The Mathematics of Pulse Compression: A Problem in System Analysis

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The Mathematics of Pulse Compression: A Problem in System Analysis By John E. Chin, Charles E. Cook Chin J. E. and Cook C. E., "The Mathematics of Pulse compression", Sperry Engineering Review, Vol. 12, Oct. 1959, pp. 11-16.

Summary The concepts of linear FM pulse compression are investigated in this paper, and system parameters defined. The techniques of pulse compression are described, and spectrum and waveform analyses given. Finally, physical considerations are discussed in terms of system capability and complexity.

Introduction As new ideas are introduced into signal transmission technology they are subject to the twin tests of physical realizability (from the mathematical viewpoint) and physical practicality. Although practice often outstrips theory, (e.g., the wheel was in use long before the mathematical analysts derived the equations describing the locus of a point on the rim of a rolling wheel), their eventual balance usually results in better applications of techniques which are not predicted by practice alone. This paper explores the analysis of a new technique in which theory was in advance of practice, the object being to demonstrate that system analysis may not only prove the theoretical feasibility of unknown techniques, but also establishes the physical basis for continued development.

The Role of System Analysis Before the examination of pulse compression as a signal technique, a short digression may be in order to give the authors’ concepts of what should be accomplished by one attempting the role of a system analyst. Certainly he should not be restricted to the development a Particular piece of hardware. On the other hand, should not be so divorced from reality that he cannot appreciate the problems of engineers responsible for system hardware. He should not be so involved with mathematical niceties that he adopts the attitude that the problem is solved once the mathematical theory is developed, leaving to others the mundane practical problems. The system analyst will, with experience, develop the insight that leads to systems synthesis, other new ideas which he will relay to his colleagues interested in practical equipment. Operating in this fashion, this type of engineer will blend analytical and laboratory provide techniques for others to build on. An outstanding example of this at Sperry was the basic equipment of the pulse compression technique by a research group committed to no specific product but rather to general investigations of possible areas of improvement in the transmission of radio signals. In turn, the engineers associated with specific product development have been able to implement improvements suited to their specific requirements. The flow of ideas with this arrangement has been, happily, most fruitful. The following sections examine, in order, the basic concepts of pulse compression, spectrum and waveform analyses of this technique, and the physical considerations involved in its implementation.

Basic Concepts of Pulse Compression Pulse transmission systems, in many instances, have suffered from the lack of necessary peak power levels in the face of operational requirements that do not permit a broadening of the pulse width with its attendant increased 1

Figure 1: illustrates idealized pulse compression characteristics: (a) wide pulse envelope, (b) carrier frequency modulation, (c) filter time-delay characteristics (d) compressed pulse envelope (e) input, output waveforms of compression filter. average power. These operational requirements would normally include a specification of maximum pulse width or a necessary data rate for a particular transmission system. Proposals made by S. Darlington and R. H. Dicke adopted essentially equivalent ideas of solving this problem by transmitting broad pulses and converting them to narrow pulses in the receiving apparatus[1, 2]. The basic idea proposed by Dicke is described below. If the carrier frequency of a transmitted pulse whose envelope is shown in Fig. 1a were linearly swept, as shown in Fig. 1b, a pulse compression filter with the time delay vs. frequency characteristic of Fig. 1c could be used to delay one end of the pulse relative to the other. This would produce, at the filter output, the narrower pulse of Fig. 1d which would be of greater peak amplitude. The linear time delay characteristics of the filter would act to delay the high frequency components at the start of the input pulse more than the low frequency components at the end of the pulse, as shown in Fig. 1e, with frequency components between experiencing a proportional delay. The net result would be a time compression of the pulse. Since a passive linear filter is postulated, the principle of conservation of energy applies and the buildup in peak power of the compressed pulse would be proportional to the ratio of the widths of the filter input and output pulses. Thus: Pˆ0 T = τ Pˆi where: Pˆi = peak power input pulse Pˆ0 = peak power compressed pulse If the pulse width τ represents the desired resolution, it can be seen that if this technique is feasible, a pulse of width T , representing an increase in average power, may be transmitted with an associated frequency modulation that contains the information necessary to construct the desired compressed pulse of greater effective peak power. However, the actual peak power limitations of a pulse system are bypassed, thus opening another avenue for extending system performance. Starting with the fundamental idea presented by Dicke, the purpose of the analytical study program was to put this idea on a more formal basis, defining the relationships among the various parameters of a system making use of the pulse compression technique. In essence, the task was to provide information leading to the development of practical applications of the pulse compression concept.

