• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 Abstract
 Definition of systems and...
 Resolution and shaping of the range...
 Shaping the system range respo...
 Inverse filtering
 Systems using vectors of infor...
 Development of closed-form...
 Statistical derivation of...
 Directional doppler processors
 Computer algorithms
 Conclusions
 Appendix
 Reference
 Biographical sketch
 Copyright






Title: Improvement of the range response of short-range FM radars
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082472/00001
 Material Information
Title: Improvement of the range response of short-range FM radars
Physical Description: xviii, 222 leaves. : illus. ; 28 cm.
Language: English
Creator: Mattox, Barry Gray, 1950-
Publication Date: 1975
 Subjects
Subject: Radar   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 219-221.
Statement of Responsibility: by Barry Gray Mattox.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082472
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000580854
oclc - 14090967
notis - ADA8959

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
        Page xiv
        Page xv
        Page xvi
    Abstract
        Page xvii
        Page xviii
    Definition of systems and goals
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
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        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Resolution and shaping of the range response
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
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        Page 33
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        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
    Shaping the system range response
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
    Inverse filtering
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
    Systems using vectors of information
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
    Development of closed-form relationships
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
    Statistical derivation of the relationships
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
    Directional doppler processors
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
    Computer algorithms
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
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        Page 190
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        Page 192
        Page 193
    Conclusions
        Page 194
        Page 195
        Page 196
    Appendix
        Page 197
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        Page 218
    Reference
        Page 219
        Page 220
        Page 221
    Biographical sketch
        Page 222
        Page 223
        Page 224
    Copyright
        Copyright
Full Text














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



















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 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


















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS * . . . . .... iii

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


KEY TO SYMBOLS


ABSTRACT .. . . . . . . . . . xvii

CHAPTER

I. DEFINITION OF SYSTEMS AND GOALS .* * * * * 1


1.1. Operational Constraints * *

1.2. Structural Constraints * .

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 . * * *

1.5.1. The Assumption of Small T *

1.5.2. Assumption of High Dispersion


* a a * a

* a * *
F ac *
* * *

* * * *

* * * *



* * * *










Factor *


1.5.3. The Quazi-Stationary Target Assumption

II. RESOLUTION AND SHAPING OF THE RANGE RESPONSE * .

2.1. The Resolution Problem *

2.1.1. Accuracy . . .

2.1.2. Ambiguity and Resolution . .

2.1.3. .Parameters of Resolution . .

2.1.3.1. The time ambiguity constant *


. . . . . . . . * 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
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(T) . . . . . . . 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 * * * * * e. * . 91








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 n-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 . 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 . . . . . . 197

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

REFERENCES . . . . . . . . . . . . . 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 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 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 imm 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 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









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

E1(W,T) the continuous counterpart of the IF voltage spectrum E(w,T)

Ea,B(') expected value of the argument with respect to a and B

E(.) expected value of argument average over all random variables

f frequency (Hz.)

fo center frequency (Hz.)

fm frequency of sinusoidal modulation (Hz.)

FA frequency resolution constant










F[].,F-1 [] 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
-CO


f(t) = F-1[F(~)] = -1 I F(w)ejWtd
2T
00ft
C(f) = F[c(t)] = / c(t)e j2nftdt
-m
-00


c(t) = F[C(f)] = f C(f)ej2Wftdf
-00

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

L(Tr) loss in transmission media

L2(T) space loss

m(t) modulation voltage


xiii









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

9 matrix used to transform independent elements into orthog-
onal elements

p.d.f. probability density function

p(m(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 S T

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

T 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)

V modulation index

v velocity of target in direction of antenna

v propagation velocity of signal in the transmission medium


xiv










W(
w(A)

Wa
o0

wc(t)

Wm(t)

Mreturn(t)

x(t)

Xreturn(t)

X(T,O)

Xp(T,()

y

Yo

Yh


z(T),z(T,t)


zE(T)'

z+(T)


zi(T)



zR(T)


zI(T)




J (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 targets)

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(r)

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 6

demodulation as a function of time; also written t(t)

a linear combination of elements of I









a vector of demodulations defined for a number of subsystems

usually a frequency variable (Hz.)

