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Improvement of the range response of short-range FM radars

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Title:
Improvement of the range response of short-range FM radars
Creator:
Mattox, Barry Gray, 1950-
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Language:
English
Physical Description:
xviii, 222 leaves. : illus. ; 28 cm.

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Subjects / Keywords:
Ambiguity ( jstor )
Autocorrelation ( jstor )
Bandwidth ( jstor )
Demodulation ( jstor )
Fourier transformations ( jstor )
Linear programming ( jstor )
Modulated signal processing ( jstor )
Signals ( jstor )
Spectral bands ( jstor )
Waveforms ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Radar ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 219-221.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Barry Gray Mattox.

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University of Florida
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University of Florida
Rights Management:
Copyright Barry Gray Mattox. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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022780180 ( ALEPH )
14090967 ( OCLC )
ADA8959 ( NOTIS )

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IMPROVEMENT OF THE RANGE RESPONSE OF SHORT-RANGE FM RADARS












BY

BARRY GRAY MATTOX


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1975




















The author proudly dedicates this dissertation to his parents, Mr. and Mrs. Dana Brooks Mattox, and to his wife Debbie.
















ACKNOWLEDGEMENTS


The author wishes to express gratitude to his chairman, Dr. Leon W. Couch, for being the outstanding teacher that he is and for his constructive criticisms and to Mr. Marion C. Bartlett for the many discussions invaluable to an understanding of the systems studied.

Thanks are also due to Mr. James C. Geiger, who constructed most of the figures and to Miss Betty Jane Morgan, who typed the bulk of this dissertation.

The author is indebted to the Department of Electrical Engineering and to Harry Diamond Laboratories for supporting research pertinent to this dissertation, and to Professor R. C. Johnson for allowing work in this area of study.


iii
















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS e* * i.

LIST OF FIGURES * * * * . . . . . . . . . . . * viii


KEY TO SYMBOLS


ABSTRACT . xvii

CHAPTER

I. DEFINITION OF SYSTEMS AND GOALS . . . .


1.1. Operational Constraints. 1.2. Structural Co nstraints **-*-*
1.3. Assumptions. . . . . . .
1.4. Models.*.
1.4.1. Envelope Detection *****
1.4.2. Coherent Detection .* 1.4.3. Linearity of the System.
1.4.4. The IF or Beat Waveform.
1.4.5. The Linear Processor .
1.4.6. The Non-linear Section . 1.5. Assumptions . . . * 0 0 0 *0 a
1.5.1. The Assumption of Small T *
1.5.2. Assumption of High Dispersion


Factor


1.5.3. The Quazi-Stationary Target Assumption II. RESOLUTION AND SHAPING OF THE RANGE RESPONSE .
2.1. The Resolution Problem *a aa
2.1.1. Accuracy aa a aaa.aa
2.1.2. Ambiguity and Resolution . . . . .
2.1.3. ,Parameters of Resolution . . . . .
2.1.3.1. The time ambiguity constant*


. 0 . . . . * . . . . . . . . 0 . 0 0 * . 0 . xii










Page

2.1.3.2. The frequency resolution constant- 29 2.1.3.3. The ambiguity function . . . . * 29
2.2. Shaping of the Range Response by Windowing . . * 35
2.2.1. The Importance of the Autocorrelation
F unction . . 35
2.2.2. Windowing a Bandlimited Spectrum . .* 39
2.2.3. Specific Windows . * . . . * 40
2.2.4. Effect of Windowing the Power Spectrum .* 43

III. SHAPING THE SYSTEM RANGE RESPONSE. . . . . . . . 46
3.1. Autocorrelation Systems .46 3.2. Delay-line IF Correlator Systems. . . . . 50
3.3. Harmonic Processor Systems . . . . . . . . . . .56
3.4. General Coherent Demodulator Systems . . 61

IV. INVERSE FILTERING . . . . . . . . . . . *.o. . . . . 62
4.1. Application to the T Domain . .* . . . . . * 65
4.1.1. The Autocorrelation System. . .65 4.1.2. The Delay-line IF Correlator System . . * 68
4.1.3. The Filter, h(T). . . . . . . . . . . . 6
4.1.4. Harmonic and General Coherent Demodulation
Systems . . . . . . . * . . . . . . . . . . 69
4.2. Summary and Conclusions about Inverse Filtering - 70

V. SYSTEMS USING VECTORS OF INFORMATION. . . 72
5.1. Alternative Information Vectors . .74 5.2. Equivalent Single-Channel System . . . . . .76 5.3. Existence and Dimension of H . * . . . . . . . . 77
5.3.1. Dimension of H Based on IF Waveform . . .77 5.3.2. Dimension of H Based on the Range Response '78
5.4. Choice of Constraint Times~ . . . . . . . . . .9 79

VI. DEVELOPMENT OF CLOSED-FORM RELATIONSHIPS . 9 .* 83
6.1. Methods for Prediction of the Range Response . . . . 83
6.2. Derivation of the Relation for Predicting the Range
Response . * . . . . . . . . . . . . . . . . 85
6.2.1. Physical Interpretation of the Relation . *89
6.2.2. Summary 9 * e 9 a* * * * * e e * * . 91








Page


6.3. Solving for the Demodulating Function . . . . . . . 92 6.4. Solving for the Modulating Function . . . . . . . . 94 VIT. STATISTICAL DERIVATION OF THE RELATIONSHIPS . . . . . . . 97 7.1. Description of Signal and Reference . . . . . . . . 97 7.2. The Assumption of Ergodicity and Notes on Averaging 100
7.3. Transformation to a Convenient Argument Space
Before Averaging . . . . . . . . . . . . . . . . . 103
7.4. Statistically Derived Range Response Transform - - 105
7.4.1. The Periodic Case . . . . . . . . . . . . . 106
7.4.2. Range Response Transform Statistically
Derived for Multi-dimensional Reference 107 7.5. A Convenient Graphical Method , . . . . . . . . . . 108

VIII. DIRECTIONAL DOPPLER PROCESSORS . . . . . . . . . . . . . 112
8.1. SSB Directional Doppler Techniques at RF . . . . . 112 8.2. SSB Directional Doppler Techniques at IF * * - - * 113 8.3. A More General Directional Doppler Processor . . . 116
8.3.1. Analysis of the SSB System in the P-Domain 117
8.3.2. The Form of the General Two-Channel
Processor - * * , * * . . . . . . . . . . . 118

IX. COMPUTER ALGORITHMS . . . . . . . . . . . . . . . . . . . 123
9.1. Program One - Solution of the Range Response . . . 124
9.1.1. Program Flow . . . . . . . . . . . . . . . 125
9.1.2. Precautions and Assumptions - - * . . . . . 129
9.2. Program Two - Solution of the Demodulation Function 131
9.2.1. Program Flow . . . . . . . * - - * . . . . 132
9.2.2. Precautions and Assumptions . . . . . . . . 135
9.3. Program Three Solution of the Monotonic
Modulation * 9 o o * - * # * . * - o - * - - 140 9.3.1. Program Flow 9 * - 9 e . . . . . . . . . . 144
9.3.2. Precautions and Assumptions . . . . . . . . 144
9.4. Examples of Computer Solutions 147

X. CONCLUSIONS . . . . . . . . . . . . . . . . 194









Page

APPENDIX

A. EXAMPLES OF CLOSED-FORM SOLUTION . . . . . . . . . . . . 0 197

B. COMPUTER PROGRAM LISTINGS . . . . . . . . . . . . ! . . . 203

REFERENCES . . . . . . . * . . . . . . . . . . . . . . . . . . 0 219

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . 222


vii
















LIST OF FIGURES


FIGURE Page

1.1 RELATIONSHIP BETWEEN MODULATION DENSITY AND POWER SPECTRUM 4

1.2 SYSTEM DIAGRAMS 6

1.3 I F SPECTRA 1

1.4 LINEAR PROCESSOR BLOCK 14

1.5 INSTANTANEOUS FREQUENCIES AND BEAT PATTERN 16

2.1 COMPARISON OF ACCURACY FOR HIGH AND LOW AMBIGUITY SYSTEMS 22

2.2 CW RESPONSES WITH AND WITHOUT ATTENUATION FACTORS 23

2.3 RANGE AMBIGUITIES 24

2.4 OBSCURING OF SMALLER RESPONSE BY LARGER 25

2.5 SIGNAL AND MATCHED RESPONSE 33

2.6 PERIODIC AND NON-PERIODIC EXAMPLES 38

2.7 POWER SPECTRA AND CORRESPONDING RANGE RESPONSES 40

3.1 THE AUTOCORRELATION SYSTEM 47

3.2 MODIFIED AUTOCORRELATION SYSTEM 49

3.3 I F CORRELATOR SYSTEM 51

3.4 PLOTS OF Z2(T,TR) 52

3.5 AMBIGUITY FUNCTION (MAGNITUDE) FOR LINEAR FM SIGNAL 60

3.6 GENERAL COHERENT DEMODULATOR SYSTEM 61

4.1 EXAMPLE OF PROBLEM SPECTRA 64

4.2 PARALLEL IMPLEMENTATION OF CONVOLUTION 67

4.3 TRANSLATION OF CONVOLUTION IN DELAY TO A CONVOLUTION IN TIME 67 5.1 HYPOTHETICAL RANGE RESPONSE INDICATING CONSTRAINT POINTS 73


viii










FIGURE Page

5.2 INFORMATION ELEMENTS AS A FUNCTION OF.DELAY 80

6.1 INSTANTANEOUS FREQUENCY VERSUS TIME, SHOWING THE TIME ROOTS 87 7.1 GENERAL SYSTEM DIAGRAM 97

7.2 FUNCTIONAL REFERENCE GENERATOR 100

7.3 OSCILLOSCOPE CONNECTION FOR DISPLAY OF * w 109

7.4 SIMPLE GRAPHICAL MAPPING TECHNIQUE 110

7.5 MULTIPLE MAPPING FOR NON-MONOTONIC MODULATION 110

8.1' PHASING-TYPE SSB PROCESSING AT RF 114

8.2 TWO-CHANNEL PROCESSOR MODEL MODELED IN THE T DOMAIN 114

9.1 COMPUTATIONAL FLOW CHART FOR PROGRAM ONE 126

9.2 TWO TYPES OF LINEAR INTERPOLATION 128

9.3 A RESPONSE AND ITS ALIASED COUNTERPART 130

9.4 COMPUTATIONAL FLOW CHART FOR PROGRAM TWO 133

9.5 INCREASING THE NUMBER OF SPECTRAL POINTS 137

9.6 COMPARISON OF ALIASED HANNING AND RECTANGULAR TRANSFORMS 140

9.7 HAMMING SPECTRUM, SHOWING NO SIGNS OF RINGING 141

9.8 RECTANGULAR SPECTRUM, SHOWING RINGING 141

9.9 CO1MPUTATIONAL FLOW CHART FOR PROGRAM THREE 145

9.10 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION DC DEMODULATION 154

9.11 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION, THIRD-HARMONIC COSINE DEMODULATION 155

9.12 RESPONSE OF TWO-CHANNEL SYSTEM USING SAWTOOTH MODULATION, THIRD-HARMONIC COSINE AND SINE DEMODULATION 156

9.13 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION, HALF-COSINE DEMODULATION 157

9.14 RESPONSE OF TWO-CHANNEL SYSTEM USING SAWTOOTH MODULATION, HALF-COSINE AND HALF-SINE DEMODULATION 158









FIGURE Page

9.15 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION, HALF-COSINE DEMODULATION 159

9.16 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION AND DEMODULATION 160

9.17 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION AND DEMODULATION 161

9.18 SPECIFIED SHORT-PULSE RESPONSE 162

9.19 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE USING RECTANGULAR WINDOW 163

9.20 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.19 FOR SAWTOOTH MODULATION 164

9.21 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE USING HANNING WINDOW 165

9.22 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.21 FOR WASTOOTH MODULATION 166

9.23 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE
USING HAMMING WINDOW 167

9.24 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.23 FOR SAWTOOTH MODULATION 168

9.25 SPECIFIED BANDLIMITED SIN(X)/X RESPONSE 169

9.26 RESPONSE OBTAINABLE BY BANDLIMITING SIN(X)/X USING RECTANGULAR WINDOW 170

9.27 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.26 FOR SAWTOOTH MODULATION 171

9.28 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE AT BT=6.4 USING RECTANGULAR WINDOW 172

9.29 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28 FOR SAWTOOTH MODULATION 173

9.30 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SAWTOOTH MODULATION 174

9.31 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28 FOR SINE MODULATION 175

9.32 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SINE MODULATION 176










FIGURE


9.33 SPECIFIED THREE-UNIT EVEN PULSE RESPONSE'

9.34 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE
RESPONSE USING RECTANGULAR WINDOW

9.35 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.34
FOR SAWTOOTH MODULATION

9.36 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE
RESPONSE USING HAMMING WINDOW

9.37 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.36
FOR SAWTOOTH MODULATION

9.38 SPECIFIED THREE-UNIT ONE-SIDED PULSE RESPONSE

9.39 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT ONE-SIDED
PULSE RESPONSE USING HAMMING WINDOW

9.40 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.39
FOR SAWTOOTH MODULATION

9.41 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.39 FOR SAWTOOTH MODULATION


TRANSFORM OF RANGE RESPONSE GIVEN MODULATION FUNCTION OBTAINED WITH DC DEMODULATION

TRANSFORM OF RANGE RESPONSE GIVEN MODULATION FUNCTION OBTAINED WITH DC DEMODULATION

TRANSFORM OF RANGE RESPONSE GIVEN

MODULATION FUNCTION OBTAINED WITH DC DEMODULATION

TRANSFORM OF RANGE RESPONSE GIVEN MODULATION FUNCTION OBTAINED WITH DC DEMODULATION


BY FIGURE 9.19 ZE OF FIGURE 9.42 AND


BY FIGURE 9.21 ZE OF FIGURE 9.44 AND


BY FIGURE 9.23 ZE OF FIGURE 9.46 AND BY FIGURE 9.26 ZE OF FIGURE 9.48 AND


Page 177 178 179 180 181 182 183


184


185 186 187 188 189 190 191 192 193


9.42 9.43 9.44 9.45 9.46 9.47 9.48 9.49















KEY TO SYMBOLS


A(T) attenuation factor

B peak-to-peak instantaneous frequency deviation (Hz.)

BW RF bandwidth as defined by Carson's rule

BL bandlimited

c(T) the complex autocorrelation function of complex envelope u(t)

D dispersion factor

DMS distance measuring system

d distance to target

e(t,T) various forms of the intermediate frequency signal (various
subscripts)

eR the IF reference signal

E signal energy (no arguments)

ET signal energy in one period of the modulation

E(W,T) IF voltage spectrum for a target delay of T El(W,T) the continuous counterpart of the IF voltage spectrum E(w,T) Eas(.) expected value of the argument with respect to a and 0 E(.) expected value of argument average over all random variables

f frequency (Hz.)

f0 center frequency (Hz.)

fm frequency of sinusoidal modulation (Hz.)

FA frequency resolution constant









F[.J,F- N Fourier and inverse-Fourier transforms.

Depending on the arguments of the functions to be transformed

or the arguments of the transform, the transformations are

defined as

F(w) = F[f(t)] = f f(t)e-jwtdt f(t) = F-I[F( )] =-- f F(w)ejOtdw



00 _ 7f
C(f) F[c(t)] = I c(t)eJ2 ftdt
-w
-00

c(t) = F[C(f)] = f C(f)eJ2 ftdf Notation of frequency-domain functions will be consistent;

i.e. C(.) will not be expressed alternately as C(f) and C(w).

Unless stated otherwise, upper-case functions are the Fourier

transforms of corresponding lower-case functions. g(t) desired output waveform when inverse filtering G antenna gain

h(t) filter impulse response H inverse matrix for constraining range response points

I identity matrix

K(f) frequency dual of c(T) Kf frequency modulation constant (radians/volt-sec,)

KR overall target reflectivity

LP low-pass

Lj(T) loss in transmission media L2(T) space loss m(t) modulation voltage


xiii









1N usually the dimension of a vector/matrix or the number of
terms in-a series

matrix used to transform independent elements into orthogonal elements

p.d.f. probability density function

pcO(a) the p.d.f. of the instantaneous modulation wm

P(f) the signal power spectrum

PP(f) the line power spectrum of a periodic signal

Pave. average transmitted power

Ppeak env. peak envelope power transmitted T T
R(T) autocorrelation function Rp(T) over one period - T S

Rp(T) autocorrelation function defined for periodic signals

Sx(x,T) IF signal as a function of delay T and vector x t time, as measured from origin to

ttrue some hypothetical absolute time

T the modulation period

TA the time resolution constant

round-trip delay time to target To a specified delay time

Tn round-trip delay time to the nth target

TR a reference delay, usually of a delay line

Tmax the maximum delay time under consideration

u(t) complex envelope of x(t)

up(t) complex envelope of periodic x(t)

modulation index v velocity of target in direction of antenna

v propagation velocity of signal in the transmission medium.


xiv










w

w(W) Wa

wc(t) Wm (t) Return (t)

x(t)

Xreturn(t)
x(T,O)




y

Yo

Yh


z(T) ,z(T,t) zE(r)' z (T)



zi(T)


zt(T) zI(T)







C(t)
OMCt


the vector of the FM modulation and its derivatives a window function (various subscripts) radian frequency

center frequency (radians/sec.) instantaneous radian frequency instantaneous frequency modulation wm= wc-wo instantaneous frequency of the return signal the RF signal

the delayed RF signal from the target(s) ambiguity function (subscripts indicate variations in form) ambiguity function defined for periodic signals the output of a coherent single-sideband processor the output of the third mixer in a delay-line IF correlator the output of the second channel of directional doppler processor

range response (various subscripts indicate different systems)

complex envelope of the range response z(T) the upper-half-plane-analytic range response;
Z(T) = Re{z+(T)}

an information element which varies with delay the vector of information elements zi(T) output of the first channel of a directional doppler processor

input to the second channel filter in a directional doppler processor

the demodulation written as a function of a and 8 demodulation as a function of time; also written tt(t) a linear combination of elements of i









a vector of demodulations defined for a number of subsystems

usually a frequency variable (Hz.) O usually a phase angle

(') Hilbert transform in the domain of interest
-) complex conjugate

* convolution operator

zT the transform of z

[gmn] the matrix G containing elements gmn















Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




IMPROVEMENT OF THE RANGE
RESPONSE OF SHORT-RANGE FM RADARS



By

Barry Gray Mattox

March, 1975

Chairman: Leon W. Couch
Major Department: Electrical Engineering

The problem of range resolution of a class of periodically modulated FM radars is approached using assumptions of a quazi-static target, high modulation index, and a modulation period much longer than signal return times. Various systems of the class are examined with an emphasis on resolution improvement. System consideration begins with the simplest, for which the range response and RF power spectrum are Fourier transforms. Modification of this system to a system with a range response displaced from the origin yields the delay-line intermediate-frequency (IF) correlator. These systems are then related to the harmonic systems or n-systems, for "nth-harmonic" systems. Conventional windowing techniques are reviewed as-used in conjunction with some of these systems.

Briefly, the process of inverse fi ltering in the range or delay

domain is considered. It is shown that the technique can, indeed, be employed on some of these (essentially) bandlimited (BL) systems by approximating the filtering convolution by a discrete point summation or by xvii








artificially transforming the problem to the time domain. The similarity to windowing the power spectrum in the ideal case of filtering is noted.

Harmonic n-systems are examined with regards to optimum use of the ensemble of harmonic information elements which form a vector space. The concept is generalized to include any set of independent information elements. Using n of these elements, a scheme is devised to constrain any n points of the range response. Problems associated with this method are investigated, including problems of behavior between constrained points.

A general IF coherent detector system is investigated, and three new functional relationships are derived involving the range response transform, the modulation function, and the demodulation function. Under the given assumptions the relationship is shown to be a more general closed-form relation than other types which apply to more specific systems. An inverted form of the relationship defines a demodulation function to be used for obtaining BL range responses with a given FM modulation. The third relation is an integral equation for the modulation function whose explicit solution may or may not be accessible in closed form, depending upon the range response desired and the demodulation function given.

Although research was aimed primarily at periodic systems, it becomes apparent that stochastic systems may be analyzed or synthesized by suitable re-derivation of the relationships. A generalization of IF signal representation allows for the elimination of most constraints with a resulting elevated complexity of solution. A chapter on directional doppler processors describes a two-channel processor which allows synthesis of any desirable BL range response. Finally, computer solutions are developed to solve those problems for which closed-form solution is incon venient or impracticable.
xviii
















CHAPTER I
DEFINITION OF SYSTEMS AND GOALS


This dissertation is primarily concerned with a class of highindex, periodically modulated FMIradars or distance measuring systems (DMS) which are to operate at relatively short distances. Items of consideration include resolution, simplicity/cost, and immunity to noise. In this first chapter the class of systems will be defined and models developed, using assumptions consistent with the problem.


1.1 Operational Constraints

The class of systems will be defined, both in purpose and in

structure. The DMS considered here are to operate within these specifications:

Al The system will primarily measure distance; or,
more basically, the DMS will detect the presence
of an object at a pre-specified distance(s).
A2 The distance to be measured will be small comared to the wavelengths of major FM modulation
components.
A3 The resolution (ability to distinguish between
targets of various amplitudes at similar distances) must be "good." Alternately, we should
be able to design the range response.
A The target to be detected is to be either
stationary or moving "slowly."
A5 The system should be simple and cost-effective.
A6 The system must be able to operate in the noise
and signal environment for which it is designed.


1.2 Structural Constraints

There may be many possible system structures which satisfy the









the above requirements, but we restrict ourselves to the following structural framework:

Bl The system will be a continuous-duty type FM
radar.
B2 A single antenna will be employed.
B3 Envelope detection of the RF voltage at the
antenna terminals will yield the information
to be processed.
B4 The "transmitter" will be periodically frequency-modulated using high-index modulation.
B5 Modulation rate will be slow relative to signal return times.

The list of requirements are consistent with applications such as aircraft altimeters or low-height warning devices. Some of these requirements will now be discussed as they relate to one another and to the assumptions to be employed in this dissertation. The brief discussion is, of course, in no way intended to represent all of the considerations involved in a choice of systems.

Requirement Bl results from more than one factor, among which are simplicity of construction and energy (or, more correctly, average power) transmitted. A continuous-duty oscillator is usually simpler to design than one that is amplitude-modulated or pulsed. The energy of a signal is the time integral of the signal magnitude squared:

CO 2
E f jx(t)i dt
-00

If the radar is periodically modulated (either in amplitude or in angle) we may speak of the energy per period: T/2 2
ET f 1 (01 dt
-T/2

where T is the period of the modulation. It can be shown that accuracy and range capability of a radar (influenced









by additive noise considerations) are monotonically increasing functions of signal energy [1]. Regarding requirement A6, we would like to maximize energy by maximizing both the magnitude of x and the duration of x over (-T/2, T/2). Since the signal is periodic, its average power may be expressed as

Pave I x(t)I dt =ET (1.2-3)
av. T T/2

If peak envelope power is a limitation, we may design the signal-envelope to be constant at that peak power; thus, for maximum energy, the duty cycle is increased to 100% so that

ET = T Pave. "T Ppeak env. (1.2-4)

The DMS is to have good resolution in distance (requirement A3). Inherent signal resolution is dependent upon bandwidth; to achieve high resolution, a power spectrum of large bandwidth is necessary [1,2,3]. If the signal envelope is constant, bandwidth must be achieved by frequency modulation (requirements Bl and B4). The requirement of large bandwidth, together with requirement A2 or B5 (A2 and B5 are equivalent requirements), indicates high-index modulation (BM). When the modulating waveshape is sinusoidal, the modulation index is defined as B/2 (1.2-5)
hfm

where B is the instantaneous peak-to-peak frequency deviation in Hiz, and fm is the frequency of the sinusoidal modulation. Applying Carson's rule for FM bandwidth [4] we see that

BW = 2fm(l+p~) ?fm' B for P' >> 1 (1.2-6)

Furthermore, for "high-index" modulations the shape of the power spectrum will be that of the probability density function of the FM modulation









it process" [5]; if the modulation voltage, m(t), is voltage limited, then x(t) will be essentially bandlimited (BL) to BW = B (Figure 1.1).






Pm (a) PMf





a
Vmin Vma X-1BK




FIGURE 1.1 RELATIONSHIP BETWEEN MODULATION DENSITY AND POWER SPECTRUM




When the modulation is not a simple sinusoid, the index is not

really defined by (1.2-5). A more general parameter than modulation index is the dispersion factor, commonly defined for chirp radars [6]: D =B -T .(1.2-7)

For sinusoidal modulation of frequency fm =l/T D = 2p(1.2-8)

The requirements of a single antenna and envelope detection stem from the desire to keep the DMS simple and cost effective. Two isolated antennas and a multiplier will give essentially the same simplified math model, as will be shown.

Finally, since our DMS will be processing a periodic signal containing range information, it becomes necessary for the target to remain at approximately the same position during one period T. From an energy standpoint, and with regard to signal-to-noise ratios, we should like to









process the return signal over as long a time as practical since the energy per decision is proportional to the time per decision; target movement and a minimum rate of decisions are limiting factors.


1.3 Assumptions

Throughout most of the dissertation three assumptions w ill be adhered to in the analysis:

1. The target return times T (signal propagation
times to and from the target) are small compared to the periods of major modulation components.
2. The modulation index or dispersion factor is
large.
3. The target will be assumed quazi-stationary,
i.e., almost static over a modulation period T.

The assumptions are, of course, supported by-and are re-statements of-system requirements (A2 and B5, B4, and A4, respectively).


1.4 Models

The system to be studied is illustrated in block diagram form in Figure 1.2(A). The voltage controlled oscillator is modulated by voltage m(t) producing signal x(t) which is fed to the antenna (or transducer as the case may be). The signal propagates through the medium and is reflected, in part, by the target. On returning to the antenna, the signal will be delayed by 2d(141
V

where d is the distance to the target, and v is the propagation velocity of the medium.
















e(t,T)


I z (t)


(A) ENVELOPE DETECTION SYSTEM


e(t,T)


z (T)


- -I


(B) MULTIPLIER SYSTEM EMPLOYING TWO
ANTENNAS


FIGURE 1.2 SYSTEM DIAGRAMS


!
!








The signal voltage is also attenuated by a factor of A(T) = G2LI(T)L2(T)KR (1.4-2)

where G2 is the contribution of antenna gains,
2d
Lj(2) is the loss in the media at a distance d,
2d
L2(2-) is the space loss,

and KR is the target reflectivity (overall).

Assuming the antenna behaves as a point radiator, and the target, something intermediate to the extremes of a point reflector and an infinite plane, the space loss will vary between (K/T4) and (K/T2) with (K/T3) often taken as a design estimate for the "average" target.


1.4.1 Envelope Detection

The signal is normalized and written as

t
x(t) = cos(w0t + Kf f m(X)dX) (1.4.1-1)
-00o

where Kf is the FM modulation constant in rad./sec-volt. Then the voltage at the antenna is the sum of the signal and the delayed, attenuated return:

t t t-T
x(t)+A(T)x(t-T) = cos(w0t+Kf fm(X)dX)+A(T)cos(w0t+Kf fm(X)dX-w0T+Kf f m(X)dX)
-0 -0 t

(1.4.1-2)
(A)
t t t
cos(w0t+Kf fm(X)dX)+A(){cos(0t+Kff m(X)dX)cos(W0T+Kf f m(X)dX)
-00 -00 t-T

t t
. sin(wot+Kffm(X)dX)sin(W0T+Kf f m(X)dX)}
-0 t-T








t t 19
{[+A(T)cos(W0T+Kf f m(X)dX)]2+A2(T)sin2(0o+ f m(X)dX)} t-T' tt
cos(wt+Kf fm(X)dX+e) (C)
-00
t t
[1+A(T)COS(W0T+Kf f m(X)dX)]cos(wot+Kff m(X)dX+O) t-T -00


if IA(T)I << 1 (D)


where t
[-A(T)sin(W0T+Kf f m(X)dX)
0 Tan-1 t- T
-1+A(T)cOs(W0T+Kff m(X)dX)I t-T


An envelope detector yields the intermediate-frequency (IF) signal e, which is a function of time and delay. A basic form will be denoted el and is explicit as the envelope of the expression in (1.4.1-2D):

t
el(t,T) = l+A(T)cos(W0T+Kf f m(X)dX) (1.4.1-3)
t-T

At this time we shall not be interested in the DC term of unity as it carries no information about the target. We must remember, however, that the term is derived from the amplitude of the oscillator, and that any AM noise or modulation will be directly demonstrated in this term. The signal of interest, denoted e2, is gotten by blocking the DC (no AM assumed):

t
e2(t,T) = A(T)cos(W0T+Kf f m(X)dX) . (1.4.1-4)
t-T

Under the assumption of a slow modulation with respect to return times, we may consider m constant over (t-T,t) such that, for the integrand of (1.4.1-4),

M(m) Z m(t-T) Z m(t) for X over (t-T,t) . (1.4.1-5)








Then the convenient mathematical approximation e3 is derived as

t
e3(t,T) = A(T)cos(W0r+Kf f m(t)dX) t-7T

t
= A(T)cos( OT+Kfm(t) f dX) t-T

= A(T)COS Wc(t)T


where


wc(t) = w0+Kfm(t)

We see that wc(t) is the instantaneous frequency of x(t): d t
WC t) = t- (wojt+Kf fm(X)dX) = wo+Kfm(t)


1.4.2 Coherent Detection

The signal from the multiplier of Figure 1.2(B), assuming completely isolated antennas, is

t
2x(t)"A(t)x(t-t) =2A(T)cos(W0t+Kff m(X)dX)

t-T
COS(WO(t-T)+Kf f m(X)dX) (1.4.2-1
-0 (A)

t
A(T)cos(woT+Kf f m(X)dX) t-T
t t-T
+ A(T)cos(2w0t-W0T+Kf fm(X)dX+Kf f m(X)dX) . (B)
-00 -C0

The second term is centered at 2w0 in frequency. The low-pass (LP) filter is designed to pass


t
e2(t,T) = A(r)cos(W0T+Kf f m(X)dX) (1.4.2-2
t-r


(1.4.1-6)
(A)

(B)



(C)


(1.4.1-7)


)


)









Again, we denote the simplified math form as e3 using (1.4.1-5):


e2(t,T) z e3(t,T) = A(T)cos[Wc(t)T] (1.4.2-3)


where wc(t) =0+Kfm(t) as before. This is the simplified result obtained when the DC term was dropped from

-the expression derived for the envelope detector system. Note, however, that any AM problems associated with the DC term are not present in the multiplier system unless an imbalance occurs in the physical multiplier.


1.4.3 Linearity of the System

We define the system to be linear if the principle of superposition applies. The systems of Figure 1.2 do allow superpositioning of target influences: let a system of multiple targets be modeled by denoting the returned signal to be

N
x return(t) I An(Tn)x(t-Tn) (1.4.3-1)
n=l (A)

N t-Tn
I An(Tn)cos(o0t-wOT+Kf f m(X)dX) (B)
n=l

where N is the number of targets. Envelope detection of the sum of a "large" signal plus small signal(s) has been shown to equivalent to coherent detection using the large signal as a reference. Then the product of the return(s) and reference is given as


t N t-Tn
2x(t).xreturn(t) =2cos(w0t+Kf fm(X)dX). I An(Tn)cOs(wo(t-Tn)+Kf f m(X)dX)
-00 n=l -CO
(1.4.3-2)
N t (A)
I An(Tn)cOS(woO0n+Kf f m(X)dX) n=l t-Tn
+ double frequency terms . (B)









Then
N t
e2(t,t) = Y An( n)cOS(WoTn+Kf f m(X)dX) (1.4.3-3)
n=1 t-Tn

where T is the vector [T, T2, . Tn]T and TT indicates the transpose of T and, as before,
N
e3(t,!_) I An(Tn)COS[c(t)Tn] (1.4.3-4)
n=1 (A)

N
= Y e3(t,Tn) (B)
n=l

Clearly the response of all targets appear superpositioned at the output of the envelope/product detector. The system remains linear by definition until the non-linear (NL) processing block (Figure 1.2).


1.4.4 The IF or Beat Waveform

A great deal of signal processing has already occurred in the

envelope detector (or multiplier/LP filter) to yield a signal e which has a bandwidth on the order of 2/T for many types of modulation [7] from a signal x(t), having a bandwidth of B, which,by assumption 2, must be much greater than l/T. For both envelope and multiplier systems the information signal e is a result of producting the signal and its return and is sometimes called the "beat" waveform. The frequency components of e3(t,T) depend not only on wc(t) but on the value of T, power being constrained to spectral lines generally clustered about BT/T for sawtooth modulation. Figure 1.3 shows two sample spectra. Both represent the transform of e3(t,T) of a linearly modulated DMS with bandwidth B and period T. Note that, when the target is assumed stationary, the spectrum consists of lines or delta functions which have areas as outlined by the
















(A) BT = 6
/
/
/
-


\EJ2Tf,T)1 l' 7 '\, _ -


0 1 2 3 4 5 6 7 8 9
T T T T T T T T T


JE(27Tf,T) l


(B) BT = 5.5


'ZI I Ef27fT) I


FIGURE 1.3


I F SPECTRA


dotted envelope. This envelope is found by taking the magnitude of EI(w,T), the Fourier transform of e3 over one period: T/2
E(W,T)= f e3(t,T)e-Jtdt (1.4.4-1)
-T/2 (A)

T/2
f f A(T)cos[wc(t) ]e-itdt. (B)
-T/2

The continuous spectrum El(W,T) describes the magnitude and phase of the lines of IF voltage spectrum E(w,T); spectral lines occur at zero and at all multiples of l/T Hz. The beat waveform will also be called the


IE(2rf,T) I


0 1
T








intermediate frequency waveform even though it may appear to be baseband in the sense that the line spectra extend all the way to zero, or DC, for the stationary target model. When the target is perturbed from the stationary situation, sidebands appear about each of the lines in the spectrum of e3(t,T) because of a change--or modulation--of spectral intensity of each line. This modulation may be modelled as AM in nature. There are, however, no restrictions that the envelope be real in a complex notation; i.e. SSB-AM is possible. Using the example of linear sawtooth modulation, let

MW - Lvolts, T T
T 2 2

Kf = 2irB radians per second per volt then
Wc(t) = 2TB t + wo rad/sec.


T/2 2nBTt
E(w,T)= f A(T)cos eT e e tdt (1.4.4-2)
-T/2 (A)
27IBT 27TBT
A(T) T/2 [ (.--- )t -j(- + )t
-2 A( f e + e Iat (B)

2 -T/2

A(T)T sin(7BT-wT/2) sin(TrBT+wT/2) (C)
4 L (7BT-wT/2) (7BT+wT/2)


The IF spectrum for the linear system is atypically simple; the spectral envelope is not easy to compute in general [7].


1.4.5 The Linear Processor

Looking again at Figure 1.2, one sees that the block which first operates on e3 is the (non-stationary, in general) linear processor. The systems under consideration contain (stationary) linear filters and timevarying gains as shown in Figure 1.4. The filter n(t) will typically be















- - - - - -


z(T )


I LL k L)


FIGURE 1.4 LINEAR PROCESSOR BLOCK


a band-pass or a low-pass filter or integrator. When the target is not moving, z is a function Of Tr only; but if the target is slowly-moving, h(t) must permit variations that occur in z(T) as Tr changes in time. The bandwidth of h(t), then, depends primarily on the rates of change in Tr
d
(secondarily, of course, on T z(T)).


1.4.6 The Non-linear Section

The last part of the system will generally be non-linear in some respect because a judgement or decision (which must be discrete choices) will be made by the electronics or by a human observer (very non-linear). This block may contain items such as rectifiers, squarers, and comparators.


e ( t, T)








1.5 Assumptions

1.5.1 The Assumption of Small T

The transmitted signal is written, using no simplifying assumptions, as

t
x(t) = cos(w0t+Kffm(X)dX) . (1.5.1-1)


It is somewhat instructive to observe the instantaneous frequency, written (as before)

Wc(t) = w0+Kfm(t) � (1.5.1-2)

Similarly, the return signal A(T)x(t-T) has an instantaneous frequency of


Wreturn(t) = W0+Kfm(t-T) � (1.5.1-3)


Plots of Wc and return are given in Figure 1.5(A) for the linear modulation case:

t T T
M .W. < t < -(1.5.1-4)

m(t) periodic such that m(t+T)=m(t), all t.

It is easy to see that "turn-around time" (indicated by the section of Figure 1.5(A) measured as T) detracts from the effectiveness of any system which might simply count cycles of the IF waveform (Figure 1.5(C)) or measure power in the IF spectral lines. This effect would have to be compensated or at least accepted as error. Tozzi [7] has done a good deal of work in the analysis of the IF spectrum for the cases of triangle, sawtooth, and sinusoidal modulations. He uses the amplitude of a single line--coherently detected--to provide range information and analytically developes the results for various ranges of r, which are not restricted to being small. Even for these three simple cases, manipulation becomes












wreturn(t)


w (t)


Wo1TB{


w0-B











27TB/T





2 BI T


(A) INSTANTANEOUS
FREQUENCIES


w-Ureturn


(B) BEAT FREQUENCY



I 1
I I

T
I2




I (C) BEAT WAVEFORM
e(t,T)


FIGURE 1.5 INSTANTANEOUS FREQUENCIES AND BEAT PATTERN









drawn-out and Tozzi offers no easy or closed-form solution for the general-case modulation.

Using the assumption of "small T," we will develope relations which will predict the range response of a more general coherent system. The single line response becomes a special case; likewise, general modulation functions present less of a problem to analysis (Chapter VI).


1.5.2 Assumption of High Dispersion Factor

As noted previously, the dispersion factor is defined as

D = B *T (1.5.2-1)

where B is the signal bandwidth and T is the modulation period. Resolution is determined, by-and-large, by B. Woodward gives good intuitive reasoning in the choice of the time-resolution factor, which he shows to be a measure of the signal's frequency "spread" or "occupation" [3]. In examining the magnitude of D, we will first assume that we desire resolution of at least 10% of maximum range. As bandwidth is inversely related to resolution we have


B > 1 10 (1.5.2-2)
.10 Tmax Tmax (A)

As stated previously, the period of modulation must be large--at least ten times,say--with respect to the largest return times expected: T > 10 Tmax (1.5.2-2)
(B)

Then
D = B T > - 10 Tmax = 100 (1.5.2-3)
Tmax

For periodic modulation, spectral lines are spaced l/T apart. Then in an









RF spectrum of bandwidth B there are


B
B-= BT = D (1.5.2-4)


lines in the spectrum. In a later section on ambiguities, we shall show that total time ambiguity (as defined by Woodward) is minimized in a bandlimited spectrum by requiring that spectrum to be flat (or rectangular). Although any periodic signal produces a line spectrum rather than a continuous flat spectrum, these lines become smaller and closer as T increases, so that, in the limit as T =, the spectrum approaches a continuous power density spectrum. Thus, within limits, large T seems desirable; and larger T implies larger D. Another factor which advocates large T is the occurrence of a periodic autocorrelation for periodic signals. In special cases, the range response differs from the autocorrelation only by the factor A(T). In any event, the returned signal of delay T + nT is
t-nT-T
Xreturn(t) = A(nT+T)cos[W0(t-nT-T)+Kf f m(X)dX] (1.5.2-5)
- (A)
t-T
= A(nT+T)cos[W0(t-T)+Kf f m(X)dX+e] (1.5.2-5)
-0 (B)

where
t-T-nT
6 = w0nT+Kf f m(X)dX = wonT (1.5.2-6)
t-T

since the integral of zero-mean m(X) over any number of periods is zero. Thus a signal return from a target at T+nT varies from that at T in an amplitude factor and a constant phase, wonT. This undesirable quaziperiodic effect will be minimized by letting T be large enough so that A(+nT) is very small (n 0 0), so that the periodicte may ieatrd fA(T)
ity may be disregarded for all practical purposes. An alternative method








of suppressing these ambiguities is the addition of IF noise modulation.

We shall usually consider, not the line spectrum associated with the signal, but the envelope of that line spectrum which would occur if we let T - -. (See derivation of the Fourier integral from series [81.) Thus presented is the argument for large values of T and thus D, supporting assumptions 1 and 2.


1.5.3 The Quazi-Stationary Target Assumption

The assumption that the target is stationary or slowly moving is demanded by our periodic processing model. It also supports a certain freedom of design of the ambiguity function (see next chapter), which will not have to be tightly controlled along the frequency axis [9].

Motion of the target will be treated as a perturbation of the stationary problem, somewhat in the same manner as one might treat the amplitude modulation of a carrier as a perturbation of the carrier. In the end, of course, one must consider the rate of change in terms of doppler frequencies, especially in the consideration of noise, as the signal-to-noise ratios must depend on the doppler bandwidth chosen. In those cases we shall consider the target motion to be linear in time and space.

Note that to assume such a simplified model is to neglect the compression/decompression of the entire return signal spectrum [101 by the factor

V v


where v is the propagation velocity in the medium

v is the velocity of the target in the direction of the
pickup antenna

+ H indicates positive (negative) relative motion.






20


This compression is usually taken to be a simple shift if 2ffB <<� o This case has been studied for example modulations when the shift has been substantial [ill.
















CHAPTER II
RESOLUTION AND SHAPING OF THE RANGE RESPONSE


The subjects of resolution and shaping of the range response

have been handled, not.only in the field of radar, but also as functional design in such fields as data communications and computer transform "windowing." This-chapter will introduce the basic ideas of resolution and the ambiguity function as they apply to the problem and will briefly examine some of the popular windowing techniques.


2.1 The Resolution Problem

We begin by a brief discourse on the qualities of a radar system relating to "accuracy," "ambiguity,11 and "resolution."


2.1.1 Accuracy

The accuracy of the system depends on the range response as well as the signal strength and noise power. The accuracy for a given signal energy and noise power is dependent on how peaked the output response is. This response peak is maximized relative to the noise power when the "receiver" is matched to the signal. The solid curves of Figure 2.1 represent measured range responses designed to peak at a delay TO; ideal noiseless responses are shown in dotted curves. We see that the wider response of (A) system leaves opportunity for more error due to noise than the narrower response of (B) system as shown by the measured ranges TJ in each diagram. It will be shown in a later section that the shape of the matched filter response is that of an "ambiguity function."









I z(T) I


(A) NOISY AND IDEAL RE ONSES%'F SYSTEM WITH HIGH
AMBIGUITY



Iz(-T)I





V T' 0 T
(B NIY N IEL EPOSS FSYTM'IHIO

AMBIUIT




FIGU (B) N. OIS NO CUAYFRHG AND IDEA RESPONSESTO SYSTEMSWT O











Thus, as Key et al [9) have noted, inherent accuracy depends on the signal-to-noise ratio and the shape of the ambiguity function.