Pulse Compression Spectrum Analysis From the previous section, one would feel intuitively that the spectrum of frequencies contained in the narrow (i.e., compressed) pulse would tend to have a bandwidth inversely proportional to the pulse width, and that the wide (i.e., 2

uncompressed) pulse spectrum would be related to it in a specific manner. Confirmation of the inference is given in the following analysis. If a signal is described by the function: f (t) = cos[φ (t)] (1) where φ (t) is the signal phase as a function of time, then what is termed the instantaneous radian frequency function is determined by: d ω = [φ (t)] (2) dt Analyzing the system under study, the transmitted signal function is:
f (t) = A cos ωct + 21 µt 2 − T2 < t < T2 (3) φ (t) = ωct + 21 µt 2

(4)

Differentiating the expression yields: d (5) [φ (t)] = ωc + µt dt This is the postulated linear progression of frequency vs. time about a center frequency ωc . If µ is positive, ω sweeps from a low to a high value, and if µ is negative, ω sweeps from a high to a low value of frequency. The units of µ are, of course, radians/sec2 .

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A signal spectrum F(ω) may be found by Fourier integration, where: F(ω) =

Z ∞

−∞

f (t)e− jω t dt

(6)

Neglecting the constant A and recognizing that f (t) = 0 outside the interval −T /2 < t < T /2, the spectrum of the frequency swept signal is: F(ω) = =

Z T /2 1 2


cos ωct + 12 µt 2 e− jω t dt

−T/2 Z T /2

−T/2

e

i h 1 j (ωc −ω )t+ 2 µ t 2

dt +

1 2

i Z T /2 h 1 j (ωc +ω )t+ 2 µ t 2

e

(7) dt

−T/2

The second integral essentially defines the spectrum at negative frequencies and has a negligible contribution at positive frequencies, provided the ratio fc /∆ f is sufficiently large, which would be the case in any practical application of pulse compression. The spectrum expression, after a suitable change of variables, becomes: h i Z √ µ T + ωc −ω q j(ωc −ω )2 π 2 µ − j π χ2 2µ F(ω) = 21 µπ e (8) √ µ h T ω −ω i e 2 dχ π

−2+

c µ

This integral yields: F(ω) =

1 2

q

π µ

− j(ωc2−µω )

e

2

"

µ T2 + (ωc − ω) C √ πµ

!

µ T2 + (ωc − ω) + jS √ πµ

!

µ T2 − (ωc − ω) +C √ πµ

!

µ T2 − (ωc − ω) + jS √ πµ (9)

where: C(χ) =

Z χ 0

S(χ) =

Z χ 0

cos π2 y2 dy

(10a)

sin π2 y2 dy

(10b)

are the Fresnel integrals and define the real and imaginary components of the pulse compression spectrum (ie., the amplitude and phase characteristics). The phase term: e−

j(ωc −ω )2 2µ

(11)

is the parabola shaped phase term that must be removed to permit the pulse compression phenomenon to take place, as will be shown in the next section. The practical implication of this is that the input and output spectra of the compression filter are identical except for this phase term, and thus the filter need only be a device that operates on the phase characteristic of the input spectrum. If the output spectrum notation is abbreviated: i q h F(ω) = 21 µπ C1 +C2 + j(S1 + S2 ) (12)

the spectrum amplitude characteristic can be computed: |F(ω)| =

1 2

i1/2 q h 2 2 π µ (C1 +C2 ) + (S1 + S2 ) 4

(13)

!#

Figure 2: indicates the spectra of compressed pulses of width τ for different input pulse widths. and the phase characteristic: arg[F(ω)] = tan−1

S1 + S2 C1 +C2

(14)