6 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 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








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
















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:

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 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









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:



E = f Ix(t)dt (1.2-1)
-00

If the radar is periodically modulated (either in amplitude or in angle)

we may speak of the energy per period:

T/2
ET = f Ix(t) dt (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









by additive noise considerations) are monotonically increasing functions

of signal energy [I]. 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 ET
Pave. f Ix(t) 2dt =- (1.2-3)
-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 (B4). When the modulating waveshape is

sinusoidal, the modulation index is defined as

B/2 (1.2-5)
fm

where B is the instantaneous peak-to-peak frequency deviation in Hz,

and fm is the frequency of the sinusoidal modulation.

Applying Carson's rule for FM bandwidth [4] we see that

BW = 2fm(l+p) = 2fml' B for >> 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








"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) P(f)






Vmin Vmax
-)I B IK-


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 = 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 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









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 T (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 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 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
T = -- (1.4-1)


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


I









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
L1(-2) 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/Tr4) and (K/T2), with

(K/T 3) 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(wot + Kf f m(A)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(T)x(t-T) = cos(mot+Kf fm(X)dX)+A(T)cos(mot+Kf fm(X)dX-o0T+Kf / m(A)dX)
-0 -0 t

(1.4.1-2)
(A)
t t t
= cos(wot+Kf fm(X)dX)+A(T){cos(o0t+Kff m(A)dX)cos(moT+Kf / m(X)dA)
-0o -00 t-T

t t
sin(0ot+Kf m(X)dX)sin(wT+Kf f m(X)dX)}
--o t-T








t t 1
= {[l+A(T)cos(WoT+Kf f m(X)dX)]2+A2(T)sin2(&ot+ f m(X)dX)}
t-T t-T

t
Scos(wot+Kf m(A)dXA+) (C)
-00
t t
= [1+A(T)cos(0T+Kf f m(X)dX)]cos((mwt+Kf m(X)dX+O)
t-T -co


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


where t
--A(T)sin(w0T+Kf f m(X)dX)
6 = Tan1 t---
-1+A(T)cos(wgT+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 el

and is explicit as the envelope of the expression in (1.4.1-2D):

t
el(t,T) = 1+A(T)cos(w0T+Kf / 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 sig-

nal of interest, denoted e2, is gotten by blocking the DC (no AM assumed):

t
e2(t,T) = A(T)cos(WOT+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(A) : 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(0WT+Kf f m(t)dX)
t-T

t
= A(T)cos(wgo+Kfm(t) / dA)
t-T

= A(T)COS WC(t)T


where


wc(t) = w0+Kfm(t)

We see that c.(t) is the instantaneous frequency of x(t):

d t
Wc(t) = j (wOt+Kf fm(X)dX) = wo+Kfm(t)


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-T) =2A(T)cos (wt+Kff m(X)dX)
-00
t-T
Scos(mo(t-T)+Kf / m(X)dX) (1.4.2-1
-0 (A)
t
= A(T)cos(woT+Kf / m(X)dX)
t-T
t t-T
+ A(T)cos(2wot-moT+Kf fm(X)dX+Kf / m(X)dX) (B)
-CO -CO


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

ter is designed to pass


t
e2(t,T) = A(T)cos(w0T+Kf / m(X)dX) .(1.4.2-2
t-T


(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[oc(t)T] (1.4.2-3)


where wc(t) = o0+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

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

N t-Tn
= An(Tn)cos(wot-iwO+Kf m(X)dX) (B)
n=l -m

where N is the number of targets.

Envelope detection of the sum of a "large" signal plus small signals)

has been shown to equivalent to coherent detection using the large signal

as a reference. Then the product of the returns) and reference is given

as


t N t-Tn
2x(t)*xreturn(t) =2cos(wot+Kf fm(X)dX). An(Tn)cos(wo(t-Tn)+Kf f m(X)dX)
-00 n=l -CO

(1.4.3-2)
N t (A)
= An(Tn)cos(o rn+Kf / m(X)dA)
n=l t-Tn
+ double frequency terms (B)








