north direction N . The distances D1 and D2 indicated in Fig. 18.2b, are calculated according to the following formulae [6] D1 = d0

sin(ϕ2 ) , sin(ϕ1 + ϕ2 )

D2 = d0

sin(ϕ1 ) sin(ϕ1 + ϕ2 )

(18.1)

Subject to (18.1), ϕ1 and ϕ2 are complementary angles which are marked in Fig. 18.2b. Considering the navigational situation as in Fig. 18.2b, the angles ϕ1 and ϕ2 are related to the obtained bearings α1 and α2 as follows: ϕ1 = 180◦ − ψ − α1 and ϕ2 = 180◦ − ψ − |α2 |. The position lines corresponding to the ground distances D1 and D2 are arcs of circles with radii r1 = D1 and r2 = D2 whose centers coincide with the phase centers of the radio beacon antennas, respectively V1 and V2 . A common point of these arcs corresponds to a designated point P ' . These considerations show that in order to determine the aircraft’s position (P ' , H ), the flight altitude H and two bearings α1 and α2 , are required. In the initial period of air navigation development, these bearings were most frequently performed with an on-board radio compass and broadcast stations, located within the range area, were the equivalent of the radio beacon [3, 7]. The accuracy in determining the azimuth angle α depends not only on the operating principle of the implemented radio beacon, but also on the amount of reflections from terrain obstacles, reaching the on-board receiver. Thus, this issue is discussed in parallel with the principles of phase, Doppler and pulse radio beacons, as follows.

18.1 Phase Radio Beacon The main antenna of a directional radio beacon operating according to phase method is characterized by an omnidirectional decentric directivity pattern in the horizontal plane. This antenna rotates around a vertical axis with angular velocity Ω = 2π 30 rad/s, Fig. 18.3a. Assume that the following signal is fed to antenna rotating correspondingly u A (t) = U A sin(ω0 t)

(18.2)

The carrier frequency f 0 of signal (18.2) is within the range 108–118 MHz. The course of an electric force of the wave transmitted through this antenna towards an aircraft, point P(α), is expressed by the following formula: E A (t, α) = E Am [1 + m sin(Ω · t − α)] sin(ω0 t)

(18.3)

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18 Radio Beacons and Distance Measuring Equipment Supporting …

Fig. 18.3 Diagram illustrating the principle of operation of a radio phase beacon

Formula (18.3) shows an amplitude modulation of the carrier wave effect caused by rotary motion of the decentric characteristic [1, 7]. Considering the north (α = 0), the phase of an envelope strength signal (18.3), is equal to Ω t. The antenna set of the discussed radio beacon also includes a second, stationary an omnidirectional antenna emitting frequency modulation reference signal, containing information regarding phase Ω t. The electric force of the wave generated by this antenna at a point P(α), is equal to Er e f (t) = Er m sin[ω p t − m P F cos(Ω · t + 0)] sin(ω0 t)

(18.4)

Subject to (18.4), ω p = 2π · 9960 rad/s corresponds to the subcarrier pulsation and m P F is the phase modulation index. The signal which reaches the on-board receiver antenna from the radio beacon, is the sum of signals (18.3) and (18.4), and can be expressed as follows: E(t, α) = E Am { 1 + m sin(Ω · t − α) + m p sin[ω p t − m P F cos(Ω · t + 0)]} sin(ω0 t)

(18.5)

where m p = Er m /E Am . Once the signal (18.5) is amplified and amplitude detected (elimination of the carrier wave) at the receiver output, the following voltage is obtained { } u out (t, α) = Uout 1 + m sin(Ω · t − α) + m p sin[ω p t − m P F cos(Ω · t + 0)] (18.6)

18.1 Phase Radio Beacon

397

Fig. 18.4 The block diagram of an on-board receiver receiving the signals emitted by the radio phase beacon

which is subsequently processed in a system outlined with a dashed line in Fig. 18.4. The first signal component (18.6) is filtered out by a low-pass filter (Filter-30). A sinusoidal signal is obtained at the output of this filter u α (t, α) = Uα sin(Ω t − α)

(18.7)

The frequency of this sinusoidal signal is equal to F = Ω/(2π ) = 30 Hz. The second signal component (18.6) is filtered out by a narrowband band-pass filter (Filter-9960) and is subsequently subjected to a frequency detection. The following signal is obtained at the output of the frequency detector u F D (t) = k F M

d [ω p t − m P F cos(Ωt + 0)] = k F M ω p + k F M m P F Ω sin(Ωt) dt (18.8)

for k F M corresponding to a constant of the detector. The variable component of signal (18.8) and signal (18.7) are fed to the phase detector, at the output of which occurs a signal which determines the phase shift α = Ω t − (Ω t − α)

(18.9)

This phase shift is equal to the determined azimuth angle. The radio beacon operation is controlled by a monitor system which analyzes the transmitted signals. If the parameter changes of these signals exceed the acceptable ranges, the system activates a backup set and reports a failure. In case of damage, the radio beacon is immediately disconnected because the emission of a irregular signal is unacceptable. As the landing aid system, the VOR radio beacon is periodically controlled by the aerial surveying services. This corresponds to the surrounding area as well, which should be precisely leveled within 300 m and within 1000 m no obstacles with an angular height greater than 2°, should occur. The Doppler VOR, discussed in the subsequent subsection, is more resistant to reflection from terrain obstacles.