The Fresnel functions do not represent a closed form solution, and the spectrum functions must be derived from tables of Fresnel integrals[3, 4]. The Fresnel function argument is: χ= by making the substitutions µ = ∆Tω where ∆ω = pulse width, and ωc − ω = n ∆2ω , then:

µ T2 ± (ωc − ω) √ πµ

2π τ

χ=

(15)

is the frequency deviation within wide pulse and τ the narrow q

T 2τ

(1 ± n)

(16)

The argument, χ, appears as a function of the compression ratio T /τ, and is seen to be independent of the absolute amount of the frequency deviation ∆ω. Fig. 2 illustrates spectra shapes for various values of compression ratio. Numerical examples chosen showed that the spectrum width is consistent with the narrow pulse ( τ1 ≈ ∆ω). It can be deduced that removal of the nonlinear phase term from the input signal spectrum will result in a narrower pulse width. As the compression ratio increases the spectrum amplitude tends to become, to a first approximation, rectangular in shape. One of the familiar examples used in courses dealing with Fourier analysis is that of a rectangular amplitude spectrum and a linear phase spectrum. The resultant waveform is the well known sin x/x pattern. One of the results of the next section is that the compressed waveform has precisely a sin x/x form for the system outlined.

Pulse Compression Waveform Analysis With the exponential form of notation for ease of analysis, the pulse compression input signal may be described: ( 1 2 −T0 ≤ t ≤ T0 e j(ωct+ 2 µ t ) (17) f (t) = 0 |t| > T0

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The pulse spectrum for this expression then is: F(ω) = =

Z ∞

f (t)e− jω t dt

−∞ Z T0

e

−T0

i h 1 j (ωc −ω )t+ 2 µ t 2

(18) dt

The preceding section deduced that the characteristic of a “compression” filter should be purely imaginary (i.e., phase shift only), and of the form: H(ω) = e j

(ωc −ω )2 2µ

(19)

Presuming a filter of this characteristic, the filter output spectrum is defined as G(ω) = F(ω)H(ω). Therefore, G(ω) is: h i 2

e

j (ωc2−µω )

Z T0

e

1 j (ωc −ω )t+ 2 µ t 2

dt

(20)

−T0

The output time function g(t) is the inverse Fourier transform of this spectrum: g(t) = or g(t) =

1 2π

Z ∞

e

j (ωc2−µω )

−∞

with suitable manipulation this rearranges to: g(t) = letting u =

1 2π

ω −(ω√ c + µτ − µ t) , 2µ

Z T0

g(t) = but

R∞

−∞ e

ju2 du

=

R∞

−∞ cos u

2+

e

−T0

√ 2µ 2π

j



1 2π 2

Z ∞

G(ω)e jω t dω

Z T0

e

−T0

i h 1 j (ωc −ω )τ + 2 µτ 2

ωc2 (ωc +µτ −µ t)2 1 2 2 µτ +ωc τ + 2µ − 2µ

Z T0

−T0

"

e

j



(21)

−∞



Z

∞

e



dτ e jω t dω

j 21µ [ω −(ωc +µτ −µ t)]2

−∞

ωc2 (ωc +µτ −µ t)2 1 2 2 µτ +ωc τ + 2µ − 2µ

 #"

Z ∞

e

−∞

√ jπ π e 4 and Z T0 q µ j(ωc t− 21 µ t 2 + π4 ) e j µ t τ dτ g(t) = 2π e −T0 q Z T0 1 π 2 = 2πµ e j(ωct− 2 µ t + 4 ) cos µtτ dτ 0 q 2µ T02 j(ωc t− 1 µ t 2 + π ) sin µtT0 2 4 = π e µtT0

ju2

#

(22)

 dω dτ

du dτ

(23)

(24)

j sin u2 du =

(25)

π∆ f where ∆ f = swept frequency deviation. The output-input peak power ratio is derived by squaring the now µ = 22T 0 amplitude of the output pulse, the input amplitude having been taken as unity. This yields:   2  T0 2µT02 4π∆ f = = 2T0 ∆ f (26) π 2T0 π

If the wide pulse width assumes the same dimensions as in the previous section: 2T0 = T then the output-input pulse width and peak power ratios become T ∆ f , if the convention is adopted that the output pulse is measured at the points t ± 12 ∆ f . 6

