Fourier Transformation for Signal and System Description: Compact, Visual, Intuitively Understandable 9783658338169, 9783658338176, 3658338164

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Jörg Lange · Tatjana Lange

Fourier Transformation for Signal and System Description Compact, Visual, Intuitively Understandable

essentials

Springer essentials

Springer essentials provide up-to-date knowledge in a concentrated form. They aim to deliver the essence of what counts as “state-of-the-art” in the current academic discussion or in practice. With their quick, uncomplicated and comprehensible information, essentials provide: • an introduction to a current issue within your field of expertise • an introduction to a new topic of interest • an insight, in order to be able to join in the discussion on a particular topic Available in electronic and printed format, the books present expert knowledge from Springer specialist authors in a compact form. They are particularly suitable for use as eBooks on tablet PCs, eBook readers and smartphones. Springer essentials form modules of knowledge from the areas economics, social sciences and humanities, technology and natural sciences, as well as from medicine, psychology and health professions, written by renowned Springer-authors across many disciplines.

Jörg Lange • Tatjana Lange

Fourier Transformation for Signal and System Description Compact, Visual, Intuitively Understandable

Jörg Lange Schwielowsee, Germany

Tatjana Lange Schwielowsee, Germany

ISSN 2197-6708 ISSN 2197-6716 (electronic) essentials ISSN 2731-3107 ISSN 2731-3115 (electronic) Springer essentials ISBN 978-3-658-33816-9 ISBN 978-3-658-33817-6 (eBook) https://doi.org/10.1007/978-3-658-33817-6 This book is a translation of the original German edition „Fourier-Transformation zur Signal- und Systembeschreibung” by Lange, Jörg and Lange Tatjana, published by Springer Fachmedien Wiesbaden GmbH in 2019. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of 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 Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface

As students, we had not really understood the Fourier transform. We could handle the formulas quite well, but we had not really internalized the meaning of the whole thing. This was probably because we, who live in time and space, simply could not imagine a “frequency or image domain” into which the Fourier transform would take us, and consequently could not think and “feel” in this category. The wonderful and often very helpful world of the Fourier transform was first recognized by us as young assistants, when we tried to teach it to the students ourselves, who then let us know in the evenings in the pub that for them this Fourier transform is a single horror. Already at that time, ideas arose about how the Fourier transform and its technical applications, especially in signal and system theory and in control engineering, could be clearly conveyed. In the meantime, these approaches have been further developed using today’s design possibilities and are now summarized in this booklet. They are intended to help STEM (science, technology, engineering, mathematics) students in particular, and of course also graduated professionals to better understand the subject matter and, in particular, to grasp it emotionally. We consider this to be particularly important because all modern digital technologies, be it digital sound and image recording and storage, digital radio and digital television, digital mobile telephony, digital signal transmission, without which there would be no Internet (!), the modern control technologies, without which no car can drive and no airplane can fly today, are largely based on the findings of the Fourier transformation. We are sincerely grateful to our colleague Karl Mosler for his critical review of the text.

v

vi

Preface

Our thanks naturally also go to Springer Verlag, which made the publication of this booklet possible, and especially to Ms. Iris Ruhmann and Dr. Angelika Schulz for their fruitful and always helpful and pleasant cooperation and support. We would also like to thank Ms. Snehal Surwade for her support in preparing the English-language edition of this booklet. Schwielowsee, Germany November 2022

Jörg Lange Tatjana Lange

What You Can Find in This essential

• An image-based explanation of the cosine and sine functions as functions of time • Examples for the application of the time-dependent cosine and sine functions as building blocks for the construction of other periodic functions • A vivid and example-based explanation of the Fourier series and a short derivation of the formulas for calculating the Fourier coefficients • A mainly visual representation of the transition from the Fourier series for periodic functions to the Fourier integral for aperiodic functions • Figures and formulas for the Fourier transformation of standard signals (aperiodic functions or pulses) • The most important properties of the Fourier transform and a useful approximation relationship for determining the spectral function of simple bell-like signals • The use of the Fourier transform as a tool for mathematical modeling of the interaction of signals and systems • A brief consideration of the application of the Fourier transform to stochastic signals • A largely visual and intuitive and thus unorthodox derivation of the transfer function of the optimal receive filter (Wiener filter)

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

About Sine, Cosine, and Other Periodic Functions . . . . . . . . . . . . .

3

3

Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4

From the Fourier Series to the Fourier Integral . . . . . . . . . . . . . . . 21

5

The Fourier Transformation of Standard Signals . . . . . . . . . . . . . . 27

6

Properties of the Fourier Transform and Approximation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7

The Fourier Transformation as a Tool for the Description of Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8

Application of the Fourier Transform to Stochastic Signals . . . . . . . 47

9

Optimal Receive Filter (Wiener Filter) . . . . . . . . . . . . . . . . . . . . . . 51

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

ix

1

Introduction

Is the Fourier transform, which was developed around 1822 by its namesake, Jean Baptiste Joseph Fourier (1768–1830), just a nice mathematical gimmick or does it have any practical meaning? The second part of this question must be answered with a clear and forceful “yes.” The Fourier transform is “hidden” in all modern technologies that have something to do with signal transmission and processing, such as mobile telephony, the Internet, digital controllers in vehicles, household appliances, medical equipment, and space travel. Of course “many roads lead to Rome” and it would be presumptuous to claim that all the applications mentioned in the sequel would not have been invented without the Fourier transform. For some of them, the only thing that applies is that they can be explained particularly clearly by means of the Fourier transform, as we will show with the example of modulation in Chap. 6. In other applications, such as orthogonal frequency division multiplexing (OFDM), which is used for signal reception for LTE in mobile communications, a modification of the Fourier transformation, the so-called Fast Fourier Transformation, was implemented directly. LTE and thus the fast mobile Internet are simply unthinkable without Fourier transformation. The digitization of speech and music and thus MP3 and MP4 directly use the insights of the Fourier transformation. This applies not least to the internet, which does not, as a recent loose contact keenly claimed, get by without electricity, but is dependent on the services of “very physical” wired or radio-based transmission networks, which in turn uses the Fourier transformation. It is, therefore, worthwhile to engage in this subject. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_1

1

2

About Sine, Cosine, and Other Periodic Functions

Before we begin our considerations of the Fourier transform, we need to do some work on the cosine and sine functions. We first know the sine and the cosine from the angular relationships in a rightangled triangle (Fig. 2.1) as. sin ðαÞ =

Opposite side b = Hypotenuse c

cos ðαÞ =

Adjacent side a = Hypotenuse c

However, if we represent the triangle with a hypotenuse c = 1 but in the unit circle (Fig. 2.2) and rotate this hypotenuse uniformly at a speed f0 = const. around the origin of the unit circle, then the length of the changing cathetus b describes the sine as a function of time t: b = 1  sin ðαðt ÞÞ = sin ð2π  f 0  t Þ, where the angle is given in α = α(t) = 2π  f0  t radians (rad); 2π = 360 ° ; α½rad = ðα½ °   2π Þ=360 ° . If we now imagine (mathematically abstracting) that the hypotenuse has been rotating for an infinite length of time, so to speak “since the Big Bang,” and will continue to rotate, we get a periodic function (Fig. 2.3). The period length tp corresponds to a complete rotation of 360° or 2π. Here tp = 1/f0 applies. For the sake of simplicity, we position the coordinate system in Fig. 2.3 in such a way that the time t = 0 denotes our viewing time, that is, the “now.” Negative time values stand for the past, positive ones for the future. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_2