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18 Radio Beacons and Distance Measuring Equipment Supporting …

18.2 Doppler VOR In order to discuss the Doppler VOR principles of operation (D-VOR), assume that two identical antennas A1 and A2 rotate in a circle with a radius R ≈ 2.5λ and angular velocity Ω. Furthermore, as assumed, these antennas are arranged at the opposite ends of the diameter D = |A1 A2 | = 2R ≈ 5λ, Fig. 18.5. The antenna A1 is fed with a harmonic signal u A1 (t) = U A cos[ω0 +ω p )t +π ] with a carrier frequency f 1 = f 0 + f p , for f 0 corresponding to a channel frequency and f p = 9960 Hz is referred to as a subcarrier. The second antenna, i.e. A2 is fed with a harmonic signal u A2 (t) = U A cos[ω0 − ω p )t + 0] as well, but with a frequency equal to f 2 = f 0 − f p . The frequencies of signals received by the on-board receiver antenna, located in the direction determined by the angle α, are described by the following formulae f 1d = f 0 + f p + ( f 0 + f p ) ϑc sin(Ω · t − α) sin(Ω · t − α) f 2d = f 0 − f p + ( f 0 − f p ) −ϑ c

(18.10)

where ϑ = Ω · R corresponding to a linear velocity of the rotating antennas. Considering that the condition f p ti

(E.5)

Appendix E: Determining the Dependence (7.41)

427

Fig. E.1 Graphic illustration of the ambiguity function described by formula (E.5)

Fig. E.2 Vertical cross-sections of the ambiguity function that is presented in Fig. E.1. a crosssection at FD = 0, b cross-section at τ = 0

Figure E.1 shows the “body” illustrating the function (E.5) in a orthogonal coordinate system [τ, FD , χ (τ, FD )]. The intersection of this body with a zero frequency translation plane (FD = 0), marks the line described by the equation |( | | 1− χ (τ, 0) = | |0

|τ | ti

)

for |τ | ≤ ti for |τ | > ti

(E.6)

In the case of the surface (E.5) intersection with a zero delay plane (τ = 0), one obtains

428

Appendix E: Determining the Dependence (7.41)

Fig. E.3 Horizontal cross-sections of the ambiguity function that is presented in Fig. E.1 for various values of parameter q

| | | sin(π · ti FD ) | | for any values of FD | χ (0, FD ) = | π · ti FD |

(E.7)

The intersections (E.6) and (E.7) are illustrated in Fig. E.2. The presented intersections constitute typical examples of the vertical sections. A large number of interesting information regarding the properties of the tested signal is also obtained as a result of the analysis of the horizontal sections, i.e. the line of intersection of the field of ambiguity with the parallel plane described by the equation χ (τ, FD ) = q, where 0 < q < 1. Figure E.3 shows examples of such intersections which are determined for the function (E.5). √ The contour marked with a solid line corresponds to the parameter q = 0.5. On the √ other hand, the set of contours marked with dotted lines was determined for q = 0.01 [2]. References 1. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “Soviet Radio”, Moscow (in Russian) 2. Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken

Appendix F

Indeterminacy Function χ(τ, .) of a Single Rectangular Radio Pulse with Internal Linear Frequency Modulation

Assume that the radio pulse with internal linear √frequency modulation described by 1/ T0 . The value of the analytical signal the dependence (9.1), has an amplitude A = √ assigned to this signal is equal to z(t) = 1/ T0 exp( j μt 2 /2) · exp( j ω0 t). Thus, the complex envelope of this complex signal can be expressed by the following equation √ Um (t) = 1/ T0 exp( j μt 2 /2). The two-dimensional correlation function, expressed by formula (7.40), is in this case identical to the indeterminacy function χ (τ, .) written in the integral form as follows | T.0 /2 | | 1 exp( jμt 2 /2) · exp[− j μ(t + τ )2 /2] · exp(− j.t)dt for 0 ≤ τ ≤ T0 | T0 | −(T0 /2)+τ | T0 /2−|τ χ (τ, .) = || . | | T1 exp( j μt 2 /2) · exp[− jμ(t + τ )2 /2] · exp(− j.t)dt for − T0 ≤ τ < 0 | 0 −(T /2) | 0 |0 for|τ | > T0

(F.1)

Once the elementary conversions are performed, the integrand, appearing in the integrals (F.1), can be written as the following product

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

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430

Appendix F: Indeterminacy Function χ(τ, .) of a Single Rectangular Radio Pulse …

exp( j μt 2 /2) · exp[− j μ(t + τ )2 /2] · exp(− j.t) = exp(− j μτ 2 /2) · exp(− j. L F M t) where . L F M = .+μτ . The multiplier exp(− j μτ 2 /2) of this product is independent of time t and in the integration process fulfills the role of a constant coefficient, which is described by the symbol Cμτ in the further considerations. Considering the above indication, i.e. . L F M = . + μτ and Cμτ = exp(− j μτ 2 /2), the first of the integrals (F.1) is equal to ] −Cμτ [ exp(− j. L F M T0 /2) − exp(+ j. L F M T0 /2 − j. L F M τ ) j. L F M T0 Cμτ = exp(− j. L F M τ/2) · 2 j sin[. L F M (T0 /2 − τ/2)] j. L F M T0 Cμτ exp(− j. L F M τ/2) sin[π · T0 [FD + μτ/(2π )](1 − τ/T0 )] (F.2) = π · T0 [FD + μτ/(2π )]

R(τ, .) =

where FD +μτ/(2π ) = . L F M /(2π ). The moduli of the coefficient Cμτ and function exp(− j. L F M τ/2) are equal to 1 and consequently the modulus of (F.2) can be expressed as | | | sin[π · T0 [FD + μτ/(2π )](1 − τ/T0 )] | | |R(τ, FD )| ≡ χ (τ, F D ) = || | π · T0 [FD + μτ/(2π )]

(F.3)

Due to the following conversion ) ( 1 τ 1 = 1− π · T0 [FD + μτ/(2π )] T0 π · T0 [FD + μτ/(2π )](1 − τ/T 0 ) and the dependence χ (τ, FD ) = χ (−τ, −FD ), formula (F.3) can be written in the standard form [2]. | ) | |( || |τ | sin[π ·T0 [FD +μτ/(2π)](1−|τ/T0 |)] | for |τ | ≤ T0 | | 1 − T0 π ·T0 [FD +μτ/(2π)](1−|τ/T0 |) | χ (τ, FD ) = | (F.4) |0 for |τ | > T0 For μ = 0 and T0 ≡ ti expression (F.4) reduces to expression (E.5) given in Appendix E.