Physical Considerations of Pulse Compression The two previous sections have defined the analytical spectra and time functions associated with linear FM pulse compression. A review of the results will reveal the constraints that come into being when this technique is employed. The time, or waveform, analysis showed that the use of a filter with a specified phase characteristic resulted in a pulse narrower by a factor T ∆1 f and increased in amplitude by the factor T ∆ f . The spectrum development showed that the precompression spectrum square-law phase term and that of the specified filter were conjugate. Thus if this filter is used, the conjugate phase terms cancel each other, but the spectrum is otherwise unchanged. The conclusion drawn from this is that despite the radical change in the shape of the pulse the energy content of the signals remains constant. Thus any receiver parameter that depends on signal energy and, by implication, average power will also remain constant. Since the waveform analysis postulated a lossless filter, the principle of conservation of energy applies and the deduction follows as a matter of course. The spectrum analysis shows that the system bandwidth must everywhere accommodate a signal bandwidth of ∆ f regardless of the input pulse envelope time duration. Therefore, the internally generated noise content of the system will not vary, and any system parameter that depends on the ratio of signal energy to noise energy will be unchanged whether the signal is considered before or after the compression phenomenon takes place. Before continuing, it might be well to discuss the signal resolution capabilities of pulse compression, which have been shown previously to be determined by the compressed pulse width. Intuitively, one might feel that this is inconsistent with the premise that the transmitted pulse is broad. However, the apparent inconsistency can be resolved easily when it is remembered that the transmission and reception are linear and therefore the principle of superposition is applicable. Briefly then, the receiver response to overlapping broad pulses from multiple inputs after compression will be precisely the superposition of the responses from each input when the others are absent. Fig. 3 shows the uncompressed pulses from two inputs, and permits a comparison with the effect of overlapping input signals on a pulse compression system. The points in the preceding paragraphs have been emphasized to show that despite the increase in peak signal power resulting from pulse compression, it has not been a case of buying something for nothing as far as signal energy levels are concerned. What has been bought is an increased signal resolution, and this has been at the cost of system complexity (frequency-modulation equipment and compression filters). The introductory sections offered reasons for employing pulse compression, but it must be inferred that indiscriminate use of this or any other signal processing technique would add unnecessary cost and complexity. The waveform study shows the basic shape of the compressed pulse to be of sin x/x form, meaning that there are rather high signal sidelobes or residual baseline signals (Fig. 4). In some environments these baseline residuals are not of much importance. In other environments they may be critical. Further signal processing techniques can reduce the residuals to some extent[5]. No final solution has been found for this problem, but studies in this area are proceeding. Presumably, results will be forthcoming as the appropriate contracting agencies authorize continued studies. Of prime interest is the actual requirement for realizing the compression filter characteristic. For the purposes of the analytical discussions the filter phase characteristic derived is: β=

(ωc − ω)2 2µ

The associated time delay is: td =

dβ ωc − ω = dω µ

Depending on the sign of µ this indicates a negative value of time delay over half the band. Clearly this is unrealizable since, in a passive filter, it implies an output before an input is applied. To lead to a physically realizable filter, a constant may be added to the time delay function to give it positive values at all frequencies of interest. Thus the 7

Figure 3: Fig. 3, above left, illustrates the signal resolution capability of the pulse compression technique. The uncompressed pulses, top, may be compared with the overlapping signal waveforms below. Fig. 4, above right, indicates the high residual baseline signals (sidelobes) which characterize the sin x/x form. Fig. 5, below, is a system block diagram which includes the pulse compression filter network in the receiver section.

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modified time delay term is: td =

ωc − ω +k µ

This leads to a phase term of: β=

(ωc − ω)2 − kω + c 2µ

where c is the constant of integration. What this accomplishes is the removal of the vertex of the phase parabola outside of the assumed band limits of the signal spectrum, thus excluding phase slopes leading to negative time delays. This now permits the compression filter to be designed by standard phase-shift approximation techniques. The net result on the signal is a delay of k seconds. Since this is a known delay it can easily be accounted for in system operation. Inasmuch as this filter is operating on the phase characteristic of the received spectrum, the actual pulse compression filtering must take place in the r-f or i-f sections of the receiver. Fig. 5 illustrates a possible system for the use of this method of signal processing. Both the spectrum and waveform analyses indicate the optimum relationship among the system parameters. If one of the parameters changes, such as transmitted pulse width or total frequency deviation, the requirements for the optimum compression filter will also change. This implies some restriction on system flexibility if various pulse compression modes or resolving capabilities are desired. This particular problem is definitely the concern of the system builder, who must be able to justify any added complication of his radio transmission and reception apparatus.