Then
N t
e2(t,t) = Y An(Tn)cos(0oTn+Kf f m(X)dX) (1.4.3-3)
n=1 t-Tn

where T is the vector [11, T2, `* Tn]T

and T_ indicates the transpose of T

and, as before,
N
e3(t,T) = An(Tn)cos[WC(t)Tn] (1.4.3-4)
n=l (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 1/T. For both envelope and multiplier systems the infor-

mation signal e is a result of producing 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 con-

strained 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
/
/
/
-. S


\ |EJ2Tf,T)

\
l1 W '\


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


IE(2nf,T)


(B) BT = 5.5


I EI 2' ft) I


FIGURE 1.3


I F SPECTRA


dotted envelope. This envelope is found by taking the magnitude of

E1(w,T), the Fourier transform of e3 over one period:

T/2
El(m,T) = / e3(t,T)e- Jdt (1.4.4-1)
-T/2 (A)

T/2
= f A(T)cos[wc(t)r]e-Jtdt (B)
-T/2

The continuous spectrum E1(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 1/T Hz. The beat waveform will also be called the


IE(27Tf,T)


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 sta-

tionary situation, sidebands appear about each of the lines in the spec-

trum 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 no-

tation; i.e. SSB-AM is possible. Using the example of linear sawtooth

modulation, let

t T T
m(t) -Lvolts, < t <
T 2 2

Kf = 2iB radians per second per volt

then
tc(t) = -7T t + 0O rad/sec.


T/2
T/2 2nBTt -jt
EI(,T) = f A(T)cos T e -tdt (1.4.4-2)
-T/2 (A)
27BT 27rBT
T/2 [-j (--- )t -j( + W)t
A(T) T T
= f2 e + e dt (B)
S-T/2

A(T)T sin(rBT-wT/2) sin(7BT+mT/2) (C)
4 L (7BT--T/2) (7Br+T/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 time-

varying gains as shown in Figure 1.4. The filter n(t) will typically be















-I -- - - -


z(t)


I (t)
p- t -- -LI I - -- -II - -.


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 T only; but if the target is slowly-moving,

h(t) must permit variations that occur in z(T) as T changes in time. The

bandwidth of h(t), then, depends primarily on the rates of change in T
d
(secondarily, of course, on ~z(r)).


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.


I
e(t,T)l









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(o0t+Kf m(X)dA) (1.5.1-1)
-00


It is somewhat instructive to observe the instantaneous frequency,

written (as before)

Wc(t) = og+Kfm(t) (1.5.1-2)

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


Wreturn(t) = -0+Kfm(t-T) (1.5.1-3)


Plots of wc and return are given in Figure 1.5(A) for the linear modu-

lation case:

t T T
m(t) < t < (1.5.1-4)
T2

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 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

line--coherently detected--to provide range information and analytically

developed the results for various ranges of T, which are not restricted

to being small. Even for these three simple cases, manipulation becomes












wreturn(t)


w (t)


Wo0+TB


WO "














2nBT/T
0--



2TBB (1-)
T


(A) INSTANTANEOUS
FREQUENCIES-


I








W-wreturn
__ I


(B) BEAT FREQUENCY



I I
I I

I 1
T
I 2




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 gen-

eral-case modulation.

Using the assumption of "small T," we will develop 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 > 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

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

For periodic modulation, spectral lines are spaced 1/T apart. Then in an









RF spectrum of bandwidth B there are


B
1/= 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 m, 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(T). In any event, the returned signal of delay

T + nT is
t-nT-T
Xreturn(t) = A(nT+T)cos[mW(t-nT-T)+Kf f m(X)dX] (1.5.2-5)
-o (A)
t-T
= A(nT+T)cos[WO(t-T)+Kf / m(X)dA+6] (1.5.2-5)
-00 (B)

where

t-T-nT
6 = m0nT+Kf / 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 quazi-

periodic effect will be minimized by letting T be large enough so that

A( T+nT)
the amplitude factor A(T) is very small (n # 0), so that the periodic-

ity may be disregarded for all practical purposes. An alternative method









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 [10] by

the factor

vv


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.






20


This compression is usually taken to be a simple shift if 2nrB << wg.

This case has been studied for example modulations when the shift has

been substantial [ll].
















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 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

T1 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."