Appendix F: Indeterminacy Function χ(τ, .) of a Single Rectangular Radio Pulse …

431

Fig. F.1 Graphic illustration of the ambiguity function described by formula (F.4)

Such conclusion does not surprise, since in this particular case one deals with a single radio pulse, which corresponds to a segment of a cosine function with a length ti = T0 . Figure F.1 shows the body which illustrates the function (F.4) in a rectangular coordinate system [τ/T0 , FD T0 , χ (τ, FD )]. References 1. Gonorowskij IS (1977) Radio circuits and signals, 3rd edn. Publishing House “Soviet Radio”, Moscow (in Russian) 2. Levanon N, Mozeson E (2004) Radar signals. Wiley, Hoboken

Appendix G

Costas Loop as a BPSK Signal Demodulator

A functional diagram of the phase loop used for BPSK signals demodulation, is shown in Fig. G.1 [1]. Assume that the detected BPSK signal is described by the following formula u in (t) = D(t) sin[ωc t + θc (t)]

(G.1)

where D(t) = ±1 corresponds to a sequence of values representing the phase modulating code, and θc (t) is an alleged (hidden) time function describing detuning of an input signal from a pulsation ωc . Furthermore, assume that the local oscillator V C O generates a signal

Fig. G.1 The block diagram of the Costas loop

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

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434

Appendix G: Costas Loop as a BPSK Signal Demodulator

Fig. G.2 The signal obtained at the output of the Costas loop

u ν (t) = 2 cos(ωc t + θν )

(G.2)

Due to the multiplication of signals (G.1) and (G.2) one obtains u (u) 1 (t) = D(t) sin[ωc t + θc (t)] · 2 cos(ωc t + θv ) = D(t) sin[2ωc t + θc (t)] + D(t) sin[θc (t) − θv ]

(G.3)

When the signal (G.3) passes through the low-pass filter (L P F), a differential signal is obtained at its output u (u) 2 (t) = k F D(t) sin[θc (t) − θv ]

(G.4)

Analogically, the following signals are obtained in the lower channel of the analyzed loop u (l) 1 (t) = D(t) sin[ωc t + θc (t)] · 2 sin(ωc t + θv ) = D(t) cos[θc (t) − θv ] − D(t) cos[2ωc t + θc (t)]

(G.5)

u (l) 2 (t) = k F D(t) cos[θc (t) − θv ]

(G.6)

When signals (G.5) and (G.6) are multiplied, one obtains an error signal (l) u err (t) = u (u) 2 (t) · u 2 (t) =

=

1 1 (k F )2 D 2 (t) sin[2θc (t) − 2θv ] + (k F )2 D 2 (t) sin(0) 2 2

1 (k F )2 sin{2[θc (t) − θv ]} 2

(G.7)

which does not contain the modulating signal D(t) because D 2 (t) ≡ 1. The dependence of the error signal on the argument θc (t) − θv is shown in Fig. G.2.

Appendix G: Costas Loop as a BPSK Signal Demodulator

435

The error signal fed to the oscillator VCO results in such re-tunning, that the error signal (G.7) approaches zero. In other words, the difference θc (t) − θv → 0 or θc (t) − θv → π . When the loop is tunned-in, i.e. the error signal is equal to zero, a signal u (l) 2 (t) = ±k F D(t), mapping the modulating code, is obtained at the loop output. References 1. Costas JP (1956) Synchronous communications. Proc IRE 44(12):1713–1718 2. Costas JP (1962) Receiver for communication system. US Patent, 3,047,659

Appendix H

Goniometric Radio Direction-Finder

The mechanical rotation of the frame antenna or the linear antenna array may be rendered difficult in numerous situations, due to the large geometric dimensions of antennas this type, which are used in the long and medium wave ranges. In order to eliminate this inconvenience, antenna systems with two stationary frame antennas perpendicularly-oriented to each other, have been developed, see Fig. H.1. The signals received by these antennas are fed to two rectangular coils (1 and 2), respectively, which generate electromagnetic fields perpendicular to each other. Between these two coils, a third one (also rectangular) is placed, rotating around a vertical axis. Such set of three coils is referred in the literature to as a goniometer and the system shown in Fig. H.1, is the Bellini-Tosi system. Considering the results of the analysis discussed before, one can assume that the current induced in the antenna 1 is described by the following formula i 1 (t) = I 0 sin(ϕ) cos(ωt)

(H.1)

Taking (H.1) into consideration, the current induced in the antenna 2 is expressed by an analogous equation i 2 (t) = I 0 sin

(π 2

) − ϕ cos(ωt) = I 0 cos(ϕ) cos(ωt)

(H.2)

The complex amplitudes of the currents i 1 (t) and i 2 (t) are equal respectively to I1 = I0 sin(ϕ) and I2 = I0 sin(ϕ)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

(H.3)