2.1.2 Ambiguity and Resolution

Ambiguity and resolution are two very related, but distinct,

qualities of a signal. The ambiguity of our signal in range implies to what extent the range of the target can or cannot be determined with a









degree of certainty. A spread range response would lead to a high degree of range uncertainty when parameters, such as target strength or system gains, are unknown, or when noise is present. Woodward [21 has indicated that the ambiguity function describes the probability of a target being at a given range.

The range response for a continuous-wave (CW) radar is given by Figure 2.2. The envelope of the ambiguity func tion is simply the





Response with No Attenuation


Attenuated Response




FIGURE 2.2 CW RESPONSES WITH AND WITHOUT ATTENUATION FACTORS




constant response shown by the dotted line, uninfluenced by the attenuation due to distance, A(T). The dotted response shows no discrimination in range, and there are equal conditional probabilities of the target being at any range, even when conditioned on knowledge of all system parameters.

Figure 2.3 shows the range responses of various systems; these same figures correspond to types of range ambiguity inherent in various transmitted signals. For the time being, we shall consider only ambiguities in range, as, indeed, our quazi-stationary model precludes the necessity of high resolution in velocity.





z (T)


(A) WIDE MAINLOBE


T









(B) HIGH SECONDARY LOBES

T






(C) SPREAD, LOW-LEVEL
BACKGROUND


Z (T)


FIGURE 2.3 RANGE AMBIGUITIES


Rihaczek [121 has classified radar signals into categories solely by the type ambiguities they process. It is easy to imagine applications in which a response of Figure 2.3(B) might introduce extreme problems and the types of Figures 2.3(C) or 2.3(A) might be preferable. Range response (A) indicates poor resolution in close targets; (B) would give spurious responses for targets which are at some distance from the position of main response; the extensive "background" response of (C) opens









Lhe possibility of cumulative responses from all targets. It is Seen that the mainlobe width limits the minimum separation for which two targets may be resolved. Especially when the returns are of unequal strength, one response may be completely overshadowed by another. This effect is illustrated in Figure 2.4.







za(T)



(A) (B)


FIGURE 2.4 OBSCURING OF SMALLER RESPONSE BY LARGER




It should be stressed that, in some applications, decision circuitry may be thwarted more by sidelobe ambiguities to targets than by additive noise. As we have seen, a strong target may produce sidelobe responses which excede the mainlobe response of a weaker target. And, as is most often the case, target shape, material, size, or "complexion," all of which affect strength of signal return, may be unknown. Thus, the sidelobe responses will ultimately determine the dynamic range of targets which may be detected.

We shall not attempt to allude to the many ramifications of resolution but will, instead, refer the reader to [2] and [3].


2.1.3 Parameters of Resolution

Three parameters will now be defined for a measure of ambiguity:









the time resolution constant, the frequency resolution constant, and the ambiguity function.


2.1.3.1 The time ambiguity constant

The time ambiguity constant is defined for signals of finite energy as [3]


f jc(r) 2dt
A-00
TA = c(0)12 (2.1.3.1-1)
Ic(O) 2(A)


f IU(f) Idf
-00

[fI U(f) 2df]2 (B)
-00


where the signal is expressed as the real part of its analytic form:


x(t) = Re{u(t)ejw0t} (2.1.3.1-2)


where u(t) is called the complex envelope of the signal and has an autocorrelation function


c(T) = fu(t)u*(t+T)dt . (2.1.3.1-3)
-00

Equation (2.1.3.1-lB) is derived by applying Parsonval's theorem to the numerator and denominator of TA. Units of TA are time, and, as Woodward puts it, TA is inversely proportional to the "range of frequencies occupied by the signal [3]." This idea may be rigorously expressed for spectra with flat sections using (2.1.3.1-lB). The time ambiguity constant measures, for each signal, the total ambiguity in range when the target is stationary.

In most applications, we wish to minimize TA within the bounds









of certain system restrictions. Let us assume that we are restricted to a maximum signal bandwidth B (perhaps by a maximum FM peak deviation). Then, using the calculus of variations, we define the complex spectrum as the optimum spectrum plus a perturbation from optimum (in an arbitrary direction):


U(f) U(f)+EA(f)


(2.1.3.1-4)


where U0(f) is the optimum spectrum and A(f) is any function.

Optimum U(f) is defined by requiring a minimum of TA (with respect to E) to occur at c=O so that U(f) = U0(f). Mathematically stated


dTA

E:=0


(2.1.3.1-5)


+B/2
f [
-B/2 TA = +B/2

-B/2


(Uo+sA) (Uo*+EA*)] 2df

2
(U0+eA) (U0*+ A*)df


dTAI [fUOU0*df ]2 f2'U2(AUo*+A*U0)df C=O
-f (UoU0*)2df.2fUoU0*df.f (AU0*+A*U0)df}/D2 = 0


(2.1.3.1-7)


where D is the denominator of (2.1.3.1-6). From multiplying both sides by D2 we obtain 2fUoUo*df{fUoUo*df-fIUoI2(AUo*+A*Uo)df-f(UoUo*)2dff(AUo*+A*Uo)df} = 0 f(AU0*+A*U0){IU012flU0(a)12da-f U0(a) [4da}df = 0


(2.1.3.1-8)


(2.1.3.1-6)









Since A must be arbitrary


f{IUo(f)12"IUo(a)12-IUo(a)I4}da 0 B B
for - f & f B (2.1.3.1-9)


or
B/2 B B
f JUo(c)12[IUo(c)j2=IUo(f)j2]da = 0 - < f S 7 (2.1.3.1-10)
-B/2


which has a solution at
I B< ct 2 -2

0 otherwise


We have shown, then, that a rectangular spectrum satisfies the condition for minimizing time ambiguity defined in TA.

Tozzi has found that, when processing individual lines of the IF spectrum, linear modulation--and thus a flat spectrum under the highdispersion assumption--provides best resolution and the least spread distribution of power in the various other harmonic lines of the IF spectrum [7].

Intuitively speaking, the wider the bandwidth or occupied sections of the spectrum, the better the target resolution. Equal weighting of each section of spectrum occupied has been shown to be in the best interest of total ambiguity (in terms of TA).

Another way of regarding the BL spectrum is to consider that the rectangular spectrum is the most "versatile" transmitted spectrum which may be supported over the interval (f0 - B/2, f0 + B/2) in the sense that it may be modified to be of practically any other form by the receiver using appropriate filters [6,9]. Any departure from the uniform spectrum,










especially in the way of zeros or unoccupied sections, reduces the ease with which this may be accomplished.


2.1.3.2 The frequency resolution constant

The dual of the time resolution constant is the frequency resolution constant:

f IK(O)I2dO
FA = (2.1.3.2-1)
K(02 (A)

00 Ju(t) j4dt

= (B)
2
f Iu(t)j2dt


where

K( ) f f U*(f)U(f+O)df (2.1.3.2-2)
-00

and

U(f) = F[u(t)] (2.1.3.2-3)

Of course, it can be shown that to minimize FA, the envelope of the transmitted signal should be constant over the duration of the signal. Other properties of FA are duals of properties of TA. The frequency resolution constant will not be of extreme interest in this work because of the assumption of quazi-static targets.


2.1.3.3 The ambiguity function

We may get a good grasp of the ambiguity function by following its derivation. The ability to resolve or distinguish between two signals may be monotonically measured by their integral squared difference:

ISE = f Ix1(t)-x2(t)12dt . (2.1.3.3-1)
-0 (A)









If two signals are of the same form but differ in arrival time and are frequency shifted with respect to each other, we may write them as

J2r(f0 - 2)t
T2 A +
x1(t) = Re{u(t - 2 )e =Re{x} (2.1.3.3-1)
(B)

J27r(f0 + 2-)t A + x2(t) = Re{u(t + 2)e =Re{x2} (C)


where the basic signal form is x(t) = Re{u(t)ej27rfot} T is the difference in arrival times and

0 is the frequency shift. The signals xj and x2 may be returns from two targets whose delays differ by T and whose velocities differ by v so that relative time delay is T = 2vt/v. The doppler frequency shift is simply the time derivative of the doppler phase:
1 d 2f0 2f0v
2 dt v
The integral of (2.1.3.3-lA) is simply doubled if the analytic forms
+ +
xI and x2 are substituted for xj and x2. Then

j27T(f0 - 2)t j2r(f0 + 2)t 2
2*ISE = f u(t- )e 2 -u(t + )e 2 t
(2.1.3.3-2)

(A)


= f IU(t - T)12dt + f Iu(t + 2)12dt

CO 00

S00 u (t- u*(t + ej)ei2*tdt - u* (t - 2)u(t + T )ej2Trotdt
-CO -00

(B)

Notice that the first two terms of (2.1.3.3-2B) represent signal energy are thus constant; then to maximize the ISE, we must minimize the third and fourth terms which are subtracted from the energy terms. Since









these last two terms are complex conjugates it is sufficient to minimize the magnitude of either. The term whose magnitude is to be minimized is defined to be the ambiguity function:


Xl(T, )A f u(t 7)u*(t + -)eJ2 tdt
-0u 2 2


(2.1.3.3-3)
(A)


The function is sometimes defined


00
X2(, ) = f u(t)u*(t + T)e dt


(2.1.3.3-3)
(B)


which varies from X only in phase. As indicated above, we are generally interested in the magnitude only; clearly


(2.1.3.3-4)


Note that the ambiguity function along the T axis becomes the autocorrelation function of the complex envelope:


X2(T,O) = C(T) = f u(t)u*(t +T)dt


Likewise, along the "doppler" axis the ambiguity function is


X2(0,4) = f u*(f)u(f+4)df = K(4)


(2.1.3.3-6)


by the application of Parsonval's Theorem. If we consider a signal of unit energy (c(O) = 1), then


X 2
f X(O, )i2d =FA


(2.1.3.3-7)


-00
where there is no need to designate the particular form of X by sub00
scripts. Also, r IYt. n%12,; = T( 1 1


k .I.J.J -


-0- A
_cO


Just as integrations along each axis produce measures of total ambiguity in T or * (range or velocity), we may integrate in both directions to


IX(,) = IX(,)


(4.L.3.3-5)


_









measure a total "combination" ambiguity for the signal. The double integration yields a particularly interesting and profound result [2]:


f f IX(T,4))2dTd = 1 (2.1.3.3-9)


for signals of unit energy. Thus, the two-dimensional analogy to total ambiguity along the T-axis or 4-axis is not at our control as are TA and FA. That is, although we can control ambiguities along both of the axes, we may not define the ambiguity everywhere in the range-doppler plane because of the restriction of unity volume of total ambiguity. All that we may do in signal design is to control its distribution. Rihaczek [12] addresses the problem of signal design with emphasis on choosing the ambiguity function that best suits the application.

Oftentimes, practical design considerations will dictate the waveshape of the transmitted signal. If a filter matched to the transmitted signal is used at the receiver, the output will be of the form


x(t-T)e j274t * x*(-t) = f x(X-T)eJ2 Xx*(X-t)dX (2.1.3.3-10)
-0 (A)


= feJ2Tf0Tu(X-T)e j27r(f+) *f0 (tX)dX
(B)
00
f U(X-T)u*(-t)eJ274) eJ2 fo (t-T)dX (C)
-00


= e-j2Tfo(T-t) x2(T-t,) (D)


which is the ambiguity function times an RF phase function. Note that the matched receiver is designed for a maximum peak-signal to averagenoise-power and does not necessarily provide the best resolution, especially if the signal has not been optimally designed [13]. A simple









example using the pulse radar signal of Figure 2.5(a) will be used to illustrate this point. The matched response to the signal appears in







L t
'VV- -(A) * .- (B)

FIGURE 2.5 SIGNAL AND MATCHED RESPONSE



Figure 2.5(B), having some ambiguity for a total time of 2T1. A "receive]!' consisting of unity feedthrough (just a connection) yields an output pulse which is the same as the input pulse, with ambiguity extending over a time width T1. Undoubtedly, there will exist some application for which the widening of the received pulse by the matched receiver is not satisfactory.

We establish, by this example, that it is possible to re-distribute effective ambiguity in the range-doppler plane using linear processing. (Notice that the effects of noise have not been considered yet.) This principle will be important in discussing processors in a later chapter.

We now examine the situation with which we shall be working. We desire chiefly range information from our DMS with little or no interest in velocity information. (Velocity information may be obtained from the fine-structure range data; i.e., doppler cycles may be observed. As range resolution is improved, i.e. the range "window" narrowed, the number of doppler cycles which may be counted is decreased, thus decreasing doppler resolution.) Thus we are left with one degree of freedom in









choosing our ambiguity function: we design for high range resolution and let doppler resolution fall where it will.

At this point some of the properties of the ambiguity function are summarized for unit energy signals [2,3]:


1. Xl(T,V) = Xl*(-T,-v) Hermetian symmetry with respect to the origin

2. X2(T,O) = C(T)

3. X2(0,) = K( )
00
4. f IX(t,0)I2dT = TA
-00
00
5. f JX(O,*)12d* = FA
-00
00 00
6. f f IX(T, )12drd = E regardless of signal
-00 -o

7. The two-dimensional Fourier transform of IX(T,4)12 produces
IX(t,f) 12.
00 00
f f IX(r,) 12e-J2fT22TrltdTd = IX(tf)12
-00 -00
00 CO
8. f IX(T,O) 12dT = f IX(T,0l 12e-j2"OT d-00 -0O
00 00
9. f jX(T,0) 2d = f IX(0,0)12ej2710Td
-00 -00

U (f), u(t)*<--- X (r, )

Then
10. U(f)e jrpf2<-> X(T-P,,)


11. u(t)ej7kt2 <- X(T,O+kT) 12. au(at) <--- X(T, 0)
a

where the double arrow indicates a pairing of signals and ambiguity functions.









Relationships 10., 11., and 12. are derived from simple substitution into appropriate forms of the ambiguity function. The first two are of importance in linear swept FM DMS, sometimes called "chirp" systems. We will have an opportunity to examine this "shearing" of the ambiguity function later.


2.2 Shaping of the Range Response by Windowing

In the face of our assumption of a quazi-static target and a disregard for high resolution in velocity, the importance of the signal autocorrelation function will be established. The effects of bandlimiting its transform, the signal energy/power spectrum, will be discussed and related to shaping or "windowing" the spectrum, and a few of the more common windows will be discussed with regards to their derivation, merit, and utility.


2.2.1 The Importance of the Autocorrelation Function

As we noted in previous sections, we concern ourselves with targets which may be treated as stationary to a good mathematical approximation. This leads us to examine the ambiguity function near the T-axis, for which -e < S where c is some maximum expected value of doppler frequency. If e is small enough, we may consider


X X(T,O) = c(T) (2.2.1-1)


and our "ambiguity design problem" [12] is reduced to that of one dimension.

For periodic signals of period T,


up(t) = up(t�nT) n = 0,1,2,(3.


(2.2.1-2)









and the ambiguity function may be redefined as T/2 j2T

Xp(T,-) f Up(t)u (t+T)eJ2 tdt (2.2.1-3)
-T/2

Similarly, instead of c(T), autocorrelation function for signals of finite energy, we define T/2
R(T)= f up(t)u(t+)dt (2.2.1-4)
-T/2

the periodic autocorrelation function of the periodic signal up(t). Of course,


Xp(T, )I
E<4

Z Xp(T,O) = Rp(T)


We usually normalize the volume of ambiguity to be unity (for

unit energy) as in equation (2.1.3.3-9). However, since Xp must be periodic in T, the normalization volume (the energy) becomes infinite so that it is more convenient to speak of ambiguities of one period only; i.e., to look only at values of T in the interval (-T/2, T/2). Then the energy of one period may be normalized--let the average signal power be l/T-and we make a mental note that all ambiguities occur periodically in T. Again, as noted in the first chapter, the space attenuation factor of the return signal makes this simplification perhaps more appropriate to the actual application than is our original assumption of range periodicity of X p. Periodicity may be eliminated or extended by adding lowfrequency random or pseudo-random modulation, respectively [2,p.191].

To formalize our approach, we observe the relationship between Rp(T) for the periodic signal having envelope up(t) and c(T) for the "single-period" finite-energy signal u(t):


(2.2.1-5)









Let u(t) = 0 for Iti ! T/2 and


Up(t) = X u(t+nT)


= u(t) * I 6(t+nT)


T/2 Rp (T) =f
-T/2

T/2

-T/2

T/2

-T/2 -C!


Up (t)u (t+T)dt


u(t+mT) I u*(t+nT+T)dt
M=-0o 1n=-o


00
u(t)X
� n -=o


u* (t+nT+T)dt


f u(t)u*(t+nT+T)dt


See Figure 2.6 fects.


= . c (T+nT) . (E)


for an illustration of periodic versus non-periodic ef-


We have established an exact relationship of periodic and nonperiodic autocorrelations. Hopefully, c(T) will be negligible for ITI > T/2 in which case


Rp(T) = c(T)


for ITI < T
2


is a good approximation. In a well-designed system this will be the case. Our range response--A(T) neglected--can be considered {Rp(r) ITI < T
Z(T) = (2.2.1-9)
f 2
because of the space-loss weighting that will eventually be imposed; A(T) will be very small for ITI > T/2.


Then


(2.2.1-6)
(A)


(2.2.1-7)
(A)


(2.2.1-8)










u(t)
in


0 t0 t
u p (t)n




T+t 0 0 t0 T t
(A) SIGNALS


C (f)


(C) SPECTRE


P p(f)
tA]
PA-


1 0
mT


C (T)

to

-to to





-T+t0 -to to
(B) AUTOCORRELATION
VTMThTng


1
4 %s 2t0

t


1
2t0


I 1


FIGURE 2.6 PERIODIC AND NON-PERIODIC EXAMPLES Equation (2.2.1-7E) may be re-written


Rp(T) = C(T) * I Transforming both sides we obtain


Pp(f) = C(f) �
n---


6 (-r+nT)


(2.2.1-10)


6(f - i)


(2.2.1-11)


We will consider a power spectrum P(f) which is proportional to the envelope of Pp(f) and which corresponds to the non-periodic autocorrelation and range response:


__"4,___L_ - - _ .r


i


I


V,, ah'il ' ,dl. ,Jl. %.,e &'ll










P(f) =F[R('r)] = C(f) (2.2.1-12)
(A)

where


R(T) {R~) I!(B)




We must keep in mind that power is actually contained in line spectra, and problems associated with such spectra must be considered accordingly.

We see now that when the system is of the matched variety, we control the range response by controlling its transform, the power spectrum of the transmitted signal.


2.2.2 Windowing a Bandlimited Spectrum

Techniques used to control the power spectrum, especially a bandlimited spectrum, may be looked upon as "viewing" the spectrum

through a "window" so that the spectrum is weighted at every point by the characteristic of the window at that point. Mathematically expressed


Fw(x) = W(x) .F(x) (2.2.2-1)

where F(x) is the original function
W(x) is the window function
Fw(x) is the resulting windowed function.

Such windowing techniques have been applied to smoothing transformed data in which the time record was limited in duration (time domain); to concentrating power in preferred directions in antenna arrays (space domain); and, of course, to range response shaping of radar systems (frequency domain), an application which we will discuss here.

Consider the power spectrum and corresponding range response

given by Figure 2.7(A,B). Because of the high sidelobes of z(T), severe














( A ) ( B )





f
(C) (D)



FIGURE 2.7 POWER SPECTRA AND CORRESPONDING RANGE RESPONSES





ambiguities exist (see Section 2.1.2). In contrast the transform of the power spectrum in Figure 2.7(C) has much lower sidelobes at the expense of a wider mainlobe.

It will not be our objective in this dissertation to make specific judgement at to which range response is 'best." Indeed, the a propo response must be a function of many system design considerations which include, but are not limited to, dynamic range of target reflectivity, noise power, desired resolution, acceptable error rates, and the expected doppler band of frequencies. For this reason I shall mention some of the various ways of shaping the bandlimited spectrum by windowing, but we shall study just a few of these windows as examples.


2.2.3 Specific Windows

The simplest window is the rectangular window. We see that this is "no window at all" when used on a function that is already zero outside the window bounds. Thus we may consider any (essentially) bandlimited function as a function extending to all frequencies times a









rectangular window. Since multiplication in one domain implies convolution in the other, the window transform (a time function)--always somewhat other than a single delta function--tends to "smear" the windowed function's transform through convolution. This rectangular window is definded as
1 -7B
W0(W) A (2.2.3-1)
0 otherwise


In the transform domain this window is


w0(t) B � sin fBt (2.2.3-2)
7TBt

All of our windows will be defined to be zero outside the interval (-7B,7B). Then it is obvious that W0(W) �Wi(w) = Wi(&) (2.2.3-3)

where Wi is any window.

Transforming (2.2.3-3) we obtain the convolution (*) w0(t) * wi(t) = wi(t) . (2.2.3-4)

These interesting results are BL analogies of multiplication by unity and convolution with a delta function, respectively.

A more "active" window was first put into use for smoothing meteorological data by Julius Von Hann. (Von Hann actually smoothed in the transform domain by discrete convolution with the respective coefficients 11 21 4.) The Hanning window is given as [8, p.14]

1 + cosA 2 2 Bos -7B:Sw <7B
W1(M) 9 (2.2.3-5)
0 otherwise


which has transform









1 1 1 w1(t) w(t) w(t + w0(t
2~ w4t F


(2.2.3-6)


In contrast to the -14dB maximum sidelobe of w0(t), sidelobesof wl(t) are down by at least 30dB.

The Hanning window is a variation on the Hanning window which yields lower (-40dB) maximum sidelobe levels. (Sidelobe levels do not fall off as fast, however, as do those of the Hanning window.) This window is given by
.
.54 + .46 cos -TrB< < B


w2< ) = I


(2.2.3-7)


otherwise


having, of course, a transform of


w2(t) = .54w0(t) + .23w0(t + 1 ) + .23w0(t - 1 ) (2.2.3-8)

While having lower sidelobes, windows W1 and W2 have the effect of widening the mainlobe compared with window W0. The distance between the first nulls of wl(t) or w2(t) is double that of w0(t).

The Dolph-Tchebycheff weighting [15] yields an optimum transform in the sense that sidelobes are reduced with a minimum broadening of the mainlobe. In fact, one is able to specify sidelobe levels, which, in turn, determine the mainlobe width. The Dolph-Tchebycheff window transform assumes the form


w3(t) = cos Ir/(Bt)2-A2 (2.2.3-9)
cosh nA

The sidelobe level for this window is a uniform [cosh irA]-, which makes the window not very interesting for most purposes. As a signal, w3(t) contains infinite energy.

A more applicable approximation of W3 has been developed by Taylor [6]:











N
+ ~ aCos nw -TB < wj < 7rB W4 Mn=1(2 .2. 3-10)
0 otherwise


The coefficients, {azn}, of the above Taylor "weighting" have been calculated for values of desired sidelobe attenuation and N. The number of terms (N+l) determines how well W4 approximates W3. A typical design value for N might be five [11].

Among-other windows are these, which we but mention here:

W5 Papoulis'window [16] was designed for windowing
time domain finite-length data records. This
windowing of the sample record yields a minimum bias in the spectral estimate and a "low" variance of those estimates.

W6 The prolate spheroid wave function [17] is the
optimum window function to constrain a maximum
amount of transform 1'energy" to be within a
specified interval. Sidelobe levels, although
correlating somewhat with sidelobe "energy" are
not considered in the criteria. The prolate
functions are quite complicated in form and are
usually approximated by other functions.

W7 The Kaiser window is an approximation to the
prolate window. Kaiser has recognized the difficulties of working with such functions and purposes a window expressed in more familiar
zero-order Bessel functions [18].


2.2.4 Effect of Windowing the Power Spectrum

We wish now to focus attention on windows WO rectangular
W1 Ilanning
W2 Hamming

W4 Taylor

These windows have been defined as zero outside (-ITB*irB). If we consider each of these windows to be a weighting of an existing band-










limited spectrum, then there is no need to set Wi(w) = 0 outside (-rrB,rB); indeed, there exists no need for specification at all outside this interval. Then, just as smoothing the transform of a bandlimited spectrum by convolution with w0(t) is equivalent to convolution with 6(t), so is smoothing with wi(t) equivalent to smoothing with


(2.2.4-1)


where


W(w)= ) [Wi(w+2nnB)]
n=-oo


(2.2.4-2)


The transform of periodic W!(w) is, of course, a series of delta functions in the t domain. All of the above windows--and, indeed, all real, even windows, if N is large enough may be expressed


N
nw
Wi = ni cos
n=0

for the appropriate frequency range, having transform N a I

wi'(t) = a0i6(t) + Z , [6(t + n) + 6(t -n n=l

where {ani} are appropriately specified:


Rectangular a00 = 1; ano = 0, n 0 0
=
Hanning a01 = 1;a11 = a; ni = 0, n > 1

Hamming a02 =.54; all = .46; an2 = 0, n > 1

Taylor (6 terms, 40dB sidelobes) [11]

a04 = 1; a14 = 0.7782308; a24 = -0.0189046

a34 = 0.0097638; a44 = -0.003221; a54 = 0.0006948


(2.2.4-3) 2.2.4-4)


The effect of the weighting or windowing in the frequency domain is


wi'(t F -[wi' (m)










reflected in a convolution of the original response with the delta functions of (2.2.4-4). Applying a window to power spectrum P(f) = F[R(T)] yields the following autocorrelation (range response for matched systems with slowly varying targets): N a
z(r) = R(T) * aoi6(r) + 2 -- (6[r + + 6[T -B]) (2.2.4-5)
n=l (A)
N ani + n
R I LR(T+) + R(T--)] (B)



Thus we see windowing as the appropriate addition of suitably weighted, advanced and delayed replicas of the original transform.
















CHAPTER III

SHAPING THE SYSTEM RANGE RESPONSE


In this chapter we shall detail more specifically the types of

systems which were described in general terms in Section 1.4. The breakdown will include

autocorrelation systems
delay-line IF correlator systems
harmonic processor systems
general coherent demodulator systems.

We will discuss, in conjunction with each system, methods which might be employed in controlling the range response. Our examination of systems will not extend into the non-linear sections shown in the general sysstems of Figure 1.2.


3.1 Autocorrelation Systems

A block diagram and model of the autocorrelation system is given in Figure 3.1. The linear processor is simply the doppler-pass filter. This filter has a bandwidth which'is very low compared with l/T since the target is assumed to be moving slowly, implying low doppler frequencies. Thus the IF signal will be averaged over many periods giving

1 M/2
Zatten.(T) Z lim j f e(t,T)dt (3.1-1)
M->o -M/2 (A)


1 M/2
im A(T)x(t)X(t-T)dt (B)
M_ -M/2

A(T)
T RP(T) (C)















TO NL CIRCUITS


(A) BLOCK DIAGRAM


(B) MODEL


x (t) Xreturn(t)


FIGURE 3.1 THE AUTOCORRELATION SYSTEM


Zatten (T) ~ A(T) R(r)
�n T


(3.1-1)
(D)


where Rp(T) is the periodic autocorrelation function of x(T) and ~jT
R(r) = Rp(T) ITI 2 f 0 otherwise

The symbol z will be used to denote the range response with subscripts differentiating various systems or mathematical forms. Since close-in










resolution will be determined by R(T), we shall be discussing the range response z(T), which does not include the amplitude characteristic, where, for the autocorrelation sysLem,
T
zO(T) A(T) Zatten(T) z R(t) . (3.1-2)


The range response z0 being the transform of the signal power spectrum in this system does not depend on any of our previously stated assumptions, except that the target be assumed (quazi-) stationary. The range response must conform to all of the properties of a realizable autocorrelation function, some of which are listed below [19,20]:


1. R(O) IR(T)
bb
2. f f g(t)R(t-s)g*(s)dtds > 0 for any g
aa
(R(T) is a positive definite function).


3. P(f) = F[R(T)] a 0
(The power spectrum must be non-negative and real.)


4. R(T) = R*(-T)

One of the most obvious and also the most serious objections to such a range response is that its maximum occurs at T=O or zero range. There is the possibility of implementing a range response that has other peaks in addition to the zero part, but the high peak at zero, coupled with the high gain A(O) may present problems from close targets of relatively insignificant physical size.

A variation of the autocorrelation system employs an RF delay line to produce a displaced autocorrelation function, zI(T,TR)=R(T-TR) where TR is the delay of the delay line (Figure 3.2). Such a range

















e(t,T-TR)


TO NL CIRCUITS


x(t--ER)


m(t)


FIGURE 3.2 MODIFIED AUTOCORRELATION SYSTEM


response may be ideal, but it requires two isolated antennas, a multiplier, and the delay line. If one is willing to surmount these disadvantages, the range response may be shaped (whereever it is centered in delay) by prescribing the desired window function to be the shape of the

power spectrum.

Windowing of the RF power spectrum may be accomplished at the "transmitter" in three ways:

1. Amplitude modulate the output stage as the frequency is swept through the band.
2. Pass the transmitted signal through a linear
filter. For high dispersion, linearly modulated signals, the output power spectrum will
be given by K"IH(2rf)12 where H(w) is the filter transfer function.
3. Use non-linear frequency modulation. For high
dispersion signals, the p.d.f. of m(t) will
describe P(f) [5].









Amplitude modulation is often undesirable because of the necessity for a modulator section (expensive and bulky for high-power systems) and because of a reduced average power when maximum peak power is fixed. Filtering also reduces average power of the signal, and if the center frequency of the transmitter is altered from that for which the filter was designed, the shaping of the spectrum will be other than desired. Thus, filters require extra stability measures on the oscillator. If nonlinear FM is used, a high average power is preserved; the power is distributed according to the desired window through selection of the suitable modulation, m(t).

Shaping may also be accomplished in the "receiver" (if the system is so separable--i.e., the system must have separate antennas) by filtering methods similar to 2. above. Obviously, the system is no, longer "matched" and the signal-to-noise will suffer accordingly. However it has been shown [6] that degradation is slight (on the order of a dB) for common windows.


3.2 Delay-line IF Correlator Systems

The delay-line IF correlator system produces translated range responses and uses an RF delay line but does not require two antennas. The block diagram is given in Figure 3.3. The "first mixer" consists of the envelope detector which has the effect of a multiplier (see Chapter I); both first and second "mixers" inherently suppress double-frequency terms and successive LP filters are not separate physical components.

Consistent with our assumptions we may express the IF signal as


e 3 (t, _U) = COSEW W TI -


neglecting amplitude factors.


(3.2-1)



































FIGURE 3.3 I F CORRELATOR SYSTEM





Similarly, the output of the second mixer is expressed eR(t,TR) = cos[(t)TR] (3.2-2)

The output of the third mixer is

yO(tTR) = ~- cos[mw(t)(T+TR)] + - cos[t (t) ]-TRA (3.2-3)
20 2

and after time-averaging (effected with the last filter) we obtain
1 1
Z2(T,TR) = - R(T+TR) + - R(T-TR). See (3.1.-lA) . (3.2-4)


This sum of delayed and advanced responses is the response to a matched system: the beat waveform, e(t,T), is perfectly correlated with the ref-i erence IF signal, eR(t,TR) for T = TR. Since neither signal (e3 or eR) distinguishes between positive delay and negative delay (which, of course,








corresponds to non-causal situation), responses appear for T of either sign. Any response z2(T,TR) for T < 0 is of no importance to us. The
1
portion of the advanced response, 1 R(T+TR) which affects for T > 0 is the response of 1 R(T) for T > TR. For most responses, as TR increases,
1
the effect of - R(T+TR) falls off rapidly enough to approximate z2 by Z1(T,TR) of the previous section. For R(T) monotonically decreasing in ITI, we may always find a value of TR such that for any desired c,


fo IZ (,TR)-2Z2(T,TR) 12dT < 6(TR)
0
- provided R(r) is square integrable.


(3.2-5)


Figure 3.4 shows the effect of interfering "positive" and "negative" responses for large and small TR. Of course, when TR = 0,


Z2(T,O) = R(T) = ZI(T,O) = Z0(T)


(3.2-6)


and the reference beat waveform is eR(t,O) = 1.


z2


T (A) LARGE TR


T (B) SMALL ZR


-TR


z 2


(C) TR = 0


FIGURE 3.4 PLOTS OF Z2(T,TR)









Since z2 is composed of shifted R(T), the obvious way to shape z2 would be to shape the autocorrelation through manipulation of the power spectrum as discussed in previous sections.

The RF delay for this system must faithfully reproduce a delayed version of the signal, thus requiring considerable bandwidth (B). Such high-capacity channels may be constructed either of distributed parameters (as in a transmission line) or of a series of lumped sections. Transmission lines are long and bulky and high bandwidth lumped parameter delays must contain many sections, thereby increasing their complexity and cost.

It is apparent that, if the signal x(t) is deterministic, then x(t-TR), and thus also eR(t,TR), will be deterministic. In fact, from (3.2-2) and (1.4.1-6)


eR(t, R) = cos[w0TR+KfTRm(t)] . (3.2-7)

The delay line and the second mixer achieve construction of a beat waveform which could be constructed solely from knowledge of the modulation, m(t). In a periodic system the modulation, and thus the RF signal, signal autocorrelation and beat signals eR and e (T constant) must all be periodic. The exponential (Fourier) series representation of the IF signal is
.21nt
e(t,T) I an(T)e T (3.2-8)
n=-oo

where T is the modulation period and

an could have been written an[(T, 0,Kfm(t)] Similarly, let
.2Trmt
eR(t,TR) I Z m(TR)e (3.2-9)
= � ( .2-9









For a given system the Fourier coefficients contain the range information. Furthermore, extraction of this information takes place as


Z2(TTR) =


Ai2
lim A yO(t,T,TR)dt
A-)- -A/2


A/2
Alim f eR(t,TR)e(t,T)dt A-o- -A/2


A/2 C
limA f I
A-)- -A/2 n=-- m=-


an(T)am(TR)ej 27 (n+m) tdt


n0

n=-00

CO

n=-o


The system we have


is matched when e = eR; i.e., when an = an. Then at T = TR


Z2(TR,TR) =
n=-oo


Ian (TR) 12


(3.2-11)


As expected, the summation above is the total power of the line spectrum of e. The coefficients are computed:

1 T/2 2rnt
an(T) = f e(t,T)e T dt (3.2-12)
-T/2 (A)


T/2.2 nt
1T/ [e [OT+KfTm(t)] + -j[w0T+KfTm(t)]] T dt.
2T -+e T
_T/2


For sawtooth modulation, m(t) = t for Iti < 2T


(3.2-10)
(A)


an (T) $_n (T R) an (T) an (R)











1 T/2
an(T) = T e
-T/2


j[W0T+(KfTt - n t]


-j [ w0 T+ (Kf - t


2Tn
dt (3.2-13)
(A)


Kf TT
sin(-T- - ?rn)

KfTT
(2- Tn)


2e-jW0T
+ -2 )


sin KfTT + 7n)
KfTT
(-- + un)


For the reference or the signal e when T = TR, we have


an(TR) = n(Tf) =


KfTRT
eJW0TR sin ( 2 -T-n)
2 KfTRT
2 -'n)


KfTRT
e-jW0TR sin (--2 + 7fn)
2 KfTRT
+ 7n)

(3.2-14)


For


KfTRT = 27k


all an vanish except for


e j0 TR(sgnk)
ak 2


sgn k =


k=0


+1 k>0


-1 k< 0


ak = a0 = cos WoTR


For the choice of KfTR = 2wk the IF reference spectrum is concentrated


in a single frequency:


(3.2-15)


where


(3.2-16)
(A)


For


(3.2-16)
(B)


e j WOT 2)


, k = 0 , ti , t2 , '''











27rkt .2rkt
eR(t,TR) = ak e + a-k e (3.2-17)
(A)


= cos (-t +(B) T + OTR)(B Under such circumstances, there is no need to construct eR using the delay technique: a simple coherent sinusoidal generator will suffice. These types of systems, for which eR is a coherent sinusoid, are called harmonic processing systems [21].


3.3 Harmonic Processor Systems

We have just seen that, given a linearly modulated FM system,it is possible to design for a range response displaced to TR by detecting a single harmonic line of the IF [21] such that 2n
TR n K (3.3-1)


where the harmonic detected is at f = � n/T. It is seen that n is the number of cycles contained in the signal beat pattern, i.e. in e(t,T), during a period T when T = TR. Also, the instantaneous difference frequency in transmitted and received signals is

Wc(t)-Wc(t-T) n
2T (3.3-2)

for sawtooth modulation.

Such harmonic systems or "n-systems," as they are often called, are versatile in a number of ways:

1. A displaced range response may be obtained
without the use of delay lines.
2. The n-system is equivalent to a delay-line
IF correlator system when the modulation is
linear.









3. The n-system becomes an autocorrelation
system or, equivalently, a matched filter
system when n=O. R(T) is sometimes referred to as the zero-response for this
reason.

As per 2. above, the equivalence of the n-system and IF correlator (for discrete values of TR) no longer exists when m(t) is not linear. For this reason we may shape the harmonic system range response by shaping the RF spectrum, but if we try to obtain that shape by non-linear FM, the problem becomes much more complicated.

Filtering or AM shaping of the RF power spectrum are still

available techniques with a harmonic system. It can be shown that, since both signal and reference beat signals are derived from the W(T) windowed RF signal in the delay-line system, the equivalent window for the n-system must be W(w) because a sinusoidal "reference signal" is like a beat waveform derived from an unwindowed (flat) spectrum. Window relationship are summarized below. The desired response shape is

R(T) = F[Wx]:

Wautocorr"(w) = Wx(M) (3.3-3)
(A)

WIF corr. M = Wx(w) (B)


Wn-syst. (w) = Wx(w) � (C)
(Linear)

Note that we did not prescribe the manner in which m(t) might be designed for non-linear FM windowing to shape the harmonic system response. Many techniques are easily applied to the zero response, such as windowing or the optimum addition of non-linearities to the modulation, but these techniques do not apply for n > 0. However, the computer can be used in a Monte Carlo approach to obtain the non-linear waveshape for n > 0 [22].










Just as the RF signal may be represented in complex envelope form


x(t) = Re{u(t)ej0t} , (3.3-4)

we may represent the IF signal as


e(t,T) = cos[(4m(t)+W0)T] = Re{eE(t,T)ejOT} (3.3-5)


where eE is the complex envelope of e. Then the range response of the harmonic system, z3, may be written as T/2
z3(T) = Re [.L f eE(t,T)eR(t,rR)dt� ej 0T (3.3-6)
-T/2 (A)


= Re{z3E(T)ejWOZ} (B)


where
1 T/2 E
Z3E(T) = -rf eE(t,T)eR(t,TR)dt (C)
-T/2

is the complex envelope of the range response. It is easy to see that Iz3El is the envelope magnitude of the response whileZz3E is the doppler phasing. Let us now speak of the set of coefficients {Yn} used to describe
S2Trnt
e E(t,T) I Yn yn(e T(3.3-7) n=-oo


Couch [22] has shown that the Fourier coefficients, {yn}, relate to the ambiguity function as

*TnT
Yn(T) = e X2(-T, n (3.3-8)
n(A)

and


1Yn1 = IX2(-r T)









Thus the range response of the n-system using coherent demodulation when the reference is defined as
(27nt
eR(t) = cos ( T + 0) (3.3-9)

with reference phase 0

becomes
27Tnt
E T/2 3 T2tnt
z3 T Yn e cos (T + O)dt (3.3-10)
T -T/2 n=-- (A)


1~ T/2 27Tnt 27Tnt
= TOS2 (2I_ t- + LYn) cos (T- + 0)dt (B)
rT-/2


= lYn1 cos (LYn- 0) (C)



SIX2(-T, T) cOs (/X2 -T - ) (D)


If 8 0 and T << T, then


z3E(T) z Re{X2(-T, T)} (3.3-11)

and any departure from the terms of the small T assumption represents a phase error of fnT/T radians.

For the special case of linear modulation we obtain from (2.2.1-5), (3.2-4), (3.3-1) and (3.3-11)

z3E(T) z Re X2(-T, R) (3.3-12)
(A)

1L X2(T + 2frn 0 X2(T - 0) (B)
2 KfT 2 KfT

One sees that the ambiguity function at = n/T is the sum of displaced = 0 functions. This property comes from properties 10./11. of the ambiguity function (Section 2.1.3,3); the ambiguity function is "sheered'






60


with the introduction of quadradic phase. This fundamental appears to be the basis of obtaining displaced responses without the use of delay lines. Figure 3.5 displays this sheering behavior for a linear-FM (quadratic phase in the time domain) signal. In contrast to the linear systems the range response of a sinusoidally modulated harmonic system is the signal










T















FIGURE 3.5 AMBIGUITY FUNCTION (MAGNITUDE) FOR LINEAR FM SIGNAL



ambiguity function at 4 = n/T, which is a Bessel function of order n, not so simply related to the zero response (which is a zero-order Bessel function).

In a later chapter, a relation will be developed which will predict the displaced range response, regardless of modulation non-linearities, provided they are known.










3.4 General Coherent Demodulator Systems

At this point we shall not detail the coherent demodulator system of Figure 3.6 except to define its structure [23]. It should become obvious that all of the preceding structures are contained as subclasses of this general system.

The dotted line in Figure 3.6 indicates a linkage or coherency between m(t) and i(t), the demodulating waveform. The linear processor of Figure 1.4 has been shown separated into a time-variable gain and a time-invarient-filter.

In future sections all proposed systems will be of a type which may be represented in the form given by Figure 3.6.


e (t ,t)


TO NL CIRCUITS


FIGURE 3.6 GENERAL COHERENT.DEMODULATOR SYSTEM
















CHAPTER IV

INVERSE FILTERING


The process called "inverse filtering" came about from a desire to improve definition of waveforms arriving at various times and with various amplitudes. Such a situation exists in a pulse radar situation. The composite return signal is of the form

N
Xreturn(t) Y Ynx(t-Tn) � (4-1)
n=l

As we have seen, resolution problems occur when delay times differ by amounts comparable to the signal duration. The signal may be written as a filter response to an impulse: x(t) = x(t) * 6(t) . (4-2)

The object in the inverse filtering approach is to find a filter, h(t), such that

h(t) , x(t) = 6(t) . (4-3)

Then one may filter the composite return to obtain

N
Xreturn(t) * h(t) I Yn6(t-Tn) , (4-4)
n=l

for which no resolution problems or ambiguities exist. In the frequency domain
H(w)X(w) = 1 ; (4-5)

h(t) =F (4-6)










Naturally, signal-to-noise may be expected to suffer because we employ a filter with transfer function i/X(w) rather than the matched transfer function [24],

H(w) = X*(w) (4-7)

assuming white noise.

Actually, since the delta function is not realizable, requiring infinite bandwidth and power, the inverse filter is compromised so that x(t) , h(t) = g(t) (4-8)
(A)

H(w) X(w) = G(w) (B)

where g(t) is to be specified, hopefully being a very peaked, low-sidelobe type function. Since
H(w) = G(w) (49)

X(w)

We must be very careful to provide zeros of G(w) to coincide with those of X(w). An example of the problems associated with nulls is given by the spectra of Figure 4.1(B) [25]. Especially if X(w) were to be bandlimited, G(w) would have to be bandlimited to the frequency interval on which X(w) was supported. Otherwise H(w) must have infinite gain at a set of connected Points, yielding an infinite noise power out of the filter.