Iz(T) I


(A) NOISY AND IDEAL RE ONSES bF SYSTEM WITH HIGH
AMBIGUITY


Iz( )I







ST' 0I



(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









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






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 attenu-

ation 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 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.













(A) WIDE MAINLOBE


T









(B) HIGH SECONDARY LOBES









(C) SPREAD, LOW-LEVEL
BACKGROUND


FIGURE 2.3 RANGE AMBIGUITIES


Rihaczek [12] 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


z(T)


z(T)









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.





za(T) za(T)+zb(T)


Z Za ^T) \a(-c)+ 2b(T)
-- -- zb(T)

(A) T (B)


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,"

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:









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(T) 2dT
A -w
TA = c(0 (2.1.3.1-1)
Ic(0)2 (A)


f IU(f) df
-00

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


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


x(t) = Re{u(t)ew0Ot) (2.1.3.1-2)


where u(t) is called the complex envelope of the signal and has an auto-

correlation function


c(T) = f u(t)u*(t+T)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 TA. 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









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) g(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 e=0 so that U(f) = U0(f). Mathematically stated


dTA
de
e=O


(2.1.3.1-5)


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

-B/2


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

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


dTA = {[fUU0*dfr12 flU02(AUo*+A*Ug)df

C=0
-f(UoU0*)2df.2fUoU0*df/f(AUo*+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/fUoo*df{/UoUo*dff*/UoI2(AUo*+A*Uo)df-f(UoUo*)2dff(AUo*+A*Uo)df} = 0


f(AU+A*U){IU0 12flUo(a) 2da-IUg (a)I 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-Uo(a)4}da = 0

B B
for - f 6 B (2.1.3.1-9)
2 2

or
B/2
/ Uo(ca)2[jIU(ca)2-I U(f)12]da = 0 f (2.1.3.1-10)
-B/2


which has a solution at
B B
2 -2
IUo(a) = (2.1.3.1-11)
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 high-

dispersion assumption--provides 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 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 (fo 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,










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:

f IK(O)i2d4
-w
FA (2.1.3.2-1)
K(0)2 (A)


0 lu(t)14dt
= ------- (B)
m 2
f Ju(t)j2dt


where

K(<) = / 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 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 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

J2T(fo 2)t
T 2 A +
x1(t) = Re{u(t 2-)e } =Re{x} (2.1.3.3-1)
(B)

j27T(fo + )t +
2(t) = Re{u(t + 2)e } = Re{x2} (C)


where the basic signal form is x(t) = Re{u(t)ejj2fOt}

T is the difference in arrival times and

0 is the frequency shift.

The signals xl 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 2fov
2= dt 2v,
The integral of (2.1.3.3-1A) is simply doubled if the analytic forms
+ +
xl and x2 are substituted for xl and x2. Then

j27T(fo -)t j27T(fo + )t 2
2-ISE = / lu(t -)e -u(t + -)e I dt
(2.1.3.3-2)
(A)


00o O
= f |u(t 1)]2dt + f Ju(t + ) 12dt



I u(t- -)u*(t + )ej27tdt I u*(t -)u(t + j)e-j2 Ttdt
-co -o00

(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,) = f u(t -)u*(t + )eJ2tdt .
-00 2 2


(2.1.3.3-3)
(A)


The function is sometimes defined


00
X2(T,<) = f u(t)u*(t + T)eJ2 dt


(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


(2.1.3.3-4)


Note that the ambiguity function along the T axis becomes the autocorre-

lation function of the complex envelope:


X2(T,0) = 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(0) = 1), then


0 o,)2d FA
f |X(O,<))2d> =FA


(2.1.3.3-7)


-00
where there is no need to designate the particular form of X by sub-
00
scripts. Also, r lyt. %2,; = T ( 1 1


\ I. J-U


-~A
_co


Just as integration along each axis produce measures of total ambiguity

in T or (range or velocity), we may integrate in both directions to


|Xl(T,< )| = lX2(T,) |


(k.1. 3.3-5)


3_Q\









measure a total "combination" ambiguity for the signal. The double inte-

gration yields a particularly interesting and profound result [2]:



f f IX(T,4))2dTd4 = 1 (2.1.3.3-9)
-00 -W0

for signals of unit energy. Thus, the two-dimensional analogy to total

ambiguity along the T-axis or )-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 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 [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 trans-

mitted signal is used at the receiver, the output will be of the form


x(t-T)ej27ft x*(-t) = f x(A-T)ej27ax*(X-t)dA (2.1.3.3-10)
-0 (A)