437

438

Appendix H: Goniometric Radio Direction-Finder

Considering that no major error is made, one can assume that currents with identical complex amplitudes, i.e. I1 = I0 sin(ϕ) and I2 = I0 sin(ϕ) respectively, flow in the coils 1 and 2. Assume also that the mutual inductance between the stationary and moving coils is proportional to cos(ψ), for ψ corresponding to the deflection angle, marked in Fig. H.1. Furthermore, the maximum value of the mutual inductance between these coils is equal to M. The second of the assumptions also applies to the second pair of coils, i.e. a stationary coil 2 and a moving coil. Considering these assumptions, a voltage u(t) will be induced in the moving coil, the amplitude of which will be dependent on the currents i 1 (t) and i 2 (t) as follows U = ωMI1 cos(ψ) + ωMI2 sin(ψ) = ωMI0 [sin(ϕ) cos(ψ) + cos(ϕ) sin(ψ)] = ωMI0 sin(ϕ − ψ)

(H.4)

The directivity pattern of a goniometer with a given position of a moving coil ψ = ψ0 = const, is expressed by the following formula f (ϕ) = k sin(ϕ − ψ0 )

(H.5)

The shape of this directivity pattern is similar to the corresponding pattern of a singular frame antenna, discussed before. When the receiving direction of the electromagnetic wave (ϕ = ϕ0 = const) is given, the minimum of the signal induced in the moving coil occurs when ϕ0 − ψ = 0 lub ϕ0 − ψ = π

(H.6)

Each of the discussed conditions can be satisfied by appropriate moving coil setting, the position of which indicates (unfortunately ambiguously) the direction of the wave reception. This ambiguity is usually eliminated by adding to the signal of the moving coil, the signal received by the omnidirectional wire antenna, placed as in Fig. 13.16a likewise. This way, a goniometric system with a cardioid directivity pattern is implemented. A previous century rapid development of electronics contributed to replacement of the set of coils (a goniometer) with a proper electronic system.

Appendix H: Goniometric Radio Direction-Finder

Fig. H.1 The crossed-loop antenna with a goniometer

439

Appendix I

Time Calculation of a Satellite Radio Visibility Above the User’s Horizon Line

Figure I.1 shows the satellite which is located above the user’s horizon line while flying along an arc S1 S2 leaning against the triangle apex (S1 , 0, S2 ) The apex angle ψ of this triangle can be calculated using the following formula ( ψ = 2 arccos

RE RE + h

) (I.1)

for R E ≈ 6378 km and h = 1112 km. Substituting these data into formula (I.1), ψ ≈ 2 · 31.62◦ = 63.24◦ is obtained. In reality, the angle ψ is slightly smaller because the satellite is clearly visible (in terms of radio visibility) when located approximately .ψ = 4◦ above the horizon plane. When considering this condition, ψr = ψ − 2.ψ = 63.24◦ − 8◦ ≈ 55◦ is obtained. Ultimately, the satellite visibility time above the horizon can be estimated according to the following formula TV = (ψr /360◦ ) · 107 min ≈ 16.4 min

(I.2)

As the Eq. (I.2) indicates, 107 min corresponds to time required to fully orbit the Earth in a circular orbit. Considering Fig. I.1, one can simply demonstrate that the Doppler frequency shift f D (t) = f P (t) − f S = f S ϑr (t)/c has extreme values when the satellite is close to the horizon line, i.e. at the left (S1 ) and right (S2 ) edges of the radio visibility area. Considering both of these utmost positions, the radial velocity is parallel to the horizon plane and can be estimated according to the formula |ϑ r | = |ϑ cos(ψr /2)|

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

(I.3)

441

442

Appendix I: Time Calculation of a Satellite Radio Visibility Above the User’s …

According to (I.3), ϑ corresponds to the linear orbital velocity of the satellite, the vector of which is always perpendicular to a position vector in the center of the Earth. In the case of TRANSIT satellites, which travel at speed ϑ ≈ 7300 m/s and transmitting navigation signals of a carrier frequency f S ≈ 400 MHz, this shift had extreme values equal to f D∗ = 4 × 108 (±7.3 × 103 ) cos(±63.24/2)/(3 × 108 ) ≈ ±8.299 kHz

(I.4)

The above considerations have been made on the assumption that the Earth’s surface is spherical. In fact, the Earth is flattened at the poles. Thus, its cross sections in meridional planes are close to the ellipse with the flattening coefficient f = (a − b)/a, where a is the semi-major axis and b is the semi-minor axis of the ellipse, respectively. The orbits of the TRANSIT satellites also lie in these meridional planes, see Fig. 15.6. Various coordinate systems are used in the satellite navigation to describe an ellipsoidal shape of the Earth. So far, the most commonly used are WGS 84 and PZ–90, which are briefly described below. The World Geodetic System—1984 WGS 84 is the standard U.S. Department of Defense definition of a global reference system for geospatial information and is the reference system for the Global Positioning System (GPS). It is based on a consistent set of constants and model parameters that describe the Earth’s size, shape, and gravity and geomagnetic fields. As the global coordinate system it is compatible with the International Terrestrial Reference System (ITRS). The definitions of its main parameters are given in the Technical Note 21 of the International Earth Rotation Service (IERS) [1, 2]. The main features of this orthogonal and right-handed coordinate system are shown in Fig. I.2 and listed below. 1. It is geocentric, the center of mass being defined for the whole Earth including oceans and atmosphere. The coordinate origin is placed in the Earth’s center of mass with maximum error 2 × 10−2 m. 2. The z-axis shows the direction of the IERS Reference Pole (IRP). This direction corresponds to the direction of the Bureau International de l’Heure (BIH) Conventional Terrestrial Pole (CTP). 3. The x-axis is an intersection of the IERS Reference Meridian (IRM) and the plane passing through the origin and normal to the z-axis. The IRM is coincident with the BIH Zero Meridian. 4. As shown in Fig. I.2 the y-axis completes a right- handed, Earth-Centered EarthFixed (ECEF) orthogonal coordinate system. The WGS 84 origin also serves as the geometric center of the WGS 84 Ellipsoid and the z-axis serves as the rotational axis of this ellipsoid of revolution. The rotating ellipsoid WGS 84 is defined by four independent parameters. These are the semimajor axis, the Earth’s flattening factor, the Earth’s nominal mean angular velocity, and the geocentric gravity constant. The values of these parameters are given in the third column of Table I.1