3

4 Fig. 2.1 Angular relationships

Fig. 2.2 Sine as a function of time

Fig. 2.3 Periodic sine function

2

About Sine, Cosine, and Other Periodic Functions

2

About Sine, Cosine, and Other Periodic Functions

5

Fig. 2.4 Cosine function as shifted sine function

Let us take a closer look at the rotational speed f0. It indicates the number of rotations of the hypotenuse per second, where a complete rotation corresponds to 360° or 2π in radians. Thus [1/s] would be the unit of measurement of f0. On the other hand, f0 is identical with the frequency, that is, the number of polarity changes or cycles of a periodic sine function per second (see Fig. 2.3). It is, therefore, measured in [Hz] = [Hertz] (where 1 Hz = 1/s). In the literature, the so-called angular frequency ω0 = 2π  f0 is often used instead of the frequency f0. Since the factor 2π is dimensionless, both quantities actually have the same unit of measurement, namely [1/s]. To avoid confusion, the unit of measurement for f0 is usually [Hz] and the unit of measurement for the angular frequency ω0 is [1/s]. The previous statements apply in the same way to the cosine function, because the cosine function is nothing more than a sine function shifted to the left by tp/4 (Fig. 2.4). Sine functions and all shifted sine functions describe harmonic oscillations. We will therefore also call them harmonic functions in the following. In the following, we will mainly work with the cosine function, because as an even function it offers certain advantages in many considerations, which will become clear later. In this context, it should be remembered that even functions (e.g., the cosine function) are axisymmetric to the ordinate and odd functions (e.g., the sine function) are point-symmetric to the coordinate origin. The following applies • to even functions feven(t) = feven(-t) • to odd functions fodd(t) = - fodd(-t)

6

2

About Sine, Cosine, and Other Periodic Functions

Interim Statement Any function can be broken down into its even and odd components: f ðtÞ = f even ðtÞ þ f odd ðtÞ These two components can be determined as follows: f even ðt Þ =

f ðt Þ þ f ð - t Þ ; 2

f odd ðtÞ =

f ðt Þ - f ð - tÞ 2

Now we want to “play” a little with the cosine functions. As shown in the left part of Fig. 2.5, we add a constant (DC component) and six cosine functions whose frequencies are multiples of the frequency f0 of the first cosine function: up ðt Þ = 0:2 þ 0:353 cos ð2π  f 0 t Þ þ 0:242 cos ð2π  2f 0 tÞ þ 0:129 cos ð2π  3f 0 tÞ þ0:054 cos ð2π  4f 0 tÞ þ 0:017 cos ð2π  5f 0 tÞ þ 0:004 cos ð2π  6f 0 tÞ: The resulting function up(t) is again a periodic but not a harmonic function consisting of bell-shaped impulses. The period of this new periodic function up(t) is equal to the period tp = 5[ms] of the first cosine function, which is also called the fundamental wave. Now we perform this experiment again, but this time with different values of the DC component A0 and the amplitudes Ak of the six cosine functions, as shown in the right part of Fig. 2.5. As a result we get a periodic function of pulses, which reminds us a bit of rectangular pulses. We can improve the approximation shown in Fig. 2.5 on the right side by adding another 24 cosine functions. Thus we can reproduce a periodic rectangular sequence quite well (see Fig. 2.6). This suggests the assumption that almost any non-harmonic periodic function can be generated by adding a sufficient number of (infinitely many) cosine functions. At the beginning of the nineteenth century already (!), the French scientist Jean Baptiste Fourier (1768–1830) proved that this is true. In the following we will now introduce the mathematical apparatus based on it – known as the Fourier series and Fourier integral, and some of its practical applications.

2

About Sine, Cosine, and Other Periodic Functions

7

Fig. 2.5 Approximation of a periodic Gaussian function (left) and a periodic, roughly approximate rectangular function (right), each by summing six cosine functions

8

2

k

About Sine, Cosine, and Other Periodic Functions

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Ak

0.450 0.318 0.150 0.000 -0.090 -0.106 -0.064 0.000 0.050 0.064

Ak

0.041 0.000 -0.035 -0.045 -0.030 0.000 0.026 0.035 0.024 0.000

Ak

-0.021 -0.029 -0.020 0.000 0.018 0.024 0.017 0.000 -0.016 -0.021

k

k

Fig. 2.6 Approximation of a periodic rectangular sequence by the sum of 30 cosine functions

3

Fourier Series

Fourier has shown that non-harmonic periodic functions1 (Fig. 3.1) can be decomposed into infinite sums of cosine functions of different amplitude Ak, phase shift φk, and frequency (kf0): up ðtÞ = A0 þ

þ1 X

Ak cos ð2πkf 0 t þ φk Þ

k=1

This notation of the Fourier series uses only cosine functions that can be made visible by measurement, for example, with the aid of an oscilloscope (Fig. 3.1, right). We, therefore, also speak of a “physical” notation of the Fourier series. A second notation uses axisymmetric cosine functions and point-symmetric sine functions: up ðtÞ =

þ1 a0 X þ ½a cos ð2πkf 0 t Þ þ bk sin ð2πkf 0 t Þ 2 k=1 k

Here, the cosine components reflect the even part and the sine components reflect the odd part which are generally contained in each function (see intermediate note in Chap. 2). The connection between the two notations can be established using the following trigonometric function and substitutions:

1

Strictly mathematically, it must be possible to integrate the periodic functions absolutely. Furthermore, they must not have discontinuities of the second type, and the number of discontinuities of the first type and the extreme values within a period must be finite. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_3

9

10

3 Fourier Series

Fig. 3.1 Periodic, non-harmonic and neither even nor odd function (mixed function)

Ak cos ð2πkf 0 t þ φk Þ = Ak cos ð2πkf 0 t Þ  cos ðφk Þ - Ak sin ð2πkf 0 tÞ  sin ðφk Þ A0 = a0 =2 and

Ak cos ðφk Þ = ak

or

- Ak sin ðφk Þ = bk , k > 0,

2 3 þ1 X 6 7 up ðtÞ = A0 þ 4Ak cos ðφk Þ cos ð2πkf 0 tÞ - Ak sin ðφk Þ sin ð2πkf 0 tÞ5 |{z} |fflfflfflfflfflffl ffl {zfflfflfflfflfflffl ffl } |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} k=1 ak

a0 =2

bk

The following applies to the coefficients of the “physical” notation: A0 = a0 =2;

Ak =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2k þ b2k ,

φk = - arctan ðbk =ak Þ,

for

k>0

In case of even functions, and only these we will consider in our examples for the sake of simplicity, are φk = 0 or bk = 0 and Ak = ak, k = 1, 2, 3, . . . . We will, however, prefer a third notation in the following, which uses complex2 Fourier coefficients (while the coefficients Ak or ak and bk are real numbers). For this we recall Euler’s formula based on a series development, the “most beautiful formula in the world” or “pearl of mathematics”: ejx = cos ðxÞ þ j sin ðxÞ;

j=

pffiffiffiffiffiffiffiffi -1

The representation of complex numbers in the complex plane is based on this formula. From the above formula, simple transformations can be made to derive the two notations that we need in the following:

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi We use here the notation j = - 1 (instead of i = - 1 ), that is common in electrical engineering to indicate the imaginary unit of complex numbers.