One sees that the design problem here rests entirely with the specification of g(t). Gaussian forms have been suggested for use with time signals which are not bandlimited. Childers and Senmoto [13] specified a measure of performance based on signal-to-noise ratio and resolution, with a (standard deviation of the output pulse) and minimum epoch times, T (the separation between return signals), as parameters.













X (Ws)














(A) SIGNAL SPECTRUM



H( o)
II I II I



















(B) INVERSE FILTER FOR X(w)


FIGURE 4.1 EXAMPLE OF PROBLEM SPECTRA









One may specify one of the parameters to obtain an "optimum" in terms of the performance measure of the other parameter.


4.1 Application to the T Domain

The concept of inverse filtering applies to the FM periodic radar in the filtering of range response z(r) in the T or delay domain. The desired response Zout(T) corresponds to the output pulse, g(t), in the previous section, while z(T) corresponds to x(t) and the desired filter has impulse response h(T). The obvious difference in the real-time analogy and filtering in the delay domain is that z(r) is not available as a function of time, t. Our system may have to make decisions about the value of T while T is completely static.

The question arises as to what real, physical form such a filter would take. The filtering operation is given by the convolution integral
00
Zout(T) = f h(r)-)z(X)dX (4.1-1)
o(A)
-00

= f h(X)z(T-X)dX . (B)
-00

We see by (4.1-1) that the value of Zout(T) depends upon values of z(X) for all arguments except those for which h(T-X) = 0. We shall now examine each system type for devices by which we might obtain the convolution of (4.1-1) even though T is fixed.


4.1.1 The Autocorrelation System

In the autocorrelation system, the output variable z0(T) = R(T) is fixed for a fixed value of T; so is e(t,T) for given T. We must assume, then, that we cannot, without some modification to the basic system, obtain a new variable Zout(T) conforming to (4.1-1).






66


The modified delay-line autocorrelation system yields a response zl(t,TR) = R(T-TR) . (4.1.1-1)

Then
Co
Zout(T) = f R(T-TR-X) h(X)dX (4.1.1-2)
-0 (A)

o
- f zl(t,TR+X) h(X)dX (B)
-00


Z z (T,TR+X) � h((,*d; (C)


where the main energy contribution of h(T) is between -T0 and To, T0
AX
Z zl(T,nAX+TR)h(nAX) - AX (D)
n O- To

by substitution of an approximating summation for integration. Using such an approximate convolution, a response may be shaped and made to peak at displaced delays. If h(T) is even (Hermetian) about zero, rhe delay will be TR. Notice that care must be taken so that nAX+TR a TR-TO > 0 (4.1.1-3)

to avoid negative delays. A block diagram of this implementation is given by Figure 4.2.

An alternative to the parallel implementation would be the "artificial" translation of the convolution operation to real time by scanning through TR in time. The basic system would be quazi-fixed, TR being allowed to vary slowly in time as illustrated in Figure 4.3. The delay is varied as


Tx = TO+at .


(4.1i.1-4)

































FIGURE 4.2 PARALLEL IMPLEMENTATION OF CONVOLUTION


T = + at


sample and hold at
"TR- *o+td
t =R
a


FIGURE 4.3 TRANSLATION OF CONVOLUTION IN DELAY TO A CONVOLUTION
IN TIME









Then, in order to approximate a non-causal filter response in r by a realizable one in t, we scale and translate (delay) the response of h to be
h (To+at-td) (4.1.1-5)

where td allows truncation of the response for t j 0. Zout is a sample of the output of the filter at a time corresponding to a delay of TR.

The objection to this scheme is the slow rate of information obtainable at the output because of the necessary quazi-fixed system. All other things equal (bandwidth, modulation, range), this system places more stringent restraints upon the stationarity of the target and is, perhaps, not practical for many applications. There is also a problem of obtaining variable delays Tx.


4.1.2 The Delay-line IF Correlator System

Since the delay-line IF correlator system yields a response of

Z2(T,TR) =1 R(T+TR) + 1 R(T-TR) (4.1.2-1)
2 p 2 (A)


7 R(T-TR) For "large" TR , T > 0, (B)
2

1()
f Zl(T,TR), (C)


we may use the same procedure of filtering as was used in the case of the RF delay-line system, provided the contribution of R(T-TR) is sufficiently small for T > 0.


4.1.3 The Filter, h(T)

Let us now pause and measure the effect of a perfect inverse filter h(T) on the range response. The filter has the transform










Zout (1)
H(Q) = Z(fl) (4.1.3-1)


Our system gives a response z(T) which is bandlimited; therefore, we must design the most appropriate bandlimited function Zout(P2). A probable choice might be one of our windowing functions Wi(Q). Thus we establish the equivalence of windowing the transmitted power spectrum with Wi(2nf) (or /Wi as the case maybe) and choosing an inverse filter of



H(O) = F-I[h(T)]= Wi() (4.1.3-2)


The output range response may be viewed as the sum of translated responses, as shown below, where the function H(Q) has been represented by an exponential Fourier series for w0-nB 0 & w0+nB:

Zout (T) = F-1 [Zout()] = F-[H(R)Z(Q)] (4.1.3-1)

(A)
f.nio

= F-I an e Z( ) (B)
n=-oo

= [ an z(T-) , (C)
n=-co

a result corroborated by the parallel implementation of Figure 4.2. Both the sampling theorem [23] and the results of (4.1.3-C) would suggest that the maximum value of AX (to assure complete expression of the desired response by 1/B.


4.1.4 Harmonic and General Coherent Demodulation Systems

It has been shown that, for integer BT and sawtooth modulation, the harmonic systems correspond to a delay line system with

TR = (4.1.4-1)









Then the parallel convolution construction of Figure 4.2 is applicable to harmonic n-systems with the restriction that AX=1 (4.1.4-2)


which is the maximum spacing which allows implementation of any window function. This is the "spacing" inherent in the range responses of linear harmonic systems operating on adjacent harmonics, n and n+l.

When the modulation of a harmonic system is not linear, the response is given by equation (3.3-12A), but equation (3.3-12B) is no longer valid. It is certainly not clear that a convolution of z(r) with h(T) may be made in this case. The same uncertainty must, of course, be true of a more general coherent system.


4.2 Summary and Conclusions about Inverse Filtering

It was shown that inverse filtering in its ideal form de-convolves a signal to yield a delta function output. The approach must be modified in the case of a bandlimited signal to yield an output which is similarly bandlimited, such as the transform of a window function. When filtering in the delay domain, the problem becomes one of obtaining Z(T) for varying values Of T when the target is stationary. For the autocorrelation system with an RF delay line and for the delay-line IF correlator system, the problem was solved by using an approximating point-bypoint convolution using parallel implementation. Alternately, by using a variable delay and a scaled time-domain filter, it is possible to obtain a time-domain inverse filter which simulates the convolution in T. The delay must be swept slowly, and sampled once per sweep, imparting a slow rate of information output to the system, and placing more stringent requirements of stationarity on the target.





71



The harmonic systems using non-linear modulation and the more general coherent systems do not betray any obvious opportunities for inverse filtering.















CHAPTER V

SYSTEMS USING VECTORS OF INFORMATION


Systems such as the harmonic system which demodulates a single line of the IF spectrum are simple and yield a certain quantity of information about range. But additional information is to be had in the other lines of the IF spectrum. We have seen how, in the linear modulation case information may be manipulated by a linear combination of these lines to produce a desired range response whose transform is a window function. In this chapter we shall designate each harmonic sub-system output as zn(T) and the ordered collective of all outputs as

T
Z(T) = [z1(T),z2( ),. ZN(T)] (5-1)


For convenience, the arguments will often be omitted: z = z(T). Each output, zn(T), will be called an information element and the vector z will be called the information vector. Let us now define an NxN matrix


G = [Z(TI)z(T2)'.Z(TN)] (5-2)
(A)

= [gmn] where gmn A Zm(Tn) . (B)

This matrix uniquely defines each of the N elements of information for N values of delay, assuming no noise in the system [26,p.287]. We shall denote the set of times TI, T2,-"TN as {Ti}. If an inverse, H, exists for G such that











1 0 . 0

HG = GH = I= 0 i 0 (5-3)


- 0 0 . . i

we now have a linear transformation of G which contains new N-dimensional information vectors at N points in T:


I = GH [. OUt(T)Z.out(T2).OUt(.TN)] (5-4)

Each row of I is the value of a system output defined at N points in T and at no other points. Figure 5.1 describes the range response of one such system output. The times {Ti} do not necessarily have to be ordered such that

Ti < Ti+i

as Figure 5.1 also illustrates. Of course, there is no ambiguity evident




I '(T) # out3



%
I
I
I I
I i
5 I I
g / , I
OZ,.,,1W" " . C.

IT2 T T6 T3 TI5 T4 " ".


FIGURE 5.1 HYPOTHETICAL RANGE RESPONSE INDICATING CONSTRAINT POINTS



in the response zout3(T) as defined at points {Ti}. But each system output (row) comes from the transformation
N
Zoutm(T) = I hmnzn(T) (5-5)
n=l









A less deceiving picture of ambiguity is given by the dotted response zout3 for continuous values of T. Intuitively one might think that the more points constrained, the better. This reasoning turns out not to be necessarily true. The number of constraint points N is limited by the highest rank G may be. And the rank of G depends on the number of linearly independent vectors z(Ti) that exist. Alternately, the rank may be expressed as the number of linearly independent row vectors of G: if an information element Zj(T 0 gives additional'information not contained collectively in {Zi(Tk)l for iOj and for all k, then it creates a vector linearly independent of the other vectors.


5.1 Alternative Information Vectors

The information elements {Zi(T)l need not be the outputs of harmonic sub-systems, but may be any set of variables, each of which is linearly independent of the others as a function Of T [27,p.29]. We shall consider only variables which are derived from linear operations upon the IF waveform. e. Examples of these elements might be

1. equally or unequally spaced samples of the
IF waveform taken at specific times in the
period T,
2. trigonometric series coefficients (of the
IF signal expansion),
3. Walsh-function expansion coefficients,
4. the outputs of square-wave demodulators, 5. other (generalized) Fourier series coefficients.

The linear operation takes the form shown in Figure 1.4; Table 5.1 gives the demodulating function for each element set listed above. The filter h(t) time-averages over one or more periods of the modulation and is, in practice, a LP filter with a cutoff frequency much lower than l/T.











INFORMAT ION DEMODULATION
ELEMENTS WAVEFORM, *n(t) ORTHOGONAL?

1. Time Samples c(t-tn) Yes

2. Trig. Coeff. sin 2 ornt 2rnt Yes
__ __ _ __ _ __ __ _ _T_ T

3. Walsh Coeff. wal (n , t) see [26] Yes
T

4. Squarewave Demod. Squarewave of period - No
n

5. Gen. Four. Coeff. 4n(t) (orthogonal set) Yes


TABLE 5.1. SAMPLE DEMODULATION FUNCTIONS


Then
1 T/2
Zn(T) = f e(t,T)*n(t)dt (5.1-1)
-T/2

All of the demodulating waveforms {*n} form orthogonal sets except for case 4., for which {n} are squarewaves. Squarewaves are linearly independent but not orthogonal. This set was included because of the appealing possibility of using choppers or switching inverters instead of true analog multipliers. The squarewave functions (or any other linearly independent set) may be added to form an orthogonal set using the Schmidt orthogonalization process [26,p.ll;8,p.458]. This is unnecessary however, as orthogonality of the demodulating functions in time do not guarantee orthogonality, or even independence, of {zn(T)} in T. Moreover, the orthogonalization process is a linear transformation which may be represented in matrix form. If we call the transformation matrix e, then we first orthogonalize {1n(t)} by


(5.1-2)


-rt.(t) = o *(t) .










In the next section, we will show that this represents a linear transformation of

z rth. z . (5.1-3)


Then if we transform using H' (derived from zorth.) we obtain


Zout = H' zorth. = H' 9 z = Hz (5.1-4)

so that we are still performing one transformation by H, determined as


H = G-1 = [Z(T1)'''z(TN)I� . (5.1-5)

Thus we perceive no generalizable advantage of orthogonal sets over nonorthogonal ones.


5.2 Equivalent Single-Channel System

Up to this point we have considered a linear combination, defined by H, of information elements, each element derived from a separate subsystem as in Figure 1.4. We shall now observe that it is easy to express the entire system in the form of Figure 1.4 when a single output element is desired

N
Zoutm Ihmnzn (5.2-1)
n=l (A)
N T/2

I hmn y f e(t,T) �n(t)dt (B)
n=l T/2

1 T/2 (
y f e(t,Tr) m(t)dt (C)
-T/2

where
N
c WA hmnn(t) (D)
m n=l









This simple result is due, of course, to the linearity of the system, which allows an interchange of the summation and integration in (5.2-1). We simply coherently demodulate with a time function defined as the appropriate linear combination of {1in(t)}.


5.3 Existence and Dimension of H

Whenever one speaks of an inverse matrix, such as H = G-1, the existence of that matrix comes into question. That existence depends upon G having a non-zero determinant, which, in turn, implies linear independence of all vectors zi(T) _ [zi(T1)zi(T2) . zi(T N)] (or alternately, linear independence-of z). Thus the size of G for which G1exist is limited to the number of independent information vectors which may be obtained.


5.3.1 Dimension of H Based on IF Waveform

To determine a maximum value for N we will consider the case of linear sawtooth modulation and define a T max such that we are interested only in the case when T -< Tmax. Then the instantaneous difference (IF) frequency will vary between 0 and mTx. The IF waveform will be a si nusoid windowed by the modulation period T. This windowing of the time waveform. will spread the IF bandwidth somewhat, but if BTmaxthe maximum number of cycles over a period T--is more than a few cycles, this spreading effect is negligible compared with the frequency range or bandBTmax
width T. By the sampling theorem we know that e(t,-r) is defined by its sample points uniformly spaced by T , there being a total of

2BTmax points for period T. Since these points uniquely define e(t,T) (disregarding the fact that we approximated the bandwidth), any more would be redundant. And, just as the information out of a channel cannot









excede that into the channel [28,p.106], we may model a transformation from the time domain to other domains as a "channel" and realize that no more independent elements may exist in one domain than in another. Thus the maximum rank of G and the maximum size of H is approximately N = 2BTmax (5.3.1-1)

When the modulation is monotonic but not linear, we may consider a piecewise-linear approximation to the modulation. It is easy to see that, neglecting the windowing effect of each section, we will obtain the same number of independent points as before, the points being spaced unequally according to the changing slope of the modulating waveform throughout the period.


5.3.2 Dimension of H Based on the Range Response

Since we may describe the system as the single channel of Section 5.2, the range response becomes T/2 (
Zoutm(T) T f e(t,t)lm(t)dt (5.3.2-1)
-T/2 (A)



1 T/2
= f cos[Wc(t)T]lc(t)dt � (B)
-T/2


There can be no more independent points in zoutm (T) from T 0 to T = Tmax than there are independent vectors z(Ti) for 0 < Ti S Tmax because zoutm is a linear combination of zn. Then the rank of G can be found through using the sampling theorem on z(T). To do this we find the bandwidth of zoutm (T) by transforming:










Z outm (n) = f Zoutm(T)e~j-r dT (5.3.2-2)
-0 (A)

1T/2
= I Y o [Cc(t)T]I (t)dteJdt (B)
-T -T/2 m

T/2
f dt 'P(t) f Cos Wc(t)T eJ T dT (C)
-T/2 -0


1T/2 c
f T2 (t) 2 [6(o-wc(t)) + 6(Sl+(c(t))]dt (D)
T T/2


The delta functions are non-zero only when 0 = �wc(t). Thus if wc(t) is limited to (S21,02), then Zoutm(Q) is bandlimited to (Q1,Q2). For the essentially bandlimited signal such that 02-Ri = 2rB, Zoutm(T) is bandlimited to a bandwidth B, and the number of independent points per unit time is 2B for a total of

N = 2BTmax (5.3.2-3)

independent points over (0,Tmax).


5.4 Choice of Constraint Times

One sees that the system may be defined in terms of its transformation matrix H = G operating on information vector z. Once the set of information elements have been determined, G--and thus H--depends upon the {Ti} by (5-2A)

To illustrate the problem concerning an optimum choice of {Ti} we advance the following example: The information vector contains four elements each of which is given as a function of T by Figure 5.2.
























Z3


0 1 2 3 4
B B B B


FIGURE 5.2


INFORMATION ELEMENTS AS A FUNCTION OF DELAY


Now we "arbitrarily" choose Ti = i/B; then


Zl(2) ZI(3) Z(4)




4 B ' Z4(4)


= I from Figure 5,2,


1 1T


Z 3

B


(5.4-1)
(A)


=


0 0 0 0









and

H =G-1 =1 (5.4-2)


z ot=Hz z .(5.4-3)


Here, {Til were chosen so that the transformation H is completely ineffective, and the range response is not improved. Other sets {Tij would constrain other points to be zero in each response, and at least a different response would be obtained. It thus appears that {Ti} will have to be "optimized." However, the manner in which one might optimize the set is illusive. One method used to attempt such an optimization was a recursive gradient algorithm, implemented in APL. Results were not satisfying, however; the algorithm failed to converge to reasonable values of Ti. Part of the problem may have been the use of a NL "measure" of sidelobe levels. That is, we used as a measure the highest of the sidelobe magnitudes at designated "test points" in Tr. Since a gradient technique will converge to any minimum, including local minimums, there is no guarantee of ever finding the absolute minimum. There is no reason to believe that many local minima do not exist as one progresses through the Hilbert space that describes {Ti}.

Finally, by equation (5.3.2-2D) we know that the range response, Zoutm(T) must be bandlimited. The uncompromising practice of constraining individual points to be nulls in the response does not appear consistent with most signal design methods. Perhaps a more profitable course to pursue is that of confining the range response to be that of a window transform, in which case we would require


H' = R(5.4-4)


H'G = R









so that

H' = RH = RG-1 (5.4-5)


where R is the desired range response matrix and H' defines the appropriate linear combinational matrix for the
system.

However, the above-mentioned problems would still apply, and we see that this process does not lead to a very systematic method of solution.

In the next section we developed relationships which are easier to use and are thus of more value and importance.




Full Text
FIGURE Page
9.33 SPECIFIED THREE-UNIT EVEN PULSE RESPONSE 177
9.34 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE
RESPONSE USING RECTANGULAR WINDOW 178
9.35 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.34
FOR SAWTOOTH MODULATION 179
9.36 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE
RESPONSE USING HAMMING WINDOW 180
9.37 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.36
FOR SAWTOOTH MODULATION 181
9.38 SPECIFIED THREE-UNIT ONE-SIDED PULSE RESPONSE 182
9.39 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT ONE-SIDED
PULSE RESPONSE USING HAMMING WINDOW 183
9.40 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.39
FOR SAWTOOTH MODULATION 184
9.41 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.39 FOR SAWTOOTH MODULATION 185
9.42 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.19 186
9.43 MODULATION FUNCTION OBTAINED WITH ZE OF FIGURE 9.42 AND
DC DEMODULATION 187
9.44 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.21 188
9.45 MODULATION FUNCTION OBTAINED WITH ZE OF FIGURE 9.44 AND
DC DEMODULATION 189
9.46 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.23 190
9.47 MODULATION FUNCTION OBTAINED WITH ZE OF FIGURE 9.46 AND
DC DEMODULATION 191
9.48 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.26 192
9.49 MODULATION FUNCTION OBTAINED WITH ZE OF FIGURE 9.48 AND
DC DEMODULATION 193
xl


Response Magnitude
0.38
0
-16
0
6.4 Normalized Delay 16
FIGURE 9.28 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE AT Bx=6.4 USING RECTANGULAR WINDOW
ho


-12 -10 -8 -6 -4 -2 0 2 4
6
8 10 1
Normalized Dla
FIGURE 9.17 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION AND DEMODULATION
CM >>


53
Since Z2 is composed of shifted R(x), the obvious way to shape
Z2 would be to shape the autocorrelation through manipulation of the
power spectrum as discussed in previous sections.
The RF delay for this system must faithfully reproduce a delayed
version of the signal, thus requiring considerable bandwidth (B). Such
high-capacity channels may be constructed either of distributed parame
ters (as in a transmission line) or of a series of lumped sections.
Transmission lines are long and bulky and high bandwidth lumped parame
ter delays must contain many sections, thereby increasing their com
plexity and cost.
It is apparent that, if the signal x(t) is deterministic, then
x(ttr), and thus also eR(t,xR), will be deterministic. In fact, from
(3.2-2) and (1.4.1-6)
eR(t,xR) = cos[w0xR+KfxRm(t)] (3.2-7)
The delay line and the second mixer achieve construction of a
beat waveform which could be constructed solely from knowledge of the
modulation, m(t). In a periodic system the modulation, and thus the RF
signal, signal autocorrelation and beat signals eR and e (t constant)
must all be periodic. The exponential (Fourier) series representation of
the IF signal is
. 27rnt
CO j
e(t,x) = l an(x)e T (3.2-8)
n=-oo
where T is the modulation period and
an could have been written an[(x,w0,Kf,m(t)] .
Similarly, let
. 2irmt
CO j
eR(t,xR) = l $m(xR)e T (3.2-9)
m=-oo


C(t)t
0.595
FIGURE 9.32 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9
28 FOR SINE MODULATION
176


121
Equation (8.3.2-6B) Is recognized as the inverse Fourier transform
of the product \p pw :
c
Z+(fi) = 2n (fi)pUc()
> t, R +jf 1
0) w
c c
b)
(8.3.2-8)
(A)
or
ZE(fi)
2ir ip (Si) p (2)
U) J
m m
i i ^ j, I
\p = \p +j\p
) to 0)
mm m
(B)
Relation (8.3.2-8) reflects the high-index assumption. For the
periodic case this relation becomes
ze(q) (fi)
m
dto _1(n)
mi
di2
(8.3.2-9)
since p
4 i
to
m
(fi) = h 1
dto _1(fi)
mi
dti
These two forms are exactly those of (7.4-3C) and (7.4.1-4B) but with
the understanding that iji may be complex with the real and imaginary
R X
parts, ip and ip corresponding to the two references of the two-
channel processor of Figure 8.2.
It is seen that the linear two-channel system expressed in
the model of equation (7.4-3C) (i|> considered complex) is a general
representation which includes all linear single-channel systems as
a special case when ip is purely real. (Consider Figure 8.2 for
the case ip* =0.) It is extremely versatile, because any
bandlimited range response envelope may be designed. That is, the
transform of z (t) is not restricted in form except that it must be


119
I e1
z (t) --Reiz* (x)e }
= Reiz^ (x)e }
where is the doppler radian frequency.
The Hilbert transformer produces a 90 shift so that the output
yh is
yh(T) = i
E1, J
= Re{z (x)e
I j (u0t+- sgn u )
= Re{z (T)e 1 d
I j ( = [sgn Re{z (x)e
For Wo>'rrB the product theorem [5, p.19] gives
yh(t) = [sgn zI(t)* ~
= zI(x)*h(x)
where h(i) = [sgn
Thus we may model the filter as a Hilbert transformer in the
delay domain preceded by a sign changer for negative doppler
frequencies. The output of the processor is
R /vl
z(t) = z (t) z (t) where the + indicates receding and (8.3.2-1)
approaching targets respectively


Response Magnitude
0.0 __
-16
-3 0 3
FIGURE 9.33
SPECIFIED THREE-UNIT EVEN
PULSE RESPONSE
Normalized Delay
16
177




149
t by -T and negating the phase response. Thus, for directional
doppler systems, the right half-axis is valid for receding targets,
and the left half-axis, for approaching targets.
Figure 9.11 is the magnitude plot of the response of a
so-called n=3 harmonic system, with sawtooth modulation and co
sinusoidal (single-channel) demodulation. The response is a good
example of an autocorrelation response (Figure 9.10) being translated
positively and negatively in normalized delay. (Figure 9.10 is also
the range response plot obtained when Sawtooth modulation is employed.)
Since the modulation is linear, both ip and ^ are cosinusoidal,
t u)
m
providing multiplication in the 0-domain by
j 30 30
ip (0) = cos(3 ^)=-^{e B+e B). (9.4-1)
hi B
m
The positive and negative responses are phased so as to
interfere destructively near the origin and constructively for
¡Bt | >4. It is seen that phasing may play a part in designing systems
with low sidelobes. Notice in this example that the sidelobes
tend to level off near the ends of the plot rather than continue to
fall off. This higher degree of aliasing is due to responses closer
to the ends of the plots.
When a singly-translated response is desired, a two-channel
(directional doppler) processor is used to effect a demodulation of
. 3fi
2 ~R
(fi) e (9.4-2)
m
Such a system, using sawtooth modulation and third-harmonic cosine


\¡> a vector of demodulations defined for a number of subsystems
<|> usually a frequency variable (Hz.)
0 usually a phase angle
() Hilbert transform in the domain of interest
() complex conjugate
* convolution operator
ZT the transform of z
[gmn] the matrix G containing elements gmn
xvi


104
will be pursued in detail in the next sections. The transform is
that of equation (7.1-2F), restated here:
t
0.(x) = / u (A)dA (7.3-1)
l m
t-x
The product S ip may be represented as S . ip . This
91(t)x Sj(t^)
representation is not ideal because 0^(x) and 0j(x^) are two
dependent random variables. The problem can be alleviated if one
assumes that to (t) is slowly-varying with respect to delay times,
m
thus simplifying 0^. That is
to (t) = to (t-x) for all x to be encountered. (7.3-2)
m m
Then,
t
6, (x) / to (t)A = to (t)x (7.3-3)
1 m m
t-x
and the transformation is from t to 0=0(x)=to (t)x. The utility
m
of the transformation comes from the fact that the signal may be
expressed as a sinusoid:
sn T ~ Sfl t (9,T) = cos(0+to x). (7.3-4)
Dpi U T O
The usefulness of sinusoidal representation will be observed shortly.
If the reference is generated by the delay-line/multiplier
of Figure 3.3, we may express ip as
ip (t) = cos [to (t)x +to x] (7.3-5)
t m R o ...
(A)
= cos[to (t)x r +to x ] (B)
m t o R
= cos[6(t) ~ +)0tR3 (C)
%(0) = cos[0^ +tooxR] (7.3-6)


64
(A) SIGNAL SPECTRUM
(B) INVERSE FILTER FOR X(u)
FIGURE 4.1 EXAMPLE OF PROBLEM SPECTRA


oooooonoooooooooooooooooooooo
215
SUBROUTINE I NTRP(0RD,ABSC, IMIN, IMAX, INTP, Y,X, JillN, JMAX, YP,H)
THIS SUBROUTINE TAKES A GENERAL INPUT TABLE, ORD/ABSC, FROM
INDICES IMIN TO 1 MAX, AND AN INPUT VECTOR, XIN (WITH INDICES JMIN
TO JMAX), AND CREATES AN INTERPOLATED OUTPUT VECTOR YOUT, ALONG
WITH THE CORRESPONDING DERIVATIVE VECTOR YP FOR ALL INDICES
J BETWEEN JMIN AMD JMAX. ALL ARGUMENTS ARE INPUT ARGUMENTS
EXCEPT FOR YOUT AND YP AND ARE AS FOLLOWS:
ORD IS THE VECTOR OF INPUT ORDINATES (DIMENSIONED AT
LEAST TO IMAX IN THE MAIN PROGRAM).
ABSC IS THE VECTOR OF INPUT ABSCISSAS (DIMENSIONED AS IS ORO)
IMIN IS THE INDEX OF THE BEGINMING SET OF DATA PAIRS,
ORI) | ABSC.
IMAX IS THE INDEX OF THE LAST SET OF DATA PAIRS, ORDIABSC.
INTP DETERMINES HOW MANY POINTS WILL BE USED FOR THE
CONTINUED-FRACTION INTERPOLATION.
YOUT IS THE VECTOR OF INTERPOLATED OUTPUTS CORRESPONDING TO
THE INPUT VECTOR XIN.
XIN IS THE INPUT VECTOR OF ARGUMENTS (DIMENSIONED TO AT
LEAST JMAX IN THE MAIN PROGRAM).
JMIN|JMAX ARE THE BEGINNING AND ENDING INDICES OF THE INPUT
ARGUMENTS FOR WHICH OUTPUTS ARE DESIRED.
YP IS THE VECTOR OF DERIVATIVES.
H IS THE INCREMENT USED IN THE COMPUTATION OF DERIVATIVES.
NOTE: THIS SUBROUTINE CALLS TWO SUBROUTINES WHICH ARE CONTAINED
IN THE SCIENTIFIC SUBROUTINE PACKAGE ATSM AND ACPI.
DIMENSION Z(101),F(101),0RD(IMAX),ABSC(IMAX),Y(JMAX),X(JMAX),
*ARG(101),VAL(101),WORK(101),YP(JMAX)
DO 10 J=IMIN,IMAX
Z(1+J-1 MlN)=ABSC(J)
F(1+J-IMIN)=ORD(J)
10 CONTINUE
EPS=lE-5
l-IMAX-IMIN+1
DO 1 J=JMIH,JMAX
CALL ATSG(X(J),Z,F,WORK,l,l,ARG,VAL,INTP)
CALL ACFI ( X( J), ARG,VAL, Y( J), INTP, EPS, I F.R)
CALL ATSG(X(J)*H,Z,F,WORK,1,1,ARG,VAL,INTP)
CALL ACFI(X(J)*H,ARG,VAL,YH,INTP,EPS,IER)
YP(J)(YH-Y(J))/H
1 CONTINUE
RETURN
END


C(t)
FIGURE 9.20 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.19 FOR SAWTOOTH MODULATION
164


15
1.5 Assumptions
1.5.1 The Assumption of Small t
The transmitted signal is written, using no simplifying assump
tions as
t
x(t) = cos(wot+Kf / m(X)dX) (1.5.1-1)
00
It is somewhat instructive to observe the instantaneous frequency,
written (as before)
u)c(t) = )0+Kfm(t) (1.5.1-2)
Similarly, the return signal A(x)x(t-x) has an instantaneous frequency of
return^ = w0+Kfm(tr) (1.5.1-3)
Plots of coc and treturn are given in Figure 1.5(A) for the linear modu
lation case:
m(t) = | £ .< t < | (1.5.1-4)
m(t) periodic such that m(t+T)=m(t), all t.
It is easy to see that "turn-around time" (indicated by the section of
Figure 1.5(A) measured as x) detracts from the effectiveness of any sys
tem which might simply count cycles of the IF waveform (Figure 1.5(C)) or
measure power in the IF spectral lines. This effect would have to be com
pensated or at least accepted as error. Tozzi [7] has done a good deal
of work in the analysis of the IF spectrum for the cases of triangle,
sawtooth, and sinusoidal modulations. He uses the amplitude of a single
linecoherently detectedto provide range information and analytically
developes the results for various ranges of x, which are not restricted
to being small. Even for these three simple cases, manipulation becomes


2
the above requirements, but we restrict ourselves to the following struc
tural framework:
Bl The system will be a continuous-duty type FM
radar.
B2 A single antenna will be employed.
B3 Envelope detection of the RF voltage at the
antenna terminals will yield the information
to be processed.
B4 The "transmitter" will be periodically fre
quency-modulated using high-index modulation.
B5 Modulation rate will be slow relative to sig
nal return times.
The list of requirements are consistent with applications such as
aircraft altimeters or low-height warning devices. Some of these require
ments will now be discussed as they relate to one another and to the as
sumptions to be employed in this dissertation. The brief discussion is,
of course, in no way intended to represent all of the considerations in
volved in a choice of systems.
Requirement Bl results from more than one factor, among which
are simplicity of construction and energy (or, more correctly, average
power) transmitted. A continuous-duty oscillator is usually simpler to
design than one that is amplitude-modulated or pulsed. The energy of a
signal is the time integral of the signal magnitude squared:
oo 2
E = / |x(t)| dt (1.2-1)
If the radar is periodically modulated (either in amplitude or in angle)
we may speak of the energy per period:
T/2
ET = / |x(t)|2dt (1.2-2)
-T/2
where T is the period of the modulation.
It can be shown that accuracy and range capability of a radar (influenced


5.9
SECOND-CHANNEL DEMODULATION IS ZERO
Pi
0.
-1.23-
-T/2
Time
T/2
FIGURE 9.35 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.34 FOR SAWTOOTH MODULATION
179


218
Control Cards
The following control cards are used to enter any of the
programs and data for execution on the IBM 370 system of the North
east Regional Data Center at the University of Florida as of Decem
ber 10, 1974. The program to be run is stored in a saved file
with the name PROGRAM, and the data is similarly stored under the
name DATA. Provision for inclusion of the IBM Scientific Subroutine
Package is made by the specification SUBLIB='GATOR.SSPLIB.FORT' as
shown below. These "cards" are entered from a time-sharing ter
minal as a remote job entry (RJE); output will appear at the main
computing center.
//MATJOB JOB (1006/2003,009/10/0/,,,0), 'MATTOX 1 C LASS=M
/PASSWORD 004,BARRY
/ROUTE PRINT LOCAL
// EXEC FORTGCG,SUBLIB1='GATOR.SSPLIB.FORT'
//FORT.SYSlN DD
/INCLUDE PROGRAM
/*
//GO.SYS IN DD
/INCLUDE DATA
/*
If the program were to be entered as cards on a card reader, the
user would simply replace the /*INCLUDE PROGRAM by the Fortran source
deck and replace the /*INCLUDE DATA by the data deck.


Specified Real Response
1.0
FIGURE 9.25
SPECIFIED BANDLIMITED SIN(X)/X RESPONSE
Bandwidth
Normalized Delay
16
169


24
(C) SPREAD, LOW-LEVEL
BACKGROUND
FIGURE 2.3 RANGE AMBIGUITIES
Rihaczek [l2] has classified radar signals into categories solely
by the type ambiguities they process. It is easy to imagine applications
in which a response of Figure 2.3(B) might introduce extreme problems and
the types of Figures 2.3(C) or 2.3(A) might be preferable. Range re
sponse (A) indicates poor resolution in close targets; (B) would give
spurious responses for targets which are at some distance from the posi
tion of main response; the extensive "background" response of (C) opens


The author proudly dedicates this dissertation to his parents,
Mr. and Mrs. Dana Brooks Mattox, and to his wife Debbie.


BIOGRAPHICAL SKETCH
Barry Gray Mattox was born April 16, 1950, in Pittsburgh,
Pennsylvania. He was brought up in South Carolina, where he gradu
ated from Irmo High School, Irmo, South Carolina, in 1967. He at
tended Clemson University and received a Bachelor of Science degree
in Electrical and Computer Engineering in May, 1971. In the fall
of 1971 he entered graduate work at the University of Florida on a
University Fellowship and received the Master of Engineering degree
in December, 1972. After interruption for two quarters for Active
Duty for Training as a Reserve Army Officer, the author continued
graduate study towards his doctorate, holding a research assistant-
ship in the Department of Electrical Engineering.
Mr. Mattox is married to the former Deborah McCullough.
He is a member of Tau Beta Pi, Phi Kappa Phi, and a student member
of the Institute of Electrical and Electronics Engineers. He holds
a First Class Radiotelephone license and an Amateur Extra Class
license as issued by the Federal Communications Commission,

222


133
FIGURE 9.4. COMPUTATIONAL FLOW CHART FOR PROGRAM TWO


130
(2) The input function consists of sample points inplying
a transform which is periodic. In the case of the
DFT, the transform is also made up of sample points.
(3) Using (1) and (2) we see that some aliasing of the
transform is inevitable.
E
The function to be transformed is the spectrum Z (Q) which
E
has been computed at equally spaced sample points. Since Z (ft)
is, by assumption, bandlimited, it is unnecessary to truncate the
function to accomplish a DFT. Along with this blessing, however,
comes the problem that it is impossible to adequately sample in
ft to prevent aliasing of the transform in the r domain. An example
of this problem is illustrated in Figure 9.3, showing true and
aliased transforms.
FIGURE 9.3 A RESPONSE AND ITS ALIASED COUNTERPART
The usual means of alleviating the problem of transforming finite-
record-length input data is windowing. But in our case windowing
would be entirely inappropriate because windowing is designed to
reduce sidelobes, the very phenomenon of our system in which we are


94
Equation (6.3-5B) yields ij>(t) when Z+ and w(t) are given. The solution
is given piecewise in N equations when there are N solutions a)c The
difficulty of the solution depends mainly upon the modulation and its
inverses. Modulations with some sort of symmetry are usually easier
to handle than others.
There is certainly no unique solution to (6.3-5B), N > 1, since
'l(t) for t = tn is solved in terms of (t)} for i ^ n. It is not hard
to visualize that for a section of the ft axis for which M solutions
o)ci ^ exist, ii>(t) for M 1 of these times may be chosen arbitrarily and
independently, thus constraining Examples of closed-form solution of the demodulation function
are given in Appendix A. Computer solutions are given in Chapter 9.
6.4 Solving for the Modulating Function
It should be very useful to solve for wc(t) so that one might
be able to design the modulation to give a desired response using a given
demodulation function.
We begin this solution first solving for the modulation func
tions with a single inverse. Referring to (6.2-12) or (6.3-5B), we see
that the demodulation or spectrum (respectively) are specified, usually
by "shape" and without much regard for multiplicative constants. Note
that in (6.2-14) and Z+ may not both be specified arbitrarily, since
their magnitudes must be reflected in aic(t);
'dt
vI 2n ^(t)
Wc(t)l = T Z+Uc(t)]
(6.4-1)
If i|j(t) were replaced by Ki|>(t), ()c(t) would be constrained to be Kuc(t).
But this would increase the bandwidth of the system to K B, which is


artificially transforming the problem to the time domain. The similarity
to windowing the power spectrum in the ideal case of filtering is noted.
Harmonic n-systems are examined with regards to optimum use of
the ensemble of harmonic information elements which form a vector space.
The concept is generalized to include any set of independent information
elements. Using n of these elements, a scheme is devised to constrain
any n points of the range response. Problems associated with this method
are investigated, including problems of behavior between constrained
points.
A general IF coherent detector system is investigated, and three
new functional relationships are derived involving the range response
transform, the modulation function, and the demodulation function. Under
the given assumptions the relationship is shown to be a more general
closed-form relation than other types which apply to more specific sys
tems. An inverted form of the relationship defines a demodulation func
tion to be used for obtaining BL range responses with a given FM modu
lation. The third relation is an integral equation for the modulation
function whose explicit solution may or may not be accessible in closed
form, depending upon the range response desired and the demodulation
function given.
Although research was aimed primarily at periodic systems, it be
comes apparent that stochastic systems may be analyzed or synthesized by
suitable re-derivation of the relationships. A generalization of IF sig
nal representation allows for the elimination of most constraints with a
resulting elevated complexity of solution. A chapter on directional dopp-
ler processors describes a two-channel processor which allows synthesis
of any desirable BL range response. Finally, computer solutions are de
veloped to solve those problems for which closed-form solution is incon
venient or impracticable.
xviii


75
INFORMATION
ELEMENTS
DEMODULATION
WAVEFORM, ifin(t)
ORTHOGONAL?
1. Time Samples
6(t-tn)
Yes
2. Trig. Coeff.
, 2imt 2irnt
sin T or cos T
Yes
3. Walsh Coeff.
wal (n see [26]
Yes
4. Squarewave Demod.
T
Squarewave of period
No
5. Gen. Four. Coeff.
n(t) (orthogonal set)
Yes
TABLE 5.1. SAMPLE DEMODULATION FUNCTIONS
Then
T/2
zn(-r) B h f e(t,T)ij;n(t)dt (5.1-1)
-T/2
All of the demodulating waveforms {tj;n} form orthogonal sets except for
case 4., for which {tyn} are squarewaves. Squarewaves are linearly inde
pendent but not orthogonal. This set was included because of the ap
pealing possibility of using choppers or switching inverters instead of
true analog multipliers. The squarewave functions (or any other linearly
independent set) may be added to form an orthogonal set using the Schmidt
orthogonalization process [26,p.11;8,p.458]. This is unnecessary however,
as orthogonality of the demodulating functions in time do not guarantee
orthogonality, or even independence, of {zn(i;)} in r. Moreover, the or
thogonalization process is a linear transformation which may be repre
sented in matrix form. If we call the transformation matrix 9, then we
first orthogonalize {if>n(t)} by
*orth.(t> 9 *(t) (5-1-2>


195
In addition, the possibility of inverse filtering presupposes that
the response Z(t) be available for various values of t, even when
the target is stationary. For some types of systems this availability
is not obvious.
The approach of Chapter V is to note the existence of a
large vector of information elements, such as the harmonic coefficients *
of the IF waveform. Many such vectors exist, but the independence
of information elements is restricted. Using a vector of N independent
elements, N points of the range response may be constrained. It
is shown that, given a suitable set of constraint points, the system
which produces the satisfying response involves a simple algebraic
combination of information elements, leading eventually to a reduction
of N channels to one channel. Unfortunately, no way was discovered
for choosing the proper constraint points so that behavior between
those points is desirable.
In Chapters VI and VII a fruitful relationship for range
response is derived in terms of the modulation and demodulation
functions. The relationship may be easily inverted to yield solution
of demodulation function or modulation function in many cases. A
"physical interpretation" of the relationship shows the similarity
to controlling the response by windowing the power spectrum. Normal
restrictions of a positive real power spectrum are not implied,
however, making the relationship very general. Although most of the
work was done on periodic systems, a statistical derivation of the
relationship is performed, showing complete applicability to stochastic
systems as well. Although the relationship in its simplest form
assumes small t, quazi-static targets, and high-dispersion modulations,


89
assuming t independent of time. This is valid when x is relatively small
with respect to T, as indicated in our assumption in Chapter I.
We now have a functional relation for the transform of the range
response in terms of the modulation and demodulation functions. The rela
tion may often be solved in closed form.
If the modulation is monotonic over the modulation period,
(6.2-12) becomes
z+(n) = -y ^[wc_1()] ^ .c"1 (.)
(6.2-14)
6.2.1 Physical Interpretation of the Relation
If \¡> is always non-negative, we may consider the demodulation
operation as windowing in the receiver. Any window that one might im
pose on the RF power spectrum using AM may be effected in the receiver
by using the window function (neglecting noise effects)
t|>[a>c-1(fi)] = w(n)
(6.2.1-1)
(A)
where W(ft) is the window function of the power spectrum,
or by
4>(t) = W[aic(t) ]
(B)
In the case of AM windowing the signal would be
(6.2.1-2)
The return signal, neglecting A(x), would be
(6.2.1-3)
where mc(t-x) ~ mc(t). This yields an IF signal