Appendix I: Time Calculation of a Satellite Radio Visibility Above the User’s …

443

Table I.1 Primary parameters of the rotating ellipsoids WGS 84 and PZ-90.02 Parameters

WGS 84

PZ-90.02

1

Earth’s rotation rate (. E )

7,292,115,146 × 10−5 rad/s

7,292,115 × 10−5 rad/s

2

Earth’s gravitational constant (G M)

3.986004418 × 1014 m3 /s2

3.986004418 × 1014 m3 /s2

3

Semi-major axis of Earth (a)

6,378,137 m

6,378,136 m

4

Flattening ( f )

1/298,257,223,563

1/29,825,784

The above basic parameters of WGS 84 in combination with the speed of light c ≈ 299,792,458 m/s are appropriate for the computation of satellite orbit (eg GPS navigation satellites) and for prediction (prognostic) purposes. Geodetic latitude ϕ of a point O is defined as angle between the normal to the ellipsoid surface and equatorial plane. Geodetic longitude λ of the O point is determined as a corner angle between a plane of a prime meridian and a meridian plane transiting through the point O. Direction of the determinate of longitudes—from a prime meridian to the east, i.e. from 0 to 360 grades. Geodetic height (altitude) H of a point O is defined as a distance from the ellipsoid surface to the point O along the normal. The PZ-90.02 Earth-Centered Earth-Fixed Reference Frame system The PZ-90.02 is an improved version of the national coordinate system PZ-90, similar to WGS 84, developed in Russia [2, 4]. This coordinate system is a basis for geodetic support of the Russian Global Navigation Satellite System, known as GLONASS. The GLONASS broadcast ephemeris describes a position of transmitting antenna phase center of given satellite in the PZ-90.02 reference system defined as follows. The origin point of this system is located at the center of the Earth’s body. According to [4] the coordinate origin is placed in the Earth’s center of mass with maximum error 5 × 10−2 m. The z-axis is directed to the Conventional Terrestrial Pole as recommended by the International Earth Rotation Service (IERS). The x-axis is directed to the point of intersection of the Earth’s equatorial plane and the zero meridian established by the BIH (Eng. International Time Bureau). The y-axis completes the coordinate system to the right-handed one. The geodetic coordinates of a point in the PZ-90.02 reference system refer to the ellipsoid, the semi-major axis and flattening of which are given in the fourth column of Table I.1. Similarly as in the WGS 84 the geodetic latitude ϕ of a point O is defined as angle between the normal to the ellipsoid surface and equatorial plane. Geodetic longitude λ of the O point is determined as a corner angle between a plane of a prime meridian and a meridian plane transiting through the point O. Direction of the determinate of longitudes—from a prime meridian to the

444

Appendix I: Time Calculation of a Satellite Radio Visibility Above the User’s …

Fig. I.1 Illustration of the TRANSIT satellite flight in the area of radio visibility

Fig. I.2 The characteristic planes, ellipses, and poles of the system WGS 84

east, i.e. from 0 to 360 grades. Geodetic height (altitude) H of a point O is defined as a distance from the ellipsoid surface to the point O along the normal. References 1. Jean-Marie Zogg GPS. Essential of satellite navigation. Compendium. https:// zogg-jm.ch/Dateien/GPS_Compendium(GPS-X-02007).pdf 2. Teunissen PJ, Montebruck O (eds) (2017) Springer handbook of global navigation satellite systems. Springer International Publishing AG, Switzerland 3. User’s handbook on datum transformations involving WGS 84. https://legacy. iho.int/iho_pubs/standard/S60_Ed3Eng.pdf 4. Zueva AN et al (1990) System of geodetic parameters. Parametry Zemli. https:// www.unoosa.org/pdf/icg/2014/wg/wgd7.pdf

Appendix J

Reproducing Methods of a Carrier Frequency Signal from a Phase Modulated Signal

The navigation signal emitted by TRANSIT system satellite transmitter is described by the general formula u T L P S K (t) = Um cos[ω S t + sn (t)π/3]

(J.1)

for: sn (t) ⊂ . According to (J.1), the phase of signal ψ(t) = ω S t +sn (t)π/3 is additionally dependent on the modulating function sn (t), which has three discrete values, i.e. −1, 0 or 1. This modulation method indicates a special case of a multistate code phase modulation, referred in the English-language literature to as Phase Shift Keying (PSK). A modulating function sn (t) produces step changes ψ(t) with respect to ω S t and equal to −60◦ , 0◦ or 60◦ . For this reason, the signal (J.1) is occasionally referred to as a Three Level Phase Shift Keying (TLPSK). The carrier frequency of this signal (crucial for a demodulation purpose) can be recovered by means of various electronic circuits. The most popular include Phase Locked Loop (PLL) circuit with a functional diagram, as shown in Fig. J.1. The signal u T L P S K (t) = Um cos[ω S t + sn (t)π/3] is fed to the input of the sextupler (N = 6), with the signal containing component u ×6 (t) = k · Um cos[6ω S t + sn (t)2π ] = k · Um cos(6ω S t) at the output, for k corresponding to a certain constant

Fig. J.1 The block diagram of the carrier recovery system from the N-state PSK signal