2

3

Fourier Series

11

2

3

6 7 ejx þ e - jx = ½ cos ðxÞ þ j sin ðxÞ þ 4 cos ð - xÞ þ j sin ð - xÞ5 = 2 cos ðxÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} 2

= cos ðxÞ

= - sin ðxÞ

3

6 7 ejx - e - jx = ½ cos ðxÞ þ j sin ðxÞ - 4 cos ð - xÞ þ j sin ð - xÞ5 = 2j sin ðxÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} = cos ðxÞ

= - sin ðxÞ

or with the substitution x = 2πkf0t cos ð2πkf 0 tÞ =

ej2πkf 0 t þ e - j2πkf 0 t 2

and

sin ð2πkf 0 tÞ =

ej2πkf 0 t - e - j2πkf 0 t 2j

Figure 3.2 shows the geometrical interpretation of Euler’s formula of the cosine. For this purpose, we represent two rotating phasors ej2πkf 0 t and e - j2πkf 0 t in the complex plane, which rotate at the same speed in different directions around the coordinate origin. interpret these two rotating phasors as vectors and form the vector sum  þj2πkf We 0 t þ e - j2πkf 0 t . e The sum vector thus formed always lies on the real axis. At uniform (constant) rotational speed f0, this vector describes the function 2 cos (2πkf0t), that is, ej2πkf 0 t þ e - j2πkf 0 t = 2 cos ð2πkf 0 t Þ. Now we insert Euler’s formula of the cosine into our first expression of the Fourier series: up ðt Þ = A0 þ

þ1 X

þ1 X

eþjð2πkf 0 tþφk Þ þ e - jð2πkf 0 tþφk Þ 2 k=1 k=1   þ1 X Ak ejφk A e - jφk - j2πkf 0 t ej2πkf 0 t þ k e = A0 þ 2 2 k=1 Ak cos ð2πkf 0 t þ φk Þ = A0 þ

Ak

For a better understanding, we will briefly leave the notation of the Fourier series with the sum sign and use the notation at length:

12

3 Fourier Series

Fig. 3.2 Illustration of Euler’s formula in the complex number plane

up ðt Þ = A0 þ

þ1  X Ak  ejφk

k=1 jφ1 A1 e j2π1f 0 t

= A0 þ e |{z} |fflffl{zfflffl} 2 C0

þ

C1

2 þ

ej2πkf 0 t þ

Ak  e - jφk - j2πkf 0 t e 2



A1 e - jφ1 - j2π1f 0 t A2 ejφ2 j2π2f 0 t e þ e 2 ffl} 2 |fflfflfflffl{zfflfflffl |fflffl{zfflffl} C-1

- jφ2

jφk

C2

- jφk

A2 e A  e j2πkf 0 t Ak  e e - j2π2f 0 t þ . . . þ k e þ e - j2πkf 0 t þ . . . , 2 2 2 |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} C-2

Ck

Furthermore, we make the following substitutions:

C-k

3

Fourier Series

A0 = C0 ; =

Ak ; 2

13

Ak ejφk = C k = jCk jejφk , 2

Ak e - jφk = C - k = jC k je - jφk , 2

jCk j

k≠0

With this we get: up ðtÞ = C0 þ C1 ej2π1f 0 t þ C - 1 e - j2π1f 0 t þ C 2 ej2π2f 0 t þ C - 2 e - j2π2f 0 t þ . . . þ Ck ej2πkf 0 t þ C - k e - j2πkf 0 t þ . . . If we now slightly change the order of the summands and take into account that e - j2πkf 0 t = eþj2πð - kÞf 0 t , we get an infinite series with a running index -1≤k≤+1: up ðt Þ = . . . þ C - k ej2πð - kÞf 0 t þ . . . þ C - 2 ej2πð - 2Þf 0 t þ C - 1 ej2πð - 1Þf 0 t þ C0 þC 1 ej2π1f 0 t þ C 2 ej2π2f 0 t þ . . . þ Ck ej2πkf 0 t þ Ckþ1 ej2πðkþ1Þf 0 t þ . . . respectively þ1 X

up ðtÞ =

Ck ej2πkf 0 t

k= -1

The coefficients C 0 = A0 ; =

Ak , 2

C k = jC k jejφk =

Ak jφk e , 2

C - k = jCk je - jφk =

Ak - jφk e , 2

jCk j

k≠0

are called the complex Fourier coefficients. For even periodic time functions, we have up(t) = up(-t) and φk = 0 for all k. Thus all Fourier coefficients are real, and the following applies: C 0 = A0 ; A Ck = C - k = k ; 2

) k≠0

when

up ðtÞ = up ð - tÞ, that is even

The Fourier coefficients are represented graphically as a line spectrum. This is quite simple in case of real Fourier coefficients, as illustrated in the following

14

3 Fourier Series

Fig. 3.3 Periodic sequence of approximately rectangular pulses and associated line spectrum

example. Thus, Fig. 3.3 shows the line spectrum of an even periodic (almost rectangular) function, which we had formed in Chap. 2 (Fig. 2.6) by adding cosine functions. Note, that in Fig. 3.3 only the first 15 spectral lines are shown on both sides. (A possibility of graphical representation of complex coefficients or functions is shown in Chap. 7 in connection with the representation of complex transfer functions). The sign ⇄ in Fig. 3.3 symbolizes the operation of the Fourier transform. Now the question arises, how the values of complex Fourier coefficients can be determined from a given periodic function up(t)? This question is answered by the following derivation, which you can skip if you trust the result.

Derivation To derive the calculation formula for the coefficients Ck, we first write the Fourier series again at length: up ðtÞ =

1 X

Ck  ej2πkf 0 t = . . . þ C - k e - j2πkf 0 t þ . . . þ C - 1 e - j2π1f 0 t þ C0

k= -1

þC1 ej2π1f 0 t þ . . . þ Ck ej2πkf 0 t þ Ckþ1 ej2πðkþ1Þf 0 t þ . . . Then we multiply both sides by e - j2πkf 0 t, whereby we achieve that the coefficient Ck on the right side stands alone without a multiplier:

Derivation

15

up ðt Þ  e - j2πkf 0 t = . . . þ C - k e - j2πð2kÞf 0 t þ . . . þ C - 1 e - j2πð1þkÞf 0 t þ C0 e - j2πkf 0 t þC1 ej2πð1 - kÞf 0 t þ . . . þ Ck þ Ckþ1 ej2πf 0 t . . . We then integrate this equation on both sides: Z

t p =2 - tp =2

Z þ

- t p =2 - tp =2

Z þ

up ðtÞ  e - j2πkf 0 t dt = . . . þ

- t p =2 - tp =2

C - 1 e - j2πð1þkÞf 0 t dt þ Z Ck dt þ

- tp =2 - tp =2

Z

Z

tp =2 - t p =2

- tp =2 - tp =2

C - k e - j2π ð2kÞf 0 t dt þ . . .

C0 e - j2πkf 0 t dt þ

Z

- tp =2 - tp =2

C 1 ej2πð1 - kÞf 0 dt þ . . .

Ckþ1 ej2πkf 0 t dt þ . . .

Let us now look at the integrals on the right: Ztp =2

t =2

Ck dt = Ck  tj -p tp =2 = C k 

ht

p

2

-

 - t i p

2

= Ck  tp :

- tp =2

For all other integrals on the right side applies with f0 = 1/tp: Ztp =2

Cn ej2πðn - kÞf o t dt =

- tp =2

=

=

Cn

t =2  ej2π ðn - kÞf o t -p tp =2 j2π ðn - k Þf o

Cn t =2  ej2πðn - kÞt=tp j -p tp =2 j2π ðn - k Þf o

h i Cj  ejπðn - kÞ - e - jπðn - kÞ , j2π ðn - k Þf o

16

3 Fourier Series

A look at exponential function in the unit circle (Fig. 3.2) shows us that e jπ(n - k) = e-jπ(n - k) for all integer (n - k). tpR=2 Cn ej2πðn - kÞf o t dt = 0 holds for all integerðn - k Þ. Thus all tp =2

This brings us to the final formula tpR=2 up ðt Þ  e - j2πkf 0 t dt = tp  Ck respectively C k = - tp =2

1 tp



tpR=2 - t p =2

up ðtÞ  e - j2πkf 0 t dt

to calculate the complex Fourier coefficients Ck from up(t). In summary, we have the following mathematical apparatus for the Fourier transform of periodic time functions:

up ðtÞ =

þ1 X k= -1

Ck e

j2πkf 0 t

1 ⇄ Ck = tp

Ztp =2

up ðt Þ  e - j2πkf 0 t dt

- t p =2

The relationship of the coefficients of the alternative notations of the Fourier series is shown in Fig. 3.4.