84
one [27,p.l4], and prediction and design, thus, straight forward. The
band limited property of the power spectrum often leads the designer to
a "standard" window. Implementation varies according to other factors
in the system, such as the waveshape of the frequency modulation.
Another subclass is that of the harmonic systems, whose response
is a "cut" of the ambiguity function. Thus, prediction of the response
is defined; but, as the shape of the ambiguity function varies with dif
ferent frequency ordinates, it has been unclear as to what types of modu
lation and demodulation will give a response which is defined as optimum
in some senseperhaps being the transform of a window functionfor a
displaced response. The general problem of designing a signal to have a
desired ambiguity function has not been solved, and such design usually
proceeds on a trial-and-error basis benefitted by experience [9]. Couch
and Johnson [2l] have developed an algorithm which approximates the FM
modulation waveshape using piecewise-linear sections and sums responses
to obtain a close approximation to the response of the single-harmonic
system. If the demodulating waveform is not sinusoidal, then it may be
represented as the sum of sinusoids (trigonometric series), and thus each
harmonic response may be superpositioned to obtain the response for the
general periodic system using any modulation/demodulation. Since the de
sired result is obtained on a trial-and-error or insight basis, however,
this algorithm is not the optimum design tool. Bartlett, Couch, and
Johnson [23] have developed formulas for prediction of a response in
series form, where, under proper assumptions, the series is composed of
weighted, displaced signal autocorrelation functions.
It would be ideal to have a simple relationship for the modula
tion, demodulation, and response. Key et al. have developed a relation


Response Magnitude
0.198
0.0
-16 0 Normalized Delay
FIGURE 9.21 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE USING HANNING WINDOW
16
165


CHAPTER VII
STATISTICAL DERIVATION OF THE RELATIONSHIPS
This section deals with the determination of a relationship
between modulation demodulation, and range response, made by
statistical methods and applicable to a class of signals which
include random processes.
7.1 Description of Signal and Reference
The system is that of Figure 7.1, with the constant amplitude
FM signal written without simplifying assumptions as
x(t) = cos[Kf/m(A)dX+)Qt] (7.1-1)
where m(t) may be a random process.
FIGURE 7.1 GENERAL SYSTEM DIAGRAM
97


28
Since A must be arbitrary
/{|Uo(f)|2-|Uo(cO|2-|uo(a)|4}dcx = 0
for | < f < | (2.1.3.1-9)
or
B/2
/ | 0 (a) 12C| u0 (a) 12~ I u0 (f) 12]da = 0
-B/2
- f < f < f (2.1.3.1-10)
which has a solution at
K(a)
i B . B
1 2 a 2
0 otherwise
(2.1.3.1-11)
We have shown, then, that a rectangular spectrum satisfies the condition
for minimizing time ambiguity defined in T^.
Tozzi has found that, when processing individual lines of the IF
spectrum, linear modulationand thus a flat spectrum under the high-
dispersion assumptionprovides best resolution and the least spread dis
tribution of power in the various other harmonic lines of the IF spec
trum [7].
Intuitively speaking, the wider the bandwidth or occupied sec
tions of the spectrum, the better the target resolution. Equal weighting
of each section of spectrum occupied has been shown to be in the best
interest of total ambiguity (in terms of T).
Another way of regarding the BL spectrum is to consider that the
rectangular spectrum is the most "versatile" transmitted spectrum which
may be supported over the interval (fq B/2, fo + B/2) in the sense that
it may be modified to be of practically any other form by the receiver
using appropriate filters [6,9]. Any departure from the uniform spectrum,


Page
6.3. Solving for the Demodulating Function 92
6.4. Solving for the Modulating Function 94
VII. STATISTICAL DERIVATION OF THE RELATIONSHIPS 97
7.1. Description of Signal and Reference 97
7.2. The Assumption of Ergodicity and Notes on Averaging 100
7.3. Transformation to a Convenient Argument Space
Before Averaging 103
7.4. Statistically Derived Range Response Transform 105
7.4.1. The Periodic Case 106
7.4.2. Range Response Transform Statistically
Derived for Multi-dimensional Reference 107
7.5. A Convenient Graphical Method 108
VIII. DIRECTIONAL DOPPLER PROCESSORS 112
8.1. SSB Directional Doppler Techniques at RF 112
8.2. SSB Directional Doppler Techniques at IF 113
8.3. A More General Directional Doppler Processor 116
8.3.1. Analysis of the SSB System in the 0-Domain 117
8.3.2. The Form of the General Two-Channel
Processor 118
IX. COMPUTER ALGORITHMS 123
9.1. Program One Solution of the Range Response 124
9.1.1. Program Flow 125
9.1.2. Precautions and Assumptions 129
9.2. Program Two Solution of the Demodulation Function 131
9.2.1. Program Flow ..... 132
9.2.2. Precautions and Assumptions 135
9.3. Program Three Solution of the Monotonic
Modulation 140
9.3.1. Program Flow 144
9.3.2. Precautions and Assumptions * 144
9.4. Examples of Computer Solutions 147
X. CONCLUSIONS 194
vi


Fourier transforms or the solution of the nonlinear integral equation
can be tedious if not impossible in closed form for many given
functions. Finally, closed-form or series-form solutions might be
readily attainable, but may give little design insight whereas a
plot of the desired waveform may be adequate for designing hardware
to generate an approximation to that waveform.
9.1 Program One Solution of the Range Response
The first algorithm implements a numerical solution of
-1.
ze ; l *t["k "l(n)l-
i=l i
(a)
(9.1-1)
where <|> may be complex,
*t = ^ + ^
(9.1-2)
and the processor is that of Figure 8.2 using complex demodulation
m (t) is the modulation function made up of N monotonic sections
t m r
and {w } are the set of N inverse functions as defined by (6.2-8A)
m.
i
The result of equation (9.1-1) is the complex envelope of the range
, e i E
response: |z (t)| is the magnitude of the response and /z (x) is the
doppler phase of the response where the actual response is
F jm t
z(t) = Re {z (x)e } (9.1-3)
(A)
= |zE(x) | cos(moT+ZzE(x)) (B)
where ojqt = w^t and is the doppler radian frequency.
Notice that m is not involved in our computer solution, which yields
o
i E i E
|z (x)I and Zz (t). The input data to the program are (1) the


H<¡P WH
6
(A) ENVELOPE DETECTION SYSTEM
(B) MULTIPLIER SYSTEM EMPLOYING TWO
ANTENNAS
FIGURE 1.2 SYSTEM DIAGRAMS


11
Then
N t
e2(t,x) = I An(Tn)cos()0Tn+Kf / m(X)dX)
n=l t-xn
where x^ is the vector [xj, X2, xn]^
and x_ indicates the transpose of x_
and, as before,
N
e3(tI.) = I An(xn)cos[wc(t)xn]
n=l
N
= l e3(t,xn)
n=l
(1.4.3-3)
(1.4.3-4)
(A)
(B)
Clearly the response of all targets appear superpositioned at the output
of the envelope/product detector. The system remains linear by definition
until the non-linear (NL) processing block (Figure 1.2).
1.4.4 The IF or Beat Waveform
A great deal of signal processing has already occurred in the
envelope detector (or multiplier/LP filter) to yield a signal e which has
a bandwidth on the order of 2/T for many types of modulation [7] from a
signal x(t), having a bandwidth of B, which,by assumption 2, must be much
greater than 1/T. For both envelope and multiplier systems the infor
mation signal e is a result of producting the signal and its return and
is sometimes called the "beat" waveform. The frequency components of
e3(t,x) depend not only on o)c(t) but on the value of x, power being con
strained to spectral lines generally clustered about Bx/T for sawtooth
modulation. Figure 1.3 shows two sample spectra. Both represent the
transform of e3(t,x) of a linearly modulated DMS with bandwidth B and
period T. Note that, when the target is assumed stationary, the spectrum
consists of lines or delta functions which have areas as outlined by the


3
by additive noise considerations) are monotonically increasing functions
of signal energy [l]. Regarding requirement A6, we would like to maxi
mize energy by maximizing both the magnitude of x and the duration of x
over (-T/2, T/2). Since the signal is periodic, its average power may
be expressed as
1 T/2 g_
Pave. "V f |x(t) | dt (1.2-3)
T -T/2 L
If peak envelope power is a limitation, we may design the signal envelope
to be constant at that peak power; thus, for maximum energy, the duty
cycle is increased to 100% so that
Ex = T PaVe. = ^ Ppeak env. (1*2-4)
The DMS is to have good resolution in distance (requirement A3).
Inherent signal resolution is dependent upon bandwidth; to achieve high
resolution, a power spectrum of large bandwidth is necessary [l,2,3]. If
the signal envelope is constant, bandwidth must be achieved by frequency
modulation (requirements B1 and B4) The requirement of large bandwidth,
together with requirement A2 or B5 (A2 and B5 are equivalent requirements),
indicates high-index modulation (B4). When the modulating waveshape is
sinusoidal, the modulation index is defined as
(1.2-5)
where B is the instantaneous peak-to-peak frequency deviation in Hz,
and fm is the frequency of the sinusoidal modulation.
Applying Carsons rule for FM bandwidth [4] we see that
BW = 2fm(l+y) 2fmy = B for v 1 (1.2-6)
Furthermore, for "high-index" modulations the shape of the power spectrum
will be that of the probability density function of the FM modulation


136
type of input response. Since the problem has been discussed,
we shall not re-iterate the statements of section 9.1.2.
In Program Two, an additional degree of freedom exists
in choosing the way in which to constrain the range response to
be bandlimited. If the specified response is originally bandlimited
to less than or equal to B, and the window is specified as ree
ls
tangular, Z (fi) will be the transform of the response specified.
Otherwise, the response will be that of the specified response
convolved with the window transform.
We see then that we must make a windowing decision in order
to insure the best type of sidelobe responses. Furthermore, one
window will not necessarily be best for use in approximating a
given input function, even if some definition of "best" or "optimum"
is imposed. For example, if we desire a response with the sidelobe
characteristics which result from a raised-cosine (Hamming) spectrum,
and the ideal response is given as a short pulse (approximating a
delta function), a Hamming window would be appropriate; but if the
input response has already been described as the transform of a
raised-cosine function, a Hamming window would not be appropriate,
but, instead, the rectangular window sould be used. In most cases,
the choice is not so obvious, and it may be desirable to compute
responses using different windows and then to compare the results.
Again, we shall not attempt to discuss or define "optimum" or
"desirable" since this will vary with application.
Originally the program was written to (1) compute the wideband
transform of the specified range response using a high sampling rate
(large number of DFT points) to reduce aliasing to a negligible


29
especially in the way of zeros or unoccupied sections, reduces the ease
with which this may be accomplished.
2.1.3.2 The frequency resolution constant
The dual of the time resolution constant is the frequency reso
lution constant:
i 2 .
where
fa
/ |k(*> I d
-00
K(0)2
/|u(t)|4dt
00 ,
/ |u(t)|2dt
K() = / U*(f)U(f+4>)df
(2.1.3.2-1)
(A)
(B)
(2.1.3.2-2)
and
U(f) = F[u(t)]
(2.1.3.2-3)
Of course, it can be shown that to minimize Fa, the envelope of the
transmitted signal should be constant over the duration of the signal.
Other properties of Fa are duals of properties of Ta* The frequency res
olution constant will not be of extreme interest in this work because of
the assumption of quazi-static targets.
2.1.3.3 The ambiguity function
We may get a good grasp of the ambiguity function by following
its derivation. The ability to resolve or distinguish between two sig
nals may be monotonically measured by their integral squared difference:
ISE = f |xj(t)x2(t)|2dt
(2.1.3.3-1)
(A)


CHAPTER II
RESOLUTION AND SHAPING OF THE RANGE RESPONSE
The subjects of resolution and shaping of the range response
have been handled, not only in the field of radar, but also as functional
design in such fields as data communications and computer transform "win
dowing." This-chapter will introduce the basic ideas of resolution and
the ambiguity function as they apply to the problem and will briefly
examine some of the popular windowing techniques.
2.1 The Resolution Problem
We begin by a brief discourse on the qualities of a radar system
relating to "accuracy," "ambiguity," and "resolution."
2.1.1 Accuracy
The accuracy of the system depends on the range response as well
as the signal strength and noise power. The accuracy for a given signal
energy and noise power is dependent on how peaked the output response is.
This response peak is maximized relative to the noise power when the "re
ceiver" is matched to the signal. The solid curves of Figure 2.1 repre
sent measured range responses designed to peak at a delay tq; ideal
noiseless responses are shown in dotted curves. We see that the wider
response of (A) system leaves opportunity for more error due to noise
than the narrower response of (B) system as shown by the measured ranges
ti in each diagram. It will be shown in a later section that the shape
of the matched filter response is that of an "ambiguity function."
21


108
Z (ti) = 2it(ti,x)p (ft,x)dx ti > 0
+ oi >x to ,x
c c
= 2ir(Q,x)p (x)dx p (ft)
i) X X/ CO u)
c c c
= 2ir E {i|i .x^^ }pw (ft)
c c
= 2ml> (ft) p (fi)
CO CO
c c
Equation (7.4.2-2D) is important because it indicates that, as long
as the assumption of small t holds, we need only know the p.d.f.
of the modulation and the "average" demodulation as a function of
instantaneous frequency (or modulation voltage). Both of these
functions may be determined empirically if not easily available
from a math model.
7.5 A Convenient Graphical Method
For the most part, we have considered deterministic systems
in a deterministic manner. This is oftentimes highly desirable and
avoids the many drawbacks associated with stochastic systems.
However, in our case, we often cause ourselves considerable difficulty
by treating signals and references as deterministic instead of
stochastic as in section 7.2. The stochastic representation tells
us much less about the process than does a deterministic description.
For many purposes, though, this is both sufficient and advantageous
in approach [29]. For example, compare relation (7.4-3C) with
relation (6.2-12). Practically speaking, it is usually a simple
matter to measure p.d.f. while it may be difficult to describe the
deterministic function, determine inverse functions, and find
(7.4.2-2)
(A)
(B)
(C)
(D)
derivatives.


60
with the introduction of quadradic phase. This fundamental appears to be
the basis of obtaining displaced responses without the use of delay lines.
Figure 3.5 displays this sheering behavior for a linear-FM (quadratic
phase in the time domain) signal. In contrast to the linear systems the
range response of a sinusoidally modulated harmonic system is the signal
FIGURE 3.5 AMBIGUITY FUNCTION (MAGNITUDE)
FOR LINEAR FM SIGNAL
ambiguity function at

so simply related to the zero response (which is a zero-order Bessel
function).
In a later chapter, a relation will be developed which will pre
dict the displaced range response, regardless of modulation non-linear
ities, provided they are known.


Response Magnitude
0.643
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.13 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION, HALF-COSINE DEMODULATION
157


76
In the next section, we will show that this represents a linear trans
formation of
4rth.=0 S'1
Then if we transform using H' (derived from z^th.) we obtain
(5.1-4)
so that we are still performing one transformation by H, determined as
H = G 1 = [z/t!)*,z(tn)] 1
(5.1-5)
Thus we perceive no generalizable advantage of orthogonal sets over non-
orthogonal ones.
5.2 Equivalent Single-Channel System
Up to this point we have considered a linear combination, de
fined by H, of information elements, each element derived from a separate
subsystem as in Figure 1.4. We shall now observe that it is easy to ex
press the entire system in the form of Figure 1.4 when a single output
element is desired
n=l
(5.2-1)
(A)
N T/2
l hmn t f e(t>T) I'nCOdt
n=l -T/2
(B)
b f e(t,-r)^(t)dt
-T/2
(C)
where
^m(t) l bmnJ'nOO
n=l
(D)


146
Program Three also. In addition, we must consider satisfaction of
(9.3-5B) a pre-requisite to using Program Three. If this assumption
is violated, ^ is non-positive, and an indication of t, (t) ,
E dm
to Z (a) ) and is recorded on the output record for that
mm dt
iteration of the integration routine. In this way, execution continues,
and the user may judge from the output how far the system deviates
from satisfying the requirement.
One special case in which (9.3-4) may easily be met is the
£
case such that z (x) is always a realizable autocorrelation function
£
so that Z (q) is always non-negative. K^(t) is simply specified to
remain positive. Within these restrictions any response shape may
be specified and a modulation obtained to yield such response, which
must necessarily peak at zero (a property of realizable autocorrelation
functions). However, the response may be shifted to peak at any
desired delay by specifying the response to be
zE(x) = z E(x-x ) (9.3.2-1)
a o
where t is the desired shift
o
E
and Re {z (x)} is a realizable autocorrelation response.
In the frequency domain this response becomes
p ~ j fix -r*
z (fl) = e Z (J) (9.3.2-2)
a
where Z E(fl) = F {z E(x)}
a a
From (9.2-1) it is easy to see that the necessary demodulation for



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86
oo x/2
Z.(fl) / £ / ej,cVt)dt e"jftTdx (6.2-5)
_oo 1 -t/2
where we are omitting the argument of ic(t) for convenience.
Since t is assumed to be independent of time for purposes of analysis
at this point, we may write
Let
T/2 00
Z, (ft) = -^ / ip(t) / e^UcTe ^dxdt
1 -T/2
T/2
= h f ^(t)2ir[-)c(t)]dt .
-T/2
y = wc(t)
t = u)c ^(y) = wc"1[wc(t)] .
(6.2-6)
(A)
(B)
(6.2-7)
(A)
(B)
-1.
If a)c (y) has more than one solution then we write the set of solutions
{)c 1(y)}, and a transformation of variables must be made for each
section of time that a separate solution exists. That is, the time axis
will be segmented into sections corresponding to each of the solutions.
Each solution t = w ^(y) will be said to exist on the interval (tmin,
timax) See Figure 6.1. Then
A -1, *
fci = wCi (y)
_i -i t
dti = a)cl (y)dy where ci (y)
(6.2-8)
(A)
(B)


ooooooooooaooooooooooooooo
209
**************************************************************************
* PROGRAM THREE SOLUTION OF THE MODULATION FUNCTION *
* DATA FORMAT FOR THIS PROGRAM IS AS FOLLOWS
*
* *
* WINDOW TYPE, NUMBER OF B-TAUS, NO. OF PTS. PER B-TAU (313) *
* (O-RECTANGULAR, 1-HAMMING, 2-HANNIIIG) *
* RANGE RESPONSE MAGNITUDE "INPUT" FORMAT *
* RANGE RESPONSE ANGLE "INPUT" FORMAT *
* DEMODULATION #1 "INPUT" FORMAT *
* (MAY INCLUDE ADDITIONAL SETS OF R.R. MAG./R.R. ANGLE/DEM. t1) *
* BLANK CARD (INDICATES END OF THE DATA) *
* *
* "INPUT" FORMAT: *
* *
* NUMBER OF DATA PAIRS, ALPHANUMERIC DATA FORMAT (13,77A1) *
* (IF THE FIRST FIELD CONTAINS A NEGATIVE INTEGER, *
* ADDITIONAL CARDS OF THIS FORMAT WILL BE ACCEPTED. *
* OTHERWISE, THE FIELD DESCRIBES THE NUMBER OF DATA *
* PAIRS TO FOLLOW.) *
* ORDINATE, ABSCISSA (ONE CARD PER PAIR) (2F3.3) *
* *
**************************************************************************
REAL MOD(101),TMOD(101)
REAL*8 ALPHA(G)
DIMENSION RL(101),ARL(lOl),TARL(101),T(1024),Y(1024),YP(1024),
*YA(1024),IMIN(16), II1AX( 16), NA( 3 ), I N( 2048), S( 2048) ,
*DM0D1(101),TDMOD(101),
*Z(1024), ZWT( 1024 ),D( 1024),TRL( 101),HZ(3),R(409G),ABSC( 1025)
EXTERNAL FUN
READ(5,102) HWIND,N8T,NPTBT
102 FORMAT(3l3)
NB=2*NBT
NFFT=NBT*NPTBT
H=lE-3
NA( 1) = 3.32192 8095*ALOG10(NFFT-0. 0) + 0.1
HFFT= 2**NA(1)
N=NFFT-1
NA(2) = 0
NA(3)=0
C INPUT THE RANGE RESPONSE MAGNITUDE AND ANGLE
80 CALL IHPUT(RL,TRL,NRL)
IF (NRL.EQ.O)' STOP
CALL I NPUT(ARL, TARL, HARD
C CREATE THE VECTOR OF EQUALLY-SPACED TIME VALUES
DEL=1./N
DO 1 J=l,N
1 T(J) 0.5 + (d-0.5)*DEL
C INTERPOLATE VALUES OF RL AND ARL FOR THE VECTOR OF TIMES
CALL I NTRP(RL, TRL, 1, NRL, 2, Y, T, 1, II, YP,H)
CALL I NTRP( ARL, TARL, 1, NARL, 2, YA, T, 1, N, YP, II)
CALL MATPLT(Y,ALPHA<2),N,1,N,1)
CALL MATPLT(YA,ALPHA(3),N,1,N/l)
C CONVERT THE MAGNITUDE AND ANGLE TO REAL AND IHAG. PARTS OF VECTOR R
KF1=NFFT+1
KF2=NFFT+2
KK*=N/2
KL=KK+1
DO 2 J=l,KK
I =2* J
K=KL+J
R(I+l)=Y(K)*COS(YA(K))
R(l+2)=Y(K)*SIN(YA(K))
R(KF1*I)=Y(J)*COS(YA(J))
2 RKF2+I)=Y(J)*SIU(YA(J))
R(1)-Y(KL)*C0S(YA(KL))
R(2)*Y(KL)*SIN(YA(KL))


70
Then the parallel convolution construction of Figure 4.2 is applicable
to harmonic n-systems with the restriction that
AX = i (4.1.4-2)
which is the maximum spacing which allows implementation of any window
function. This is the "spacing" inherent in the range responses of lin
ear harmonic systems operating on adjacent harmonics, n and n+1.
When the modulation of a harmonic system is not linear, the re
sponse is given by equation (3.3-12A), but equation (3.3-12B) is no
longer valid. It is certainly not clear that a convolution of z(x) with
h(x) may be made in this case. The same uncertainty must, of course, be
true of a more general coherent system.
4.2 Summary and Conclusions about Inverse Filtering
It was shown that inverse filtering in its ideal form de-con-
volves a signal to yield a delta function output. The approach must be
modified in the case of a bandlimited signal to yield an output which is
similarly bandlimited, such as the transform of a window function. When
filtering in the delay domain, the problem becomes one of obtaining z(x)
for varying values of x when the target is stationary. For the autocor
relation system with an RF delay line and for the delay-line IF correla
tor system, the problem was solved by using an approximating point-by
point convolution using parallel implementation. Alternately, by using
a variable delay and a scaled time-domain filter, it is possible to ob
tain a time-domain inverse filter which simulates the convolution in x.
The delay must be swept slowly, and sampled once per sweep, imparting a
slow rate of information output to the system, and placing more stringent
requirements of stationarity on the target.


42
wx (t) = y w0 (t) + J W0 (t + ^ Wq (t |-)
(2.2.3-6)
In contrast to the -14dB maximum sidelobe of wgCt), sidelobes .of (t)
are down by at least 30dB.
The Hanning window is a variation on the Hanning window which
yields lower (-40dB) maximum sidelobe levels. (Sidelobe levels do not
fall off as fast, however, as do those of the Hanning window.) This
window is given by
W2-((o)
.1-
54 + .46 cos
B
itB < to < ttB
otherwise
(2.2.3-7)
having, of course, a transform of
w2(t) = .54w0(t) + .23w0(t + j) + .23w0(t ~) (2.2.3-8)
While having lower sidelobes, windows W^ and W2 have the effect of wid
ening the mainlobe compared with window Wo The distance between the
first nulls of wi (t) or w2(t) is double that of wo(t).
The Dolph-Tchebycheff weighting [15] yields an optimum transform
in the sense that sidelobes are reduced with a minimum broadening of the
mainlobe. In fact, one is able to specify sidelobe levels, which, in
turn, determine the mainlobe width. The Dolph-Tchebycheff window trans
form assumes the form
<*> <2-2-3-9>
The sidelobe level for this window is a uniform [cosh irA] \ which makes
the window not very interesting for most purposes. As a signal, W3(t)
contains infinite energy.
A more applicable approximation of W3 has been developed by
Taylor [6]:


115
After SSB processing to recover only the upper (lower) sidebands
about the nth harmonic, the output becomes [21]
y(t,x) =<
3C -S|ZL + u,)t
Re[cn e ajd > 0
J (t + w,)t
Re[c- e ] a),- < 0
+n d
(8.2-2)
where the upper (lower) set of signs is used for upper (lower) side
band processing. The range response envelope then will be
Z*(T) = {
C J > 0
+n d
(8.2-3)
c a), < 0
;n d
. T
That is, for a target moving in one direction c (x) describes the
range response while cn(T) describes the response to the target
moving in the opposite direction. Reference [21] also shows that
cn(x) is a cut of the ambiguity at f = |:
c (x) = X(-T, ^) (8.2-4)
n 1
It is easy to show that, if the (periodic) modulation is even
(i.e., there exists some value a such that m (a+t)=w (a-t) for all
m m
t), then the magnitude of the ambiguity function is even in x :
|X(x,)| = |X(r,) | (even modulation) (8.2-5)
Because of the Hermetian symmetry of the ambiguity function with
respect to the origin (property 1, section 2.1.3.3),


101
the product of the IF signal and the IF reference. The system is
assumed to be an ergodic one in which time averages equal ensemble
averages. The time average of waveform y(t) is denoted by
Et {y(t)} = ,
A
(7.2-1)
or, for y(t) periodic with period T,
T ,
pr + a
iy(t)} = / y(X)dA a arbitrary (real). (7.2-2)
T ,
-~2 + a
Time is measured frcm an arbitrarily defined origin. That is,
measurement time t may be related to "true" time by
t = t^ -t
true o
(7.2-3)
where t represents the origin of time t in terms of true time.
When dealing with periodic systems, it is desirable to treat t as
a uniformly distributed random variable so that, for any function of
time y(t), the first-order p.d.f. of y(t) may be determined from
y = y(t).
(7.2-4)
That is, the p.d.f. of y may be found by standard techniques applicable
to an instantaneous transformation of variables [19, p.34]. Since
t may be considered to be uniformly distributed over a large interval,
and y is periodic in t, we might effectively consider time to be a
uniformly distributed random variable with a p.d.f. of
Pt(t)
1
T 5
2 < t <
T
2
(7.2-5)


14
I
*?>
h(t)
z(t )
5
FIGURE 1.4 LINEAR PROCESSOR BLOCK
a band-pass or a low-pass filter or integrator. When the target is not
moving, z is a function of x only; but if the target is slowly-moving,
h(t) must permit variations that occur in z(x) as x changes in time. The
bandwidth of h(t), then, depends primarily on the rates of change in x
(secondarily, of course, on^-z(x)).
1.4.6 The Non-linear Section
The last part of the system will generally be non-linear in some
respect because a judgement or decision (which must be discrete choices)
will be made by the electronics or by a human observer (very non-linear).
This block may contain items such as rectifiers, squarers, and comparators.


65
One may specify one of the parameters to obtain an "optimum" in terms of
the performance measure of the other parameter.
A.1 Application to the x Domain
The concept of inverse filtering applies to the FM periodic radar
in the filtering of range response z(x) in the t or delay domain. The
desired response zout(x) corresponds to the output pulse, g(t), in the
previous section, while z(t) corresponds to x(t) and the desired filter
has impulse response h(x). The obvious difference in the real-time anal
ogy and filtering in the delay domain is that z(x) is not available as a
function of time, t. Our system may have to make decisions about the
value of x while x is completely static.
The question arises as to what real, physical form such a filter
would take. The filtering operation is given by the convolution integral
00
zQUt(x) = f h(T-X)z(X)dX (A.1-1)
(A)
00
= f h(X)z(x-X)dX (B)
CO
We see by (A.1-1) that the value of zQUt(x) depends upon values of z(X)
for all arguments except those for which h(x-X) =0. We shall now exam
ine each system type for devices by which we might obtain the convolution
of (A.1-1) even though x is fixed.
A.1.1 The Autocorrelation System
In the autocorrelation system, the output variable zq(x) = R(x)
is fixed 'for a fixed value of x; so is e(t,x) for given x. We must
assume, then, that we cannot, without some modification to the basic sys
tem, obtain a new variable zout(x) conforming to (A.1-1).


C(t)
SECOND-CHANNEL DEMODULATION IS ZERO
FIGURE 9.22
FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.21 FOR SAWTOOTH MODULATION
166


81
and
H = G
-1
I
(5.4-2)
z = Hz = z
out
(5.4-3)
Here, {t} were chosen so that the transformation H is completely inef
fective, and the range response is not improved. Other sets {t} would
constrain other points to be zero in each response, and at least a dif
ferent response would be obtained. It thus appears that {t^} will have
to be "optimized." However, the manner in which one might optimize the
set is illusive. One method used to attempt such an optimization was a
recursive gradient algorithm, implemented in APL. Results were not sat
isfying, however; the algorithm failed to converge to reasonable values
of t. Part of the problem may have been the use of a NL "measure" of
sidelobe levels. That is, we used as a measure the highest of the side-
lobe magnitudes at designated "test points" in t. Since a gradient tech
nique will converge to any minimum, including local minimums, there is
no guarantee of ever finding the absolute minimum. There is no reason
to believe that many local minima do not exist as one progresses through
the Hilbert space that describes {t}.
Finally, by equation (5.3.2-2D) we know that the range response,
zoutm(T) roust be bandlimited. The uncompromising practice of con
straining individual points to be nulls in the response does not appear
consistent with most signal design methods. Perhaps a more profitable
course to pursue is that of confining the range response to be that of a
window transform, in which case we would require
H'G = R
(5.4-4)


147
the system must be
~jw(t)T
Jt) = e (t) (9.3.2-3)
C 3t
E
where \p and Z satisfy (9.3-4) in the solution of to (t).
a a m
9.4 Examples of Computer Solutions
This section includes figures depicting plot outputs from
the three programs and comments to clarify concepts that have been
presented in previous sections. Figures 9.10 and 9.17 show the
range responses computed by Program One. The specified input responses,
obtainable bandlimited responses, and necessary first- and second-
channel demodulations from Program Two are given by Figures 9.18
through 9.41. Computed modulation functions of Program Three are
given along with the real parts of the spectra from which they were
computed by Figures 9.42 through 9.49. (The spectra are not
ordinarily outputted from the program.) In the figures the top
boundary of the shaded area indicates the value of the function.'
Figure 9.10 gives the magnitude and phase plots of the
range response envelope computed for a system using triangular
modulation and single-channel DC demodulation. The system is an
autocorrelation system, the response being a realizable autocorrelation
function which peaks at zero. Since there is no second channel,
E
Z (ti) is real and the response is thus even with an odd phase response.
The peculiar-looking phase response is due to a slight variation in
computed angle for 180 phase: the phase switches from nearly +tt
to nearly -ir radians with no significance. This and other phase
responses are not of particular interest to the system designer


93
Z+^(ft) is the part of Z+ contributed during the ith time segment (see Fig
ure 6.1). These segments in time do not overlap, but the intervals in ft
over which the solutions o)ci ^ exist and contribute do overlap. For ex
ample, the contributions to Z+ for ft in the interval (mc(t7max) ,mP (t9nvf n))
in Figure 6.1 come from times t2, parts of ti and parts of t3.'
Letting
ft = aic(t^) dft = da)c(t^),
tf = coci 1(ft), dti = dmc(ti),
z+iE^Ct)]
dtj
dmc(tj[)
2 IT ^Etj]
T | ' dti
27r ^C**ci_1 <**c Ct) ) ]
T l[¡r wc[mci-1(wc(t))]|
(6.3-3)
(A)
(B)
Then
Z+(ft) = Z+n + l Z+1(ft)
i^n
Z+n = Z+ £ Z+i(ft)
i^n
(6.3-4)
(A)
(B)
Thus, for tnmin < t < t
nmax
2tt
idF <:<*>
z+[mc(t)] £
i^[coci 1(mc(t))]
i^n (o[oc (u>c(t))]
(6.3-5)
(A)
or
I|>[uci (mn(t))J
z+[uc(t)3- J ^
i^n |p)c[wci (o)c(t))]j
tnmin< t< tnmax
(B)
Notice that, for tnmin
tnmin < t < tnmax, mci"1[a3c(t)] 4 t for tfn..


137
*
W
r^]7
A
Iii,
(A) ORIGINAL RESPONSE
TRANSFORM
i/f-
(B) WINDOW TO OBTAIN
BL TRANSFORM
(C)INVERSE TRANSFORM
FOR BL RESPONSE
(D)COMPRESS BY LOWER
SAMPLING RATE
S
window
T
(E)WINDOW TO REDUCE
RINGING
_L
KT
FIGURE 9.5 INCREASING THE NUMBER OF SPECTRAL POINTS


138
effect; (2) window the transform to bandwidth B, thus confining the
spectrum to a smaller number (2*NBT in the program) of DFT points;
R I
(3)use these remaining points to compute \p and tp However,
using linear interpolation in computation of the demodulation
function, this number of points proved insufficient for good
representation of the required demodulation. There are two ways to
overcome this problem. One is to use a different type of inter
polation scheme or to convolve the data with a digital filter response
to obtain the necessary smoothing. Other researchers may wish to
examine these possibilities. Another method is to increase the number
of data points by the method indicated pictorially by Figure 9.5
and described below:
(1) Recognize that the range response (Figure 9.5(C)),
which is the transform of the windowed spectrum
(Figure 9.5(B)), is oversampled by a factor of
K=W/B. (This assumes that the original response
was adequately sampled and no aliasing has occurred.
We know that this can be only approximately true.)
(2) Sample the range response at one-Kth the original
rate, compressing the time (delay) axis by a factor
of K. Add zeros (Figure 9.5(D)).
(3) Realize that the response must be aliased to some
degree (a bandlimited function cannot be strictly
time-limited) and that there exists the problem
of transforming a truncated (perhaps somewhat
aliased) function.
(4) Window the compressed function to reduce the effects
of transforming a chopped-off, finite-record-length,
function. (We have used a 30% Hanning window.)
(5) Transform to obtain a spectrum which is expanded in
frequency, containing K times as many points as the
original bandlimited function (Figure 9.5(F)).


1X3
and a summing circuit, the system becomes a phasing-type SSB receiver
[21], and a doppler output will be greater for one direction (the
direction depending on whether the channels are added or subtracted,
i.e., depending on which sideband is detected). A filter-type SSB
processor (assuming feasibility at RF) would provide equivalent
results. Figure 8.1 illustrates the signal spectrum and the
phasing-type (Kalmus) processor employed for detection.
8.2 SSB Directional Doppler Techniques at IF
The IF signal in a periodic system is composed of the various
harmonics of the basic modulation frequency ^ (or the doppler sidebands
of those harmonics). Couch and Johnson [21] have shown that a harmonic
SSB processor operating about the nth harmonic of the IF signal will
produce a range response which, under certain conditions, will produce
directional doppler information. Also, Bartlett et al. [23] describe
an IF-correlator directional doppler processor in terms of references
which are sums of waveforms matched to signals at integer multiples
of a delay time tq. Their results are very general and, with
simplifying assumptions, yield range responses in terms of sums of
displaced autocorrelation functions.
The periodic IF signal is modeled as
CO
(8.2-1)
(A)
00
Re c, e
k
(B)
where x is time varying such that oi^t = )qt
Tj is the doppler radian frequency
and c, is a function of x.
k


109
One sees the problem most naturally expressed as in (7.4-3C);
other representations such as (6.2-12) compute the factors of
(7.4-3C). Since ^ is perhaps the most revealing form of the
to
m
demodulation we note here a simple way of obtaining this form.
(A) SETUP
m
to
m
(B) MAPPING
m(t)
FIGURE 7.3 OSCILLOSCOPE CONNECTION FOR DISPLAY OF ip
to
If an oscilloscope is connected as shown in Figure 7.3(A), the
function ip (to) will be displayed; the function is, by definition,
0)
m
i{i versus to (The modulation voltage m may be used on the
horizontal axis if the modulator has a linear voltage-to-frequency
characteristic.) When the modulation is not monotonic, the demodulation
may be a function of variables other than to and the trace will contain
m
the functions 4) (to) for the entire range of x- A graphical
m
technique may be used to map the demodulation' as a function of time,
through the inverse modulation function(s), into the function ip
m
as shown in Figure 7.4. Obviously, if the modulation is linear,
and ip will have the same form. If the demodulation is generated
m
by a delay-line reference generator, it will map into a sinusoidal
ip regardless of the modulation functionsee [29] When the
(0


50
Amplitude modulation is often undesirable because of the necessity for a
modulator section (expensive and bulky for high-power systems) and be
cause of a reduced average power when maximum peak power is fixed. Fil
tering also reduces average power of the signal, and if the center fre
quency of the transmitter is altered from that for which the filter was
designed, the shaping of the spectrum will be other than desired. Thus,
filters require extra stability measures on the oscillator. If non
linear FM is used, a high average power is preserved; the power is dis
tributed according to the desired window through selection of the suit
able modulation, m(t).
Shaping may also be accomplished in the "receiver" (if the sys
tem is so separablei.e., the system must have separate antennas) by
filtering methods similar to 2. above. Obviously, the system is no
longer "matched" and the signal-to-noise will suffer accordingly. How
ever it has been shown [6] that degradation is slight (on the order of
a dB) for common windows.
3.2 Delay-line IF Correlator Systems
The delay-line IF correlator system produces translated range
responses and uses an RF delay line but does not require two antennas.
The block diagram is given in Figure 3.3. The "first mixer" consists of
the envelope detector which has the effect of a multiplier (see Chapter
I); both first and second "mixers" inherently suppress double-frequency
terms and successive LP filters are not separate physical components.
Consistent with our assumptions we may express the IF signal as
e3(t,t) = cos[m (t)x]. neglecting amplitude factors. (3.2-1)


FIGURE 1.5 INSTANTANEOUS FREQUENCIES AND BEAT PATTERN
N3|H


140
FIGURE 9.6 COMPARISON OF ALIASED HANNING AND RECTANGULAR TRANSFORMS
9.3 Program Three Solution of the Monotonic Modulation
For reasons mentioned in section 6.4 we restrict solution
to that of a monotonic (increasing) modulation function by numerically
solving the relation
db)
m
dt
*t ZE(o) )
m
(9.3-1)
(A)
where
is the given demodulation ,
ZE(fi) F (zE(t)},
and K is a real constant, to be determined.


5
process the return signal over as long a time as practical since the en
ergy per decision is proportional to the time per decision; target move
ment and a minimum rate of decisions are limiting factors.
1.3 Assumptions
Throughout most of the dissertation three assumptions will be
adhered to in the analysis:
1. The target return times x (signal propagation
times to and from the target) are small com
pared to the periods of major modulation com
ponents.
2. The modulation index or dispersion factor is
large.
3. The target will be assumed quazi-stationary,
i.e., almost static over a modulation period T.
The assumptions are, of course, supported byand are re-statements of
system requirements (A2 and B5, B4, and A4, respectively).
1.4 Models
The system to be studied is illustrated in block diagram form
in Figure 1.2(A). The voltage controlled oscillator is modulated by
voltage m(t) producing signal x(t) which is fed to the antenna (or trans
ducer as the case may be). The signal propagates through the medium and
is reflected, in part, by the target. On returning to the antenna, the
signal will be delayed by
= 2d
v
where d is the distance to the target,
(1.4-1)
and v is the propagation velocity
of the medium.


31
these last two terms are complex conjugates it is sufficient to minimize
the magnitude of either. The term whose magnitude is to be minimized is
defined to be the ambiguity function:
Xi (t,d>) = / u(t y)u*(t + £)ej2irtdt (2.1.3.3-3)
Z (A)
The function is sometimes defined
X2(t,<}>) = / u(t)u*(t + T)ej2lItdt (2.1.3.3-3)
(B)
which varies from X only in phase. As indicated above, we are gener
ally interested in the magnitude only; clearly
|Xi (t,4>) I = |X2(t,<|>)|
(2.1.3.3-4)
Note that the ambiguity function along the t axis becomes the autocorre
lation function of the complex envelope:
CO
X2(t,0) = c(x) = / u(t)u*(t +i)dt 3.3-5)
00
Likewise, along the "doppler" axis the ambiguity function is
CO
X2 (0, <(>) = / u*(f)u(f+(J>)df = K ) (2.1.3.3-6)
CO
by the application of Parsonval's Theorem. If we consider a signal of
unit energy (c(0) = 1), then
00
/ |x(0,(j>) |2d =FA (2.1.3.3-7)
oo
where there is no need to designate the particular form of X by sub
scripts. Also, / |x(T,0)|2dx = Ta (2.1.3.3-8)
Just as integrations along each axis produce measures of total ambiguity
in T or (range or velocity), we may integrate in both directions to


54
For a given system the Fourier coefficients contain the range information.
Furthermore, extraction of this information takes place as
1
A/2
z2(t,tr) = lim j
f yQ(t,x,TR)dt
(3.2-10)
A-x
-A/2
(A)
1
A/2
= lim -
f eR(t,TR)e(t,r)dt
(B)
A-*
-A/2
A/ 2 00 00
= lim j
/ l l
(C)
A-*
-A/2 n=- m=-
I n(T)6_n(TR) (D)
n=-
00
= l an(x)B*(TR) (E)
n=-
The system is matched when e = eR; i.e., when an = 8n. Then at t = tr
we have
00 2
z2(TR> TR> = I Ian(xR)I (3.2-11)
n=-<
As expected, the summation above is the total power of the line spec
trum of e. The coefficients are computed:
! T/2 -j-
an(T) = t f e(t,x)e
1 -T/2
.2imt
dt
(3.2-12)
(A)
T/.2 r- ^ filln T+K-C'
(B)
For sawtooth modulation, m(t) = t for |t
<
T
2*


22
(B) NOISY AND IDEAL RESPONSES OF SYSTEM WITH LOW
AMBIGUITY
FIGURE 2.1 COMPARISON OF ACCURACY FOR HIGH AND LOW AMBIGUITY SYSTEMS
Thus, as Key et al [9] have noted, inherent accuracy depends on the
signal-to-noise ratio and the shape of the ambiguity function.
2.1.2 Ambiguity and Resolution
Ambiguity and resolution are two very related, but distinct,
qualities of a signal. The ambiguity of our signal in range implies to
what extent the range of the target can or cannot be determined with a


110
modulation is not monotonic, there are more than one inverse functions
cd ^ and more than one mappings from ip to ip for at least some
m. l 10 > i
i m
parts of to (t) This represents the case of a reference ^
m
where x is simply the element i, and the demodulation is averaged
with respect to i as per (7.4.2-2). Figure 7.5 depicts an example
of such a modulation and mapping.
P
i/cd
m
(2)
0
2
3
CD <0
m
CD >0
m
1
-1
t
k" i=l5f*-i=2 >1
K r x
FIGURE 7.5 MULTIPLE MAPPING FOR NON-MONOTONIC MODULATION


o o
208
COMPUTE AND PLOT THE FIRST DEMODULATION FUNCTION
INTERPOLATE TO FIND Z(MOD(T))
CALL INTRP(R,ABSC,1,NK,2,ZUT,Y,1,N,T,H)
DO 5 J=1,N
5 D( J) =ZWT (J)*ABS(YP(J))
CALL MATPLT(D/ALPIIA(1)/N/1,N/1)
DATA ALPHA(1)/ALPHA(G)/'DEMOO #1','DEM0D #2/
C COMPUTE AND PLOT THE SECOND DEMODULATION FUNCTION
DO G R(J+KK3)=Z(2*J+2)
6 R(J)=Z(2*J+1+KK2)
R(KK3)sZ(2)
C INTERPOLATE TO FIND Z(MOD(T))
CALL INTRP(R,ABSC,1,NK,2,ZUT,Y,1,N,T,H)
DO 0 J=l,N
9 D(J)=ZWT(J)*ABS(YP(J))
CALL MATPLT(D,ALPHA(G)/N/1/N/1)
GO TO 80
END


202
The modulation is a solution of
uc(t) t
/ z+(0)dn K / I|(t)dt
o>c (-T/2) -T/2
where K is found as
(oc(T/2) oiq+ttB
/ z+(n)dn / i dt
wc(-T/2) u)0-ttB ^ tiBwj
T/2 T/2 2 u)jT
/ ^(t)dt / cos o>it sin t sin
-T/2 -T/2 1 L L
Substituting into the integral equation, we obtain the desired modu
lation.
wc(t)
/ I d0
ujo-tvB
ttB>i
sin
o)jT
t
/ cos w^t dt
-T/2
(j)c(t)-u)o+rrB
ttBui
sin
till
0)1
1 T
(sin o>it + sin eapjj-)
ojc(t) = sin it + wo
0)jT
sin
2
Our solution is valid for o)c(t) monotonic. We know for oic(t) to be
monotonic we must consider such that, over (-T/2,T/2) jtoj11 < ir/2.
Then we have the restriction
1
<
TT
" T


205
17 IF(NFRST.EQ.O) GO TO 20
NFRST=0
GO TO 15
20 CALL MATPLTCZ, ALPIK8),1024,1,2*NUMBER,1)
CALL FFTCZ, 2 NUMBER, B, t.'FFT, I)
CONVERT TO MAGNITUDE AND ANGLE FORM
CALL MAC.ANG(B,NFFT)
CALL MATPLTCB,ALPlK 3),NFFT,1,NFFT,2)
CALL MATPLTC B,ALPH(4)/NFFT,2,NFFT,2)
GO TO 51
END


151
and computes and plots the necessary demodulation functions.
Figures 9.18, 9.19, and 9.20 are, respectively, the specified input
range response, the obtainable bandlimited response (using rec
tangular windowing), and the necessary first-channel demodulation
for a system using sawtooth modulation. The input phase was
specified as a constant zero and is thus not plotted while the
obtainable response phase is the uninteresting zero- and pi-valued
phase of a sin(x)'/x function. It is known that this response is
obtained when DC demodulation is used. The Gibb's phenomenon is
present in Figure 9.20, producing ringing at each end of the record.
The second-channel demodulation is zero because the response is
Hermetian-symmetric.
Figures 9.21 and 9.23 show the obtainable responses from the
same system when Hanning and Hamming windowing, respectively, are
specified instead of rectangular. Figures 9.22 and 9.34 give the
first-channel demodulations necessary to obtain these responses.
Second-channel demodulations are zero.
The next series of plots, Figures 9.25 through 9.27, illustrate
solution of the demodulation function for an input response which
is already (essentially) bandlimited. The truncated sin(x)/x is
shown closely reproduced in magnitude as the obtainable BL response
with the rectangular function being the obvious window choice. The
first-channel demodulation effectively windows the rectangular
spectrum (produced by the linear modulation) to produce a rectangular
spectrum which has a bandwidth consistent with the specified BL
function. Here again, Gibb's phenomenon occurs at the sharp dis
continuities of the rectangular demodulation.