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

445

446

Appendix J: Reproducing Methods of a Carrier Frequency Signal from a Phase …

Fig. J.2 The block diagram of the carrier recovery system from the BPSK signal

amplitude coefficient. This component, filtered with a narrow-band (highly selective) band-pass filter (B P F), is fed to one of the inputs of the phase detector (P D). The signal obtained due to a sextupler (N = 6) of a signal generated by the local generator V C O, is fed to the second input of this detector. In other words, the signal pulsation which is fed to the second of the P D inputs, is equal approximately to 6ω S . The error signal obtained at the output of the phase detector (P D), is directly proportional to the sine of signals’ phase angle difference, fed to its two inputs. The following signal retunes V C O until the phase difference of the fed signals is zero. The described situation is referred to as the PLL synchronism, including phase detector (P D), lowpass filter (L P F), V C O generator and sextupler circuit, marked with the symbol ×N in Fig. J.1. Considering PLL synchronism, the signal u out (t) = k · Um cos(ω S t) is obtained at the output of the V C O generator, for k < 1 corresponding to a certain constant amplitude coefficient. Another, broadly used version of PSK modulation, is the Quadrature Phase Shift Keying (QPSK), (0, 90◦ , 180◦ , 270◦ ). The QPSK signal can be expressed as follows u Q P S K (t) = Um cos[ω S t + sn (t)π/2)

(J.2)

for sn (t) ⊂ . As described, the signal carrier frequency (J.2) can be recovered assuming the coefficient N = 4, see Fig. J.1. In the case of Binary Phase Shift Keying (BPSK), which is the simplest case of PSK modulation, the harmonic signal with a carrier frequency can also be reproduced with a circuit as shown in Fig. J.1, assuming the coefficient N = 2. However, in this case, the most frequently implemented is the carrier recovery system, consisting of a squared unit ()2 , narrow-band filter (B P F) and a frequency divider (D/2) with a quotient equal to 2, Fig. J.2. A signal u B P S K (t) ≡ u in (t) = Um cos[ω S t + sn (t)π ], for sn (t) ⊂ , when squared in a block ()2 , is expressed by the following formula [u in (t)]2 = k · Um2 cos2 [ω S t + sn (t)π ] 1 1 = k · Um2 + k · Um2 cos[2ω S t + s n (t)2π ] 2 2 1 1 2 ≡ k · Um + k · Um2 cos(2ω S t) 2 2

(J.3)

Appendix J: Reproducing Methods of a Carrier Frequency Signal from a Phase …

447

According to (J.3), k is a certain constant amplitude coefficient. The signal (J.3) component, with a pulsation equal to 2ωs , is filtered out with a narrow band-pass filter (B P F) and then processed in a (D/2) block in order to obtain a signal with half of the original pulsation, i.e. equal to ωs . Examples of telecommunication systems which use BPSK modulation signals include the GPS-NAVSTAR and GLONASS satellite navigation systems.

Index

A Agency for Aerial Navigation Safety in Africa (ASECNA), 344, 351 Altimeter, 15, 16, 19–21, 355, 394, 413 Ambiguity function, 119, 153, 426 Amplitude detector, 183 Analytic signal, 116, 117, 429 Angle aspect, 59 azimuth, 3, 21, 24, 30, 54, 79, 175, 223, 227, 228, 259, 261, 383, 384, 394, 395, 397, 400, 404–406 determination, 21 elevation, 406 Antenna array, 4, 23, 94, 196, 201, 213, 233–237, 239, 241–246, 248–250, 253, 266, 274–278, 357, 359–361, 367, 368, 372, 381, 388, 389, 393, 422, 437 frame (loop), 266–271, 279–281, 437, 438 magnetic (ferrite), 238, 271, 272 Wullenweber, 277 Autocorrelation function, 113–117, 120, 121, 157–160, 320 B Barker’s codes, 157, 160 Beacon doppler, 393, 395, 397, 398, 401 phase, 393, 395–397, 400, 401 pulse, 376, 393, 395, 402, 406, 408, 409, 411

Beamformer analog (Blass and Butler matrices), 195 digital, 253 BeiDou (Compass), 335 BeiDou Satellite Based Augmentation System (BDSBAS), 350 Binary Phase Shift Keying (BPSK), 157–159, 337, 433, 446, 447 Bistatic radar system, 45, 77, 78, 81, 82, 84 Blind speeds, 184, 189, 190

C Clutter Attenuation (CA), 190, 238 Coherent signals, 178, 304 Conical scanning, 205–207, 223 Continous Wave/Frequency Modulation (CW/FM), 16, 20, 21 Convolution, 127–130, 133, 163 Cross-correlation function, 107, 115, 116, 121, 122, 131, 134

D Decca Navigator, 264, 266, 291, 293 Delay line canceller, 184–186, 190 phase, 14, 140, 154, 185, 230, 232, 233, 242, 243, 277, 408, 413 time, 8, 13, 14, 16–18, 57, 89, 116, 121, 138, 144, 150, 152, 155, 157, 203, 204, 206, 242, 277, 284–286, 289, 308, 320, 326–328, 340, 351, 390, 410

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Rosłoniec, Fundamentals of the Radiolocation and Radionavigation, Springer Aerospace Technology, https://doi.org/10.1007/978-3-031-10631-6

449

450 Difference frequency, 8, 11, 17–19, 93, 94, 102–105, 157, 184, 189, 230, 260, 293, 295, 297, 364, 366, 370 time, 17, 27, 33, 34, 80, 81, 93, 97, 101, 103–105, 180, 214, 260, 283–285, 292, 296, 318, 341, 361, 415 Differential Global Positioning System (GPS), 339 omega, 297 Digital beam forming, 242 down converter (DDC), 239 signal processing (DSP), 124, 157, 238 Dilution of precision, 33, 342 Direction finding, 223, 225, 226, 228, 230, 231, 233, 266, 280, 362, 365–367, 370, 384–386 of arrival (DOA), 27, 28, 33 Discrete Fourier transform, 168, 169 Distance Measuring Equipment (DME), 357, 374, 390, 393, 394, 406, 408–411 Doppler frequency, 12, 17, 18, 81, 93, 94, 100, 119, 153, 178, 184, 187, 222, 265, 273, 275, 299–302, 307, 309–311, 313, 336, 362, 399, 441 phenomenon (effect), 8, 10, 11, 17, 18, 93, 102, 153, 178, 179, 181, 184, 188, 191, 197, 265, 273, 299, 300, 308, 310, 362