= /0e-J 2f0Tu(X-T)ej 27r(f 0+) Xu*(-t)e j27O (t-X)dX

(B)

= u(X-r)u*(X-t)eJ 27r eJ2 fO (t-T)dX (C)
-00


= e-j2fg(T-t) X2(T-t,4) (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









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


S-T-------- 2T ----




t t
(A) (B)

FIGURE 2.5 SIGNAL AND MATCHED RESPONSE



Figure 2.5(B), having some ambiguity for a total time of 2T1. 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 T1. 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










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) = X *(-T,-v) Hermetian symmetry with respect to
the origin

2. X2(T,0) = c(T)


3. X2(0,4) = K( )
00
4. f IX(T,0)I2dT = TA
-00
00
5. f |X(0,4)12dp = FA
-00
00 00
6. f f IX(T,() 12drd = E regardless of signal
-00 -00


7. The two-dimensional Fourier transform of IX(T,) 12 produces
IX(t,f) 12.


of IX(T,4) 12e-j2nfT2j2TtdTd1 = IX(t,f)12
-00 -00


8. f IX(T,) 12dT = I IX(T,0) 2e-j27OTd-
-00 -00

00 03
9. f IX(T,0)j2d = f IX(0,0)12ej2TOT d
-00 -00

U(f),u(t)<-- X(Tr,o)

Then

10. U(f)e jTf2<--> X(T-pO,,)


11. u(t)ej7Tkt2 <- X(T,4+kT)


12. au(at) <--> X(aT, 0)


where the double arrow indicates a pairing of signals and ambiguity

functions.









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 T-axis,

for which -e < 5S e where c is some maximum expected value of doppler

frequency. If E is small enough, we may consider


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


and our "ambiguity design problem" [12] 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)









and the ambiguity function may be redefined as

T/2
Xp(T,) = f up(t)up(t+T)e 21tdt .(2.2.1-3)
-T/2

Similarly, instead of c(T), autocorrelation function for signals of

finite energy, we define
T/2
Rp() = up(t)u*(t+r)dt ,(2.2.1-4)
-T/2

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

course,


Xp(T,4)
E<

: Xp(T,0) = 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 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 T in the interval (-T/2, T/2). Then the energy

of one period may be normalized--let the average signal power be 1/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 period-

icity of Xp. Periodicity may be eliminated or extended by adding low-

frequency 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 Itl T/2 and


up(t) = X u(t+nT)
n=-m

= u(t) 6(t+nT) .
n=-


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

T/2
= /
-T/2

T/2
= I
-T/2


n=--oo


up(t)u (t+T)dt


oo oo
Su(t+mT) I u*(t+nT+T)dt
m=-00 n=-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 non-

periodic autocorrelations. Hopefully, c(T) will be negligible for

ITI > T/2 in which case


Rp(T) = c(T)


for ITI <
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) MI < I
z(T) = 2 (2.2.1-9)
o Iri > I

because of the space-loss weighting that will eventually be imposed; A(r)

will be very small for IT| > T/2.


Then


(2.2.1-6)
(A)


(2.2.1-7)
(A)


(2.2.1-8)










u(t)




0 to t
up(t)




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


C(f)


(C) SPECTI


P p(f)
A -A

,t-il


1 0
T


c(T)




-to to r


Rp (T)


-T+to -to to
(B) AUTOCORRELATION
UNTTITTTnMS


1
* % 2t0

%ii


1
2t0


S1 .r


FIGURE 2.6 PERIODIC AND NON-PERIODIC EXAMPLES






Equation (2.2.1-7E) may be re-written


Rp(T) = C(T) I
n=-aa

Transforming both sides we obtain


Pp(f) = C(f) *
n=--


6(T+nT)


(2.2.1-10)


6(f )


(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 autocorrela-

tion and range response:


-- - ..._ .. r


i


L-CC r


I


V I, hIVL ,L. Jl ., IY










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

where


R(T) = (. (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 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(T), severe









Pl(f) Z



(A) f (B)

P2(f) z2



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 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









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 function--tends to "smear" the windowed

function's transform through convolution. This rectangular window is de-

finded as
1 -7B < w( < rB
WO(W) = (2.2.3-1)
0 otherwise

In the transform domain this window is


sin irBt
w0(t) = B s Bt (2.2.3-2)

All of our windows will be defined to be zero outside the interval

(-nB,7B). Then it is obvious that

Wo(w) *Wi(w) = Wi(&) (2.2.3-3)

where Wi is any window.