Fig. 3.4 Relationship between the coefficients of the different notations of the Fourier series

Derivation

17

Example We calculate the (complex) Fourier coefficients Ck for the periodic rectangular function shown in the figure. Periodic rectangle function

This periodic function is described by the following analytical expression: up =

þ1 X

  u t - mt p ;

 uðtÞ =

m= -1

U0 0

for

jt j ≤ T=2 else

To determine the complex Fourier coefficients we use the expression

1 Ck = tp

Ztp =2

up ðt Þ  e - j2πkf 0 t dt,

- t p =2

where T < tp is assumed. In the integration area -tp/2 ≤ t ≤ tp/2, the periodic function up(t) is identical to its generating function u(t), while u(t)  0 for |t| > T/2 and u(t) = U0 for |t| ≤ T/2. So we can write 1 Ck = tp

ZT=2 - T=2

U 0  e - j2πkf 0 t dt =

T=2 U0 1   e - j2πkf 0 t - T=2 tp - j2πkf 0

18

3 Fourier Series

Ck =

h i U0 1 U 0 eþjπkf 0 T - e - jπkf 0 T   e - j2πkf 0 ðT=2Þ - e - j2πkf 0 ð - T=2Þ =  tp - j2πkf 0 t p πkf 0 2j

With Euler’s formula sin ðπkf 0 T Þ =

ejπkf 0 T - e - jπkf 0 T 2j

we finally get the expression Ck =

U0 U T sin ðπkf 0 T Þ U 0 T  sin ðπkf 0 T Þ = 0  =  sincðπkf 0 T Þ πkf 0 T t p πkf 0 tp tp |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} sincðπkf 0 T Þ

for calculating the Fourier coefficients of a periodic rectangular sequence with the parameters U0 (amplitude of the rectangular pulses), T (pulse width, rectangle width) and tp (period). Note The function sin(x)/x = sinc (x) is called the sinc-function (or sinus cardinalis or sampling function). Its typical course is shown in the figure. sinc-function

For x = 0, by means of the rule of de l’Hospital we get:

Derivation

19

lim sincðxÞ = lim

x→0

x→0

0

ð sin xÞ sin x cos x = lim =1 = lim 0 x x → 0 ðxÞ x→0 1

Example (Continued) Since the periodic rectangular function shown in the figure of the previous example box is an even function, its Fourier coefficients Ck are real coefficients. The transition to the non-complex notation of the Fourier series is therefore particularly easy: up ðt Þ = A0 þ

þ1 X

Ak cos ð2πkf 0 tÞ

k=1

with A0 = C0 and Ak = 2|Ck| = 2Ck, k > 0. The coefficients Ak shown in the table in Fig. 2.6 were calculated for the parameters U0 = 1 [V], T = 1.25 [ms], tp = 5 [ms] using the formula derived above, which is shown here as an example with k = 3:    U 0 T sin ðπ3f 0 T Þ U 0 T sin π3 T=tp     = C3 = π3f 0 T tp tp π3 T=t p

mit

f0 =

1 tp

3π sin 4 1 ½V  1:25 ½ms sin ðπ  3  ð1:25 ½ms=5 ½msÞÞ 1 ½V  C3 =  = 4 5 ½ms π3ð1:25 ½ms=5 ½msÞ 3π 4 1 ½V 0:7071  = 0:075 ½V → A3 = 2  C 3 = 0:15 ½V C3 = 4 2:3562

Figure 3.3 shows the coefficients graphically Ck as a line spectrum.

4

From the Fourier Series to the Fourier Integral

Now we want to “play” a little with the numerical example from Chap. 3. There we have applied the formula for calculating the Fourier coefficients to a periodic rectangular sequence and found that for this case Ck =

U0
T sinc ðπkf 0 T Þ; tp

f0 =

1 : tp

We have represented the values of the coefficients as a line spectrum, where the height of the lines is proportional to the coefficient values and the position of the lines on the frequency axis f is determined by k
f0. If we connect the upper ends of the lines, an envelope is created, which is described by the following expression: fenvelope over Ck g =

U0
T sincðπfT Þ = U 0
T
f 0
sincðπfT Þ tp

If we now let the period tp of the rectangular sequence become larger while keeping the pulse width T and amplitude U0 constant, we see that the spectral lines move closer and closer together and at the same time the amplitude of the envelope becomes smaller and smaller, but the shape of the envelope remains unchanged (Fig. 4.1). However, when we divide the Fourier coefficients Ck, which become smaller and smaller with increasing period, by the distance f0 between the spectral lines, which also becomes smaller and smaller, then the amplitude of the envelope does not change over the modified coefficients Ck/f0 (Fig. 4.2). This modified envelope is now described by the function # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_4

21

22

4

From the Fourier Series to the Fourier Integral

Fig. 4.1 Relationship between period and “density” of the spectral lines (1)

fenvelope over Ck =f 0 g = U 0
T
sincðπfT Þ: If we now (as a thought experiment) let the period go toward infinity, only a single pulse remains of the original periodic rectangular sequence in the coordinate origin, that is, an aperiodic function. In the spectral domain, the spectral lines are now inseparably close to each other, but the modified envelope over Ck/f0 does not change. It is transformed by passing to the limits tp → 1 or f0 → 0 merely into a continuous function, which we will refer to in the following as spectral amplitude density U( f ) (or spectral function or simply spectrum):

4

From the Fourier Series to the Fourier Integral

23

Fig. 4.2 Relationship between period and “density” of the spectral lines (2)

U ðf Þ = lim ðCk =f 0 Þ f0 →0

In our example, the aperiodic function u(t) consisting of a single rectangular pulse thus has a sinc-function as the corresponding spectral function (Fig. 4.3): ( T U 0 for jtj ≤ uðtÞ = 2 ⇄ U 0
T
sincðπ
f
tÞ 0 In general, the Fourier coefficients can be calculated:

24

4

From the Fourier Series to the Fourier Integral

Fig. 4.3 Aperiodic time function and corresponding continuous spectral function

1 Ck = tp

þt Zp =2

up ðt Þ
e - j2πkf o t dt

- tp =2

and with tp = 1/f0 we get: Ck = f0

þt Zp =2

up ðtÞ
e - j2πkf o t dt

- t p =2

Temporarily, we want f0 to be replaced by the interval Δf, which is usual in limit value considerations, that is, Δf = f0. This brings us to the following notation: Ck = Δf

þt Z p =2

up ðt Þ
e - j2πkΔft dt:

- tp =2

Furthermore, we can replace up(t) by its forming aperiodic function u(t), since within the integration limits both functions are identical. If we now go to the limits tp → 1 or f0 = Δf → 0, we obtain the following expression for the calculation of the continuous spectral function U( f ) from the corresponding aperiodic time function u(t):

4

From the Fourier Series to the Fourier Integral

C U ðf Þ = lim k = lim Δf tp → 1 tp → 1 Δf → 0

Δf → 0

þt Z p =2

up ðtÞ
e

25

- j2πkΔft

Zþ1 dt =

uðtÞ
e - j2πft dt

-1

- tp =2

whereby as result of this limit value consideration the set of points {k
Δf} transforms into the continuous variable f, that is fk
Δf g → f

for

Δf → 0:

Now the next question is how we can determine the corresponding aperiodic time function u(t) from a given continuous spectral function U( f ). We start with the complex Fourier series: up ðtÞ =

þ1 X

Ck
eþj2πkf 0 t

k= -1

Also for the following limit value consideration, we replace f0 by Δf, thus Δf = f0, and carry out a small reformulation: up ðt Þ =

þ1 X Ck þj2πkΔft
Δf :
e Δf k= -1

If we now go again to the limit value consideration Δf → 0 or tp → 1, and take into account that. uðtÞ = lim up ðtÞ and lim tp → 1

Ck

Δf → 0 Δf

= U ðf Þ,

we get finally Zþ1 þ1 X Ck þj2πkΔft
e
Δf = U ðf Þ
eþj2πft df : Δf Δf → 0 k= -1

uðtÞ = lim up ðtÞ = lim tp → 1

-1

The relations for the Fourier transform of aperiodic functions can thus be summarized by the following Fourier integrals:

26

4

Zþ1 uðt Þ =

U ðf Þ
e -1

þj2πft

From the Fourier Series to the Fourier Integral

Zþ1 df

⇄ U ðf Þ =

uðt Þ
e - j2πft dt

-1

If we now consider both integrals, we see a clear symmetry, which we will illustrate with the help of Figs. 5.1 and 5.2: • A rectangle function in the time domain has a sinc-function in the frequency domain. • A rectangular function in the frequency domain has a sinc-function in the time domain. Similar symmetrical relationships also apply to other functions.