Index magnetic, 8, 60, 62 Forward Scattering Radiolocation (FSR), 5, 61, 62, 81, 213 Fourier integral, 108–110, 425 series, 108, 109, 167, 169 transform, 108–110, 114, 150, 166, 168, 169, 425 Frequency angular, 109, 111, 119, 138, 184, 199, 273, 405, 413 carrier, 4, 6, 12, 72, 94, 159, 177, 178, 189, 190, 239, 253, 285, 287, 299, 304, 306, 308–310, 312, 313, 317, 318, 330, 331, 336–338, 359, 360, 366, 367, 384, 388, 389, 393, 395, 398, 402, 408, 413, 421, 442, 445, 446 diffrences of arrival (FDOA), 93, 94, 103, 105 repetition, 184, 187, 189, 201–204, 239, 384, 409, 411, 422 spectrum, 12, 15, 21, 131, 133, 140, 152, 166, 169, 179, 181, 200, 204, 287

E Elevation plane, 62, 195, 204, 224, 228, 236, 237, 275, 319, 320, 367, 370, 372, 373, 375, 376, 378, 380–382, 388 Ellipse equation, 84, 442 semi-axis, 442 European Geostationary Navigation Overlay Service (EGNOS), 334, 342–346, 351, 390

G Galileo, 266, 316, 329, 332–336, 338, 346 Geometrical Dilution of Precision (GDOP), 33, 34, 342 Glidepath, 359 Globalnaja Navigacjnaja Sputnikowaja Sistiema (GLONASS), 266, 299, 316, 329–333, 346, 349, 350, 355, 443, 447 Global Positioning System - Navigation Signal Timing and Ranging (GPS NAVSTAR), 299, 315, 442, 447 Goniometer, 266, 437, 438 Gravitation, gravity, 10, 208, 210, 211, 214–216, 259, 304, 329, 346, 393, 442 Ground Based Augmentation System (GBAS), 339, 342, 391 Gyroscope, 210, 211

F Fast Fourier Transform (FFT), 109 Field electric, 8–10, 45, 46, 48, 49, 56, 60, 155, 267, 268, 272, 382 electromagnetic, 372, 437

H Homing methods, 205, 216, 219, 220 Hyperbola, 32, 264, 283, 284, 289, 292, 294, 311 Hyperbolic navigation systems, 284

Index I Identification Friend –Foe system (IFF), 6, 72, 75, 266, 384 Incoherent signals, 179 Indian Regional Navigation Satellite System (IRNSS), 329, 335, 337 Instrument Landing System (ILS), 342, 357–359, 362, 366, 367, 372, 373, 380, 382–385, 390–393 Integrator, 73, 121, 140, 144, 145 Interference, 27, 56, 59, 61, 65, 67, 83, 107, 120, 122, 175, 176, 190, 258, 283, 291, 295, 306, 318, 333, 334, 391, 400, 405 Interferometer, 383, 385, 387–389 International Civil Aviation Organization (ICAO), 75, 355, 357, 390 Interrogator, 6, 72–75, 79, 142, 159, 383, 384, 388, 389, 406–411, 421 Intersection, 28, 32, 33, 88–90, 261, 262, 265, 289, 302, 311, 312, 359, 427, 428, 442, 443 Inverse Discrete Fourier transform (IDFT), 169 Ionosphere, 281, 290, 291, 303, 306, 313, 318, 339–341, 390

K Korean Augmentation Satellite System (KAAS), 344

L Landing, 15, 20, 259, 346, 355–359, 373, 384, 389–391, 397, 408, 411 Latitude, 10, 257, 290, 295, 319, 329, 337, 443 Least squares method, 35 Linear Frequency Modulation (LFM), 12, 15, 149–154, 156, 196, 199, 203, 429 Loop antenna, 267–270 Costas, 160, 433 Loran – C, 264, 290 Loxodrome, 262, 263

M Master station, 95, 285, 286, 288, 293, 294, 296, 390

451 Matched filter, 120, 122–124, 130–135, 138–140, 143, 144, 149, 152, 157, 159, 161, 163, 169, 170, 200, 201 Microwave Landing System (MLS), 357, 373–375, 377, 379, 381–383, 390, 392, 393 Modulation amplitude, 21, 151, 152, 189, 205, 207–210, 360, 361, 366, 367, 370, 383, 396, 408, 414, 429, 446, 447 binary, 157, 334, 335, 447 biphase, 157, 158, 287 digital, 306, 335 frequency, 12, 15, 149, 152, 156, 157, 209, 274, 331, 334, 362, 364, 366, 396, 413, 414, 429, 446 Monopulse methods, 205, 223 radiolocation, 205, 223 Moving Target Detector (MTD), 94 Moving Target Indicator (MTI), 184 MTI – improvement factor, 190 Multi - functional Satellite Augmentation System (MSAS), 342, 344, 347, 348 Multistatic radar system, 94

N Navy Navigation Satellite System (NNSS), 265, 266, 299 Newton method, 35–37, 39, 41, 42 NOVA, 304

O Omega navigation system, 266, 291, 295, 297 Orbit earth, 71, 299–301, 304, 306, 315, 316, 329, 333, 337, 344, 441, 442 elliptic, 299, 337 geostationary, 337, 339, 344 Orthodrome, 262