Conclusion The analysis of the technique of linear FM pulse compression has been explored and the system concepts defined. This technique holds promise for transmission-reception systems that are peak power limited, for the final pulse width may be designed into the system after pulse energy requirements are established. In addition, pulse compression appears capable of producing very narrow pulses that offer difficulties for conventional methods. The price to be paid for employing pulse compression is additional complexity, and the project engineer must exercise objectivity in balancing the advantages and disadvantages of a more complex system. Actual results and capabilities of pulse compression operation are still of a classified nature.

Acknowledgment Messrs. W. W. Mieher and C. E. Brockner were responsible for the initiation and planning of pulse compression research at Sperry. Among those responsible for systems applications, Messrs. C. F. Chubb, A. N. Fine, A. E. Hylas, B. Hulland, and R. F. Schreitmueller have made valuable contributions to the study and implementation of pulse compression. Part of this effort has been supported by Air Research and Development Command, Rome, New York.

Authors John E. Chin, coauthor of the foregoing paper, is an engineer the Passive Systems and Subcontracting Engineering Depart of the Countermeasures Division. Chin joined the Company as an engineering aide during the summer months of 1953 while attending the University of Illinois. He was awarded the bachelor’s degree in engineering physics in June 1954 and returned to Sperry as a junior engineer assigned to the radar research group of the Armament Radar Engineering Department. He was promoted to associate engineer in November 1955 and to his present

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position in May 1957. At that time he was engaged in the development of pulse compression techniques for radar applications. Since his transfer to the Countermeasures Division in February 1958 he has been concerned with antenna subcontractor liaison and laboratory studies of ECM antenna state of the art. Chin is coauthor of a paper which was presented at a pulse compression symposium in 1957. He has one patent application pending.

Charles E. Cook, coauthor of the foregoing paper, is a senior engineer in the Advanced Detection Systems Engineering Department of the Air Armament Division. Upon joining Sperry in 1951, Cook was assigned to radar research projects as an assistant project engineer. In 1953 he advanced to project engineer and became a senior engineer in 1956. He was engaged in the basic investigation of pulse compression and coded transmission techniques, and has served as consultant in this field to other divisions of the Company. At present he is associated with further development studies involving the use of pulse compression and correlation techniques applied to radar and communication systems. From January 1945 to July 1946 he served in the Navy as an electronics technician. Harvard University awarded him the S.B. degree in physics in 1949 and he received the M.E.E. degree in 1954 from the Polytechnic Institute of Brooklyn. Before coming to Sperry he was an electrical engineer specializing in pulse and electronic control circuits. Cook is a member of Sigma Xi and the I.R.E. He has written a paper on Fresnel integral pulse shape modification which appeared in the June 1958 issue of the Sperry Engineering Review, and a paper on pulse compression waveforms delivered at the 1958 National Electronics Conference and published in the conference Proceedings. He has also authored a number of papers for delivery at classified symposia on pulse compression. Cool has three patent 10

applications pending.

References [1] D. Sidney, “Pulse transmission,” May 18 1954, uS Patent 2,678,997. [2] R. H. Dicke, “Object detection system,” Jan. 6 1953, uS Patent 2,624,876. [3] E. Jahnke, Tables of functions with formulae and curves. New York: Dover Publications, 1945, pp. 34–37. [4] A. Van Wijngaarden and W. Scheen, Table of Fresnel integrals. Report R49, Computation Dept. of Mathematical Centre at Amsterdam: North-Holland Publ. Co, 1949. [5] C. E. Cook, “Modification of pulse compression waveforms,” in National Electronics Conference Proceedings, vol. 14, 1958, pp. 1058–67.

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