5

The Fourier Transformation of Standard Signals

In the following, the Fourier transformation is shown for typical standard signals that are frequently used in (measurement) technology.

Aperiodic Rectangle Function
uðtÞ =

U 0 , jtj ≤ T H =2 0, else

 ⇄ U ðf Þ = U 0 T H

sin ðπT H f Þ = U 0 T H sincðπT H f Þ ðπT H f Þ

Aperiodic Sinc-Function sin ðπBH tÞ uðt Þ = U ðf = 0ÞBH ⇄ U ðf Þ = ðπBH tÞ




U ðf = 0Þ, jf j ≤ BH =2



0, else

Gaussian Function uðtÞ = e - πt ⇄ U ðf Þ = e - πf 2

2

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_5

27

28

5 The Fourier Transformation of Standard Signals

Fig. 5.1 Rectangle signal u(t) and corresponding spectral function U( f )

Fig. 5.2 Sinc-function u(t) and corresponding spectral function U( f )

The transformation of the original Gaussian function is reciprocal, that is, for the functions u(t) and U( f ) the same functional relation (dependence of t and f, respectively) applies.

Compressed and Stretched Gaussian Functions f 2 2 1 uðtÞ = e - πðatÞ ⇄ U ðf Þ = e - πðaÞ a

Figures 5.3, 5.4, and 5.5 demonstrate an important general correlation: • The narrower the time function, the wider its spectral function. • The wider the time function, the narrower its spectral function.

Dirac Function (Delta Function)

29

Fig. 5.3 Gaussian function

Fig. 5.4 Compressed Gaussian function (here a = 2)

Fig. 5.5 Stretched Gaussian function (here a = 0.5)

Dirac Function (Delta Function) The Dirac function is a mathematical abstraction that does not exist in nature, but often plays an important role in model-based considerations, for example, as an abstract model of test signals. Calculations usually become easier if real test signals are replaced by the Dirac function. If the necessary boundary conditions are taken into account (see below), the calculation results are sufficiently accurate for practical applications. The Dirac function is defined as follows: Zε δðtÞ  0 for

δðtÞdt = 1

t ≠ 0; -ε

for

ε>0

30

5 The Fourier Transformation of Standard Signals

Fig. 5.6 Dirac function and corresponding spectral function

Fig. 5.7 Approximate realization of the Dirac function

This means that the Dirac function is a pulse in the point t = 0 whose pulse width tends toward zero and whose amplitude tends toward infinity. It is symbolized by a vertical arrow (Fig. 5.6). uðtÞ = C 0 δðtÞ ⇄ U ðf Þ = C0 The spectral function of the Dirac function C0δ(t) is defined as U( f ) = C0 = const. In (metrological) practice, the Dirac function is replaced by a very narrow pulse (of any shape), since the half-width BH of the spectral function is inversely proportional to the pulse half-width TH, that is, BH ≈ 1/TH (see also Chap. 6), and thus the course of the spectral function is approximately constant in the frequency range of interest (Fig. 5.7).

Cosine Function up ðtÞ = U 0 cos ð2πf 0 tÞ ⇄ U ðkf 0 Þ =

U0 U δðf þ f 0 Þþ 0 δðf - f 0 Þ 2 2 |{z} |ffl{zffl} C-1

Cþ1

Cosine Function

31

Fig. 5.8 Cosine function and corresponding spectral function

We have shown here in the figures units of measurement which should help us to understand the physical sense of the spectral functions: If we assume that the signals in the time domain (on the left side of the figures) are electric voltages with the unit of measurement [V], then the corresponding spectral functions (on the right side of the figures) have the unit of measurement [V] in the case of periodic time signals (Fig. 5.8) and the unit of measurement [V/Hz] or [Vs] in the case of aperiodic time signals (Figs. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, and 5.7). The unit of measurement [V/Hz] or “volt per 1 Hz bandwidth” results from the transition from periodic to aperiodic signals (Chap. 4): As the period of the time function increases, the lines of the spectrum move closer together and their amplitude decreases. We compensate for the reduction in amplitude by multiplying it by the period. This results in the unit of measurement [Vs] or, equivalently, [V/Hz]. If we square the spectral amplitude density, the result shows us the distribution of the energy of the signal over the frequency. The unit of measurement of this spectral energy density is then [(V/Hz)2] or [V2s/Hz]. Taking the units of measurement into account also helps to avoid errors when solving practical tasks. Note Due to the symmetry properties (see Chap. 4), the functional relationships shown here can also be exchanged in the time domain and in the frequency domain, as for example in Figs. 5.1 and 5.2.

6

Properties of the Fourier Transform and Approximation Relations

In the following some important properties of the Fourier transform (without further proof) are listed. 1. If u(t) ⇄ U( f ) and k is a constant independent of time and frequency, then k
uðtÞ ⇄ k
U ðf Þ 2. If u1(t) ⇄ U1( f ) and u2(t) ⇄ U2( f ), then it holds: ½u1 ðt Þ þ u2 ðtÞ ⇄ ½U 1 ðf Þ þ U 2 ðf Þ

3. Even real time functions have even real spectral functions and odd real time functions have odd imaginary spectral functions. 4. Real time functions, which generally consist of an even and an odd component, that is u(t) = ueven(t) + uodd(t), have spectral functions with an even real component and an odd imaginary component. 5. Time and frequency shifting: The shift of a time function means multiplication of the spectral function by a rotating phaser: uðt - t0 Þ ⇄ U ðf Þ
e - j2πt0 f :

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_6

33

34

6

Properties of the Fourier Transform and Approximation Relations

The same applies to the shift of a spectral function: uðtÞ
eþj2πf 0 t ⇄ U ðf - f 0 Þ

This is the conclusion that is important in practice: 1 ½uðt - t0 Þ þ uðt þ t0 Þ ⇄ U ðf Þ
cos ð2πt0 f Þ 2 uðtÞ
cos ð2πf 0 t Þ ⇄

1 ½U ðf - f 0 Þ þ U ðf þ f 0 Þ 2

The last expression explains the basic principle of amplitude modulation (analog transmission technology, broadcasting), since it shows how the spectrum of a frequency-limited desired signal is shifted to a higher frequency band by multiplying it by a cosine carrier (Fig. 6.1).

Fig. 6.1 Amplitude modulation as an application example for the frequency shift

6

Properties of the Fourier Transform and Approximation Relations

35

6. Time and frequency scaling: The compression or stretching of a time function u(t) by multiplying its variable t by a positive real factor a corresponds to a stretching or compression of the corresponding spectral function, as shown for example in Figs. 5.3, 5.4, and 5.5: uðat Þ ⇄


 1 f U , a > 0, real a a

The same applies to the frequency domain:   1 t u ⇄ U ðaf Þ, a a

a > 0, real

7. The area under the spectral function U( f ) is equal to the value of the time function u(t) at the position t = 0: Z1 U ðf Þdf

uðt = 0Þ = uð0Þ =

≙

area under U ðf Þ!