P Parseval’s identity, 111 Phase centre, 3, 14, 15, 77, 194, 207, 225, 227, 250, 251, 302, 319, 331, 386, 387, 443 code, 335, 445 detector, 159, 182, 185, 191, 230, 232, 236, 280, 397, 400, 446 error, 20, 297, 443, 446

452 modulation, 21, 117, 149, 157, 158, 209, 287, 306, 335, 371, 414, 445 Polarization circular, 176, 177, 333, 351 linear, 177, 362 Power density, 7, 8, 21, 46, 65, 66, 72, 73, 82, 83, 130, 419 radiation, 258, 362, 368, 388 Poynting’s vector, 8 Precise Positioning Service (PPS), 318, 327, 328, 337 Propagation in free space, 7, 27, 73, 300 in troposphere, 69, 74, 339, 419 velocity, 155, 283, 300, 322, 419 Pseudo-random codes, 318 Pulse modulation, 12, 149, 152, 156, 157, 196, 199, 429 Q Quadrature, 115, 162, 209, 210 Quadrature phase detector, 182, 209, 210 Quadrature Phase Shift keying (QPSK), 334, 335, 446 Quasi Zenith Satellite System (QZSS), 329, 335, 337–339 R Radar primary surveillance (PSR), 3, 5, 11 secondary surveillance (SSR), 5, 421 Radar Cross Section (RCS), 45, 46, 52–55, 59, 61, 65, 66 Radial velocity, 11, 12, 100, 178, 181, 189, 222, 273, 300, 301, 303, 362, 441 Radiation electromagnetic, 7, 45, 47, 367, 386, 388 pattern, 239, 362, 367, 368, 384–386, 388 Radiocompass, 279 Random codes, 318 Range equation of the bistatic radar system, 82, 84 multistatic radar system, 93, 94, 101 primary radar system, 5, 6, 65, 67, 68, 77 secondary radar system, 65, 72, 75, 383, 384 Rayleigh formula, 110, 111, 135

Index Receiver multichannel, 357, 384 Reflection coefficient, 194, 198, 371 normal, 9 Refraction (normal, critical, sepercritical, sub-reflaction), 70 troposperic, 69, 341 Reverse Fourier transform, 169

S Satellite clock, 306, 308, 311, 318, 319, 322, 331, 332, 334, 336, 341, 346 constellation, 316, 329, 332–335, 337, 349 message, 317, 318, 321, 322, 332, 336, 341, 344, 347–349 navigation, 258, 265, 266, 295, 299, 303, 306–309, 312, 313, 315, 318, 321, 322, 332, 334–337, 339, 341, 349, 351, 357, 390, 391, 442, 443, 445, 447 orbit, 71, 265, 299–301, 303–306, 311, 312, 315, 316, 329, 330, 333–338, 344, 347, 348, 441–443 Satellite Based Augmentation Systems (SBAS), 338, 339, 342, 343, 347–351, 390 Side looking, 196 Signal linear frequency modulation (LFM), 12, 149, 151–154, 156, 429 processing, 4, 80, 120, 121, 124, 133, 146, 202, 204 reflection, 12, 21, 59, 117, 124, 136, 156, 175, 180, 185, 203, 237, 342, 370, 371, 401, 409 Space Surveillance System (SPASUR), 94, 95 Standard Positioning Service (SPS), 315, 317, 327, 337, 339, 342 Steepest descent method, 35, 39, 41 Surface Acoustic Waves (SAW) technology, 155, 156 Synchronization, 78, 89, 95, 190, 296, 319, 320, 331, 334, 344, 375 Synthetic Aperture Radar (SAR), 197, 202, 334 System for Differential Correction and Monitoring (SDCM), 344, 349

Index

453

T Three Level Phase Shift Keying (TLPSK), 445 Time Differences Of Arrival (TDOA), 27, 31–34, 93, 95, 105 Tracking, 21, 79, 193, 205, 210, 213, 216, 217, 220, 303, 331, 335, 410, 411 TRANSIT (NAVSAT), 266, 299 Transmit/Receive module (T/R), 238–240, 242, 244, 253 Transmitter, 6, 7, 10, 12, 15, 27, 57, 59, 62, 72, 73, 77–80, 82, 84, 86–89, 93–96, 123, 181, 212, 213, 233, 236, 238, 265, 266, 290, 295, 299, 300, 306, 310, 317, 340, 345, 384, 389, 391, 410, 445 Transponder, 6, 72–75, 383, 384, 406–410, 421, 422 Transponder Landing System (TLS), 266, 357, 382–384, 388–390, 393 Transverse Electromagnetic Wave (TEM), 7, 8, 27, 59, 65, 73, 235, 318 Troposphere, 8, 10, 65, 68–71, 74, 75, 85, 86, 339, 341, 419

V Velocity angular, 23, 58, 91, 205, 210, 214, 215, 220, 272, 274, 376, 378, 395, 398, 401, 402, 405, 442 linear, 197, 272, 274, 300, 302, 304, 316, 398 radial, 11, 100, 178, 181, 189, 222, 273, 300, 301, 303, 362, 441 Very High Frequency (VHF), 3, 4, 10, 78, 339–342, 360, 383, 384, 388–390, 391, 393 VHF Omni directional Range (VOR), 393, 394, 397, 398, 400, 401 Voltage Controlled Oscilator (VCO), 433, 435, 446

U Universal Time Coordinated (UTC), 331, 334

Z Zenith, 337 Zero crossing, 287, 309

W Wave TEM, 8, 65, 73 WGS 84, 443 Wide Area Augmentation System (WAAS), 334, 342, 344, 346, 347, 390 World Geodetic System (WGS) 84, 442, 443 Wullenweber antenna, 277