74
A less deceiving picture of ambiguity is given by the dotted response
zout3 fr continuous values of t. Intuitively one might think that the
more points constrained, the better. This reasoning turns out not to be
necessarily true. The number of constraint points N is limited by the
highest rank G may be. And the rank of G depends on the number of lin
early independent vectors jz(x) that exist. Alternately, the rank may
be expressed as the number of linearly independent row vectors of G: if
an information element Zj(x^) gives additional' information not contained
collectively in {z^ix^)} for i^j and for all k, then it creates a vector
linearly independent of the other vectors.
5.1 Alternative Information Vectors
The information elements (z(x)} need not be the outputs of har
monic sub-systems, but may be any set of variables, each of which is lin
early independent of the others as a function of x [27,p.29]. We shall
consider only variables which are derived from linear operations upon the
IF waveform e. Examples of these elements might be
1. equally or unequally spaced samples of the
IF waveform taken at specific times in the
period T,
2. trigonometric series coefficients (of the
IF signal expansion),
3. Walsh-function expansion coefficients,
4. the outputs of square-wave demodulators,
5. other (generalized) Fourier series coef
ficients.
The linear operation takes the form shown in Figure 1.4; Table 5.1 gives
the demodulating function for each element set listed above. The filter
h(t) time-averages over one or more periods of the modulation and is, in
practice, a LP filter with a cutoff frequency much lower than 1/T.


S%cfS\
IP UlPH 1 UNIVERSITY of
IJFI FLORIDA
The Fouinlation for The Gator Notion
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Mattox, Barry
TITLE: Improvement of the range response of short-range fm radars
PUBLICATION 1975
DATE:
I, Barry Mattox as copyright holder for the aforementioned dissertation,
hereby grant specific and limited archive and distribution rights to the Board of Trustees
of the University of Florida and its agents. I authorize the University of Florida to digitize
and distribute the dissertation described above for nonprofit, educational purposes via the
Internet or successive technologies.
This is a non-exclusive grant of permissions for specific off-line and on-line uses for an
indefinite term. Off-line uses shall be limited td those specifically allowed by "Fair Use"
as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code)
as well as to the maintenance and preservation of a digital archive copy. Digitization
allows the University of Florida to generate image- and text-based versions as appropriate
and to provide and enhance access using search software.
This grant of permissions prohibits use of the digitized versions for commercial use or
Barry Mattox
Printed or Typed Name of Copyright Holder/Licensee
Personal information blurred
April 21. 2008
Date of Signature


CHAPTER IX
COMPUTER ALGORITHMS
Three algorithms were written in the Fortran IV language
to implement the solution of the three equations for range response
E I R
z (t), demodulation functions ^ and <|* and modulation function w .
m
This chapter will present the application of each program, the
mechanisms involved in the computation of solutions, and the
assumptions and precautions which are to be observed when employing
each program. In addition to the textual instruction included for
the use of each program, a complete program listing is included in
Appendix B with comment statements inserted for rough explanation
and data format.
The necessity or convenience of using a computer solution
appears when (1) the input data is not available conveniently in
closed-form (such as is the case with numerical data pairs) pr
(2) when closed-form solution is not mathematically tractable.
Case (1) often occurs as a result of other numerical programs
generating output data pairs which are then to be used as input data
for the programs described here. Also, sectioned functions (i.e.,
those functions which must be described in a piecewise fashion)
are sometimes more conveniently entered via sample points (e.g.,
the segment endpoints in a piecewise-linear function) than by
specification of the multiple component functions and respective
domain intervals. As mentioned in section 6.4, solution of certain
123


0.33
FIGURE 9.42 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.19
186


CHAPTER VIII
DIRECTIONAL DOPPLER PROCESSORS
In many systems, such as in collision avoidance systems,
it is desirable to distinguish those targets which are approaching
the antenna from those which are traveling away from the antenna.
Such systems have, in the past, used single-sideband techniques
either at RF or at IF. The signals to be processed have thus been
primarily single-sideband in nature either at RF or at IF. In this
chapter a very brief outline of SSB processing techniques will be
presented as background, and then a more general directional doppler
processor will be developed with the aid of previously derived
relations. It will be shown that systems may be constructed, using
general modulations, for which range responses may be designed with
versatility exceding that allowed in designing SSB systems.
8.1 SSB Directional Doppler Techniques at RF
One of the first direction-sensitive doppler devices was
developed by Kalmus and employed two channels developed from the
mixing of the CW signal with shifted and unshifted versions of the
returned RF signal [30]. This 90 relative RF phasing created
quadrature detection at RF. The Kalmus system employed a 2-phase
motor driven by the two channels whereby the direction of rotation
indicated an approaching or receding target. With the addition of
a 90 doppler-frequency phase-shift network to one of the channels
112


47
(B) MODEL
FIGURE 3.1 THE AUTOCORRELATION SYSTEM
"atten
(t)
- A(t)
R(t)
(3.1-1)
(D)
where Rp(x) is the periodic autocorrelation function of x(t) and
T
R(x) =<
Ro I T 2
otherwise
The symbol z will be used to denote the range response with subscripts
differentiating various systems or mathematical forms. Since close-in


response magnitude and phase are entered as data along with a datum
designating scale in normalized delay and a datum which specifies
the type of windowing to be performed in order to assure a bandlimited
response. The input data for jz (x)| and Zz (x) are interpolated
at a vector of equally-spaced delay values to yield a set of sampled
data points for |z (x)| and Zz (x). These sets are then converted
E
to the real and imaginary parts for z (t), which is Fourier trans
formed to obtain the spectrum before bandlimiting. The number of
sample points in x is large so that negligible aliasing of the
computed spectrum occurs. This (potentially) wideband spectrum is
then bandlimited to bandwidth B using the specified window function to
E
give Z (ft), the transform of the actual range response to be produced.
The modulation is now read as data pairs and checked for
monotonicity. com(t) is interpolated to yield sample values of co^
and its derivative at equally-spaced values of time, and w is
E
plotted. The vector containing Z (ft) is Fourier transformed to
E
yield sampled values of z (x). Here, as in Program One, the transform
E
of Z (ft) will be aliased, the degree depending upon the amount of
energy present outside the range response definition interval. After
conversion to magnitude and phase and necessary re-arrangement for
plotting, this range response is (eventually) plotted. (The seemingly
strange sequence of execution indicated by the flow chart at this
point was dictated by a concern to reduce storage size.) The values
E
of z (x) are sampled at a lower rate (every Kth point is stored in
another vector) to compress the response in delay by a factor of K.
Zeros are then added to the record length, the non-zero section being
windowed slightly (30% Hanning windowed). This creates an expansion


117
ambiguity functions which the signal may possess. Bartlett, Couch,
and Johnson [23] have analyzed both the general linear system to be
discussed here as well as a NL system which multiples the responses
of two channels to produce the range response. The analysis to follow
will show that, with very few assumptions and with the BL restriction
of the range response, any desired range response or directional
response may be obtained by suitably designing the demodulations
of a system.
8.3.1 Analysis of the SSB System in the Q-Domain
Section 8.2 and reference [21] state that the range response
complex envelope of an USSB processor will be
ZE(t)
cn(T)
-r> 27rnt
-j
/ u(t- ^)u*(t+ -j)e T dt
~2
(8.3.1-1)
In the small-T case we may make the approximation
ju(t)x
u(t- -)u*(t+ ) = e
(8.3.1-2)
so that
ZE(t) = i / e m T dt
T
Taking the Fourier transform of the range envelope yields
T
F 1 2
ZE(!)) | / / e1 m
T
CO
2
2irt
-fix
dt dx
xr 2irt >
n 2 -J T -j(J-) )x
= ^/e T/e m dx dt
1 T
(8.3.1-3)
(8.3.1-4)
(A)
(B)


43
W4 (o)
N
- V neo
1 + I an cos
J n=l
0
-ttB < otherwise
(2.2.3-10)
The coefficients, {an), of the above Taylor "weighting have been calcu
lated for values of desired sidelobe attenuation and N. The number of
terms (N+l) determines how well W4 approximates W3. A typical design
value for N might be five [ll].
Among'other windows are these, which we but mention here:
W5 Papouliswindow [16] was designed for windowing
time domain finite-length data records. This
windowing of the sample record yields a minimum
bias in the spectral estimate and a "low" vari
ance of those estimates.
Wg The prolate spheroid wave function [17] is the
optimum window function to constrain a maximum
amount of transform ''energy" to be within a
specified interval. Sidelobe levels, although
correlating somewhat with sidelobe "energy" are
not considered in the criteria. The prolate
functions are quite complicated in form and are
usually approximated by other functions.
W7 The Kaiser window is an approximation to the
prolate window. Kaiser has recognized the dif
ficulties of working with such functions and
purposes a window expressed in more familiar
zero-order Bessel functions [18].
2.2.4 Effect of Windowing the Power Spectrum
We wish now to focus attention on windows
W0 rectangular
Wi Hanning
W2 Hamming
W4 Taylor f
These windows have been defined as zero outside (-ttB,ttB). If we con
sider each of these windows to be a weighting of an existing band-


129
SUBROUTINE FFT uses the (IBM Scientific Subroutine
Package) subroutine HARM to perform a fast discrete
Fourier transform (DFT) on an input vector. Provision
is made for inverse or forward transform and for
adding zeros on the ends of the record. Additionally,
the output data are re-arranged so that the most
negative frequency (time) component is the first point,
the zero frequency (time) component being in the center
of the data.
SUBROUTINE MAGANG converts the real and imaginary parts of
a vector argument to magnitude and phase. The odd elements
of the vector are considered to contain the real part of
the number while the even elements originally contain the
imaginary part of the number. After execution, odd elements
contain the magnitude of the number, and even elements, the
phase.
9.1.2 Precautions and Assumptions
As with any sampled data system, sampling rates and aliasing
properties need to be understood before meaningful data may be collected.
And when using a discrete Fourier transform (DFT) the inputs and
outputs must be recognized for what they represent.
Two constants fixed at the beginning of Program One are used
to scale the output range response: NUMBER specifies the overall
span of normalized range, Bt (or r, for B normalized to 1). NFFT
specifies (twice) the number of complex DFT points and thus the
E
number of points to be plotted in a (T). Then, for NUMBER=25 and
NFFT=2048, the range/delay axis will extend from -12.5 to +12.5 and
there will be approximately 41 points per unit on the delay axis.
Since the range response is the result of a DFT, the
properties of sampled data systems apply including these:
(1) A finite-record-length input is Fourier transformed;
such a bandlimited function cannot also be time-limited.


Re {Z (0)>
0.308 -
p
0.3
-nB
0
FIGURE 9.46 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.23
VO
Frequency irB


Response Magnitude
1.0 -
0.0 -
-16 0
FIGURE 9.38 SPECIFIED THREE-UNIT ONE-SIDED PULSE RESPONSE
Normalized Delay
16
182


95
inconsistant with the original specification of Z+. Since we wish to
specify the form of iJj and Z+, we introduce a multiplicative constant, K,
to be determined later:
Z+(n) |~| = K*(t)
(6.4-
(A)
Z+(n) || = Ktp (t) if Wc(t) is
monotonically increasing
(B)
t=b ,0 t=b
S Z+(n) -T dt = / Kijj(t)dt
t=3 t=3
(C)
2=mc(b) t=b
5 Z+(fi)dfi = f KiJ/(t)dt
Q =mc(a) t=a

(D)
realize that wc(#) is the unknown in
this integral equation.
The
value of t = a would normally be -1/2 with u)c(-T/2) = uq-ttB. Then the
upper limits would be the variables which define the function wc and its
argument. It is necessary (in most cases) to solve for K before solution
is attempted. This is easily accomplished by substituting known bound
ary conditions for limits giving
u>0+ttB
f z+(n)dn
ioq-ttB
K = (6.4-3)
1/2
f iKt)dt
-1/2
Depending on the functional forms, especially Z(£2), the computation of
ojc(t) may be straightforward or very difficult, e.g. a solution of
transcendental equations.
The multiple root case does not have to apply to the solution
of wc(t), since we may simply define u)(t) to be monotonic (as in the


C(t)
1.96 -
0. ~ -
-T/2
Time
T/2
FIGURE 9.41 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.39 FOR SAWTOOTH MODULATION
00
Ui




IMPROVEMENT OF THE RANGE RESPONSE OF SHORT-RANGE FM RADARS
BY
BARRY GRAY MATTOX
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975


Modulation Function
0
miiiniii


26
the time resolution constant, the frequency resolution constant, and the
ambiguity function.
2.1.3.1 The time ambiguity constant
The time ambiguity constant is defined for signals of finite
energy as [3]
f Ic(x)| dx
T A ^
2
c(0)
(2.1.3.1-1)
(A)
/ |U(f)| df
[/|U(f)|2df]2
(B)
where the signal is expressed as the real part of its analytic form:
x(t) = Re{u(t)e^t0t}
(2.1.3.1-2)
where u(t) is called the complex envelope of the signal and has an auto
correlation function
c(x) = f u(t)u*(t+x)dt (2.1.3.1-3)
00
Equation (2.1.3.1-1B) is derived by applying Parsonval's theorem to the
numerator and denominator of T^. Units of Ta are time, and, as Woodward
puts it, Ta is inversely proportional to the "range of frequencies occu
pied by the signal [3]." This idea may be rigorously expressed for
spectra with flat sections using (2.1.3.1-1B). The time ambiguity con
stant measures, for each signal, the total ambiguity in range when the
target is stationary.
In most applications, we wish to minimize Ta within the bounds


91
Comparing (6.2.1-10) and (6.2.1-7) we see that
irpQ (w) J > 0
P() =
I TTPft (-) K 0
(6.2.1-11)
when o)c(x) is monotonic. Equation (6.2.1-11) states that the power spec
trum is proportional to the p.d.f. of the FM modulation for high-index
(dispersion) case.
We have intuitively broken Z+(i2) into one part which represents
spectral power density and another which represents a windowing effect
for the case i|>(t) £ 0* However, we are not limited to non-negative
values of ip and Z+. \p may be a sinusoid, for example, yielding Z (Q)
which cannot be a realizable power spectrum. Neither would z(t) be a
realizable autocorrelation function, but, rather, a displaced type of
range response.
Some examples of computer solutions are given in Chapter 9.
6.2.2 Summary
In this section we have examined methods for predicting the
range responses of special case systems and the general coherent system.
In the general case, piecewise approximation algorithms can provide the
response. A closed-form solution, however, is available under the as
sumptions of small delay, quazi-static targets, and high dispersion
factor. Examples of closed-form solution of the range response are in
cluded in Appendix A. The relation, equations (6.2-12) and (6.2-14), has
been related to windowing in the receiver and the approximation of power
spectra by the probability density function of the FM modulation. The
general form extends to yield functions which do not correspond to
realizable power spectra thus including responses which are not


oooooooooooooooooooooooooo
206
********************************************* **************-***************
PROGRAM TWO SOLUTION OF THE DEMODULATION FUNCTIONS
* DATA FORMAT FOR THIS PROGRAM IS AS FOLLOWS:
* WINDOW TYPE, NUMBER OF B-TAU'S, NO. OF PTS. PER B-TAU (313)
* (O-RECTANGULAR, 1-HAMMING, 2-HANNING)
* RANGE RESPONSE MAGNITUDE "INPUT" FORMAT
* RANGE RESPONSE ANGLE "INPUT" FORMAT
* MODULATION FUNCTION "INPUT" FORMAT
* (MAY INCLUDE ADDITIONAL SETS OF R.R. MAG./R.R. NGLE/MOD. FN.)
* BLANK CARD (INDICATES END OF THE DATA)
*
* "INPUT" FORMAT:
*
* NUMBER OF DATA PAIRS, ALPHANUMERIC DATA FORMAT (I3,77A1)
* (IF THE FIRST FIELD CONTAINS A NEGATIVE INTEGER,
* ADDITIONAL CARDS OF THIS FORMAT WILL BE ACCEPTED.
* OTHERWISE, THE FIELD DESCRIBES THE NUMBER OF DATA
* PAIRS TO FOLLOW.)
* ORDINATE, ABSCISSA (ONE CARD PER PAIR) (2F8.3)
* *
ft*************************************************************************
*
*
*
*
*
*
*


*
*
*
*
*
*
*
*
*
*
*
*
REAL MOD(101),TMOD(101)
REAL*8 ALPHA(G)
DIMENSION RL(101),ARL(101),TARL(101),T(1024),Y(1024),YP(1024),
*YA(1024),IMIM(16),IMAX(16),R( 4096),NA(3),I IK 2048),S(2048),
*ABSC(1024),Z(1024),ZWT(1024),D(1024),TRL(101),NZ(3)
EQUIVALENCE (MOD(1),RL(1)),(TMOD(1),TRL{1)),
*(YA(1),ZWT(1)),(IN(1),IMIN(1)),(IN(17),IMAX(1)),(D(1),T(1)),
*(ARL(1),R(1)),(TARL(1),R(102))
READ(5,102) NWIND,NBT,NPTBT
102 FORMAT(313)
NB-2*NBT
NFFT=NBT*NPTBT
H-1E-3
NA(1)-3.321928095*ALOG10(NFFT+0.0)+0.1
NFFT- 2**NA(1)
N-NFFT-1
NA(2)=0
NA(3)"0
C INPUT THE RANGE RESPONSE MAGNITUDE AMD ANGLE
80 CALL INPUT(RL,TRL,NRL)
IF (NRL.EQ.0) STOP
CALL INPUT(ARL,TARL,NARL)
C CREATE THE VECTOR OF EQUALLY-SPACED TIME VALUES
DEL-1./N
DO 1 J=l,N
1 T(J)0.5 + (J-0.5)*DEL
C INTERPOLATE VALUES OF RL AND ARL FOR THE VECTOR OF TIMES
CALL INTRP(RL,TRL,1,NRL,2,Y,T,1,N,YP,H)
CALL intrp(arl,tarl,i,narl,2,ya;t,i,n,yp,h)
CALL MATPLT( Y,ALPHA( 2 ),N, 1,11,1)
CALL MATPLT YA, ALPIIA( 3), N, 1, N, 1)
C CONVERT THE MAGNITUDE AND ANGLE TO REAL AND IMAG. PARTS OF VECTOR R
KF1-NFFT+1
KF2-NFFT+2
KK-N/2
KL-KK+1
DO 2 J-l,KK
1-2*0
K-KL+d
R( I +1)-Y(K)*C05(YA(K))
R(I+2)=Y(K)*SIN(YA(K))
R(KF1+I)-Y(J)*COS(YA(J))
2 R(KF2*I )-Y(J)*Slf'(YA(J))


57
3. The n-system becomes an autocorrelation
system or, equivalently, a matched filter
system when n=0. R(t) is sometimes re
ferred to as the zero-response for this
reason.
As per 2. above, the equivalence of the n-system and IF correlator (for
discrete values of Tg) no longer exists when m(t) is not linear. For
this reason we may shape the harmonic system range response by shaping
the RF spectrum, but if we try to obtain that shape by non-linear FM,
the problem becomes much more complicated.
Filtering or AM shaping of the RF power spectrum are still
available techniques with a harmonic system. It can be shown that, since
both signal and reference beat signals are derived from the /w(w) win
dowed RF signal in the delay-line system, the equivalent window for the
n-system must be W(w) because a sinusoidal "reference signal" is like a
beat waveform derived from an unwindowed (flat) spectrum. Window rela
tionship are summarized below. The desired response shape is
R(x) = F[WX]:
^autocorr. ~ ^x())
(3.3-3)
(A)
WIF corr.(w) =
(B)
^n-syst. = Wx(m)
(C)
(Linear)
Note that we did not prescribe the manner in which m(t) might be designed
for non-linear FM windowing to shape the harmonic system response. Many
techniques are easily applied to the zero response, such as windowing or
the optimum addition of non-linearities to the modulation, but these
techniques do not apply for n > 0. However, the computer can be used in
a Monte Carlo approach to obtain the non-linear waveshape for n > 0 [22].


211
WF-l.E+30
EPS=RMAX/5 00.
CALL MATPLT!R,ALPHA!4),HK,1,NK,1)
COMMON R,ABSC,DMOD1, TDMOD, NIC,NPTD,AK, EPS, KNOT
KNOT=0
CALL RK1!FUN,HI ,TI ,WI TF,WF, ANST, ANSI/, I ER)
Al/=ANSV/
KNOT=l
CALL RK1!FUN,HI,TI,WI,TF,WF,ANST,ANSW, IER)
AK~! AV/+ 0. 5) / (ANSW+0.5)
KNOT=2
IF(IER.NE.O) V/RI TE! G, 110) IER
110 FORMATC ERROR TYPE',13,' MAS OCCURRED IN THE RK1 SUBROUTINE.')
DO 17 J=1,NK
CALL RK1(FUN,HI ,TI ,WI ABSC! J),V/F,ANST, Z!2*J-1), IER)
TI=AMST
Wl =Z!2*J-1)
17 IF(IER.NE.O) V/RI TEC C, 110) IER
CALL MATPLT(Z,ALPHA!5),2*NK1,1,2*NK-1,2)
DO 19 J=l,NK
19 RCJ)=Z(2*J-1)
CALL I NTRP(R, ARSC, 1, NK, 2, Y, T, 1,N, YP,II)
DATA ALPHA!1),ALPHA!6)/'DEMOD #l','DEMOD #2'/
C COMPUTE AND PLOT THE SECOND DEMODULATION FUNCTION
DO 6 J=l,KK1
R J+KK3)=Z!2*J + 2)
6 R!J)=Z(2*J+1*KK2)
R KK3)=Z(2 )
C INTERPOLATE TO FIND ZMODT))
CALL I NTRP!R, ABSC, 1, NK, 2, ZI7T, Y, 1,N, T,H)
CALL MATPLTZWT,ALPHA!4),N,1,N,1)
CALL MATPLT!YP,ALPHA!4),N,1,M,1)
DO 9 Jl,N
9 D(vJWWTJ)*ABSYPJ))
CALL MATPLT!D,ALPHA!6),H,1,N,1)
GO TO 80
END


Re {Z (fl)}
4.35
0.
-0.63
-tB 0
FIGURE 9.48 TRANSFORM OF RANGE RESPONSE GIVEN BY FIGURE 9.26
Frequency irB
192


20
This compression is usually taken to be a simple shift if 2ttB << (oq.
This case has been studied for example modulations when the shift has
been substantial [ll].


Page
2.1.3.2. The frequency resolution constant* 29
2.1.3.3. The ambiguity function 29
2.2. Shaping of the Range Response by Windowing 35
2.2.1. The Importance of the Autocorrelation
Function 35
2.2.2. Windowing a Bandlimited Spectrum 39
2.2.3. Specific Windows 40
2.2.4. Effect of Windowing the Power Spectrum 43
III. SHAPING THE SYSTEM RANGE RESPONSE 46
3.1. Autocorrelation Systems 46
3.2. Delay-line IF Correlator Systems 50
3.3. Harmonic Processor Systems 56
3.4. General Coherent Demodulator Systems 61
IV. INVERSE FILTERING 62
4.1. Application to the t Domain 65
4.1.1. The Autocorrelation System 65
4.1.2. The Delay-line IF Correlator System .... 68
4.1.3. The Filter, h(x) 68
4.1.4. Harmonic and General Coherent Demodulation
Systems 69
4.2. Summary and Conclusions about Inverse Filtering 70
V. SYSTEMS USING VECTORS OF INFORMATION 72
5.1. Alternative Information Vectors 74
5.2. Equivalent Single-Channel System 76
5.3. Existence and Dimension of H 77
5.3.1. Dimension of H Based on IF Waveform .... 77
5.3.2. Dimension of H Based on the Range Response 78
5.4. Choice of Constraint Times 79
VI. DEVELOPMENT OF CLOSED-FORM RELATIONSHIPS 83
6.1. Methods for Prediction of the Range Response .... 83
6.2. Derivation of the Relation for Predicting the Range
Response 85
6.2.1. Physical Interpretation of the Relation 89
6.2.2. Summary ..... 91
v


128
SUBROUTINE INPUT reads and prints alphameric data as
does SUBROUTINE MODUL. It reads the number of data pairs
to follow and then those pairs. Only the abscissa
vector is normalized to and centered on the interval
(-0.5,0.5).
SUBROUTINE INTRP interpolates a table of points for a
vector of abscissas, yielding both the ordinates and
their derivatives for each element of the vector. A
continued-fraction interpolation subroutine found in
the IBM Scientific Subroutine Package is used, with
two points used per interpolation. The points are those
closest in absolute distance to the point to be inter
polated. Two-point or linear interpolation has been
specified; however, more than two points may be
specified, if desired, by making a change in the main
program call statments. An example of the interpolation
of data is given by Figure 9.2(A), where circles indicate
input data and the solid lines indicate the value of
the function given by the (two-point) interpolation
scheme used in SUBROUTINE INTRP. Note the difference
between this and linear interpolation using the two
flanking points as illustrated in Figure 9.2(B). (It
would be a simple matter for the user to write such
an interpolation subroutine, if desired. Of course,
for points equally spaced along the abscissa, the two
schemes yield the same interpolated function.)
(B) LINEAR INTERPOLATION USING TWO FLANKING POINTS
FIGURE 9.2 TWO TYPES OF LINEAR INTERPOLATION


CHAPTER I
DEFINITION OF SYSTEMS AND GOALS
This dissertation is primarily concerned with a class of high-
index, periodically modulated FM radars or distance measuring systems
(DMS) which are to operate at relatively short distances. Items of
consideration include resolution, simplicity/cost, and immunity to noise.
In this first chapter the class of systems will be defined and models
developed, using assumptions consistent with the problem.
1.1 Operational Constraints
The class of systems will be defined, both in purpose and in
structure. The DMS considered here are to operate within these specifi
cations :
A1 The system will primarily measure distance; or,
more basically, the DMS will detect the presence
of an object at a pre-specified distance(s).
A2 The distance to be measured will be small com-
ared to the wavelengths of major FM modulation
components.
A3 The resolution (ability to distinguish between
targets of various amplitudes at similar dis
tances) must be "good." Alternately, we should
be able to design the range response.
A4 The target to be detected is to be either
stationary or moving "slowly."
A5 The system should be simple and cost-effective.
A6 The system must be able to operate in the noise
and signal environment for which it is designed.
1.2 Structural Constraints
There may be many possible system structures which satisfy the
1


49
FIGURE 3.2 MODIFIED AUTOCORRELATION SYSTEM
response may be ideal, but it requires two isolated antennas, a multi
plier, and the delay line. If one is willing to surmount these disad
vantages, the range response may be shaped (whereever it is centered in
delay) by prescribing the desired window function to be the shape of the
power spectrum.
Windowing of the RF power spectrum may be accomplished at the
"transmitter" in three ways:
1. Amplitude modulate the output stage as the fre
quency is swept through the band.
2. Pass the transmitted signal through a linear
filter. For high dispersion, linearly modu
lated signals, the output power spectrum will
be given by K* |H(2irf) ¡2 where H(w) is the fil
ter transfer function.
Use non-linear frequency modulation. For high
dispersion signals, the p.d.f. of m(t) will
describe P(f) [5].


above variables to yield additional sets of variables in which to
completely express the signal. Some will be more convenient to
work with than others. We shall now adopt a specific denotation
of the particular expression of a waveform:
S
x,y,
represents the waveform S as a function
of variables
That is, in general,
Sa>b>c(8,Y> *
s
x,y,z
(a,$,y)
(7.1-4)
Some possible representations of the IF signal S are
t
S (t,r) = e(t,x) = cos[ / o> (X)dX+w x] (7.1-5)
t,x t_T m o
S0 t(6t) = costOj (X)+ojot] (7.1-6)
J) T2 C X3
S (w,x) = cos[u x+m x 2^: H - ...] (7.1-7)
w, x o m o
The reference \p may similarly be represented as a function of time
or, depending upon the method of generation, conveniently as a
function of the modulation m or of the angle 0'.
For a deterministic, periodic signal and reference the
reference may always be written as a function of time (as in
previous chapters):
^t(t) = Kt) (7.1-8)
Systems may be built in which there is a unique relationship
between the value of the reference and the value of the modulation
(i.e., no memory elements). See Figure 7.2.


ooooooooooooooooooooo
212
SUBROUTINE MODUL(MOO,TMOD,NPTM, ISEC,IMIN, IMAX)
THIS SUBROUTINE READS THE NUMBER OF MODULATION DATA PAIRS
WHICH WILL BE SUPPLIED AMD THEN READS THE PAIRS, MOD(>|TMOD().
ALL OF THE ARGUMENTS ARE OUTPUT ARGUMENTS AS FOLLOWS:
MOD IS THE ONE-DI MENTI ONAL ARRAY WHICH DESCRIBES
THE MODULATION FUNCTION AFTER NORMALIZATION
TO A BANDWIDTH OF UNITY AND TRANSLATED TO HAVE
A CENTER OF ZERO. IT MUST BE DIMENSIONED TO
A SIZE OF 101 IN THE MAIN PROGRAM.
TMOD IS THE OF ARRAY OF THE DATA PAIR WHICH DESCRIBES
THE MODULATION TIME BASE. IT IS ALSO NORMALIZED
AND CENTERED ON ZERO AND IS DIMENSIONED AS IS MOD.
NPTM DESCRIBES THE NUMBER OF DATA PAIRS SUPPLIED.
ISEC DESCRIBES THE NUMBER OF MONOTONIC SECTIONS OF
THE MODULATION VERSUS TIME.
IMIN(N}|IMAX(N) IS THE SUBSCRIPT OF THE MODULATION
PAIR WHICH BEG I NS I ENDS THE N-TH MONOTONIC SECTION
OF THE MODULATION FUNCTION.
LOGICAL SLOPE,LSLOP
REAL MOD(101)
DIMENSION TMOD(101),IMIN(16),IMAXC16),LTRL(77)
98 FORMATC 77A1)
99 FORMAT(I 3,77A1)
100 FORMATCI 3)
101 FORMAT(2F8.3)
102 FORMATC1THERE HAS BEEN AN ERROR IN ENTERING DATA POINT NO.',14,
*' OF THE MODULATION.')
103 FORMATC1THE PROGRAM HAS STOPPED.
*' PLEASE ENTER TWO OR MORE MODULATION DATA POINTS.')
15 READ(5,99)NPTM,LTRL
IF(NPTM) 80,81,81
80 WRITE(6,98) LTRL
GO TO 15
81 CONTINUE
IF (NPTM.EQ.0) RETURN
IF (NPTM.LT.2) GO TO 7
READ(5,101) MOD(l),TMOD(l)
BMAX=MOD(1)
BMIN=BMAX
DO 1 J=2,NPTM
READ(5,101) MOD(d),TMOD(d)
BMAX= AMAX1(M0D(J),BMAX)
BMIH-AMAX1(-MOD(d),-BMIN)
1 CONTINUE
BBMAX-BMIN
T=TMOD(NPTM)-TMOD(1)
TMIN-TMOD(l)
DO 2 d=l,NPTM
MOD(d)=(MOD(d)-BMIN)/B-0.5
TMOD(d)=(TMOD(d)-TMIN)/T-0.5
2 CONTINUE
IMIN(l)l
ISEC=1
IF(NPTM-2) 7,8,9
7 WRITE(6,103)
RETURN
9 SLOPE=(MOD(2).GT,MOD(1))
dUMP=0
DO 10 d=3,NPTM
LSLOP-SLOPE
SLOPE=(MOD(d).GT.MOD(d-1))
IF(TMOD(d)-TMOD(d-1)) 4,20,3
3 IF(SLOPE.AND.LSLOP.OR..NOT.SLOPE.AND..NOT.LSLOP.OR.
*(dUMP.EQ.l)) GO TO 88
20 II1AX( I SEC)Bd-l
ISEC-ISEC*!


98
The IF signal becomes
e(t,x) = 2*x(t)x(t-x)
(7.1-2)
(A)
t-T
= cos[Kf f m(X)dX+)nx]+cos[Kf/ m(X)dX+K£f/ m(X)dX-w0x+2w0t] (B)
= cos[ / ) (X)dX+a) x] where the double-frequency term is (C)
t-T not passed by the IF system.
= cos[0i(t,x)+w x]
o
where m (t) = K.m(t)
m. f
t
and 0.(t,x) = / w (X)dX
(E)
(F)
i
0^(t,x) may be expanded in a Taylor series about x=0 as
a) (t)x (I) (t)x2 (t)x3
0 =
1
m
m
m
1!
2!
3!
(7.1-3)
as long as OjCtjX) is analytic [8, p.686]. This requires that all
derivatives of w (t) exist; thus we must qualify our modulation or
at least recognize regions for which (7.1-3) is not valid. For
example, the Taylor expansion for 0^ is not valid at the discon
tinuities of a sawtooth modulation or at the "corners" of a triangle
modulation where the first derivative is discontinuous. It is
important to notice that the signal e(t,x) can be expressed explicitly
in terms of either t and x or and x or w and x, where w is the
vector of infinite size,
w =
CO
m
<%

m
In fact there are any number of one-to-one transformations of the


143
g
Both Z (ft) and ^ (t) are, in general, complex quantities; thus,
(9.3-lB) may be written in two parts
^tR(t) = Re {ZE(o)) } K ^
(9.3-4)
(A)
and
^(t) = Im (ZEU) } K ^
(B)
It is plain that either of the above relations defines u) (t) and
m
K and that the spectral quantity and demodulation of the other
relation are thus dependent. In Program Three the modulation is
defined by (9.3-4A) and the necessary second-channel demodulation is
then determined from (9.3-4B).
Notice that, while we were able to obtain any bandlimited
response with very few restrictions on the modulation function in
section 9.2, we may not be able to obtain an arbitrary bandlimited
response using a given demodulation. That is, there is a fundamental
constraint which must be observed, that
K <|Jt(t)
(9.3-5)
(A)
ZE[a)m(t)]
More specifically, to be able to obtain any bandlimited response,
the restriction is
K ^(t)
(9.3-5)
(B)
Under the assumptions of section 1.5, as long as (9.3-5B) is satisfied,
any BL response may be obtained by suitably designing the modulation.


61
3.4 General Coherent Demodulator Systems
At this point we shall not detail the coherent demodulator sys
tem of Figure 3.6 except to define its structure [23]. It should become
obvious that all of the preceding structures are contained as subclasses
of this general system.
The dotted line in Figure 3.6 indicates a linkage or coherency
between m(t) and of Figure 1.4 has been shown separated into a time-variable gain and a
time-invarient- filter.
In future sections all proposed systems will be of a type which
may be represented in the form given by Figure 3.6.
FIGURE 3.6 GENERAL COHERENT DEMODULATOR SYSTEM


APPENDIX A
EXAMPLES OF CLOSED-FORM SOLUTION
Prediction of the Range Response
Example 1
= t
I ^ V>
Determine the range response
Modulation is linear:toc (t) = a>o + ^ < t <
T
Demodulation is sinusoidal: 4i(t)=cos u>i,t+0, < t <
(2o)q )T
r is the inverse of the modulation function
2ttB
T
2irB
The relationship used is
Z+(0) = ~ ^[(Dc1^)] |^)¡1(n)|
2tt
= -y" cos
[mi
(Q-ujo)T -p
2ttB +9^ ZÂ¥
1 _miT *
" B cos [2^B (nuo>+ e]
The analytic response is the Fourier transform
z+(t) = F"1 [Z+(Q)]
U1T (jdiT
jwoT+0 sin (itBt + r~) jmQt-0 sin (ttBt r)
mm 6 fc 6 .
2 >iT 2 u)jT
(ttBt + j-) (ttBt 2~)
197
to|H to|H


201
Design of the Modulation Function
Example 4 Determine the monotonically increasing modulation to yield
sin ttBt , /, \
a range response of when ^(t) = 1.
We know that Z+(ii) =
1
0
)q ttB < < (jjq + ttB
otherwise
Then
wc(t) t
/ Z+(0dfi = K / i|>t(t)dt
coc(-T/2) -T/2
where K is found as
wc(T/2)
/ Z+(n)d
uc(-T/2)
T/2
/ xht (t) dt
-T/2
a)o+irB
f 1 dt
(J0-1TB
T/2
/ 1 dt
-T/2
2ttB
T
Substituting into the integral equation above, we obtain our solution.
coc(t)
/ i dn
tOg-ltB'
wc(t)-m0+TTB
u>c(t)
2ttB
T
2ttB
T
2ttB
T
t
/ 1 dt
-T/2
(t + |)
t + U)Q
Example 5 Determine the modulation to obtain a range response of
sin ttBt
ttBt
when ij>(t) cos wit.
As in the previous example
Mn)
l
=<
0
WO irB < 2 < mg + irB
otherwise


Response Magnitude
0.212
0.0
-16
0
Normalized Delay 16 m
FIGURE 9.23
RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE USING HAMMING WINDOW


102
If the signal is stochastic, one may choose a typical record of
length T to characterize the system. That is, the typical record
is defined to be one period of a periodic function whose statistics
will approach those of the stochastic process as T becomes large.
We will assume that the process meets the requirements for convergence
in probability [19, p.69-71]. We may thus replace time averages
with the equivalent ensemble average by treating time as a random
variable. In general, E {y} will be used to denote an ensemble
z
average over variable z:
Ez(y} = / y*pz(z)dz
(7.2-6)
The notation E{x} indicates an average over all random variables.
Consider the average E{x(a,6,Y,x)y(3,y)}, where a,6, and y
are considered random variables. Then
E{x-y} = E {x*y}
p>Y
(7.2-7)
(A)
= fff x(cx,3,y,t) y(3,y) P
a,3,Y
(a,3,y) da,d3,dy
(B)
= ff y(3.y) [/'x(a,3Y>T)p (a)da]p (3,y)d3dy
a/ 3
(C)
= ff y(3,y) Ea/3,Y{X} d3dY
(D)
(E)
IF a is independent of 3 and y, then the average becomes
E{x-y} = E {y-E {x}}
P>Y .a
(F)
which is usually easier to handle mathematically.