-1

8. The area under the time function u(t) is equal to the value of the spectral function U( f ) at the position f = 0: Z1 U ðf = 0Þ = U ð0Þ =

uðtÞdt

≙

area under uðtÞ!

-1

The last two properties result in the following useful approximate relationships for functions that are not too strongly asymmetrical and that drop monotonically from the maximum on both sides, that is, resemble a bell (Fig. 6.2): BH ≈

1 , TH

U ðf = 0Þ ≈ uðt = 0Þ
T H , uðt = 0Þ ≈ U ðf = 0Þ
BH

36

6

Properties of the Fourier Transform and Approximation Relations

Fig. 6.2 Approximate relationships for bell-like functions

7

The Fourier Transformation as a Tool for the Description of Signals and Systems

In (electrical) engineering, one of the oldest and simplest methods of determining the transmission properties of a system is to input a harmonic (sinusoidal or cosinusoidal) signal u1(t) of a certain frequency to a system and compare it with the output signal u2(t). In the following, we will consider a linear system that can contain energy storage devices (e.g., capacitors C and/or inductors L )1 and whose behavior is thus generally frequency-dependent. If we now input a harmonic (sinusoidal or cosinusoidal) signal u1(t) of a certain frequency fk to this system, we see that the output signal u2(t) is still a harmonic signal, but it is delayed by Δtk and generally has a different (larger or smaller) amplitude. Both quantities, the amplitude and the time delay of the output signal, are frequency dependent, as shown in the comparison of cases 1 and 2 in Fig. 7.1. We describe the signals at the input and output of the system as follows: • Input signal: u1k(t) = A10 cos (2πfkt); A10 = const. • Output signal: u2k(t) = A2k cos (2πfk(t - Δtk)), A2k, Δtk - frequency dependent. The spectral functions of both signals are (see Chaps. 5 [Cosine] and 6 [time and frequency shifting]):

1

We are looking at electrotechnical systems. However, the statements are generally valid for all types of linear systems, for example, mechanical systems, where the energy stores are typically spring and mass elements. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_7

37

38 7

The Fourier Transformation as a Tool for the Description of Signals and Systems

Fig. 7.1 System response to harmonic input signals of different frequencies

u1k ðt Þ = A10 cos ð2πf k t Þ⇄U 1k ðf Þ = u2k ðt Þ = A2k cos ð2πf k ðt - Δtk ÞÞ⇄U 2k ðf Þ =

A10 ½δðf þ f k Þ þ δðf - f k Þ 2 A2k ½δðf þ f k Þ þ δðf - f k Þe - j2πf k Δtk 2

If we now compare the spectral functions of both signals, we get the frequencydependent complex transfer factor G( fk): Gðf k Þ =

U 2k ðf Þ = U 1k ðf Þ

A2k 2

½δðf þ f k Þ þ δðf - f k Þe - j2πf k tk A10 2

½δðf þ f k Þ þ δðf - f k Þ

=

A2k - j2πf k Δtk e A10

respectively Gðf k Þ =

U 2k ðf Þ = jGðf k Þjejφk , U 1k ðf Þ

with jGðf k Þj =

A2k and φk = - 2πf k Δtk : A10

This allows to calculate the amplitude of the harmonic output signal u2k ðtÞ = A2k cos ð2πf k ðt - Δt k ÞÞ for any frequency fk from the amplitude of the harmonic input signal u1k ðtÞ = A10 cos ð2πf k tÞ; using the complex transfer factor:

A10 = const:

7

The Fourier Transformation as a Tool for the Description of Signals. . .

39

A2k = A10  jGðf k Þj: The time shift can be read directly from the complex transfer factor: Δt k = -

φk : 2πf k

The set of all possible transfer factors for - 1 < fk < + 1 gives the transfer function. G( f ) = |G( f )|  e jφ( f ), see Fig. 7.3. Note: • |G( f )| is always an even function. • φ( f ) is always an odd function, that is, -φ( f ) = φ(-f ), taking φ( f ) ≤ 0 for all f ≥ 0. This follows from causality since the time delay can only be greater than (or equal to) zero, that is, Δt ≥ 0. If the structure and the values of the elements of the system are known, the transfer function can be calculated. However, if the system represents a black box with the unknown internal structure or unknown sizes of the components, the transmission factor is determined by measurement. A simple method is to change the frequency of the harmonic signal at the input in appropriate steps and measure the amplitude and time delay of the output signal for each of these frequencies (e.g., using a vector voltmeter or oscilloscope) and then display them in tabular or graphical form, as shown in Figs. 7.2 and 7.3. Note to Fig. 7.3: Due to the symmetries of the partial functions |G(-f )| = |G( f )| and φ(-f ) = - φ( f ) of the transfer function, it is sufficient for the time being to show only the right part of these functions. So far we have only described the interaction between linear systems and harmonic signals. But how do we deal with periodic but non-harmonic signals, as shown in Fig. 7.4? To answer this question, let us remember that we can represent periodic non-harmonic signals using the Fourier series:

40 7

The Fourier Transformation as a Tool for the Description of Signals and Systems

Fig. 7.2 Simple measuring arrangement for determining the transfer function point by point

Fig. 7.3 Presentation of the transfer factors and the transfer function as table and graph

Fig. 7.4 Interaction between linear system and non-harmonic periodic signal

u1p ðtÞ = A10 þ φ1k = 0

þ 1 P k=1

A1k cos ð2πkf 0 þ φ1k Þ =

for even functions

A10 þ

þ 1 P k=1

A1k cos ð2πkf 0 Þ,

Note: We are looking here at an even periodic function on the input side, as shown in Fig. 7.4. In this case all cosine functions under the sum sign have a phase shift of φ1k = 0.

7

The Fourier Transformation as a Tool for the Description of Signals. . .

41

So if we can represent the periodic input signal as the sum of cosine functions of different amplitude and frequency, then we can also represent the output signal as the sum of different cosine functions. These cosine functions have the same frequencies as the input signal, but different amplitudes and phase shifts: u2p ðtÞ = A20 þ

þ1 X

A2k cos ð2πkf 0 þ φk Þ,

k=1

where, in analogy to the above considerations, φk can be read off directly from the value of the complex transfer function at the point fk Gðf k Þ = jGðf k Þj  ejφðf k Þ = jGðf k Þj  ejφk , and the following applies to the amplitudes: A2k = A1k  jGðf k Þj: It becomes even easier if we move on to the complex notation of the Fourier series: • Input signal: u1p ðtÞ = A10 þ

þ1 X

A1k cos ð2πkf 0 Þ =

k=1

þ1 X

C 1k  ej2πkf 0 t ; C10 = A10 ; C1k

-1

= A1k =2, real for even functions

• Output signal: u2p ðt Þ = A20 þ

þ1 X

A2k cos ð2πkf 0 þ φk Þ =

k=1

=

A2k jφk e 2

With A2k = A1k  |G( fk)| we get

þ1 X -1

C 2k ej2πkf 0 t ;

C20 = A20 , C2k

42 7

The Fourier Transformation as a Tool for the Description of Signals and Systems

Fig. 7.5 Interaction between linear system and aperiodic signal

C2k =

A2k jφk A1k e = jGðf k Þj  ejφk 2 2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |{z} C1k

or

C 2k = C1k  Gðf k Þ:

G ðf k Þ

If we now finally look at an aperiodic signal (Fig. 7.5), we only need to make the transition from the Fourier series to the Fourier integral in accordance with Chap. 4: Zþ1 þ1 X C1k þj2πkΔft e  Δf = U 1 ðf Þ  eþj2πft df Δf Δf → 0 k= -1

u1 ðtÞ = lim u1p ðt Þ = lim tp → 1

-1

Zþ1 þ1 X C2k þj2πkΔft e  Δf = U 2 ðf Þ  eþj2πft df Δf Δf → 0 k= -1

u2 ðtÞ = lim u2p ðt Þ = lim tp → 1

-1

With C2k = C1k  G( fk) = C1k  G(k  Δf ) we can write: þ1 X C1k GðkΔf Þeþj2πkΔft  Δf Δf Δf → 0 k= -1

u2 ðtÞ = lim u2p ðt Þ = lim tp → 1

Zþ1 = -1

U 1 ðf Þ  Gðf Þeþj2πft df |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} U 2 ðf Þ

Finally, we obtain the following fundamental general relationship for the description of the interaction between linear systems and signals (in the frequency domain): U 2 ðf Þ = U 1 ðf Þ  Gðf Þ For practical application, the path shown in Fig. 7.6 can be followed. This is particularly useful if we perform the direct and inverse transformations

7

The Fourier Transformation as a Tool for the Description of Signals. . .

43

Fig. 7.6 Interaction between signals and system – solution using Fourier transformation

approximately, possibly using the known transformations of standard signals, or with the aid of the Discrete Fourier Transform (see e.g., Lange & Lange, 2019). Robust Example Given is the system shown in the figure (idealized low pass): Example – task definition

A narrow rectangular pulse with pulse width TH = 0.1 [ms] and an amplitude U0 = 5 [V] acts on this system. The output signal u2(t) is sought, whereby an approximate calculation is sufficient. Solution: We follow the solution path proposed in Fig. 7.6.

44 7

The Fourier Transformation as a Tool for the Description of Signals and Systems

Step 1: We determine the spectral function of the input signal as sinc-function (see Chap. 5) with U1( f = 0) = U0TH = 0.5 [V/kHz] and the first zero crossing at 1/TH = 10 [kHz] (following figure, top right). Step 2: Now we have to perform the multiplication U 2 ðf Þ = U 1 ðf Þ  Gðf Þ = U 1 ðf Þ  jGðf Þj  ejφðf Þ whereas here |U2( f )| = U1( f )  |G( f )|, as U1( f ) is a real function. We find the function |U2( f )| by multiplying the values of U1( fi) and |G( fi)| point by point for each frequency fi, as shown in the following figure. As a result we get the function |U2( f )| shown in the lower right corner of this figure, which is again approximately bell-shaped and has the following parameters: | U2( f = 0)| = 1 [V/kHz] and BH ≈ 2 [kHz]. Thus we now know, approximately, the spectral function of the output signal: U 2 ðf Þ = jU 2 ðf Þj  ejφðf Þ = jU 2 ðf Þj  e - j2πΔtf , |fflfflffl{zfflfflffl}

Δt = 0:625 ½ms

≈ bell

Step 3: For the inverse transformation of the almost bell-shaped function |U2( f )| with jU 2 ðf = 0Þj = 1 ½V=kHz and BH ≈ 2 ½kHz we can use the approximation relations from Chap. 6, Fig. 6.2. The transformation of this function, which is shown in the following figure with z2(t), that is |U2( f )| ⇄ z2(t), is again a bell-shaped function with the parameters z2(t = 0) ≈ |U2( f = 0)|  BH = 2 [V] and TH ≈ 1/BH = 0.5 [ms].

7

The Fourier Transformation as a Tool for the Description of Signals. . .

45

Example – solution

Now we still have to take into account the phase shift φ( f ) = - 2π  Δt  f with Δt = 0.625 [ms]. Fortunately, in our simple example, this is linearly dependent on the frequency so that we can apply the shifting theorem: If

z2 ðtÞ⇄jU 2 ðf Þj,

then

z2 ðt - Δt Þ⇄jU 2 ðf Þj  e - j2πΔtf = U 2 ðf Þ

As a final approximate solution, we get the bell-shaped function u2(t) = z2(t - Δt) shown in the figure above on the bottom left with the parameters u2(t = Δt) ≈ 2 [V] and TH ≈ 0.5 [ms].

46 7

The Fourier Transformation as a Tool for the Description of Signals and Systems

Note: The output signal u2(t) is significantly wider than the input signal and has a much lower amplitude and no sharp edges. This is due to the fact that the low-pass filter suppressed the higher frequencies of the input signal, thus eliminating much of the energy of the input signal. Final remark: The interaction between signals and systems can also be calculated directly in the time domain, for example, using the impulse response function g(t). This impulse response function is nothing else than the Fourier transform of the transfer function G( f ) ⇄ g(t) and it applies: Z u2 ðtÞ =

1 -1

u1 ðτÞ  gðt - τÞdτ = u1 ðtÞ  gðtÞ⇄U 2 ðf Þ = U 1 ðf Þ  Gðf Þ

The operation u1(t)  g(t) is called convolution. Unfortunately, we cannot go into this topic in the context of this essentials and have to refer the reader to further literature (e.g., Kreß & Kaufhold, 2010).

8

Application of the Fourier Transform to Stochastic Signals

In daily practice we constantly encounter random processes, for example, in economics (exchange rate fluctuations and the like) and also in technology, where such stochastic processes usually occur as disturbances. But also desired signals as speech signals and audiovisual signals can be interpreted as stochastic processes. Therefore, we need possibilities to describe these processes (signals) mathematically, in order to be able to do corresponding calculations (e.g., in Chap. 9). In the time domain, stochastic processes are typically described by autocorrelation functions (ACF), which are defined as follows 1 ψ xx ðτÞ = lim T →1 T

ZT=2 xðt Þ  xðt þ τÞdt - T=2

Figure 8.1 shows a “freehand example” of a stochastic process and its ACF. The ACF has the following characteristics: 1. The ACF is always an even function: ψ xx(τ) = ψ xx(-τ). 2. The value of the ACF at the point τ = 0 is equal to the average power m2 of the process (measured at a resistance of 1Ω): ψ xx ð0Þ = m2 = x2 ðtÞ: Here the

pffiffiffiffiffiffi m2 = xeff is the effective value of the stochastic process.

3. ψ xx(0) is always the maximum value of the ACF: |ψ xx(τ)| < ψ xx(0), |τ| ≠ 0. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_8

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48

8 Application of the Fourier Transform to Stochastic Signals

Fig. 8.1 Stochastic process and its autocorrelation function (ACF)

4. The value of the ACF for τ → 1 is equal to the power of the DC component:  2 ψ xx ð1Þ = m21 = xðt Þ ≥ 0. The Fourier transform of the ACF Z Ψ xx ðf Þ =

þ1 -1

ψ xx ðτÞ  e - j2πf τ dτ;

Ψ xx ðf Þ⇄ψ xx ðτÞ

is called spectral power density. It shows us the distribution of the process power over the frequency bands (Fig. 8.2). The spectral power density has the following properties: 1. It is always an even function: Ψ xx( f ) = Ψ xx(-f ). 2. Their functional values are always nonnegative: Ψ xx( f ) ≥ 0. 3. The area under the spectral power density is equal to the average power of the stochastic process (see Chap. 6, property 7): Z

1 -1

Ψ xx ðf Þ  df = ψ xx ðτ = 0Þ = m2

The narrower the ACF, the stronger the random character of the stochastic process. In the extreme case of so-called “white noise,” the ACF takes the form of a Dirac function and the corresponding spectral power density has a constant value for all frequency values (Fig. 8.3). This white noise is again an abstraction, because any real interference is band-limited, even if the cutoff frequency can be very high. However, real interferences (especially in telecommunications) often have an

8

Application of the Fourier Transform to Stochastic Signals

49

Fig. 8.2 ACF and spectral power density of a stochastic signal

Fig. 8.3 ACF and spectral power density of “white noise”

almost constant power density in a very large frequency band like white noise, so that this abstraction is quite helpful in the calculation (modeling) of real processes (see e.g., Chap. 9).

9

Optimal Receive Filter (Wiener Filter)

With any type of signal transmission, these signals are affected by various types of interferences (disturbances) on the transmission path. These interferences can cause signals to be received incorrectly. The Fourier transform helps us to develop a theoretical solution approach that is helpful in the construction of real technical receiving equipment. This approach is known as “optimal search filter,” “optimal receive filter,” or “Wiener Filter.” The optimal search filter is designed to ensure the reception of disturbed signals with minimum error probability. We must, therefore, find such a transfer function for the receive filter that guarantees the best possible signal reception when the form of the desired (undisturbed) signal is known. Let us consider a simple example below: At intervals of t0 pulses (or “pauses”) are transmitted via a cable. Each pulse represents a logical “1,” each pause represents a logical “0.” The cable acts like a low-pass filter. As the pulses pass through the cable, they are delayed, attenuated, and distorted (Fig. 9.1). The cable is constantly exposed to interference x(t) – caused by external electromagnetic fields and internal thermal noise (Fig. 9.2). When transmitting a series of pulses, one can therefore observe delayed, attenuated, distorted and disturbed signals at the end of the cable (Fig. 9.3). Let us now look at the area marked by the dotted circle. At the moment when the undisturbed received signal reaches its extreme value, a decision is made whether an impulse or a pause (i.e., a logical “1” or “0”) has been received. The decision threshold is usually at half of the extreme value (Fig. 9.4). Obviously the following proposition applies: The higher the signal-to-noise ratio (SNR)

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6_9

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52

9

Optimal Receive Filter (Wiener Filter)

Fig. 9.1 Digital signal transmission – schematic representation Fig. 9.2 Interference during signal transmission

Fig. 9.3 Distortions and interference in signal transmission

Fig. 9.4 Receive decision – “ONE” or “ZERO”

9

Optimal Receive Filter (Wiener Filter)

53

Fig. 9.5 SNR-maximizing receive filter (search filter) Gopt( f ) SNR =

Instantaneous value of the desired signal at the moment of sampling Expected value of the disturbance at the moment of sampling ð = effective valueÞ SNR =

ju2 ðt = 0Þj , xeff

the smaller the probability of an incorrect decision or the probability of a faulty reception. qffiffiffiffiffiffiffiffiffiffi qRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 In this case xeff = x2 ðtÞ = - 1 Ψ xx ð f Þ  df holds. So we need to design such a receiver that maximizes the SNR at the output of the receiver (Fig. 9.5). The SNR is determined by • the area below the spectral amplitude density of the desired signal, and • the square root of the area under the spectral power density of the interference. If we now consider the spectral power density of the interference in the range fg ≤ f ≤ + fg and also assume white noise as interference with Ψ xx( f ) = Ψ 0 = const. (Fig. 9.6, left), then the effective value of the interference is. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z fg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xeff = Ψ ðf Þ  df = Ψ 0  2f g -fg

and we can calculate an “equivalent mean amplitude density” of the interference Xequ( f ) with

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Optimal Receive Filter (Wiener Filter)

Fig. 9.6 Spectral densities of desired and interfering signal

X equ ðf Þ =

rffiffiffiffiffiffiffi ψ0 = const: 2f g

(Fig. 9.6, right), whereby the latter expression results from the following consideration: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ψ 0  2f g Ψ0 xeff = Ψ 0  2f g = X equ ðf Þ  2f g → X equ ðf Þ = = for jf j ≤ f g : 2f g 2f g Figure 9.7 shows that the maximum value of the (undisturbed) desired signal at the moment of decision t = 0 and the expected value of the disturbance are equal to the areas under the spectral densities of both signals. If we divide the spectral functions into frequency bands, we get a proportional SNRi for each i-th frequency band (Fig. 9.8). Obviously, the optimal receiver should have a high gain factor (transmission factor) in those frequency ranges where the proportional SNRi is high. On the other hand, the gain should be small in those frequency ranges where the proportional SNRi is also small (Fig. 9.8). Since the spectral power density of the interference is constant (independent of the frequency), the frequency-dependent gain or transmission factor G(fi) of the optimal receiver should, therefore, be proportional to the (magnitude) curve of the spectral amplitude density of the undisturbed received signal (Fig. 9.9). Mathematically strictly, however, the following applies (see e.g., Kreß & Kaufhold, 2010)

9

Optimal Receive Filter (Wiener Filter)

55

Fig. 9.7 Relationship between signal values at the moment of decision and areas under the spectral densities

Fig. 9.8 Frequency band-dependent proportional SNR

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9

Optimal Receive Filter (Wiener Filter)

Fig. 9.9 Relationship between the transmission function of the optimal receiver and the spectral density of the wanted signal

  1 Gopt ðf Þ = Gopt ðf Þe - jφ2 ðf Þ = jU 2 ðf Þje - jφ2 ðf Þ when U 2 ðf Þ = jU 2 ðf Þjeþjφ2 ðf Þ , λ λ - real factor with the dimension ½V=Hz, that is required as Gopt ðf Þ is dimensionless that is, the complex transfer function of the Wiener Filter is the complex-conjugate of the spectral amplitude density of the desired signal and thus optimally adapted to it. This corresponds to the principle of correlation reception, as it is used, for example, in mobile communications with UMTS.

What You Can Take from This essential

In this introduction to the Fourier transform, you have • recalled that the angular functions cosine and sine can also be represented as functions of time, • seen by means of examples that you can use the cosine and sine functions as building blocks for the construction of non-harmonic periodic functions and thus get a feeling for the Fourier series, • understood the transition from the Fourier series to the Fourier integral by the “trick” of making the period of a periodic function go toward infinity, • learned the Fourier transform of standard signals • learned the most important properties of the Fourier transform as well as an important approximation relationship for the transformation of bell-like signals, • understood the modelling of the interaction of signals and systems in the frequency domain and understood it by means of a simple example. It was also seen that the interaction of signals and systems in the frequency domain can be described by simple arithmetic operations and thus more easily than in the time domain, • a short presentation of the application of the Fourier transform to stochastic signals, • got to know a somewhat unusual derivation of the Wiener filter as the optimal receiver for digital signals, • seen that by transforming time sequences into the frequency domain new aspects of the behavior of these sequences can be made visible.

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Literature

Kammeyer, K.-D. (2011). Nachrichtenübertragung. Vieweg+Teubner. Kreß, D., & Kaufhold, B. (2010). Signale und Systeme verstehen und vertiefen. Vieweg +Teubner. Lange, J., & Lange, T. (2019). Mathematische Grundlagen der Digitalisierung. Springer Vieweg. Osgood, B. (2014a). Lecture notes for EE261: The Fourier transformation and its applications. Create Space Independent Publishing Platform. Osgood, B. (2014b). Lecture notes for EE261: The Fourier transformation and its applications. https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf. Zugegriffen. 17 Sept 2018. Papula, L. (2018). Mathematik für Ingenieure und Naturwissenschaftler: Bd. 1. Ein Lehr- und Arbeitsbuch für das Grundstudium. Springer Vieweg. Unbehauen, H. (2008). Regelungstechnik I: Klassische Verfahren zur Analyse und Synthese linearer kontinuierlicher Regelsysteme, Fuzzy-Regelsysteme (Studium Technik). Vieweg +Teubner.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 J. Lange, T. Lange, Fourier Transformation for Signal and System Description, Springer essentials, https://doi.org/10.1007/978-3-658-33817-6

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