68
Then, in order to approximate a non-causal filter response in t by a
realizable one in t, we scale and translate (delay) the response of h
to be
h(To+at-td) (4.1.1-5)
where td allows truncation of the response for t 5 0.
zout *s a sample of the output of the filter at a time corresponding to
a delay of tr.
The objection to this scheme is the slow rate of information
obtainable at the output because of the necessary quazi-fixed system.
All other things equal (bandwidth, modulation, range), this system
places more stringent restraints upon the stationarity of the target and
is, perhaps, not practical for many applications. There is also a prob
lem of obtaining variable delays xx.
4.1.2 The Delay-line IF Correlator System
Since the delay-line IF correlator system yields a response of
z2ttr) = \ R(t+tr) + R(t-tr) (4.1.2-1)
1 (A)
~ \ R(t-tr) For "large" tr x > 0, (B)
= \ zi(t,tr), (C)
we may use the same procedure of filtering as was used in the case of the
RF delay-line bystem, provided the contribution of R(t-tr) is suffi
ciently small for x > 0.
4.1.3 The Filter, h(x)
Let us now pause and measure the effect of a perfect inverse
filter h(x) on the range response. The filter has the transform


27
of certain system restrictions. Let us assume that we are restricted to
a maximum signal bandwidth B (perhaps by a maximum FM peak deviation).
Then, using the calculus of variations, we define the complex spectrum as
the optimum spectrum plus a perturbation from optimum (in an arbitrary
direction):
U(f) = U0(f)+eA(f) (2.1.3.1-4)
where U0(f) is the optimum spectrum
and A(f) is any function.
Optimum U(f) is defined by requiring a minimum of (with respect to e)
to occur at e=0 so that U(f) = U0(f). Mathematically stated
Ta
£Ta
de
e=0
= 0
+B/2 2
/ [(U0+eA)(U0*+eA*)] df
-B/2
+B/2 2
/ (Uq+eA) (U0*+eA*)df
-B/2
(2.1.3.1-5)
(2.1.3.1-6)
£a
d e
e=0
{[/U0U0*df]2/2|UQ|2(AU0*+A*U0)df
-/(U0uo*)2df-2/uouo*df'7(AU0*+A*U0)df}/D2 = 0
(2.1.3.1-7)
where D is the denominator of (2.1.3.1-6).
From multiplying both sides by D2 we obtain
2/U0Uo*df{/U0U0*df*/|Uo|2(AUo*+A*Uo)df-/(U0Uo*)2df/(AU0*+A*Uo)df} = 0
/(AU0*+A*U0){|u0|2/|U0(a)|2da-/|U0(a)|4da}df = 0
(2.1.3.1-8)


of the transform, Z (ft); i.e., there will be more points in B
from which to interpolate later in the algorithm. The value of
K is chosen to insure that the response is not sampled lower than
twice the Nyquist rate for a response having bandwidth B.
However, the extent to which aliasing occurred in the original
E E
transformation of Z (ft) to z (x) corresponds to the aliasing which
occurs when transforming the compressed response to obtain the
expanded spectrum; i.e., if the response is not essentially time-
limited, the ends of the response appear to be chopped-off when
zeros are added, contributing to ringing effects in the expanded
spectrum. The slight (30%) windowing is designed to reduce this
phenomenon. A DFT yields this expanded spectrum, the real part of
which is then interpolated at points w (t) for a vector of equally
m
R
spaced times, yielding Re (t)]}. Then tJj (t) is computed and
plotted as Re {Z [to (t) ] } I to 1 (t) I Likewise ^ is computed and
m m 1
plotted, starting with an interpolation of the imaginary part of the
expanded spectrum.
The computation may be repeated for a new set of data using
the same window function and input range scales.
9.2.2 Precautions and Assumptions
Section 9.2.1 gives mention to two inaccuracies due to
aliasing which occurs in computation of the demodulation. The first
aliasing problem occurs as a result of undersampling a bandlimited
E
(and, thus, not time-limited) Z (ft). This problem is the same as
that of section 9.1.2 for Program One, and its effects may be
controlled by controlling the window, the delay interval, or the


132
where =
R
i't + 3^
a) (t) is a monotonic modulation function,
m
j?
and Z (ft) is the Fourier transform of range response to be produced.
The processor is again that of Figure 8.2 using complex demodulation
. The primary inputs to the program are
(1) the desired range response (not necessarily bandlimited),
(2) designation of the type window function to be used
to insure that the response be bandlimited,
(3) the monotonic modulation function.
As discussed in section 6.3, when the modulation is not monotonic,
there is no unique demodulation function, but, rather, an infinite
set of demodulations which will yield the same response (under
given assumptions); therefore, we restrict the algorithm to systems
involving a unique solution those with monotonic modulations.
The input range response is entered as data pairs for
magnitude and for doppler phase desired. Any bandlimited response
theoretically may be obtained, providing the modulation p.d.f. is
nonzero over the entire modulation interval (no modulation dis
continuities) If the response is already bandlimited, a rectangular
window may be specified and the response will not be altered.
Otherwise, the window of choice may be specified: rectangular,
Hamming, or Hanning.
9.2.1 Program Flow
The simplified flow chart for Program Two is given by
Figure 9.4. The first step is the computation of Z (ft). Range


Response Magnitude
1.02
0.0
-16
0
Normalized Delay
FIGURE 9.39 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT ONE-SIDED PULSE RESPONSE
USING HAMMING WINDOW
16
3 83


40
FIGURE 2.7 POWER SPECTRA AND CORRESPONDING RANGE RESPONSES
ambiguities exist (see Section 2.1.2). In contrast the transform of the
power spectrum in Figure 2.7(C) has much lower sidelobes at the expense
of a wider mainlobe.
It will not be our objective in this dissertation to make spe
cific judgement at to which range response is "best." Indeed, the a.
propo response must be a function of many system design considerations
which include, but are not limited to, dynamic range of target reflec
tivity, noise power, desired resolution, acceptable error rates, and the
expected doppler band of frequencies. For this reason I shall mention
some of the various ways of shaping the bandlimited spectrum by windowing,
but we shall study just a few of these windows as examples.
2.2.3 Specific Windows
The simplest window is the rectangular window. We see that this
is "no window at all" when used on a function that is already zero out
side the window bounds. Thus we may consider any (essentially) band-
limited function as a function extending to all frequencies times a


10
Again, we denote the simplified math form as 63 using (1.4.1-5):
e2(t,f) ; e3(t,T) = A(x)cos[wc(t)x] (1.4.2-3)
where u>c(t) = a)Q+Kfm(t) as before.
This is the simplified result obtained when the DC term was dropped from
the expression derived for the envelope detector system. Note, however,
that any AM problems associated with the DC term are not present in the
multiplier system unless an imbalance occurs in the physical multiplier.
1.4.3 Linearity of the System
We define the system to be linear if the principle of superposi
tion applies. The systems of Figure 1.2 do allow superpositioning of tar
get influences: let a system of multiple targets be modeled by denoting
the returned signal to be
*return(t>
N
I An(xn)x(t-xn)
n=l
N t-xn
I An(xn)cos(J0t-)0x+Kf / m(X)dX)
n=l -<*
(1.4.3-1)
(A)
(B)
where N is the number of targets.
Envelope detection of the sum of a "large" signal plus small signal(s)
has been shown to equivalent to coherent detection using the large signal
as a reference. Then the product of the return(s) and reference is given
as
t N t-xn
2x(t) xreturn(t) =2cos(io0t+Kf fm(X)X) An(xn)cos (jq (t-xn)+Kf f m(A)dA)
N
= 1
n=l
t
An(Tn)C0S(woTn+Kf / m(A)dA)
t-xn
(1.4.3-2)
(A)
+ double frequency terms
(B)


92
restricted to being realizable autocorrelation functions. The form of
solution is one which may be inverted to yield a solution for the demodu
lating function in terns of modulation and desired range response trans
form. This process is discussed in the next section.
6.3 Solving for the Demodulating Function
We found from equation (5.3.2-2D) that the range response has a
bandlimited transform z(£i). When x(t) is (essentially) bandlimited to
(wq-ttB, jo+ttB) for to > 0, then Z^(fi) is supported on the same interval.
Assume that an appropriate BL range response shape, such as a window
function transform wx(t), has been chosen. Then we may shift this re
sponse shape in both positive and negative directions by specifying the
range response transform to be Z+(ft) = Wx(£2) cos Oto where tq is the
delay/advance from zero. We cannot specify a singly translated response
with Z+(ft) = Wx(fi)e*^T because Z+(ft) must be real if ip(t) is to be
realsee (6.2-12). The method of doubly translating a desired response
is one approach to determining an acceptable response. Any other method
of chosing suitable BL responses is also satisfactory.
Once Z+(i2) has been chosen, we choose an arbitrary wc(t) to
cover the same interval of frequency. (The modulation is sometimes
chosen to be a conveniently generated waveform.) Then we invert (6.2-12)
to a form explicit in ip(t) : Let
N
Mn) I z+i(o)
i=l
where
Z+i(fl) = for fi between wc(timln) and wc(timax).
Z+() = 0 otherwise.
(6.3-1)
(6.3-2)
(A)
(B)


CHAPTER IV
INVERSE FILTERING
The process called "inverse filtering" came about from a desire
to improve definition of waveforms arriving at various times and with
various amplitudes. Such a situation exists in a pulse radar situation.
The composite return signal is of the form
N
xreturn^^ = I Ynx(t-Tn)
n=l
(4-1)
As we have seen, resolution problems occur when delay times differ by
amounts comparable to the signal duration. The signal may be written as
a filter response to an impulse:
x(t) = x(t) 6(t) .
(4-2)
The object in the inverse filtering approach is to find a filter, h(t),
such that
h(t) x(t) = 6(t) (4-3)
Then one may filter the composite return to obtain
N
x.
return
(t) h(t) = l yn6(t-rn) ,
(4-4)
n=l
for which no resolution problems or ambiguities exist. In the frequency
domain
H(u)X(w) = 1 ;
h(t> ^'ixferl
(4-5)
(4-6)
62


71
The harmonic systems using non-linear modulation and the more
general coherent systems do not betray any obvious opportunities for
inverse filtering.


103
For prediction of range responses, one needs simply to
average the product S*i[i over all random variables. The two
quantities should be expressed in common variable space. An
example is that of time representation in which both and ^
are expressed in the one-dimensional space, t. Another repre
sentation might be S and where the signal and reference are
- m
expressed in the infinite-dimension w space (where w is an element
~ m
of w). In this particular case we might write the average as
E {S = E (i|> -E. ... {S }}
W W, T W <0 J 0) ,0) ,U) . ./t0 W, T
m mmmmm m~
= E -S }
0) 0) (0 T
m m m
where S = E* {S }
iii T (0 ,0) . /t0 W,T
m mm m -
(7.2-8)
(A)
(B)
(C)
That is, S
m
except for u
m
is the signal averaged over all random variables
7.3 Transformation to a Convenient Argument Space Before Averaging
As seen in previous chapters and in the literature [7,21,22],
analysis of the product and its average has often been carried
out in the time domain. The time domain may or may not provide a
convenient representation for mathematically expressing the average.
Certainly, the "probability density of time" (equation (7.2-4)) provides
no obstacle; however, the integral of the product ^ is not always
easy to evaluate. Sometimes we may make a transformation of variables
which will produce a more desirable form. One such transformation


150
(first channel) and sine (second channel) demodulations, produces
the computed response given by Figure 9.12.
A system which translates the sin(x)/x response by plus and
minus 0.5 units produces a response with a broad (3 units) mainlobe
and low sidelobes as shown in Figure 9.13. The +0.5 offsets allow
destructive interference of all but the mainlobes of the translated
responses. Although it may not be intuitively obvious, it is
mathematically obvious that the dual-channel counterpart to this
system produces the translatedand undistortedsin(x)/x response
of Figure 9.14.
The system which produces the response of Figure 9.15 uses
even triangle modulation with half-cosine demodulation, which can
easily be shown through graphical mapping to be quarter-cosine in
2, i. e,
^ (J) = cos(-| ^ -|) (9.4-3)
m
Notice that the responses of Figures 9.13 and 9.15 are negligibly
aliased by Program One because of the sidelobe fall-off rate.
Figures 9.16 and 9.17 show responses which are identical, occurring
in two systems which have identical power spectra (or modulation
p.d.f.) and which use the modulation function as the demodulation,
thus producing references ip which are linear in u> Under the
m
assumptions of section 1.6, the forms of and p^ alone determine
m m
the response, regardless of other parameters.
Program Two takes a specified range response, plots the
obtainable response by bandlimiting with the specified window function,


LIST OF FIGURES
FIGURE Page
1.1 RELATIONSHIP BETWEEN MODULATION DENSITY AND POWER SPECTRUM 4
1.2 SYSTEM DIAGRAMS 6
1.3 I F SPECTRA 12
1.4 LINEAR PROCESSOR BLOCK 14
1.5 INSTANTANEOUS FREQUENCIES AND BEAT PATTERN 16
2.1 COMPARISON OF ACCURACY FOR HIGH AND LOW AMBIGUITY SYSTEMS 22
2.2 CW RESPONSES WITH AND WITHOUT ATTENUATION FACTORS 23
2.3 RANGE AMBIGUITIES 24
2.4 OBSCURING OF SMALLER RESPONSE BY LARGER 25
2.5 SIGNAL AND MATCHED RESPONSE 33
2.6 PERIODIC AND NON-PERIODIC EXAMPLES 38
2.7 POWER SPECTRA AND CORRESPONDING RANGE RESPONSES 40
3.1 THE AUTOCORRELATION SYSTEM 47
3.2 MODIFIED AUTOCORRELATION SYSTEM 49
3.3 IF CORRELATOR SYSTEM 51
3.4 PLOTS OF z2(t,tr) 52
3.5 AMBIGUITY FUNCTION (MAGNITUDE) FOR LINEAR FM SIGNAL 60
3.6 GENERAL COHERENT DEMODULATOR SYSTEM 61
4.1 EXAMPLE OF PROBLEM SPECTRA 64
4.2 PARALLEL IMPLEMENTATION OF CONVOLUTION 67
4.3 TRANSLATION OF CONVOLUTION IN DELAY TO A CONVOLUTION IN TIME 67
5.1 HYPOTHETICAL RANGE RESPONSE INDICATING CONSTRAINT POINTS 73
viii


118
"o 2nt:
1 2 -J T
= fe 2tt(2-) ) dt
i. ip m
" 2
2ir
T
2ttw .
mi
T
dw "'(fi)
mi

(C)
(D)
Of course, this is the form of equation (6.2-12) with
. 2nt
-J
^t(t) = e
(8.3.1-5)
It is notable that the demodulation is complex. This is to be
E
expected, however, because a transform Z (fi) which is real implies
E
a response envelope z (x) which has Hermetian symmetry. Directional
doppler processors, then, will necessarily employ a complex
demodulation ip. Since all physical waveforms are real, the real
and imaginary parts of ip are demonstrated in the doppler phasing and
are created by sideband filters or developed in a two channel system.
8.3.2 The Form of the General Two-Channel Processor
A general directional doppler processor is shown in Figure 8.2.
R X
The two references ip and ip multiply the IF signal and the results
are filtered to effectively remove all but the variations due to
changes in x. One channel then incorporates a Hilbert transformer
which produces a 90 phase shift in time at all doppler-band frequencies.
The second-channel response into the filter may be modeled with
the doppler phase shown explicitly in terms of either t or t as
described in section 2.1.3.3:


36
and the ambiguity function may be redefined as
T/2
Xp(t,) = S Up(t)u£(t+T)e^27r<*>tdt (2.2.1-3)
-T/2
Similarly, Instead of c(t), autocorrelation function for signals of
finite energy, we define
T/2
Rp(t) = / up(t)u£(t+x)dt (2.2.1-4)
the periodic autocorrelation function of the periodic signal Up(t). Of
course,
Xp(t,<}>) ; Xp(t,0) = Rp(x) (2.2.1-5)
e<(j) We usually normalize the volume of ambiguity to be unity (for
unit energy) as in equation (2.1.3.3-9). However, since Xp must be peri
odic in t, the normalization volume (the energy) becomes infinite so that
it is more convenient to speak of ambiguities of one period only; i.e.,
to look only at values of x in the interval (-T/2, T/2). Then the energy
of one period may be normalizedlet the average signal power be 1/T
and we make a mental note that all ambiguities occur periodically in x.
Again, as noted in the first chapter, the space attenuation factor of
the return signal makes this simplification perhaps more appropriate to
the actual application than is our original assumption of range period
icity of Xp. Periodicity may be eliminated or extended by adding low-
frequency random or pseudo-random modulation, respectively [2,p.l9l].
To formalize our approach, we observe the relationship between
Rp(x) for the periodic signal having envelope up(t) and c(x) for the
"single-period" finite-energy signal u(t):


Response Magnitude
1.0
-12 -10 -8
6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.14
RESPONSE OF TNO-CHANNEL SYSTEM USING SAWTOOTH MODULATION, HALF-COSINE AND
HALF-SINE DEMODULATION
Ui
oo


FIGURE Page
5.2 INFORMATION ELEMENTS AS A FUNCTION OF DELAY 80
6.1 INSTANTANEOUS FREQUENCY VERSUS TIME, SHOWING THE TIME ROOTS 87
7.1 GENERAL SYSTEM DIAGRAM 97
7.2 FUNCTIONAL REFERENCE GENERATOR 100
7.3 OSCILLOSCOPE CONNECTION FOR DISPLAY OF ty 109
wm
7.4 SIMPLE GRAPHICAL MAPPING TECHNIQUE 110
7.5 MULTIPLE MAPPING FOR NON-MONOTONIC MODULATION 110
8.1 PHASING-TYPE SSB PROCESSING AT RF 114
8.2 TWO-CHANNEL PROCESSOR MODEL MODELED IN THE T DOMAIN 114
9.1 COMPUTATIONAL FLOW CHART FOR PROGRAM ONE 126
9.2 TWO TYPES OF LINEAR INTERPOLATION 128
9.3 A RESPONSE AND ITS ALIASED COUNTERPART 130
9.4 COMPUTATIONAL FLOW CHART FOR PROGRAM TWO 133
9.5 INCREASING THE NUMBER OF SPECTRAL POINTS 137
9.6 COMPARISON OF ALIASED HANNING AND RECTANGULAR TRANSFORMS 140
9.7 HAMMING SPECTRUM, SHOWING NO SIGNS OF RINGING 141
9.8 RECTANGULAR SPECTRUM, SHOWING RINGING 141
9.9 COMPUTATIONAL FLOW CHART FOR PROGRAM THREE 145
9.10 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION
DC DEMODULATION 154
9.11 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION,
THIRD-HARMONIC COSINE DEMODULATION 155
9.12 RESPONSE OF TWO-CHANNEL SYSTEM USING SAWTOOTH MODULATION,
THIRD-HARMONIC COSINE AND SINE DEMODULATION 156
9.13 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION,
HALF-COSINE DEMODULATION 157
9.14 RESPONSE OF TWO-CHANNEL SYSTEM USING SAWTOOTH MODULATION,
HALF-COSINE AND HALF-SINE DEMODULATION 158
ix


9
Then the convenient mathematical approximation 63 is derived as
e3(t,x) = A(t)cos(o)0T+Kf / m(t)dX)
t-T
(1.4.1-6)
(A)
= A(-r)cos(u)oT+Kfm(t) / dX)
t-T
(B)
= A(x)cos 01c(t)T
(C)
where
mc(t) = mo+Kfm(t)
We see that u)c(t) is the instantaneous frequency of x(t);
0)c(t) = (mot+Kf /m(X)dX) = uo+Kfm(t)
(1.4.1-7)
1.4.2 Coherent Detection
The signal from the multiplier of Figure 1.2(B), assuming com
pletely isolated antennas, is
t
2x(t)A(T)x(t-x) =2A(x)cos(w0t+Kf/ m(X)dX)
00
t-T
cos(a)Q(t-x)+Kf / m(X)dX)
t
= A(x)cos(toot+Kf f m(X)dX)
t-T
t t-T
+ A(T)cos(2wot-(jjQT+Kf/m(X)dX+Kf f m(X)dX)
(1.4.2-1)
(A)
(B)
The second term is centered at 2oq in frequency. The low-pass (LP) fil
ter is designed to pass
t
e2(t,x) = A(x)cos(wox+Kf f m(X)dX) (1.4.2-2)
t-x


48
resolution will be determined by R(t), we shall be discussing the range
response z(x), which does not include the amplitude characteristic, where,
for the autocorrelation system,
zo(T) = a(t) zatten. ~ (3.1-2)
The range response zq being the transform of the signal power spectrum
in this system does not depend on any of our previously stated assump
tions, except that the target be assumed (quazi-) stationary. The range
response must conform to all of the properties of a realizable autocor
relation function, some of which are listed below [l9,20]:
1. R(0) > |R(t)I
b b
2. / / g(t)R(t-s)g*(s)dtds > 0 for any g
a a
(R(t) is a positive definite function).
3. P(f) = F[R(t)] > 0
(The power spectrum must be non-negative and real.)
4. R(x) = R*(-x)
One of the most obvious and also the most serious objections to
such a range response is that its maximum occurs at x=0 or zero range.
There is the possibility of implementing a range response that has other
peaks in addition to the zero part, but the high peak at zero, coupled
with the high gain A(0) may present problems from close targets of rela
tively insignificant physical size.
A variation of the autocorrelation system employs an RF delay
line to produce a displaced autocorrelation function, zi(x,xr)=R(x-xr)
where xr is the delay of the delay line (Figure 3.2). Such a range


17
drawn-out and Tozzi offers no easy or closed-form solution for the gen
eral-case modulation.
Using the assumption of "small x" we will develope relations
which will predict the range response of a more general coherent system.
The single line response becomes a special case; likewise, general modu
lation functions present less of a problem to analysis (Chapter VI).
1.5.2 Assumption of High Dispersion Factor
As noted previously, the dispersion factor is defined as
D = B T (1.5.2-1)
where B is the signal bandwidth
and T is the modulation period.
(_
Resolution is determined, by-and-large, by B. Woodward gives good intu
itive reasoning in the choice of the time-resolution factor which he
shows to be a measure of the signal's frequency "spread" or "occupa
tion" [3]. In examining the magnitude of D, we will first assume that
we desire resolution of at least 10% of maximum range. As bandwidth is
inversely related to resolution we have
B >
10
.10 xmax Tmax
(1.5.2-2)
(A)
As stated previously, the period of modulation must be largeat least
ten times,saywith respect to the largest return times expected:
T > 10 Tmax
Then
D = B T >
10
lmax
10 Tmax = 100
(1.5.2-2)
(B)
(1.5.2-3)
For periodic modulation, spectral lines are spaced 1/T apart. Then in an


Ill
The approach has been described by Bartlett and Mattox [29]
for the expression of range response in terms of displaced auto
correlation functions when x is small. The demodulation, expressed
as a function of modulation may be expanded in an exponential
Fourier series which is valid over (-irB,7rB). Each exponential
term in the frequency domain represents a displacement in the delay
domain. That is, the complex envelope of the range response is
-i -jnx to
E, \ nlr V o m / \i
z (t) F { l cn e P(o (n)}
n m
= y c R (x-nx )
u n u o
n
jnx m
where xl> = / c e -rrB < m < xB,
r(i) n m
m n
R^(x) is the signal complex envelope autocorrelation, and the interval
- 1
of expansion is .
o
Representation in this form clearly relates the response to the
demodulation form and the modulation p.d.f. (since, under the
t r 7
m
high-index assumption R = F ^{p }).
m


Re {Z^) }
O
0.
308 ~
ttB
O
Frequency
irB
FIGURE 9.44
TRANSFORM OF RANGE RESPONSE GIVEN BY
FIGURE 9.21
188


K(t)
5.9
SECOND-CHANNEL DEMODULATION IS ZERO
0.
-0.80
-T/2
0
Time
T/2
FIGURE 9.37
FIRST-CHANNEL DEMODULATION TO OBTAIN
RESPONSE OF FIGURE 9.36
FOR SAWTOOTH MODULATION
181


S'
u.
H
c-
60
to-
S.
CO
t-
A
t:
O
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
(B) RANGE RESPONSE PHASE
FIGURE 9.10 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION, DC DEMODULATION
154


Response Magnitude
1.11
0.3
-16
-3
0
Normalized Delay
FIGURE 9.34 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE RESPONSE USING
RECTANGULAR WINDOW
16
178


oooooooooooooo
6
88
10
8
IF (TMOD(d)-TMODCd-
WRITE(6,102) J
RETURN
IMINUSEO-J
dUMP=l
GO TO 10
IMIN(ISEC)=d-l
JUMP=0
CONTINUE
IMAXC I5EC)*=NPTM
RETURN
END
1))
4, 5/G
SUBROUTINE INPUTCORD,ABSC,NPT)
THIS SUBROUTINE READS THE NUMBER OF PAIRS WHICH WILL BE SUPPLIED
AND THEN READS THE PAIRS, ORD()|ABSC(). ALL OF THE ARGUMENTS ARE
OUTPUT ARGUMENTS AS FOLLOWS:
ORD IS THE ONE-DIMENSIONAL' ARRAY WHICH DESCRIBES
THE ORDINATE OF THE INPUT PAIR AFTER NORMALIZATION.
IT MUST BE DIMENSIONED TO A SIZE OF 101 IN THE
MAIN PROGRAM.
ABSC IS THE ARRAY OF ABSCISSAS OF THE DATA PAIRS. IT MUST
ALSO BE DIMENSIONED TO 101 IN THE MAIN PROGRAM.
NPT DESCRIBES THE NUMBER OF DATA PAIRS SUPPLIED.
DIMENSION ORD(101),ABSC(101),LTRL(77)
100 FORMATO 3)
101 FORMAT(2F3.3)
102 FORMATO1THERE HAS BEEN AN ERROR IN ENTERING POINT,13,*.1)
98 FORMAT(I 3,77A1)
99 FORMAT(77A1)
15 READ(5,98) MPT,LTRL
IF (NPT) 30,35,40
35 RETURN "
30 WRI TECG,99) LTRL
GO TO 15
40 CONTINUE
READC 5,101)ORD(1),ABSC(1)
DMAX=ABS(ORD(1))
DO 10 d=2,MPT
READ(5,101)ORD(d),ABSC(d)
IF(ABSCCJ).LT.ABSCCd-1)) WRI TEC 6,102)J
DMAX=AMAX1(DMAX,ABSCORDC d)))
10 CONTINUE
T=ABSCCNPT)-ABSCC1)
TM1N=ABSCC1)
DO 20 d=l,NPT
IF(ORDCd).EQ.O.O) GO TO 19
ORD(d)=ORDCd)/DMAX
19 ABSCCd)=(ABSCCd)-TMIN)/T-0.5
20 CONTINUE
RETURN
END


32
measure a total "combination" ambiguity for the signal. The double inte
gration yields a particularly interesting and profound result [2]:
f / |x(x,) |2dxd<¡> 1
(2.1.3.3-9)
00 00
for signals of unit energy. Thus, the two-dimensional analogy to total
ambiguity along the x-axis or (¡>-axis is not at our control as are T^ and
F^. That is, although we can control ambiguities along both of the axes,
we may not define the ambiguity everywhere in the range-doppler plane be
cause of the restriction of unity volume of total ambiguity. All that we
may do in signal design is to control its distribution. Rihaczek [l2]
addresses the problem of signal design with emphasis on choosing the
ambiguity function that best suits the application.
Oftentimes, practical design considerations will dictate the
waveshape of the transmitted signal. If a filter matched to the trans
mitted signal is used at the receiver, the output will be of the form
x(t-x)e^1T<>t x*(-t) = I x(A-x)e**2lT<^x*(A-t)dA
(2.1.3.3-10)
(A)
/ e-J 2fTueJ 2* dX
-CO
(B)
= /u(A-x)u*(A-t)ej2^Xej27Tf0(tT)dA
(C)
- e-j27^^ X2(x-t,)
(D)
which is the ambiguity function times an RF phase function. Note that
the matched receiver is designed for a maximum peak-signal to average-
noise-power and does not necessarily provide the best resolution, espe
cially if the signal has not been optimally designed [13]. A simple


88
and
z+(n)
2 ^ tiraax
I f ^(ti)6[n-o)c(ti)]dti
timin
(6.2-9)
(A)
2ir N toc timax _-,i
I f '('[ci (y)]5(G-y)uci (y)dy
i-1 10 c timin
(B)
N
~ I i'Uci 1(fi)] ^[ci 1(fi^p[i5_uc(timin^'>J^-Wc(timax)]>
i=1 (C)
where
U(x) =
1 x > 0
[o x < 0
(6.2-10)
Note that if wc(t) is monotonically increasing over (timin>timax) then
)ci ^(0) is also monotonically increasing and wci ^(fi) > 0. Then
since aic(timax) > MP(t|^n), y[0-)c(timin)]-n[fi-a)c(tlmax)] > 0. Like
wise, for u>c(t) monotonically decreasing over (timinjtimax^
a)ci 1(fi) < 0 and y[-to(timin)]-y[-to(timax)3 0. Thus
^2[)ci~1(2)l^[fi)(t:imin^"lJ^-w(timax)]} > 0 (6.2-11)
and we may write (6.2-9C) as
Z (0) -4jr l
i=l
N
*[
(l)
-1
Cl
(fi)]
_d_ -1,0.
d ci W
(6.2-12)
where ip[o)c ^] = 0 where does not exist.
It is seen that the transformation to this form is conditioned on our
being able to express the IF signal as
e(t,x) = cos[mc(t)x]
9
(6.2-13)


I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Engineering.
Leon W. Couch, Chairman
Associate Professor of Electrical
Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Engineering.
Raymond C. Johnson, Jr.
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Engineering.
Donald G. Childers
Professor of Electrical Engineering


59
Thus the range response of the n-system using coherent demodulation when
the reference is defined as
eR(t) *> cos (~ffi + 8)
(3.3-9)
with reference phase 0
becomes
Z3E(t)
! T/2 3-
T f I Yn e
. 2imt
T ,2-imt ,
cos (- h 0)dt
-T/2 n
(3.3-10)
(A)
T/2
= ^ f 21Yn! cos (-^¡r^ + ZYn) cos (-^- + e)dt
1 -T/2 1 1
(B)
= 1YnI cos (zyn 0) (C)
= |x2(-T, ~) cos (X2 0) (D)
If 0 = 0 and t << T, then
z3E(t) = Re{X2(-T, S)> (3.3-11)
and any departure from the terms of the small x assumption represents a
phase error of imx/T radians.
For the special case of linear modulation we obtain from
(2.2.1-5), (3.2-4), (3.3-1) and (3.3-11)
z3E(T) s Re X2(-t, -) (3.3-12)
(A)
1 v / 2 7TX1 /\ \ jL .. / 2 Tin \ > i
~ ~2 2" >0) (B)
One sees that the ambiguity function at = n/T is the sum of displaced
= 0 functions. This property comes from properties 10./11. of the
ambiguity function (Section 2.1.3,3); the ambiguity function is sheered'1


52
corresponds to non-causal situation), responses appear for x of either
sign. Any response Z2(t>tr) for x < 0 is of no importance to us. The
portion of the advanced response, R(x+x^) which affects for x > 0 is
the response of j R('x) for x > xr. For most responses, as xr increases,
the effect of -j R(x+xr) falls off rapidly enough to approximate Z2 by
ZOt.Tr) of the previous section. For R(x) monotonically decreasing in
1x|, we may always find a value of xr such that for any desired e,
/ ¡z!(x,xr)-2z2(x,xr)|2dx < e(xR) (3.2-5)
provided R(x) is square integrable.
Figure 3.4 shows the effect of interfering "positive" and "negative" re
sponses for large and small xr. Of course, when xr = 0,
z2(t,0) = R(x) = z^XjO) = z o (x)
(3.2-6)
and the reference beat waveform is eR(t,0) = 1.
0
(A)LARGE xr
(B)SMALL xr
(C)xR = 0
FIGURE 3.4 PLOTS OF z2(x,xR)


The analytic response is the Fourier transform of Z+(ft):
trB j (w i
" TA26
e + xf> sin(T SJ1'6
(t p.)B
a2
And the real response becomes
z(x) =
0)1
ttB sin ttB(t + { a Bu
Ai cos (o)_ i + 0 t*1-)
an A1
B(T + aT>
TA,
0)1
. ttB sin ttB(t + -T)
+ -^rr A?
TA
0)1
ttB(t + j-)
cos
(o) T + 0 7^-)
o A2
1,
irB sin ttB(t 7,) Bk
- tT ^1 cos(o,oT e + jg
TA-
ttB (t 7^)
0)1
ttB sin ttB(t 7)
2 wi.
COS (0) T 0 +
O A2
ttB(t 7 )
a2
We see this response as the superposition of basic response en
velopes shifted in range and in doppler phase. This generalized
response reduces to the special cases of sawtooth modulation and
symmetrical triangular modulation when tQ = -j and tQ= 0, re
spectively. Furthermore, if oij = the system becomes a
harmonic system.


90
e(t,x) = |w[uc(t)]| cos wc(t)x
(6.2.1-4)
which is equivalent to multiplying the unwindowed IF signal by
*KO W[wc(t)].
Examining the other factor of Z+(ft) we note that
Z+ -f-
(6.2.1-5)
when the system is of the autocorrelation type (Kt) = 1) and the modu
lation is monotonic. Then, by (6.2-3)
\ z+(n) P () for ft > 0
and
p(£> *
1
T
i dfti
1 dt1
for > 0
, l/o\ 2ttB 2ttB ,
where t = mc (0), wq < y + o >
ij>(t) = 1, and o)c(t) is monotonic
(6.2.1-6)
(6.2.1-7)
P(f) = P(-f) for all f
(6.2.1-8)
Except for the multiplier, ir, (6.2.1-7) is the relation for the p.d.f. of
ft in a transformation of variables ft = wc(t) [19, p.34]. The variable t
has a uniform density of
1
T
pt(t) = <
T T
- y < t < 2
otherwise
while ft has density
Pfl(fi)
1
T
idfti ,a)c(tmin) £ ^c^max)
'dtl
(6.2.1-9)
(6.2.1-10)


8
t
cos(tOQt+Kf/m(A)dX+0) (C)
s [l+A(T)cos(a)oT+Kf / m(A)dA)]cos(cQt+Kf / m(A)dA+0)
t-T
CO
if |A(t)I << 1
(D)
where
t
-A(x)sin(iooT+Kf / m(A)dA) "
t-x
0 = Tan
-1
t
l+A(T)cos(wQT+Kf / m(X)dX)-
t-T
An envelope detector yields the intermediate-frequency (IF) signal e,
which is a function of time and delay. A basic form will be denoted e^
and is explicit as the envelope of the expression in (1.4.1-2D):
t
ei(t,t) = 1+A(x)cos (ojgT+Kf / m(X)dX)
t-T
(1.4.1-3)
At this time we shall not be interested in the DC term of unity as it
carries no information about the target. We must remember, however, that
the term is derived from the amplitude of the oscillator, and that any AM
noise or modulation will be directly demonstrated in this term. The sig
nal of interest, denoted e2, is gotten by blocking the DC (no AM assumed)
t
e2(t,T) = A(t)cos(wgT+Kf I m(X)dX)
t-T
(1.4.1-4)
Under the assumption of a slow modulation with respect to return times,
we may consider m constant over (t-T,t) such that, for the integrand of
(1.4.1-4),
m(A) t m(t-x) 2 m(t) for X over (t-T,t)
(1.4.1-5)


band limited to the interval (-ttB.ttB) radians/sec. Note that we do
not consider NL modifications of the two-channel system. The
reader is referred to [23] for a general analysis of both the
linear systems described here and systems which multiply the outputs
of the two channels.


56
. 2irkt
. 2iTkt
eR(t,TR) = ak e
-J-
+ a
-k
,2irkt x
COS + )QTR)
(3.2-17)
(A)
(B)
Under such circumstances, there is no need to construct eR using the de
lay technique: a simple coherent sinusoidal generator will suffice.
These types of systems, for which eR is a coherent sinusoid, are called
harmonic processing systems [2l].
3.3 Harmonic Processor Systems
We have just seen that, given a linearly modulated FM system,it
is possible to design for a range response displaced to tr by detecting
a single harmonic line of the IF [2l] such that
27rn
tR KfT
where the harmonic detected is at f = n/T.
(3.3-1)
It is seen that n is the number of cycles contained in the signal beat
pattern, i.e. in e(t,x), during a period T when t = rR. Also, the in
stantaneous difference frequency in transmitted and received signals is
Mt)-u>c(t-x)
2tt = T 5 1 TR
for sawtooth modulation.
Such harmonic systems or "n-systems," as they are often called,
are versatile in a number of ways:
1. A displaced range response may be obtained
without the use of delay lines.
2. The n-system is equivalent to a delay-line
IF correlator system when the modulation is
linear.
(3.3-2)


51
FIGURE 3.3 IF CORRELATOR SYSTEM
Similarly, the output of the second mixer is expressed
eR^t^R) = cos[a)(t)-rR] (3.2-2)
The output of the third mixer is
y0(t,x,TR) = j cos[c (t) (t+tr)] + j cos[w (t) (ttr)] (3.2-3)
and after time-averaging (effected with the last filter) we obtain
z2(t,tr) = -j R(t+tr) + y R(t-tr). See (3.1.-1A) (3.2-4)
This sum of delayed and advanced responses is the response to a matched
system: the beat waveform, e(t,r), is perfectly correlated with the ref
erence IF signal, eR(t,rR) for x = xR. Since neither signal (e3 or eR)
distinguishes between positive delay and negative delay (which, of course,


63
Naturally, signal-to-noise may be expected to suffer because we employ
a filter with transfer function l/X(oi) rather than the matched transfer
function [24],
H(to) = X* (to) (4-7)
assuming white noise.
Actually, since the delta function is not realizable, requiring infinite
bandwidth and power, the inverse filter is compromised so that
x(t) h(t) = g(t) (4-8)
(A)
H(u) X(w) = G (to) (B)
where g(t) is to be specified, hopefully being a very peaked, low-side-
lobe type function. Since
H(to)
G(to)
X(to)
(4-9)
We must be very careful to provide zeros of G(to) to coincide with those
of X(o)). An example of the problems associated with nulls is given by
the spectra of Figure 4.1(B) [25]. Especially if X(to) were to be band-
limited, G(co) would have to be bandlimited to the frequency interval on
which X(u)) was supported. Otherwise H(to) must have infinite gain at a
set of connected jpoints, yielding an infinite noise power out of the
filter.
One sees that the design problem here rests entirely with the
specification of g(t). Gaussian forms have been suggested for use with
time signals which are not bandlimited. Childers and Senmoto [13]
specified a measure of performance based on signal-to-noise ratio and re
solution, with o (standard deviation of the output pulse) and minimum
epoch times, x (the separation between return signals), as parameters.


w
the vector of the FM modulation and its derivatives
W(w)
a window function (various subscripts)
(0
radian frequency
u0
center frequency (radians/sec.)
wc(t)
instantaneous radian frequency
instantaneous frequency modulation (% = Wc-a)o
return^)
instantaneous frequency of the return signal
x(t)
the RF signal
^eturn^
the delayed RF signal from the target(s)
X(t,4>)
ambiguity function (subscripts indicate variations in form)
Xp ambiguity function defined for periodic signals
y
the output of a coherent single-sideband processor
yo
the output of the third mixer in a delay-line IF correlator
Xh
the output of the second channel of directional doppler
processor
z(t),z(xjt) range response (various subscripts indicate different
systems)
zE(T)
complex envelope of the range response z(t)
z+(t)
the upper-half-plane-analytic range response;
z(x) = Re{z+(t)}
Zi(r)
an information element which varies with delay
z
the vector of information elements z-l(t)
zR(x)
output of the first channel of a directional doppler
processor
zI(t)
input to the second channel filter in a directional
doppler processor
'f'a.e
the demodulation written as a function of a and 6
demodulation as a function of time; also written Ct)
m
a linear combination of elements of i{j
xv


79
zout
Lm
" f zoutme~JnTdT
oo x/2
= / 4' / COS [aic(t)T]^(t)dte^nTdT
1 -T/2 m
T/2 oo
= 4 / dt iiC(t) / cos o)c(t)x e dx
T -T/2 m -
T/2
= t ^(t) 4 [ T _T/2 m 2
(5.3.2-2)
(A)
(B)
(C)
(D)
The delta functions are non-zero only when £2 = a)c(t). Thus if wc(t) is
limited to then Zoutn (£2) is bandlimited to (£2} ,82)* For the es_
sentially bandlimited signal such that £22~£2i = 2irB, Z0utn/T^ is band-
limited to a bandwidth B, and the number of independent points per unit
time is 2B for a total of
N 2BTmax
independent points over (0,xmax).
(5.3.2-3)
5.4 Choice of Constraint Times
One sees that the system may be defined in terms of its trans
formation matrix H = G ^ operating on information vector z. Once the
set of information elements have been determined, Gand thus Kdepends
upon the (xi) by (5-2A)
To illustrate the problem concerning an optimum choice of {x-j_}
we advance the following example: The information vector contains four
elements each of which is given as a function of x by Figure 5.2.


67
FIGURE 4.2 PARALLEL IMPLEMENTATION OF CONVOLUTION
FIGURE 4.3 TRANSLATION OF CONVOLUTION IN DELAY TO A CONVOLUTION
IN TIME


Response Magnitude
1.02
0.0
-16 -303 Normalized Delay
FIGURE 9.36 RESPONSE OBTAINABLE BY BANDLIMITING THREE-UNIT EVEN PULSE RESPONSE USING HAMMING
16
WINDOW
180


141
FIGURE 9.7 HAMMING SPECTRUM, SHOWING NO SIGNS OF RINGING
~ i
FIGURE 9.8 RECTANGULAR SPECTRUM, SHOWING RINGING


xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008247200001datestamp 2009-02-24setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Improvement of the range response of short-range FM radarsdc:creator Mattox, Barry Graydc:publisher Barry Gray Mattoxdc:date 1975dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082472&v=0000114090967 (oclc)000580854 (alephbibnum)dc:source University of Floridadc:language English


Response Magnitude
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.12 RESPONSE OF TNO-CHANNEL SYSTEM USING SAVTOOTH MODULATION, THIRD-HARMONIC
COSINE AND SINE DEMODULATION
156


131
interested. In fact, responses that are desirable (in that they
£
produce low sidelobes) are already inherently "windowed" in Z (Q).
g
The aliasing of Z (£2) is due to the "energy" in the response which
falls outside the range of normalized delay, £ Bx £ ^ ,
where NUMBER (a constant to be specified in Program One) is the total
number of Bx units in the range response output plot. Thus, the
aliasing problem may be reduced by increasing NUMBER (assuming
response sidelobes ultimately continue to fall off). This is the
£
same as sampling Z (£2) at a higher rate, since NUMBER is also the
£
number of complex sample points of Z (£2).
Aside from increasing Bx range so that aliasing in the regions
of interest is negligible, we may attempt to use insight in discerning
what the effects of aliasing have been. For example, if one can
determine that aliasing is purely constructive (in-phase) and the
response is even about Bx=0, he has a good idea of the magnitude
of the response near the ends of the response
(BT= number
). Since
we are assuming a small-x model, oftentimes we can increase NUMBER
sufficiently to be able to disregard aliasing for low Bx. The
limitation, of course, on increasing NUMBER indefinitely is that
the number of points to be manipulated, transformed, interpolated,
etc., is proportional to NUMBER times the number of points per unit
Bx to be plotted.
9.2 Program Two Solution of the Demodulation Function
The second program implements a numerical solution of
*t(t) = Zc[a)m(t)]
dw (t)
m
dt
(9.2-1)


12
|E(2irf,
|E(2irf,
FIGURE 1.3 IF SPECTRA
dotted envelope. This envelope is found by taking the magnitude of
Ei(w,t), the Fourier transform of e3 over one period:
Ej (oj,t)
T/2
/
-T/2
T/2
f
-T/2
e3(t,t)e ^utdt
A(T)cos[o)c(t)x]e ^Wtdt
(1.4.4-1)
(A)
(B)
The continuous spectrum Ei(o),t) describes the magnitude and phase of the
lines of IF voltage spectrum E(u,t); spectral lines occur at zero and at
all multiples of 1/T Hz. The beat waveform will also be called the


4v(t)
4.3
SECOND-CHANNEL DEMODULATION IS ZERO
-T/2 0 Time T/2
FIGURE 9.27 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.26 FOR SAWTOOTH MODULATION
171


7
The signal voltage is also attenuated by a factor of
A(x) = G2Li(t)L2(t)Kr (1.4-2)
where G2 is the contribution of antenna gains,
2d
Li () is the loss in the media at a distance d,
2d
L2() is the space loss,
and Kr is the target reflectivity (overall).
Assuming the antenna behaves as a point radiator, and the target, some
thing intermediate to the extremes of a point reflector and an infinite
plane, the space loss will vary between (K/t y2 and (K/t )% with
(K/t3)^ often taken as a design estimate for the "average" target.
1.4.1 Envelope Detection
The signal is normalized and written as
t
x(t) = cos(o)Qt + Kf / m(X)dX) (1.4.1-1)
00
where Kf is the FM modulation constant in rad./sec-volt.
Then the voltage at the antenna is the sum of the signal and the delayed,
attenuated return:
t t t-T
x(t)+A(x)x(t-T) = cos (mgt+Kf/m(X)dX)+A(x)cos ((DQt+Kf /m(X)dX-UQT+Kf / m(X)dX)
OO 00 t
(1.4.1-2)
(A)
t t t
= cos(uiot+Kf/m(X)dX)+A(T){cos(o)ot+Kf / m(X)dX)cos(coi+Kf / m(X)dX)
oo oo t"T
t t
sin(ajQt+Kf / m(X)dX)sin(tooT+Kf / m(X)dX)}
-oo t-x
(B)


38
(A) SIGNALS (B) AUTOCORRELATION
T 2t0
FIGURE 2.6 PERIODIC AND NON-PERIODIC EXAMPLES
Equation (2.2.1-7E) may be re-written
00
Rp(t) = c(x) l 6(T+nT) (2.2.1-10)
n=-
Transforming both sides we obtain
CO
Pp(f> = C(f) Y l fi(f-f) (2.2.1-11)
n=-
We will consider a power spectrum P(f) which is proportional to the
envelope of Pp(f) and which corresponds to the non-periodic autocorrela
tion and range response:


Response Magnitude
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.16 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION AND DEMODULATION
160


21G
R(HFFT+1)0.
R(NFFT+2)=0.
DATA ALPHA(4)/'R '/
C FOURIER TRAMFORM TO OBTAIN THE UNWINDOWED "SPECTRUM."
CALL IIARMR, IIA, I N,S,1, I LR)
IF(IER.NE.O) WRITECG,100)1ER
100 FORHATC ERROR IN HARM SUBROUTINE, TYPE ',12)
C WINDOW THE TRANSFORM TO OBTAIN A BANDLIMITEO SPECTRUM
CALL VM MDOVK R,2 *NFFT,NB,NWI ND)
DATA ALPHA(5)/'MOD Fll V
CALL HARM(R,HA,IM,S,-1,IER)
IF (IER.NE.0) HR ITE(6,100) IER
C COMPRESS RANGE RESPONSE, THUS EXPANDI NO SPECTRUM FOR BETTER INTERP
K=(NFFT/(NB*2))*2
KS=1+K
KKl=2*NFFT+2
KK2=KK1+1
NFFTE=(NFFT/K)*K
MFFTM=(7*NFFTE)/10
XWIN=3.141593/(NFFTE-NFFTM)
Jl = 3
DO 7 J=KS,NFFT,K
WIMD0=1.0
IF (J.GT.NFFTM) WlNDO=0.54+0.4G*COS((d-NFFTM)*XWIN)
Z(dl)=R(d)*WI NDO
Z( Jl + 1 )=R( J + l) *171 NDO
Z(1026-J1)=R(KK1-J)*WIHDO
Z(1027-dl)=R(KK2-d)*WINDO
7 Jl=Jl+2
Z(1)=R(1)
Z( 2)=R(2 )
KK2=1027-dl
DO 8 J=Jl,KK2
8 Z(d)=0.
DO 10 J=l,NFFT
A=R(NFFT+d)
R(NFFT+J)=R(d)
10 R(d)*A
CALL MAGANG(R,2*NFFT)
CALL MATPLT(R,ALPHA(2),2*NFFT,1,2*NFFT,2)
CALL MATPLT(R, ALPIIA 3 ), 2*NFFT, 2,2*NFFT,2)
DATA ALPHA(2), ALPHAC3) / 'MAG R.L.','ANG R.L.V
NZ(1)=9
NZ(2)=0
NZ(3)=0
CALL HARM(Z,NZ,IN,S,1,IER)
IF(IER.NE.0) WRITE(6,100)IER
KK1=(NB*K*128)/NFFT
NK=KK1*'2 + 1
KK2=1024-NK
KK3=KK1+1
DEL=1./(NK-1.)
C SET UP THE TABLE OF SPECTRUM VALUES FOR THE BANDWIDTH OF INTEREST
DO 3 d=l,NK
3 ABSC(d)=-0.5+(d-l)*DEL
RMAX=Z(1)
DO 4 d=l,KK1
R(d)=Z(KK2+2*d)
RMAX=AMAXl(RMAX,R(d),Z(2*d+l))
4 R(d+KK3)=Z(2*d+l)
R(KK3)=Z(1)
CALL INPUT(DM0D1,TDMOD,NPTD)
A K=1.
HI=0.01
T1=-0.5
WI=-0.5
TF 0.5


221
28.N. Abramson, Information Theory and Coding, New York: McGraw-Hill,
1963.
29. M.C. Bartlett and B.G. Mattox, "Statistical Analysis of a Class
of IF Correlator FM Ranging Systems," University of Florida
Report No. HDL-TR-039-3, October 1974.
30. H.P. Kalmus, "Directional Sensitive Doppler Device," Proceedings
of the I.R.E., Vol. 43, pp.698-700, June 1955.


148
because the doppler output will generally be fed to a level detector
of some sort, which is insensitive to phase. For this reason,
phase response will not be given for the other examples of this
section. Upon close examination, one will notice that the tails
of the response are aliased, showing a sidelobe response which is
larger (in this case) than actual. The effect is most pronounced
near the ends of the plot, while accuracy of the small-delay response
is quite accurate. For instance, the magnitude error is 2.9% at
a normalized delay of 1.5 (first sidelobe), 2.4% at +2.5, and 57.2%
at 12.5. These errors reduce to 0.6%, 0.3%, and 1.5% respectively
when referred to the response peak at 0.
The error of Figure 9.10(A) is high compared with errors
expected for responses of higher resolution. The sin(x)/x sidelobes
fall off only as ^ and the computed response at +12.5, for example,
is made up of the actual response at +12.5 and the aliases of sidelobes
at -12.5, 37.5, +62.5,... +(12.5+n*25),.... The phasing of aliases
depends on the period of the plot (in this case 25), which determines
the points which are aliased to any particular part of the plot, and
the phase of the response at each point.
The negative portion of the delay axis results from the
implication of mathematical relations which place no restrictions on
delay. In the physical world only positive values of delay are allowed.
The negative portion of the scale does have physical significance,
however. We relate positive and negative velocities in a dual-channel
processor to a negative or positive Hilbert transformer in the second
channel (as discussed in Section 8.3.2). It is easy to show that
negating the Hilbert transform is mathematically equivalent to replacing


69
2jqi+* (n)
H(5)=iw- (4-1*3"1)
Our system gives a response z(t) which is bandlimited; therefore, we must
design the most appropriate bandlimited function Zout(Q). A probable
choice might be one of our windowing functions Wi(ii). Thus we establish
the equivalence of windowing the transmitted power spectrum with W(2Trf)
(or /wT as the case maybe) and choosing an inverse filter of
H(fl) = F_1[h(T)]
Wi(fi)
z(n)
(4.1.3-2)
The output range response may be viewed as the sum of translated re
sponses, as shown below, where the function H(ft) has been represented by
an exponential Fourier series for jqirB < Q z oiQ+irB:
zout -
F_1[Zout(^)] = F-1[H(n)Z(i2)]
- F
,-l
00 j
l an e B Z(fi)
L n=-
I an z(t -|) ,
n=-oo
(4.1.3-1)
(A)
(B)
(C)
a result corroborated by the parallel implementation of Figure 4.2. Both
the sampling theorem [23] and the results of (4.1.3-C) would suggest that
the maximum value of AX (to assure complete expression of the desired re
sponse by 1/B.
4.1.4 Harmonic and General Coherent Demodulation Systems
It has been shown that, for integer Bt and sawtooth modulation,
the harmonic systems correspond to a delay line system with
n
tR B
(4.1.4-1)


66
The modified delay-line autocorrelation system yields a response
zi(t,tr) = R(t-tr) .
(4.1.1-1)
Then
zout(x) ¡ R(t-tr-X) h(X)dX
(4.1.1-2)
(A)
= / zi(t,tr+X) h(X)dX
(B)
T0
t / ZiCtjTR+X) h(r,dX
*'0
(C)
where the main energy contribution of h(x) is between -xq and xq,
10.
AX
1 z1(x,nAX+xR)h(nAX) AX
n =
AX
(D)
by substitution of an approximating summation for integration.
Using such an approximate convolution, a response may be shaped and made
to peak at displaced delays. If h(x) is even (Hermetian) about zero,
che delay will be xr. Notice that care must be taken so that
nAX+XR > xr-xq > 0 (4.1.1-3)
to avoid negative delays. A block diagram of this implementation is
given by Figure 4.2.
An alternative to the parallel implementation would be the
"artificial" translation of the convolution operation to real time by
scanning through xR in time. The basic system would be quazi-fixed, xr
being allowed to vary slowly in time as illustrated in Figure 4.3. The
delay is varied as
Tx T0+at .
(4.1.1-4)


Page
APPENDIX
A. EXAMPLES OF CLOSED-FORM SOLUTION 197
B. COMPUTER PROGRAM LISTINGS 203
REFERENCES 219
BIOGRAPHICAL SKETCH 222
vii


120
and where
zR(x) = E{S*^R} E{cos(w (t)x)*r|*R)
z1^) = E{S*\J>^} = E{cos(oi
c
(8.3.2-2)
(A)
(B)
and t|>R and iJj* are functions of t or 2, but constant with respect to x.
Then^Cx)
E{cos(id (t)x)ij)1}*
c TTX
= E{[cos(io (t)x)* ] ij;1}
c TTX
(8.3.2-3)
(A)
(B)
= E{-sin(aic(t)x) ip }
(C)
Substituting,
R X
z (x) = E{cos(cc(t)T) \p } +E{-sin(d)c(t) x) ip }
R t T
= Re[E{/ jiHe c }]
(8.3.2-4)
'(A)
(B)
R I
Since \p and ip are not functions of x, and t and t are considered
independent variables, the analytic signal Z+(t) is
JV R I
z+(x) = E{\|/ e } where \p= ip j tp
(8.3.2-5)
Writing the demodulation as an explicit function of o¡ and another
(arbitrary) vector of random variables x,
jO) x
Z (x) = //.../ Ip e p (x)pu (co )dx do) (8.3.2-6)
T u) X A / U) CC C / v
c c (A)
jo) X
= f\p (to) e c pu () )dw (B)
C c c c
c
where \p Ex {\p }
U) ~~ oj x
(8.3.2-7)


37
Let u(t) ** 0 for |t| > T/2 and
up(t) = l u(t+nT)
n=-
(2.2.1-6)
(A)
- u(t) l 6(t+nT) (B)
n=-
Then
T/2
Rp(f) / up(t)up(t+f)dt
-T/2
T/2 00
- / l u(t+mT) l u*(t+nT+x)dt
-T/2 m=- n=-
T/2
= / u(t) £ u*(t+nT+x)dt
, -T/2 n=-ro
00 00
l / u(t)u*(t+nT+x)dt
n=00 00
00
- £ c(x+nT)
n=-
(2.2.1-7)
(A)
(B)
(G)
(D)
(E)
See Figure 2.6 for an illustration of periodic versus non-periodic ef
fects.
We have established an exact relationship of periodic and non
periodic autocorrelations. Hopefully, c(t) will be negligible for
| x | > T/2 in which case
Rp(x) = c(x) for |x| <
(2.2.1-8)
is a good approximation. In a well-designed system this will be the case.
Our range responseA(x) neglectedcan be considered
(2.2.1-9)
because of the space-loss weighting that will eventually be imposed; A(x)
will be very small for |x| > T/2.
z(x) =<
Rp(f)


144
Meeting assumption (9.3-5B) is not an easy task in general, however,
g
because the argument of Z is the unknown function to (t) ; and, in
m
fact, satisfying (9.3-5B) presumes (in general) design of the de
modulation function.
9.3.1 Program Flow
A comparison of Figures 9.4 and 9.9 shows many similarities;
the flow charts are identical through the point of computation and
windowing of spectrum Z (ft), the BL spectrum used to compute the
modulation and second-channel demodulation. At this point the order
of computation differs, with Program Three expanding the spectrum
immediately. (The steps of Program Two concerning the input modulation
function are, of course, not included in Program Three.) The first-
R RE
channel demodulation function 'I't( t) is read, and then both 4 and Z
are integrated and K is computed as in (9.3-3). The derivative
dm
function is then integrated using fourth-order Runga-Kutta
integration (found in RKl of the IBM Scientific Subroutine Package).
Care has been taken to limit the slope of the integrated modulation
function (to a value of 30) so that, using a finite step size, a very
£
small value of Z (ft) does not cause an erroneously high rate of
integration. After integration is complete, w (t) is plotted and
m
then interpolated so that both a) and its derivative to are available
mm
for a vector of equally-spaced times. Then, as in Program Two (final
three steps), the second-channel demodulation is computed and plotted.
9.3.2 Precautions and Assumptions
All of the problems due to aliasing of the BL response and
Gibb's phenomenon which were mentioned in section 9.2.2 apply to


UUOOUUOUUOOOUUUOUUOU
204
PROGRAM ONE SOLUTION OF THE RANGE RESPONSE
DATA FORMAT FOR THIS PROGRAM IS AS FOLLOWS:
MODULATION FUNCTION
DEMODULATION FUNCTION #1
DEMODULATION FUNCTION #2
BLANK CARD (INDICATES END OF THE DATA)
"INPUT" FORMAT:
NUMBER OF DATA PAIRS, ALPHANUMERIC DATA
(IF THE FIRST FIELD CONTAINS A NEGATIVE INTEGER,
ADDITIONAL CARDS OF THIS FORMAT WILL BE ACCEPTED.
OTHERWISE, THE FIELD DESCRIBES THE NUMBER OF DATA
PAIRS TO FOLLOW.)
ORDINATE, ABSCISSA (ONE CARD PER PAIR) (2F8.3)
"INPUT" FORMAT
"INPUT" FORMAT
"INPUT" FORMAT
FORMAT (13,77A1)
REAL MOD(lOl)
REAL-8 ALPH(8)
DIMENSION TMOD(lOl),IMIN(16),IMAX(16),DMOD101),TDMOD(101),
*W(1024),T(1024),Z(1024),B(4096),TP(1024),0(1024)
DATA ALPH(1)/'MOD '/,ALPH( 3)/'MAG'/,
*ALPH(4)/'ANGLE'/,ALPH(6)/'DEMOD 91'/,
-ALPH(7)/'DEMOD #2'/,ALPH(8)/Z(I)'/
NUMBER-25
NFFT-2048
51 CONTINUE
DO 30 J-1,1024
Z(J)-0.0
80 B(J)-0.0
DO 90 d-1024,NFFT
90 B(J)-0.0
DEL-1.0/NUMBER
H-1E-3
IMTP-2
CALL MODUL(MOD,TMOD,NPTM,NSEC,IMIN,IMAX)
IFNPTM.LT.2) STOP
CALL MATPLT(MOD,ALPH(1),NPTM,1,NPTM,1)
NFRST-1
IS CALL INPUT(DMOD,TDMOD,NPTD)
IF(NPTD.EQ.O) GO TO 17
CALL MATPLTDMOD,ALPH(7-HFRST),NPTD,1,NPTD,1)
DO 10 I SEC-1,NSEC
C SET UP THE W-VECTOR FOR THIS SECTION OF THE MODULATION
JL-1. +(-AMAXl(-MOD(IMIM(I SEC)),-M0D(IMAX(I SEC)))0.5)-NUMBER
JU- ( AMAXK MOD(IMindSEC)), MOD( IMAX( I SEC) ) ) + 0. 5)-NUMBER
DO 1 J* JL,JU
W(J)0.5+(d-0.5)*DEL
1 CONTINUE
C FIND THE INTERPOLATED VALUES OF T AND T-PRIME (TP)
CALL I NTRP(TMOD,MOD, IMI fl( I SEC), IMAX( I SEC), IIITP, T,W, JL, JU, TP, H)
C FIND DEMODULATION VECTOR D, CORRESPONDING TO THE T VECTOR
CALL INTRP(DMOD,TDMOD,1,NPTD,INTP,D,T,JL,JU,W,H)
C CALCULATE Z(W)
DO 3 J-JL,JU
Z(2*J-NFRST)Z(2*J-NFRST)-D(d)*ABS(TP(J))
3 CONTINUE
C THE CONTRIBUTION OF SECTION ISEC OF THE MODULATION IS NOW COMPLETE
10 CONTINUE


APPENDIX B
COMPUTER PROGRAM LISTINGS


13
intermediate frequency waveform even though it may appear to be baseband
in the sense that the line spectra extend all the way to zero, or DC, for
the stationary target model. When the target is perturbed from the sta
tionary situation, sidebands appear about each of the lines in the spec
trum of e3 of each line. This modulation may be modelled as AM in nature. There
are, however, no restrictions that the envelope be real in a complex no
tation; i.e. SSB-AM is possible. Using the example of linear sawtooth
modulation, let
t T T
m(t) volts, y < t < y
then
Kf = 2ttB radians per second per volt
mc(t)
2ttB
T
t + ojq rad/sec.
Ei(o),t) =
T/2
/ A(t)cos
-T/2
2irBxt
e-Jutdt
A(t)
2
T/2 r
' u
-T/2
-JO
2ttBt
- U))t
+ e
. ,2ttBt ^,
1 (~ + w) t J
dt
(1.4.4-2)
(A)
(B)
A(t)T f sin(trBT- 4 l (itBT-a)T/2) (ttBt+o)T/2) J
(C)
The IF spectrum for the linear system is atypically simple; the spectral
envelope is not easy to compute in general [7].
1.4.5 The Linear Processor
Looking again at Figure 1.2, one sees that the block which first
operates on e3 is the (non-stationary, in general) linear processor. The
systems under consideration contain (stationary) linear filters and time-
varying gains as shown in Figure 1.4. The filter n(t) will typically be


0.38 -
0
-.38
-T/2
Time
T/2
FIGURE 9.29 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28 FOR SAWTOOTH MODULATION
173


CHAPTER III
SHAPING THE SYSTEM RANGE RESPONSE
In this chapter we shall detail more specifically the types of
systems which were described in general terms in Section 1.4. The break
down will include
autocorrelation systems
delay-line IF correlator systems
harmonic processor systems
general coherent demodulator systems.
We will discuss, in conjunction with each system, methods which might be
employed in controlling the range response. Our examination of systems
will not extend into the non-linear sections shown in the general sys-
stems of Figure 1.2.
3.1 Autocorrelation Systems
A block diagram and model of the autocorrelation system is given
in Figure 3.1. The linear processor is simply the doppler-pass filter.
This filter has a bandwidth which is very low compared with 1/T since the
target is assumed to be moving slowly, implying low doppler frequencies.
Thus the IF signal will be averaged over many periods giving
M/2
/ e(t,x)dt
-M/2
(3.1-1)
(A)
M-x -m/2
, M/2
lim / A(x)x(t)x(t-x)dt
(B)
(C)
46


CHAPTER X
CONCLUSIONS
Formulas have been derived for analysis and synthesis of
short-range FM distance measing systems, and techniques and limitations
of improvement of range resolution have been developed. The several
methods of shaping the response that were investigated disclosed
that the basic limitation of possible range responses was the RF
peak-to-peak bandwidth B, which corresponds to the bandwidth of the
Fourier transform of the obtainable range response z(t).
Chapter I introduces the DMS, defines mathematical models,
and delimits the class of systems by statements of operational and
structural requirements and assumptions. These imply three primary
assumptions used in most of the analysis for simplification:
(1) the target is quazi-stationary, (2) range delay times are small,
(3) the modulation is high-index or high dispersion.
Chapter II introduces some of the common measures of
resolution, interrelating the time resolution constant, the frequency
resolution constant, and the ambiguity function.. Then, using special
cases, window functions are discussed as related to windowing of
the RF power spectrum. Applicability to other special cases and to
the general coherent system is attempted in Chapter III.
Chapter IV investigates the possibility of improving resolution
via inverse filtering methods. It is found that a stringent
limitation of the range response transform to bandwidth B exists.
194


modulation function, given by representative points or data pairs
(w (t),t), (2) and (3) the demodulation functions for the two
m
channels (given in the same data format as the modulation function.
In present form, the array sizes permit up to 101 points of data
input per function. If the analysis is of a single-channel system,
the second demodulation function is most economically represented
by entering zero data pairs. (There is a datum which specifies
the number of data pairs to be entered for each function.)
More information on entering the data will be given in section 9.1.2.
The solution is based on waveforms given over one period of
the periodic modulation/demodulation or for a representative record
length (as mentioned in section 7.2) of a stochastic process.
Computation time is directly related to N, to the number of points
used to describe input functions, and to whether the demodulation
is complex or real.
9.1.1 Program Flow
The algorithm was written based on equation (9.1-1) and
its implementation reflects a straightforward sequence:
(1) obtain the modulation and break it into NSEC monot'onic
sections;
(2) read the demodulation function for the first channel;
(3) for each section of the modulation
(i)obtain the inverse modulation function u) ^(f2) ,
d -1 mi
(ii)obtain the derivative w (Q.) ,
dn mi
(iii)find the value of the demodulation at points
WmjlCfi) for equally spaced values of 2 over the
range of w(t) for the modulation section under
consideration,


55
an(t)
i T/2 j [w0T+.(Kf xt -j[u0T+(KfTt +
= 07 / e + e dt (3.2-13)
"T/2 (A)
KfiT
J)0t sin( trn)
KfXT
jc0x sin + im)
- (^V-) ~ +
Kf xT Kf tT
(5 Tm) (- + irn)
(B)
For the reference or the signal e when x = x^, we have
anC^R) = M^r) =
. ,KfXRT s . /fTRT
iJ0TR sin (2 ^ eJw0TR Sln (2+ Trn)
: +
KfTRT .
(o urn)
KfTRT
(0 + nn)
(3.2-14)
For
KfxrT = 2-rrk k = 0, 1, 2,
all an vanish except for
ejo)0XR(sgnk)
where
+1 k > 0
sgn k =
-1 k < 0
(3.2-15)
(3.2-16)
(A)
For
k = 0 ak = a0 = cos )0tr
(3.2-16)
(B)
For the choice of KfXR = 2iTk the IF reference spectrum is concentrated
in a single frequency:


41
rectangular window. Since multiplication in one domain implies convolu
tion in the other, the window transform (a time function)always some
what other than a single delta functiontends to "smear" the windowed
function's transform through convolution. This rectangular window is de-
finded as
W0(u) =
1
0
7TB < (jO < ttB
otherwise
(2.2.3-1)
In the transform domain this window is
\ sin TTBt /o o ^
w0(t) = B (2.2.3-2)
All of our windows will be defined to be zero outside the interval
(-ttB,ttB). Then it is obvious that
Wo(w) *Wi(m) Wi(u) (2.2.3-3)
where Wi is any window.
Transforming (2.2.3-3) we obtain the convolution (*)
w0(t) wi(t) = w(t)
(2.2.3-4)
These interesting results are BL analogies of multiplication by unity and
convolution with a delta function, respectively.
A more "active" window was first put into use for smoothing me
teorological data by Julius Von Hann. (Von Hann actually smoothed in the
transform domain by discrete convolution with the respective coefficients
T* \ T^e Hann^n§ window is given as [8, p.14]
W!(w) =
1
1.1 0)
2 + 2 COS B
-ttB < a) < ttB
otherwise
(2.2.3-5)
which has transform


33
example using the pulse radar signal of Figure 2.5(a) will be used to
illustrate this point. The matched response to the signal appears in
K TiSH X 2TX SH
t
(A)
FIGURE 2.5 SIGNAL AND MATCHED RESPONSE
Figure 2.5(B), having some ambiguity for a total time of 2Tj. A "receiver
consisting of unity feedthrough (just a connection) yields an output
pulse which is the same as the input pulse, with ambiguity extending over
a time width T*. Undoubtedly, there will exist some application for which
the widening of the received pulse by the matched receiver is not satis
factory.
We establish, by this example, that it is possible to re-dis
tribute effective ambiguity in the range-doppler plane using linear pro
cessing. (Notice that the effects of noise have not been considered yet.)
This principle will be important in discussing processors in a later
chapter.
We now examine the situation with which we shall be working. We
desire chiefly range information from our DMS with little or no interest
in velocity information. (Velocity information may be obtained from the
fine-structure range data; i.e., doppler cycles may be observed. As
range resolution is improved, i.e. the range "window" narrowed, the num
ber of doppler cycles which may be counted is decreased, thus decreasing
doppler resolution.) Thus we are left with one degree of freedom in


196
a more general form may be written using only the quazi-static
assumption.
Chapter VII extends the applicability of the relations
derived in Chapters VI and VII to directional doppler systems.
It is shown that, using a two-channel systems and with assumptions
(1), (2), and (3) satisfied, any bandlimited (bandwidth B) range
response may be synthesized by suitably designing the demodulation.
Providing certain conditions are met (which involves restrictions
on the demodulation function), a modulation function may be designed
to yield any bandlimited response. All responses which are uneven
about zero represent two sets of directional responses: one
direction corresponds to the positive half of the delay axis, the
other direction, the negative half.
Although closed-form solution is often possible using the
relationships developed, it is often preferable to carry out a
numerical solution for the range response, the demodulation function(s),
or the modulation function. Three algorithms are presented and their
use and fallabilities discussed in Chapter IX. Several solutions
are given to illustrate the nature of the algorithms and of the
systems. Examples of closed-form solution appear in Appendix A.
The primary contribution of this work is the derivation of
the simple and concise relationships which may be used for analysis
and design under appropriate assumptions. Previous analyses have
been performed for the types of systems described here with more
difficulty, or they have been performed for special cases. More
general analyses those using fewer assumptions exist, but, they
are not as easy to perform, and they may be even more difficult to
use for synthesis.


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF FIGURES viii
KEY TO SYMBOLS xii
ABSTRACT xvii
CHAPTER
I. DEFINITION OF SYSTEMS AND GOALS 1
1.1. Operational Constraints 1
1.2. Structural Constraints 1
1.3. Assumptions 5
1.4. Models 5
1.4.1. Envelope Detection 7
1.4.2. Coherent Detection 9
1.4.3. Linearity of the System 10
1.4.4. The IF or Beat Waveform 11
1.4.5. The Linear Processor 13
1.4.6. The Non-linear Section 14
1.5. Assumptions ...... 15
1.5.1. The Assumption of Small x 15
1.5.2. Assumption of High Dispersion Factor 17
1.5.3. The Quazi-Stationary Target Assumption 19
II. RESOLUTION AND SHAPING OF THE RANGE RESPONSE 21
2.1. The Resolution Problem 21
2.1.1. Accuracy 21
2.1.2. Ambiguity and Resolution 22
2.1.3. Parameters of Resolution 25
2.1.3.1.The time ambiguity constant 26
iv


18
RF spectrum of bandwidth B there are
B
1/T
BT = D
(1.5.2-4)
lines in the spectrum. In a later section on ambiguities, we shall show
that total time ambiguity (as defined by Woodward) is minimized in a band-
limited spectrum by requiring that spectrum to be flat (or rectangular).
Although any periodic signal produces a line spectrum rather than a con
tinuous flat spectrum, these lines become smaller and closer as T in
creases, so that, in the limit as T , the spectrum approaches a con
tinuous power density spectrum. Thus, within limits, large T seems de
sirable; and larger T implies larger D. Another factor which advocates
large T is the occurrence of a periodic autocorrelation for periodic sig
nals. In special cases, the range response differs from the autocorrela
tion only by the factor A(x). In any event, the returned signal of delay
t + nT is
t-nT-x
Xreturn^) = A(nT+x)cos[io0 (t-nT-x)+Kf f m(X)dx] (1.5.2-5)
(A)
t-x
= A(nT+x)cos[wo(t-x)+Kf / m(X)dX+0]
(1.5.2-5)
(B)
where
t-x-nT
6 = w0nT+Kf f m(X)dX = to0nT (1.5.2-6)
t-x
since the integral of zero-mean m(X) over any number of periods is zero.
Thus a signal return from a target at x+xiT varies from that at x in an
amplitude factor and a constant phase, oionT. This undesirable quazi-
periodic effect will be minimized by letting T be large enough so that
a ^T"|.nT)
the amplitude factor '£({) *s very small (n ^ 0) so that the periodic
ity may be disregarded for all practical purposes. An alternative method


106
Or the one-sided transform corresponding to z+(t) is
Z+(fi) =A J 2Z(fi>
0
£2 > 0
n o
(7.4-3)
(A)
= 2iup (£2-0) )p (£2-0) ) £2 > 0
0) o r0) o
m m
(B)
In terms of the complex envelope of the range response z (T) and
its transform,
Z v£2) = 2mj) (£2)Pa. (£2) .
%
(C)
In terms of total frequency o)=oj +o) the subscript m is dropped,
mo
and (7.4-3B) becomes
Z (£2) = 2mf) (£2)p (£2) £2 > 0
+ 0) 0)
(D)
This relation is analogous to equation 6.2-14) with jjj written in
terms of instantaneous frequency instead of time, p is the p.d.'f.
w
of the instantaneous RF frequency o).
7.4.1 The Periodic Case
In the periodic case p may be obtained by computing the
0)
p.d.f. when a simple transformation of variables has been made on
t [29]:
U) =0) (t)
c c
(7.4.1-1)
and
Pt(t) =
1
T
(7.4.1-2)
0 otherwise


30
If two signals are of the same form but differ in arrival time and are
frequency shifted with respect to each other, we may write them as
T J2TT(f0 |)t +
Xi (t) = Re{u(t )e } = Reix^
T J27T x2(t) = Re{u(t + ~)e. } = Re{x2}
* 2 7T f £
where the basic signal form is x(t) = Re{u(t)e^ 0 }
t is the difference in arrival times and
<|> is the frequency shift.
(2.1.3.3-1)
(B)
(C)
The signals x^ and x2 may be returns from two targets whose delays differ
by t and whose velocities differ by v so that relative time delay is
T = 2vt/v. The doppler frequency shift is simply the time derivative of
the doppler phase:
. 1 d 2f0v
* 27 dt 2"£t
The integral of (2.1.3.3-1A) is simply doubled if the analytic forms
xf and xt are substituted for xj and x2. Then
2*ISE = / Iu(t ^)e
00
T j2tt(fo |)t T j2Tr(f0 + -|)t
-u(t + 2")e
dt
(2.1.3.3-2)
(A)
00 CO
= / |u(t -j) I 2dt + I |u(t+-j)|2dt
CO 00
CO CO
- S u(t- -|-)u*(t + ^e^ ir<^tdt / u*(t ^-)u(t + -|-)e ^27I^tdt .
00 oo
(B)
Notice that the first two terms of (2.1.3.3-2B) represent signal energy
are thus constant; then to maximize the ISE, we must minimize the third
and fourth terms which are subtracted from the energy terms. Since


217
C
C
SUnnOUTI HE Wl HDOW( R, NR, NBB,NTYPE )
DIMENSION R(MR)
IF((NTYPE.GE.3).OR.(NTYPE.LT.O)) RETURN
NB=(NBB/2)*2
SET ALL VALUES TO ZERO OUTSIDE WINDOW INTERVAL
J1-3+NB
J2=NR-NB
DO 1 J=J1,J2
1 R(J)=0.
WINDOW THE REMAINING SECTIONS
NA=2
C=3.141592G54/(NB-NA)
RECTANGULAR WINDOWING
IF (NTYPE.NE.O) GO TO 10
RETURN
RAISED COSINE HAMMING (1) OR HANNING (2) WINDOWING
10 Al=.54
A2 = .46
IF(NTYPE.NE.l) GO TO 15
Al = 5
A2 = 5
15 DO 2 d=NA,NB,2
A=A1+A2*C0S((J-NA)*C)
R(J+1)=R(J+1)*A
R(J+2)=R(J + 2 )*A
K=NR-J+1
R(K)=R(K)*A
2 R(K+l)=R(K+l )*A
RETURN
END
FUNCTION FIJN(TX,WX)
DIMENSION R(4096),ABSC(1025),DM0D1(101),TDMOD(101),
*ARG(10),VAL(10)
COMMON R,ABSC,DMODl,TDMOD,NK,NPTD,AK,EPS,KNOT
W-WX
T=f X
IF(KNOT.NE.0) GO TO 60
W=TX
T=0.0
GO TO 80
60 IF(KNOT.NE.1) GO TO 80
W=0. 0
80 CONTINUE
1F((T.LT.-0.5).OR.(T.GT.0.5)) GO TO 20
IF((W.LT.-0.5).OR.(W.GT.0.5)) GO TO 20
INTRP=2
CALL ATSMCW,ABSC,R,NK,1,ARG,VAL,INTRP)
CALL ACFI(W,ARG,VAL,ZW,INTRP;EPS,IER)
CALL ATSM(T,TDMOD,DM0D1,NPTD,1,ARG, VAL,INTRP)
CALL ACFI(T,ARG,VAL,SCI,INTRP,EPS,IER)
FUN=AK*SCI/ZW
I F(KNOT.EQ.0) FUN=ZW
I F(KNOT.EQ.1) FUN=5CI
IF (FUN.GT.O.) GO TO 30
IF (KNOT.EQ.2) WRITE(6, 555) T,SC I,W,ZW,FUN,KNOT
FORMATC ',E10.4,4(3X,E10.4),3X,I6)
FUN=-FUN
GO TO 30
20 FUN=0.0
30 IF ((FUN.GT.30.0).AND.(KNOT.CO.2)) FUN=30.0
RETURN
END


142
E
It is at once obvious that the term real (for positive K). If we can insure that this requirement is
met, the nonlinear differential equation to be solved is
do>
m
dt
*t(t)
zE(.)
m
with boundary conditions
0) (- 7) = ttB
m 2
T
and u) (+)= + irB
m 2
(9.3-1)
(B)
(9.3-2)
(A)
(B)
when to is to be monotonically increasing. Since (9.3-1B) is
separable, we may integrate using boundary conditions to provide the
limits and solve for K as
^ ZE(£2)dn
K = (9.3-3)
/ ip (t)dt
-T/2
g
Note that we assume that z (t) is bandlimited to (rtB,irB). Care must
be taken to insure that this assumption is valid.
As in Program Two, the processor is shown by Figure 8.2.
Program inputs are
(1) the desired range response envelope (not necessarily
bandlimited),
(2) designation of the type of window function used to insure
that the response be bandlimited,
(3) the first-channel demodulation.


44
limited spectrum, then there is no need to set Wi(to) = 0 outside
(-ttBjItB); indeed, there exists no need for specifiction at all outside
this interval. Then, just as smoothing the transform of a bandlimited
spectrum by convolution with wg(t) is equivalent to convolution with
6(t), so is smoothing with Wi(t) equivalent to smoothing with
Wi'(t) = i"1[wi'(m)] (2.2.4-1)
where
00
Wi'(u) = l [W (a)+2unB) ] (2.2.4-2)
n=-oo
The transform of periodic VS(u) is, of course, a series of delta func
tions in the t domain. All of the above windowsand, indeed, all real,
even windows, if N is large enough may be expressed
N
W^(io) = J an^ cos Tjp (2.2.4-3)
n=0
for the appropriate frequency range, having transform
N .
Wi'(t) = a0i6(t) + l -f- [6(t + f) + 6(t J)] 2.2.4-4)
n=l
where {an;j_} are appropriately specified:
)
Rectangular ago = 1 no = 0> n 4 0
Hanning agi = 1; an = ; ani =0, n > 1
Hamming ap2 =*54; an = .46; an2 = 0, n > 1
Taylor (6 terms, 40dB sidelobes) [ll]
oq4 = 1; am = 0.7782308; 024 = -0.0189046
034 = 0.0097638; 044 = -0.003221; 054 = 0.0006948
The effect of the weighting or windowing in the frequency domain is


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2. A.W. Rihaezek, Principles of High-Resolution Radar, New York:
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Applications to Radar, London: Pergamon, 1953.
4. M. Schwartz, Information, Transmission, Modulation, and Noise,
Second Edition, New York: McGraw-Hill, 1970.
5. H.E. Rowe, Signals and Noise in Communications Systems,
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6. J.R. Klauder, A.C. Price, S. Darlington, and W.J. Albersheim,
"The Theory and Design of Chirp Radars," Bell System Technical
Journal, Vol. XXXIX, July 1960.
7. L.M. Tozzi, Resolution in Frequency-Modulated Radars, Ph.D.
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New York: McGraw-Hill, 1966.
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Signals of Large Time-Bandwidth Product," I.R.E. International
Convention Record, Part 4, 1961.
10. C.E. Cook, "General Matched-Filter Analysis of Linear FM Pulse
Compression," Proc. IRE, Vol. 49, April 1961 (Correspondence).
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No. AD 641391, 1965.
12. A.W. Rihaezek, "Radar Waveform Selection A Simplified Approach,"
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Systems, Vol. AES-8, No.5, September 1972.
219


N
9
p.d.f.
P(Dtt(a)
P(f)
Pp(f)
p
rave.
Ppeak env.
R(t)
Rp(t)
Sx(x,t)
t
fctrue
T
ta
T
To
Tn
tr
Tmax
u(t)
Up(t)
y
v
v
usually the dimension of a vector/matrix or the number of
terms in- a series
matrix used to transform independent elements into orthog
onal elements
probability density function
the p.d.f. of the instantaneous modulation
the signal power spectrum
the line power spectrum of a periodic signal
average transmitted power
peak envelope power transmitted
T T
autocorrelation function Rp(x) over one period < x < y
autocorrelation function defined for periodic signals
IF signal as a function of delay x and vector x
time, as measured from origin to
some hypothetical absolute time
the modulation period
the time resolution constant
round-trip delay time to target
a specified delay time
round-trip delay time to the nth target
a reference delay, usually of a delay line
the maximum delay time under consideration
complex envelope of x(t)
complex envelope of periodic x(t)
modulation index
velocity of target in direction of antenna
propagation velocity of signal in the transmission medium
xiv


3.03
u
s-/
Pi H
0
0
Time
T/2
FIGURE 9.40
FIRST-CHANNEL DEMODULATION TO
OBTAIN RESPONSE OF FIGURE
9.39
FOR SAWTOOTH MODULATION
00


127
(iv) find the product fwmi_1(n) ] wmi_1(fi) | at'
the equally spaced sample data pairs to yield a
sampled data function Z?(ft) for this section of
the modulation,
(v) add zf(ft) of this section to the contributions
of other sections;
(4) repeat (2) and (3) for the second channel if indicated;
F F
(5) transform the composite spectrum Z (ft) = X Z.(ft) to
yield the range response complex envelope?- 1
Figure 9.1 is the computational flow chart for the program.
One sees that it consists of three major loops. The first (gross)
loop is that which allows the solution of multiple sets of data
without recompilation. The second loop is executed twice for each
set of data: once for the channel of the system with input ip and
E
producing Re (Z (ft)}, a second time for the second channel with
X E
input ip and contributing to Im {Z (ft)}. Within this loop is a third
loop which computes the contribution of the system during T^
g
(corresponding to the ith modulation section) to spectrum Z (ft)
for i = ISEC, where ISEC steps from 1 to NSEC, the total number of
monotonic modulation sections.
The program was written to a great extent in modular form
for economy, ease of organization, and ease of comprehension and
debugging. The user-written subroutines are described briefly below:
SUBROUTINE MODUL reads alphameric data (if any) to
identify on the output record any desirable information
and reads the number of data pairs to follow. These
pairs (a maximum of 101) are then read and the number
of monotonic sections is determined. (A maximum of 16
sections is permitted.) If an instantaneous step or
jump is encountered, the routine will not regard the
vertical section as an additional section. The routine
normalizes and centers both the time and frequency data
so that each is contained on the interval (-0.5,0.5),
regardless of input values. It is necessary to enter
the data pairs such that the time datum is always
monotonically increasing. Two output vectors identify
the indices of the endpoints of each monotonic section.


105
At this point one may well recognize that it may be more natural
to express both S and ip in terms of frequency modulation go when
m
t is considered small:
S (go )
10 m
m
ip (go )
go m
m
COS [u) T+GO t] = cos(o) +0) )t
mo mo
COS[(0 T^+W T] = COs(c +G0 )t_.
m R o R m o R
(7.3-7)
(7.3-8)
If ip is derived from the modulation voltage it will, of course, be
GO
known as the function ip ( to m m Kf
m L
7.4 Statistically Derived Range Response Transform
Now that both signal and reference are expressed in a common
argument space go^, the range response can be computed
z(t) = / S ,t(go ,t)i|) (go )p (go JdGO-,
go m go m go m U1
m mm
(7.4-1)
(A)
f cos (go +go )tg^ (go )p (go )dG0,
m o go m go m Ul
m m
(B)
Taking the Fourier transform, we have
z(2) = S S cos (go +go ) xib (go )p (go )dG0 e Tdx
m o go m co m m
-oo m m
(7.4-2)
(A)
f\p (go )p (go ) / cos(co +(o )x e dxd(o.
go m go m mo
-m m -oo
m
(B)
= f Trip (go )p (go ) [ 6 (7GO -GO )+6(iH(0 +G0 ) ] ^GOm
Gomcom mo mo
m m
(C)
=
nip (Q-go )p (fi-Go ) 2 > 0
GO O GO O
m m
trip (-Q-U) )p (2go ) < 0
GO o GO O
m m
(D)


KEY TO SYMBOLS
A(t)
B
BW
BL
c(x)
D
DMS
d
e(t,t)
eR
E
E(oj,t)
Ei(u,t)
E()
f
attenuation factor
peak-to-peak instantaneous frequency deviation (Hz.)
RF bandwidth as defined by Carson's rule
bandlimited
the complex autocorrelation function of complex envelope u(t)
dispersion factor
distance measuring system
distance to target
various forms of the intermediate frequency signal (various
subscripts)
the IF reference signal
signal energy (no arguments)
signal energy in one period of the modulation
IF voltage spectrum for a target delay of x
the continuous counterpart of the IF voltage spectrum E(),x)
expected value of the argument with respect to a and 6
expected value of argument average over all random variables
frequency (Hz.)
center frequency (Hz.)
frequency of sinusoidal modulation (Hz.)
frequency resolution constant
xii


25
the possibility of cumulative responses from all targets. It is seen
that the mainlobe width limits the minimum separation for which two tar
gets may be resolved. Especially when the returns are of unequal
strength, one response may be completely overshadowed by another. This
effect is illustrated in Figure 2.4.
(A)
FIGURE 2.4 OBSCURING OF SMALLER RESPONSE BY LARGER
It should be stressed that, in some applications, decision cir
cuitry may be thwarted more by sidelobe ambiguities to targets than by
additive noise. As we have seen, a strong target may produce sidelobe
responses which excede the mainlobe response of a weaker target. And,
as is most often the case, target shape, material, size, or "complexion,1
all of which affect strength of signal return, may be unknown. Thus,
the sidelobe responses will ultimately determine the dynamic range of
targets which may be detected.
We shall not attempt to allude to the many ramifications of re
solution but will, instead, refer the reader to [2] and [3].
2.1.3 Parameters of Resolution
Three parameters will now be defined for a measure of ambiguity


78
excede that into the channel [28,p.106], we may model a transformation
from the time domain to other domains as a "channel" and realize that no
more independent elements may exist in one domain than in another. Thus
the maximum rank of G and the maximum size of H is approximately
N = 2Bxmax (5.3.1-1)
When the modulation is monotonic but not linear, we may consider a piece-
wise-linear approximation to the modulation. It is easy to see that,
neglecting the windowing effect of each section, we will obtain the same
number of independent points as before, the points being spaced unequally
according to the changing slope of the modulating waveform throughout
the period.
5.3.2 Dimension of H Based on the Range Response
Since we may describe the system as the single channel of Sec
tion 5.2, the range response becomes
T/2
zoutm Y _T'2 e(t,T)*'(t)dt
(5.3.2-1)
(A)
1 T/2
= / cos[wc(t)t]^(t)dt (B)
There can be no more independent points in zoutm(x) from t = 0 to
t = Tmax than there are independent vectors (t) for 0 < < Tmax be
cause z0utm a linear combination of zn. Then the rank of G can be
found through using the sampling theorem on z(t). To do this we find
the bandwidth of z0Utm(T) by transforming:


200
Designing the Demodulation to Yield Desired Response
Example 3 Design the demodulation to yield the range response shown.
Modulation is linear: to. (t) to + 2ttB
E, v
z (t)
-T
We desire the response of the figure,
which has the transform
Z(fi) = t Sin(i5 o>To sin (ft + uw)t
( a )t + To 7~TH T-2-
o' o ( Q+ to )
O TO
The response for > 0 is approximately
/r.v sin(fi to )x
Z, (ft) = 2t o o
+ o
(ft to )t
o o
when to >> B.
o
We must bandlimit the above transform by multiplication with W(ft oj )
where
W(fl) = a cos
mft
z! tn) = 2t sl^n ~ )t tn
+ O (ft t0o) To 0
B
m(ft ton)
cos-
am B
Since
i d(Qc i
1 dt1
2trB
the demodulation becomes:
,2-trBToN
*(t) 2Bl Sln< T rS t
t0 m cos2,m T ,
(^)t
-ttB < Si < 7TB
, (0n 1lB < ft < to +
u o
1 < t < -
2 2
Then the actual response envelope becomes
E. E, v /v
z (t) = z (t) w(x)
where w(t) = F ^ [W(ft)]
ttB


198
Then the actual response will be
z(t) = Re[z+(T)]
1 sin ttB(t + T^)
1 sin ttB(t xd)
2 E(T + td) cs ("T+6) + 2 B(t Td) COB ("T-e)
ljT
here Td 2B
Example 2 Determine the range response
Modulation is as shown
Demodulation is sinusoidal: ij>(t) = cosiest + 0), |t|<
T
wr(t) = A. t + B. + a) for -r- < t < t
= A0t + B + a) for t < t <
2 2 o o
where A, =
2irB
1 t + T
2
2irB
A2 = t T
2
B1 "
= IPr^o) r Bl
Ai
ic^ (ft)
B2 "
and lan1 2
and
Z+l> ^ cos, fe-o) El +0]
(i) ttB < ft < (j +ttB
o o
rt /^\ 2tt r (ftton) 82 o 1
z+2(ft) = ~y cos [i)i - +0]
to irB < ft < to +tB
o o
Z, /r\\ 2tt rl r (fi_n) B] .i j 1 r (Wn) B9 i i
+ (ft) = ( cos [to 1 +0] +£ COS [to} c +0] }
for u0 irB < ft Hlc-g


100
101
102
103
104
105
106
107
108
216
SUBROUTINE MATPLT(VALUE,ALPHA,I,INDX1,INDX2,IHCR)
REAL*8 ALPHA
DI HENS I ON VALUE(INDX2)
YMAX=VALUE(INOX1)
YHIN=YHAX
DO 1' J=INDX1, INDX2,INCR
IF(VALUE(J).LT.YMIN) YMIN=VALUE(J)
IFCVALUE(J).GT.YMAX) YMAX=VALUE(J)
1 CONTINUE
IF (YMIN.GT.0.0) YMIN-0.0
WRITE(6,100) YMIN
WRITE(6,101) YMAX
WRITE(G,107) ALPHA
IF(.NOT.(YMAX.EQ.YMIN)) GO TO 3
WRITE(6,10G)
RETURN
3 CONTINUE
WR ITE(6,105)
FORMAT Cl MINIMUM ',E10.3)
FORMATC
MAXIMUM *',E10.3)
FORMATC O')
FORMATC C I 5,'
FORMATC
FORM.AT(' 1
FORMATCONO PLOT NEEDED: MAXIMUM = MINIMUM .')
FORMATC+ ,GOX, 'VARIABLE: ',A8)
FORMATC ',15,' 15 0A1)
DIFF=YMAX-YMIN
SCALE*120.0/DIFF
DATA A,B/4M' ",4H. .../
01 = 1NDX2-I NCR
INCR2e2*I NCR
DO 2 K=INDX1,Jl,INCR2
NPTS=(VALUE(K)-YMIN)*SCALE+1.5
WRITE(G,104) (A,J=l,NPTS)
NPTS(VALUE(K+I NCR)-YMIN)SCALE+1.5
WRITE(G,103)K,(B,J=1,NPTS)
2 CONTINUE
K-K+INCR2
IF(K.GT.IHDX2) RETURN
NPTS*(VALUE(K)-YMIN)*SCALE+1.5
WR ITE(6,108) K,(A,J=l,NPTS)
RETURN
END


77
This simple result is due, of course, to the linearity of the system,
which allows an interchange of the summation and integration in (5.2-1).
We simply coherently demodulate with a time function defined as the
appropriate linear combination of (i|)n(t)}.
5.3 Existence and Dimension of H
Whenever one speaks of an inverse matrix, such as H = G \ the
existence of that matrix comes into question. That existence depends
upon G having a non-zero determinant, which, in turn, implies linear in
dependence of all vectors z^Ct) = [z(ti)z(t2),z(tn)] (r alternately,
linear independence of z) Thus the size of G for which G ^ exist is
limited to the number of independent information vectors which may be ob
tained.
5.3.1 Dimension of H Based on IF Waveform
To determine a maximum value for N we will consider the case of
linear sawtooth modulation and define a x_v such that we are interested
Hid A
only in the case when t xmax. Then the instantaneous difference (IF)
BTmax
frequency will vary between 0 and ^. The IF waveform will be a si
nusoid windowed by the modulation period T. This windowing of the time
waveform will spread the IF bandwidth somewhat, but if BTmaxthe maxi
mum number of cycles over a period Tis more than a few cycles, this
spreading effect is negligible compared with the frequency range or band
width By the sampling theorem we know that e(t,x) is defined by
T
its sample points uniformly spaced by there being a total of
Tmax
2Bxmax points for period T. Since these points uniquely define e(t,x)
(disregarding the fact that we approximated the bandwidth), any more
would be redundant. And, just as the information out of a channel cannot


80
FIGURE 5.2 INFORMATION ELEMENTS AS A FUNCTION
OF DELAY
Now we "arbitrarily" choose = i/B; then
G =
m|> 1(f) 'i(f>
B
(?)
(?)
'?>
B
*2 *3(?)
*4(#>
z4£)
0
1
0
0
0
0
1
0
(5.4-1)
(A)
(B)
0
0
I
from Figure 5,2,


C(t)


73
(5-3)
we now have a linear transformation of G which contains new N-dimensional
information vectors at N points in x:
I = GH = (5-4)
Each row of I is the value of a system output defined at N points in x
and at no other points. Figure 5.1 describes the range response of one
such system output. The times {xj^} do not necessarily have to be ordered
such that
Ti < Ti+i
as Figure 5.1 also illustrates. Of course, there is no ambiguity evident
FIGURE 5.1 HYPOTHETICAL RANGE RESPONSE
INDICATING CONSTRAINT POINTS
in the response z0Ut3(T) as defined at points {x}.
put (row) comes from the transformation
N
zoutm(T) = I hmnzn(T)
n=l
But each system out-
(5-5)


114
X(o>)
4 t
1
1
1
1
i
received signal
> spectrum (for an
approaching target)
co0
\
\)0+)d
t<
J
O
a*
1
H
(A)
SIGNAL SPECTRUM
(B) KALMUS PROCESSOR
FIGURE 8,1 PHASING-TYPE SSB PROCESSING AT RF
FIGURE 8.2
TWO-CHANNEL PROCESSOR MODEL MODELED IN THE f DOMAIN


153
The Hamming and Hanning spectra of Figures 9.44 and 9.46
correspond to responses of Figures 9.21 and 9.33, and their inverses
integrate to the modulation functions of Figures 9.45 and 9.47.
The response of Figure 9.26 has a transform approximated by
Figure 9.48, which is nearly zero for some sections. These sections
cause a very rapid integration, i.e., a very high modulation slope,
which does, indeed, produce a modulation probability density which
is low in these sections. The absolute value of the function is taken
and its value is limited to 30.0 before integration so that small
values of positive or negative spectrum (as are common in Figure 9.48)
do not cause wild variations in the modulation. Error due to this
precaution is evident in the saturation of wm(t) in Figure 9.49
T
before t reaches
The examples given here illustrate many of the points discussed
elsewhere in the text and promote an understanding of the relationships
derived in Chapters VI and VII. In addition, since some of the common
range responses, modulations, and demodulations are used in these
examples, one may easily compare results given here with those which
have been worked out by other methods.


Response Magnitude
1.0
0.0 =:
-16 0 Normalized Delay
FIGURE 9.18 SPECIFIED SHORT-PULSE RESPONSE
16
162


220
14. R.B. Blackman and J.W. Tukey, The Measurement of Power Spectra,
New York: Dover, 1958.
15. C.L. Dolph, "A Current Distribution for Broadside Arrays Which
Optimizes the Relationship between Beamwidth and Sidelobe Level,"
Proc. IRE, Vol. 34, June 1946.
16. A. Papoulis, "Minimum-Bias Windows for High-Resolution Spectral
Estimates," IEEE Trans, on Information Theory, Vol. IT-19,
No. 1, January 1973.
17. H.J. Landau and H.O. Poliak, "Prolate Spheroidal Wave Functions,
Fourier Analysis and Uncertainty-Ill: The Dimension of the
Space of Time- and Band-limited Signals," Bell System Technical
Journal, Vol. 41, July 1962.
18. L.R. Rabiner, "The Theory and Approximation of Finite Duration
Impulse Response Digital Filters," Current Digital Filter Signal
Processing Techniques, NEC Professional Growth in Electronics
Seminar, 1972.
19. W.B. Davenport, Jr. and W.L. Root, An Introduction to the Theory
of Random Signals and Noise, New York: McGraw-Hill, 1958.
20. L. Weinberg, Network Analysis and Synthesis, New York: McGraw-Hill,
1962.
21. L.W. Couch and R.C. Johnson, "Range Laws for FM Radars with
Harmonic Processing and Arbitrary Modulating Waveshapes,"
University of Florida Report No. 0169-1, November 1973.
22. L.W. Couch, "Effects of Modulation Nonlinearity on the Range
Response of FM Radars," IEEE Trans, on Aerospace and Electronic
Systems, Vol. AES-9, No.4, July 1973.
23. M.C. Bartlett, L.W. Couch, and R.C. Johnson, "Directional Doppler
Detection for IF-Correlator FM Ranging Systems Using General
FM Modulations," University of Florida Report No. HDL-TR-039-5,
October 1974.
24. J.B. Thomas, An Introduction to Statistical Communication Theory,
New York: Wiley, 1969.
25. D.K. Barton, "Comments on 'Signal Resolution Via Digital Inverse
Filtering,"' IEEE Trans, on Aerospace and Electronic Systems,
Vol. AES-9, No.4, July 1973.
26. H.F. Harmuth, Transmission of Information by Orthogonal Functions,
Second Edition, Heidelberg: Springer-Verlag, 1972.
27. L.E. Franks, Signal Theory, Englewood Cliffs, N.J.: Prentice-Hall,
1969.


139
The above technique of adding zeros performs an interpolation
similar to convolution of the sample function Z (0) with a sinx/x
function to provide the addition sample values. The severity of
the aliasing mentioned in (3) above will be determined by the energy
of the range response outside the range definition limits. The
aliasing causes distortion of the response and indicates that the
function has been chopped off prematurely. In the DFT we may view
this truncation as a truncation of a Fourier series; the resulting
ringing is called Gibb's phenomenon. To illustrate this phenomenon,
we shall compare a well-defined range response, such as that produced
by a Hanning-shaped spectrum, with one not so well defined, such as
that produced by a rectangular spectrum (with sharp discontinuities).
As in Figure 9.6(A) and (B), showing aliased Hanning and rectangular
responses, cutting off the response abruptly causes more ringing in
the rectangular case. The 30% windowing of Program Two reduces, but
does not remove completely, the effect as shown by the ringing of
the nearly rectangular spectrum of Figure 9.8. Figure 9.7, the
expanded hamming spectrum, shows no observable trace of ringing,
due to a lack of sharp discontinuities in the spectrum.
The Gibb's phenomena may be controlled by reducing the
energy of the chopped-off tails (i.e., by increasing the delay
range) or by windowing as in Program Two. One may wish to vary
the percentage of the window from the 30% value, which was chosen
more or less arbitrarily.


82
so that
H' = RH = RG-1 (5.4-5)
where R is the desired range response matrix
and H* defines the appropriate linear combinational matrix for the
system.
However, the above-mentioned problems would still apply, and we see that
this process does not lead to a very systematic method of solution.
In the next section we develope relationships which are easier
to use and are thus of more value and importance.


Modulation Function
0.459
II!.IMI


Response Magnitude
0.33
16
FIGURE 9.19
RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE USING RECTANGULAR WINDOW
163


Response Magnitude
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.15 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION, HALF-COSINE DEMODULATION
159


"process" [5]; if the modulation voltage, m(t), is voltage limited, then
x(t) will be essentially bandlimited (BL) to BW = B (Figure 1.1).
4
FIGURE 1.1 RELATIONSHIP BETWEEN MODULATION DENSITY AND POWER SPECTRUM
When the modulation is not a simple sinusoid, the index is not
really defined by (1.2-5). A more general parameter than modulation index
is the dispersion factor, commonly defined for chirp radars [6]:
D = B T (1.2-7)
For sinusoidal modulation of frequency fm = 1/T
D 2y (1.2-8)
The requirements of a single antenna and envelope detection stem
from the desire to keep the DMS simple and cost effective. Two isolated
antennas and a multiplier will give essentially the same simplified math
model, as will be shown.
Finally, since our DMS will be processing a periodic signal con
taining range information, it becomes necessary for the target to remain
at approximately the same position during one period T. From an energy
standpoint, and with regard to signal-to-noise ratios, we should like to


85
analagous to that which we desire in their relation of the autocorrela
tion function, signal envelope, and phase function [9]. Two of the three
functions are separately specified, thus determining the third. Here we
are interested in the modulation waveshape, the reference demodulation
waveshape, and the range response. Given two of these functions, we want
to be able to solve for the third. The solutions to these problems are
found below.
6.2 Derivation of the Relation for Predicting the Range Response
The range response z(x) will be obtained as a function of the
demodulation reference ^(t) and the modulation waveshape u>c(t). We begin
the derivation by defining the analytic function
z+(t) = z(t) + jz(x)
where z(x) = Re|z+(x)j
and z(x) is the Hilbert transform of z(x) [5].
Likewise,
Z,(0) = F[z,(x)] =
2z(ft) n > o
| o n < o
(6.2-1)
(6.2-3)
The notation, in effect, makes the frequency domain one-sided, and we
must take the real part of all responses to be actual response. In simi
lar fashion
e+(t,x)
COS (1)CX + j[cos U)CX ~]
= ejUcT
(6.2-4)
(A)
(B)
The Fourier transform of z+(x), then, is


125
FIGURE 9.1. COMPUTATIONAL FLOW CHART FOR PROGRAM ONE


207
R(l)-Y(KL)*COS(YA(KL))
R(2)Y(KL)*SIN(YA(KL))
R(NFFT+1)=0.
R(NFFT+2)=0.
DATA ALPHA(4)/'R '/
C FOURIER TRANFORM TO OBTAIN THE UNWINDOWED "SPECTRUM."
CALL HARM(R,NA,IH,S,1,IER)
IF(IER.NE.O) UR ITE(C,100)IER
100 FORMAT(1 ERROR III HARM SUBROUTINE, TYPE ',12)
C WINDOW THE TRANSFORM TO OBTAIN A BANDLIMITED SPECTRUM
CALL WINDOW(R,2*NFFT,NB,NWIND)
C INPUT THE MODULATION FUNCTION, WHICH MUST BE MONOTONIC
CALL MODUL (MOD, TMOD, NPTM, ISEC, IMI II, IMAX)
I F(I SEC.HE.1) WRITE(G,101)
101 FORMATC THE MODULATION MUST BE MONOTONIC. ',
*' PLEASE REDEFINE THE MODULATION DATA.')
C INTERPOLATE THE MODULATION AT POINTS OF THE VECTOR T
CALL I NTRP(MOD,TMOD,1,NPTM,2,Y,T, 1,N,YP,H)
CALL MATPLT(Y,ALPHA(5),N,1,N, 1)
DATA ALPHA(5)/'MOD FN '/
CALL HARMC R,NA, I M,S, -?1, I ER)
IF (IER.NE.0) WRITEC6,100) IER
C COMPRESS RANGE RESPONSE, THUS EXPANDING SPECTRUM FOR BETTER INTERP
K(NFFT/(NB*2))*2
KS=1+K
KK12*NFFT*2
KK2=KK1+1
NFFTE=(NFFT/K)*K
NFFTM=(7*NFFTE)/10
XWIN-3.141593/(NFFTE-NFFTM)
Jl=3
DO 7 J=KS,NFFT,K
WINDO-l.O
IF (J.GT.NFFTM) WlNDO=0.54*0.46*COS((J-NFFTM)*XWIN)
Z(J1)=R(J)*WINDO
Z( J1+1)=R(J+1)*UINDO
Z(1026-J1)=R(KK1-J)*WINDO
Z(102 7-J1)=R(KK2-J)*WINDO
7 Jl*Jl+2
Z( 1) =R( 1)
Z(2)=R{2)
KK2=1027-J1
DO 8 J=J1,KK2
8 Z(J)=0.
DO 10 J=l,NFFT
A=R(NFFT+J)
R(NFFT+J)=R(J)
10 R(J)=A
CALL MAGANG(R,2*NFFT)
CALL MATPLTR,ALPHA2),2*NFFT,1,2*NFFT,2)
CALL MATPLT(R,ALPHA(3),2*HFFT,2,2*NFFT,2)
DATA ALPHA( 2 ), ALPHA(3) / 'MAG R. L.', *ANG R.L.V
NZ(1)=9
NZ(2)= 0
NZ(3)=0
CALL HARH(Z,NZ,IN,5,1,IER)
IF(IER.HE.O) WRITE(6,100)IER
KK1=(NB*K*128)/NFFT
NK*KK1*2+1
KK2=1024-NK
KK3=KK1+1
DEL*1./(NK-1.)
C SET UP THE TABLE OF SPECTRUM VALUES FOR THE BANDWIDTH OF INTEREST
DO 3 J=l,NK
3 ABSC(J)=-0.5+(J-l)*DEL
DO 4 Jl,KK1
R(J)=Z(KK2+2*d)
4 R(J+KK3)=Z(2*J+l)
R( KK3)BZ(1)


152
Figures 9.29 and 9.30 represent the quadrature demodulations
required to obtain the response of Figure 9.28 which is the rec
tangularly BL version of a pulse at Bt = 6.4 when sawtooth modulation
is used. When the modulation is sinusoidal, the first half-period
demodulations look like Figure 9.31 and 9.32 with second half
period being the mirror image about t=0 of the first. Notice that
each demodulation resembles a carrier with amplitude modulation,
but the frequency of the carrier is not constant. This is because
the sinusoids in ft of (^) map through the nonlinear modulation
m
to be non-sinusoids in time. The amplitude weighting evident towards
the ends of the demodulation plots is necessary to offset the spectrum
E
peaks caused by sinusoidal modulation, since Z (ft) must have a
rectangular envelope.
Figures 9.33 through 9.41 are self-explanatory by their
figure headings. They illustrate use of two types of windows in
specifying BL approximations and both single-channel and directional-
doppler systems.
Program Three does not normally produce an output plot of
g
Z (ft), but Figures 9.42, 9.44, 9.46, and 9.48 have been obtained to
illustrate the integration process used to obtain u> (t). Since
m
is specified as DC (unity) in these examples, K/Z (ft) represents
the function to be integrated see equation (9.3-1B). The first
example is that of integrating a nearly (and ideally) constant
function to obtain an almost linear modulation. The spectrum would
ideally be flat, corresponding to the obtainable response given by
Figure 9.19.


CHAPTER V
SYSTEMS USING VECTORS OF INFORMATION
Systems such as the harmonic system which demodulates a single
line of the IF spectrum are simple and yield a certain quantity of in
formation about range. But additional information is to be had in the
other lines of the IF spectrum. We have seen how, in the linear modula
tion case, information may be manipulated by a linear combination of these
lines to produce a desired range response whose transform is a window
function. In this chapter we shall designate each harmonic sub-system
output as zn(x) and the ordered collective of all outputs as
£(t) = [zi(t),z2(t),***zn(t)] (5-1)
For convenience, the arguments will often be omitted: z = z(x). Each
output, zn(x), will be called an information element and the vector z
will be called the information vector. Let us now define an NxN matrix
G = [z(xi)z(x2)*z(xn>] (5-2)
(A)
= [§mn] where gmn = zm(xn) (B)
This matrix uniquely defines each of the N elements of information for N
values of delay, assuming no noise in the system [26,p.287]. We shall
denote the set of times xi, x2,*,,x^ as {x}. If an inverse, H, exists
for G such that
72


FIGURE Page
9.15 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION,
HALF-COSINE DEMODULATION 159
9.16 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SAWTOOTH MODULATION
AND DEMODULATION 160
9.17 RESPONSE OF SINGLE-CHANNEL SYSTEM USING TRIANGLE MODULATION
AND DEMODULATION 161
9.18 SPECIFIED SHORT-PULSE RESPONSE 162
9.19 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE
USING RECTANGULAR WINDOW 163
9.20 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.19
FOR SAWTOOTH MODULATION 164
9.21 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE
USING HANNING WINDOW 165
9.22 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.21
FOR WASTOOTH MODULATION 166
9.23 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE RESPONSE
USING HAMMING WINDOW 167
9.24 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.23
FOR SAWTOOTH MODULATION 168
9.25 SPECIFIED BANDLIMITED SIN(X)/X RESPONSE 169
9.26 RESPONSE OBTAINABLE BY BANDLIMITING SIN(X)/X USING
RECTANGULAR WINDOW 170
9.27 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.26
FOR SAWTOOTH MODULATION 171
9.28 RESPONSE OBTAINABLE BY BANDLIMITING SHORT-PULSE AT Bt=6.4
USING RECTANGULAR WINDOW 172
9.29 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28
FOR SAWTOOTH MODULATION 173
9.30 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SAWTOOTH MODULATION 174
9.31 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28
FOR SINE MODULATION 175
9.32 SECOND-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SINE MODULATION 176
x


23
degree of certainty. A spread range response would lead to a high degree
of range uncertainty when parameters, such as target strength or system
gains, are unknown, or when noise is present. Woodward [2] has indicated
that the ambiguity function describes the probability of a target being
at a given range.
The range response for a continuous-wave (CW) radar is given by
Figure 2.2. The envelope of the ambiguity function is simply the
FIGURE 2.2 CW RESPONSES WITH AND WITHOUT ATTENUATION
FACTORS
constant response shown by the dotted line, uninfluenced by the attenu
ation due to distance, A(r). The dotted response shows n£ discrimination
in range, and there are equal conditional probabilities of the target
being at any range, even when conditioned on knowledge of all system pa
rameters.
Figure 2.3 shows the range responses of various systems; these
same figures correspond to types of range ambiguity inherent in various
transmitted signals. For the time being, we shall consider only ambi
guities in range, as, indeed, our quazi-stationary model precludes the
necessity of high resolution in velocity.




I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Engineering.
Jack R. Smith
Professor Electrical Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Engineering.
Frank F. Donivan, Jr.
Assistant Professor of Physical
Science
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Engineering.
March, 1975
Dean, College of Engineering
Dean, Graduate School


100
V
out
y
V
in
r
INSTANTANEOUS NL
FIGURE 7.2 FUNCTIONAL REFERENCE GENERATOR
That is,
ip (m) is the relationship between the demodulation
voltage and modulation voltage m.
This representation remains valid for stochastic modulations if the
hardware of the system is so designed. And when a time representation
is appropriate, we may write
(m(t)) (7.1-9)
t m
When the reference is generated by a delay line and multiplier
(Figure 3.3), then
\(t) =
t
cos[ f ) (A)dA+o> t ] = cos[ 6,(t, t )+m x ]
T m oR 1 RoR
(7.1-10)
R
t
* e,<9i>
= cos[e (x ) + w T ] where 0,(x ) = / m (A)dA
1 R o R R m
(7.1-11)
1
t- T
. 2 3 R
C T 0)
r m R m R t
COSLo) T_+W x_ 0 + ...J
o R m R 2 6
^(w) =
(7.1-12)
where is considered a constant of the system (just as m ) and is
R o .
thus not included as a variable subscript.
7.2 The Assumption of Ergodicity and Notes on Averaging
The doppler filter of Figure 7.1 has the effect of averaging


96
above equations). If for some reason we desire, for instance, a symmet
rical "multiroot" modulation, we must specify the time roots and write
the relations as
wc(tn) tn
/ Z^OOdn = K / iKOdt (6.4-4)
c^nmin) fcnmin
Hopefully could be determined from a symmetrical character of \{>(t)
and the desired modulation summetry. As an example, for a modulation
specified to be even about t = 0, with ^(t) even, we would know that
&+l
= Z
+1
Z+(fl)
2
(6.4-5)
Examples of closed-form solutions are given in Appendix A.
Computer solutions are given in Chapter 9.


19
of suppressing these ambiguities is the addition of LF noise modulation.
We shall usually consider, not the line spectrum associated with
the signal, but the envelope of that line spectrum which would occur if
we let T (See derivation of the Fourier integral from series [8].)
Thus presented is the argument for large values of T and thus D, sup
porting assumptions 1 and 2.
1.5.3 The Quazi-Stationary Target Assumption
The assumption that the target is stationary or slowly moving is
demanded by our periodic processing model. It also supports a certain
freedom of design of the ambiguity function (see next chapter), which will
not have to be tightly controlled along the frequency axis [9].
Motion of the target will be treated as a perturbation of the
stationary problem, somewhat in the same manner as one might treat the
amplitude modulation of a carrier as a perturbation of the carrier. In
the end, of course, one must consider the rate of change in terms of
doppler frequencies, especially in the consideration of noise, as the
signal-to-noise ratios must depend on the doppler bandwidth chosen. In
those cases we shall consider the target motion to be linear in time and
space.
Note that to assume such a simplified model is to neglect the
compression/decompression of the entire return signal spectrum [lO] by
the factor
v v
v
where v is the propagation velocity in the medium
v is the velocity of the target in the direction of the
pickup antenna
+ (-) indicates positive (negative) relative motion.


ACKNOWLEDGEMENTS
The author wishes to express gratitude to his chairman,
Dr. Leon W. Couch, for being the outstanding teacher that he is
and for his constructive criticisms and to Mr. Marion C. Bartlett
for the many discussions invaluable to an understanding of the
systems studied.
Thanks are also due to Mr. James C. Geiger, who con
structed most of the figures and to Miss Betty Jane Morgan, who
typed the bulk of this dissertation.
The author is indebted to the Department of Electrical
Engineering and to Harry Diamond Laboratories for supporting re
search pertinent to this dissertation, and to Professor R. C.
Johnson for allowing work in this area of study.
iii


ooooooooooooo ooooooooo
214
SUBROUTINE MAGANG(Z,IZ)
THIS SUBROUTINE SIMPLY CONVERTS THE REAL11 MAGI NARY PAIRS
GIVEN BY THE ADJACENT CELLS (REAL IN THE ODD CELL, IMAGINARY
IN THE EVEN CELLS) TO PAIRS OF MAGNITUDE AND ANGLE (MAGNITUDE
GIVEN IN THE ODD CELLS, ANGLE IN THE EVEN CELLS). IZ IS THE
DIMENSION OF Z.
DIMENSION Z(IZ)
1=1Z/2
DO 1 J=l,l
Al =Z(2*J)
R=Z(2*J-1)
Z(2*J-1)SQRT(R**2+AI**2)
IF((AI.EQ.0.0).AND.(R.EQ.0.0)) R-1.0
Z(2*J)=ATAN2(Al,R)
1CONTINUE
RETURN
END
SUBROUTINE FFT(A,IA,B,IB,INV)
THIS SUBROUTINE PERFORMS A FAST FOURIER TRANSFORM (USING HARM FROM
THE FORTRAN SSP) ON THE VECTOR A (DIMENSION IA) TO YIELD VECTOR B
(DIMENSION IB). THE FIRST HALF OF A IS CONSIDERED THE NEGATIVE-
FREQUENCY OR NEGATIVE-TIME COMPONENT WHILE THE SECOND HALF IS FOR
THE POSITIVE HALF OF THE FREQUENCY/TIME AXIS. IF INV IS 1, THEN
THE TRANSFORMATION IS THE FORWARD FFT; IF -1, IT IS THE INVERSE
TRANSFORM. NOTICE THAT IB MUST BE 2**K WHERE K IS AN INTEGER.
ALSO, IA MUST BE AN EVEN INTEGER.
DIMENSION A(IA),B(IB),NA(3),IN(2048),S(2048)
NA(1)=3.321928095*ALOG10(IB+0.0)-0.5
NA(2)=0
NA(3)=0
JI =I A/2
JB=2*(JI/2)
JC=IB-JB
DO 1 J=l,JB
B(JC+J)=A(J)
B{ J)=A(J+JB)
1 CONTINUE
IF (JB.EQ.JI) GO TO 2
B(JB*1)A(IA-l)
B(JB+2)=A(IA)
2 CONTINUE
CALL HARM(B,HA,IN,S,INV,IER)
IF(.NOT. (I ER. EQ. 0)) WRITE(G,100) IER,NA(1)
100 FORMAT('IERROR TYPE',13,' HAS OCCURRED IN THE HARM SUBROUTINE.',
*' NA(1) = *,13,*, NA(2) 1, NA(3) = 1.')
IHBIB/2
DO 3 J-l, 11 IB
TEMPB(IHB+J)
B( IIIB+J)**B( J)
3 B(J)TEMP
RETURN
END


Response Magnitude
1.0
-16
0
Normalized Delay
16
FIGURE 9.26 RESPONSE OBTAINABLE BY BANDLIMITING SIN(X)/X RESPONSE USING RECTANGULAR WINDOW
170


F[*]F *[] Fourier and inverse-Fourier transforms.
Depending on the arguments of the functions to be transformed
or the arguments of the transform, the transformations are
defined as
00
F(to) F[f(t)] / f(t)e""^,tdt
CO
00
f (t) = F-1[f(o))] / F(u))eJ,td)
00
C(f) F[c(t)] = / c(t)e"j2irftdt
00
c(t) = F[c(f)] = 7 C(f)e*^2lTftdf
00
Notation of frequency-domain functions will be consistant;
i.e. C(*) will not be expressed alternately as C(f) and C(oj).
Unless stated otherwise, upper-case functions are the Fourier
transforms of corresponding lower-case functions.
g(t) desired output waveform when inverse filtering
G antenna gain
h(t) filter impulse response
H inverse matrix for constraining range response points
I identity matrix
K(f) frequency dual of c(t)
Kf frequency modulation constant (radians/volt-sec.)
% overall target reflectivity
LP low-pass
Li(t) loss in transmission media
L2(t) space loss
m(t) modulation voltage
xiii


34
choosing our ambiguity function: we design for high range resolution and
let doppler resolution fall where it will.
At this point some of the properties of the ambiguity function
are summarized for unit energy signals [2,3]:
1. Xi(t,v) = Xj (-t,-v) Hermetian symmetry with respect to
the origin
2. X2(t,0) = c(t)
3. X2(0,<(>) = K()
00
4. / |x(x,0)|2dx = TA
00
CO
5. I |X(0,) | 2d = Fa
00
00 00
6.If |X(x,) | 2dxd<) = E regardless of signal
00 CO
7. The two-dimensional Fourier transform of IX(x,<>) I 2 produces
|x(t,f)|2.
f S |x(x,(f>) |2e"j2irfT2j27TtdTd<¡> = |x(t,f) |2
CO 00
8. f 1X (x, <{)) | 2dx = I |x(x,0)|2e
CO CO
00 03
9. / |x(x,) 12d<(> = / |x(O,<0)|2ej27TTd(0
00 00
U (f ) ,u(t) < >X(t,4)
Then
10. U(f)ejlTpf2<> X(x-p)
11. u(t)e^7T^Ct <> X(x,<|>+kx)
12. au(at) < > X(ax, ^
a
where the double arrow indicates a pairing of signals and ambiguity
functions


-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Normalized Delay
FIGURE 9.11 RESPONSE OF SINGLE-CHANNEL SYSTEM USING SA KTOOTH MODULATION, THIRD-HARMONIC
COSINE DEMODULATION


45
reflected in a convolution of the original response with the delta func
tions of (2.2.4-4). Applying a window to power spectrum P(f) = FCrCt)]
yields the following autocorrelation (range response for matched systems
with slowly varying targets):
N a
z(t) = R(t) a0i6(x) + l (6[t + £] + 6[x £]) (2.2.4-5)
n=l L * (A)
N a .
= aoiR(T) + l [R(t + f) + R(t- |)] (B)
m=l
Thus we see windowing as the appropriate addition of suitably weighted,
advanced and delayed replicas of the original transform.


i(t)
N.
v
ti
t3
FIGURE 6.1 INSTANTANEOUS FREQUENCY VERSUS TIME, SHOWING
THE TIME ROOTS


r(t)
0.376
SECOND-CHANNEL DEMODULATION IS ZERO
-T/2
Time
T/2
FIGURE 9.24 FIRST-CHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.23 FOR SAWTOOTH MODULATION
168


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
IMPROVEMENT OF THE RANGE
RESPONSE OF SHORT-RANGE FM RADARS
By
Barry Gray Mattox
March, 1975
Chairman: Leon W. Couch
Major Department: Electrical Engineering
The problem of range resolution of a class of periodically modu
lated FM radars is approached using assumptions of a quazi-static target,
high modulation index, and a modulation period much longer than signal
return times. Various systems of the class are examined with an emphasis
on resolution improvement. System consideration begins with the simplest,
for which the range response and RF power spectrum are Fourier transforms.
Modification of this system to a system with a range response displaced
from the origin yields the delay-line intermediate-frequency (IF) corre
lator. These systems are then related to the harmonic systems or n-sys-
tems, for "nth-harmonic" systems. Conventional windowing techniques are
reviewed as.used in conjunction with some of these systems.
Briefly, the process of inverse filtering in the range or delay
domain is considered. It is shown that the technique can, indeed, be em
ployed on some of these (essentially) bandlimited (BL) systems by approx
imating the filtering convolution by a discrete point summation or by
xvii


CHAPTER VI
DEVELOPMENT OF CLOSED-FORM RELATIONSHIPS
In previous chapters we have seen the need to relate the system
and the range response: to be able to predict the range response for a
given system or, conversely, to design a system to produce a specified
range response. We have seen that, for special-case systems, such as
linearly modulated, autocorrelation, or delay-line IF correlator systems,
the range response is not hard to predict and, in some cases, may be de
signed through methods such as inverse T-domain filters or by shaping the
power spectrum. There remain, however, a large group of systems for
which range responses are not easily computed, and for which design has
often been "cut-and-try."
The following sections will be devoted to developing relations
between the modulation, demodulation, and range response in the general
coherent detection system. The other systems we have discussed previ
ously remain special cases of this general system.
6.1 Methods for Prediction of the Range Response
There exists a subclass of systems for which the RF power spec
trum and the response shape are Fourier transforms:
z(t)R(t)*P(f) (6.1-1)
where the double arrow means that there is a one-to-one correspondence.
This includes systems whose response is made up of one or more displaced
functions having the shape R(t). The relationship of (6.1-1) is one-to-
83


145
FIGURE 9.9. COMPUTATIONAL FLOW CHART FOR PROGRAM THREE


39
P(f) = frt)] = C(f)
(2.2.1-12)
(A)
where
/
Rp (t )
R(x) = <
(B)
0
We must keep in mind that power is actually contained in line spectra,
and problems associated with such spectra must be considered accordingly.
We see now that when the system is of the matched variety, we
control the range response by controlling its transform, the power spec
trum of the transmitted signal.
2.2.2 Windowing a Bandlimited Spectrum
Techniques used to control the power spectrum, especially a
bandlimited spectrum, may be looked upon as "viewing" the spectrum
through a "window" so that the spectrum is weighted at every point by
the characteristic of the window at that point. Mathematically expressed
Fw(x) = W(x) F (x)
(2.2.2-1)
where F(x) is the original function
W(x) is the window function
Fw(x) is the resulting windowed function.
Such windowing techniques have been applied to smoothing transformed data
in which the time record was limited in duration (time domain); to con
centrating power in preferred directions in antenna arrays (space domain);
and, of course, to range response shaping of radar systems (frequency do
main), an application which we will discuss here.
Consider the power spectrum and corresponding range response
given by Figure 2.7(A,B). Because of the high sidelobes of z(x), severe


107
Then p (uj)
c
l
1
T
I I
I dt 1
-1
i
c.
i
(*>)
den (cd)
c.
1
dcD
where the sum is taken over
all sections of mc(t) which pass
through ordinate to.
(7.4.1-3)
(A)
u is the ith inverse modulation
c
i function.
(B)
Then Z (ft) = 2inJ> (ft) ^ I
c i
dm (ft)
c.
1
dft
(7.4.1-4)
(A)
2tt
T
l ^ ()
dm _1(ft)
c.
1
dft
(B)
and the relation resembles relation (6.2-12), but with the demodulation
related solely to instantaneous frequency.
7.4.2 Range Response Transform Statistically Derived for
Multi-dimensional Reference
For the most general reference,
of random variables, equation (7.4-2C)
ip where x is any vector
o
becomes
z(ft) = 7T/. .//ll> ( co x in o> ,x m mo mo m
m m
nf .../ i (ft-CD ,x)p (ft-cj ,x)dx
rC ,X O* ,X O
nr- m
= <
tt/
...fib (-ft-rn ,x)p (-ft-w ,x)dx
r(D ,X O ^(D ,X' O
m m
(7.4.2.-1)
(A)
ft > 0
(B)
ft < 0


116
|X(x, $)
|X(-T,
(8.2-6)
|X(T, ) | (even modulation)
and
lCn(T)l = lc_n(T)| *
(8.2-7)
However, for uneven modulations Jc^J may differ greatly from |c
and thus it may be easy to differentiate between approaching and
receding targets. A notable example is that of sawtooth modulation
using nth harmonic single-sideband detection: the response is that of
a singly-translated -sAn|x) response [21],
siniT (Bf-n)
n(BT-n)
(8.2-8)
In an effective directional doppler SSB system, the doppler sidebands
created about the harmonic line positions in the IF spectrum must
be unbalanced. .
8.3 A More General Directional Doppler Processor
In this section the SSB systems of the previous section will
be analyzed by describing them in the form of equation (7.1.1-4).
The domains will not be those of time and frequency m, but, rather,
those of delay t and its dual Bartlett and Mattox [29] have
statistically analyzed systems represented in these domains and have
shown the practical advantage of a statistical versus a deterministic
model. It will be seen in this section that the SSB method of
processing directional doppler is simply a special case of a more
general directional doppler processing which does not place restrictions
on the types of modulations which may be used or the types of


58
Just as the RF signal may be represented in complex envelope form
x(t) = Re{u(t)ejtot} (3.3-4)
we may represent the IF signal as
e(t,T> = cos[ ((% (t)+aj0)T] = Re{eE(t,x)e^wT} (3.3-5)
where eE is the complex envelope of e.
Then the range response of the harmonic system, Z3, may be written as
n T/2
z3(r) = Re / eE(t,T)eR(t,xR)dt
L -T/2
= Re{z3E(T)ejT}
(3.3-6)
(A)
(B)
where
z3e(t)
T/2
b / eE(t,T)eR(t,xR)dt
-T/2
(C)
is the complex envelope of the range response. It is easy to see that
Iz 3EI is the envelope magnitude of the response while phasing. Let us now speak of the set of coefficients {yn} used to de
scribe
m 2tmt
eE(t,x) = l Yn(x)e T
n=-
(3.3-7)
Couch [22] has shown that the Fourier coefficients, {yn), relate to the
ambiguity function as
_j ,|rnT
yn(r) e T X2(~r, f) (3.3-8)
(A)
and
IYnI = |X2 C-x, £)| .
(B)


35
Relationships 10., 11., and 12. are derived from simple sub
stitution into appropriate forms of the ambiguity function. The first
two are of importance in linear swept FM DMS, sometimes called "chirp"
systems. We will have an opportunity to examine this "shearing" of the
ambiguity function later.
2.2 Shaping of the Range Response by Windowing
In the face of our assumption of a quazi-static target and a
disregard for high resolution in velocity, the importance of the signal
autocorrelation function will be established. The effects of bandlim-
iting its transform, the signal energy/power spectrum, will be discussed
and related to shaping or "windowing" the spectrum, and a few of the more
common windows will be discussed with regards to their derivation, merit,
and utility.
2.2.1 The Importance of the Autocorrelation Function
As we noted in previous sections, we concern ourselves with tar
gets which may be treated as stationary to a good mathematical approxi
mation. This leads us to examine the ambiguity function near the x-axis,
for which -£<<£ where e is some maximum expected value of doppler
frequency. If e is small enough, we may consider
X(x ,(j>) t X(x ,0) = c(x)
(2.2.1-1)
-£<£
and our "ambiguity design problem" [l2] is reduced to that of one dimen
sion.
For periodic signals of period T,
Up(t) = Up(tnT) n = 0,1,2,3,
(2.2.1-2)