Transforming (2.2.3-3) we obtain the convolution (*)

wo(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 me-

teorological data by Julius Von Hann. (Von Hann actually smoothed in the

transform domain by discrete convolution with the respective coefficients

-,, 1.) The Hanning window is given as [8, p.14]
1i 1
S + cos -7B S m w 7B

W1(u) (2.2.3-5)
0 otherwise


which has transform









1 1 1 1 1
w1(t) M w0(t) + w0(t +) + w0(t -)
2~ w 4t F


(2.2.3-6)


In contrast to the -14dB maximum sidelobe of wo(t), sidelobes of 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 -rB < < 7B
Bn n


W2<) = I


(2.2.3-7)


otherwise


having, of course, a transform of


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

While having lower sidelobes, windows W1 and W2 have the effect of wid-

ening the mainlobe compared with window Wo. The distance between the

first nulls of wl(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


w(t) = os Bt2-A2 (2.2.3-9)
cosh nA

The sidelobe level for this window is a uniform [cosh iA]- 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
+ a cos -BB < w < rB
n= -
W4( M) = (2.2.3-10)
0 otherwise


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 [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" vari-
ance of those estimates.

W6 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

WO rectangular
W1 Hanning
W2 Hamming
W4 Taylor

These windows have been defined as zero outside (-TB,rB). If we con-

sider 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

(-7B,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 wo(t) is equivalent to convolution with

6(t), so is smoothing with wi(t) equivalent to smoothing with


(2.2.4-1)


where


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


(2.2.4-2)


The transform of periodic Wi(w) is, of course, a series of delta func-

tions 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(w) = M ani cos B-B
n=0

for the appropriate frequency range, having transform

N
wi'(t) = ao0i(t) + Z [6(t + ) + 6(t -)]
n=l

where {ani} are appropriately specified:


Rectangular a00 = 1; ano = 0, n 0 0
1
Hanning a01 = 1; a11 = ; ani = 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

as4 = 0.0097638; 044 = -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' (a)









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) = F[R(T)]

yields the following autocorrelation (range response for matched systems

with slowly varying targets):

N a
z(T) = R(T) aoi6(T) + -- (6[r + ] + 6[T ]) (2.2.4-5)
n=l (A)
N ni
= aoiR(T) + I [R(T + ) + R(r- )] (B)
m=l


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 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

SM/2
zatten.(T) Z lim M f e(t,T)dt (3.1-1)
M-0o -M/2 (A)

SM/2
lim M / A(T)x(t)x(t-T)dt (B)
M- -M/2

A(T)R ) (C)
T Rp(T) (C)















TO NL
CIRCUITS


(A) BLOCK DIAGRAM


(B) MODEL


x (t)





return(t)


FIGURE 3.1 THE AUTOCORRELATION SYSTEM


zatten(T) A(T) R()
n T


(3.1-1)
(D)


where Rp(T) is the periodic autocorrelation function of x(T) and


R(r) = Rp() ITI
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(r), we shall be discussing the range

response z(T), which does not include the amplitude characteristic, where,

for the autocorrelation sysLem,

T
z(T) A(T) Zatten.(T) 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 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 [19,20]:


1. R(0) 2 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(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, zl(T,TR)=R(T-TR)

where TR is the delay of the delay line (Figure 3.2). Such a range

















e(t,-r TTR)


TO NL
CIRCUITS


x(t-TR)


m(t)


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 whereverr 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-IH(2Tf)12 where H(w) is the fil-
ter 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 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 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. 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


neglecting amplitude factors.


e3(t, ) = cos[W(t)T]


(3.2-1)

































FIGURE 3.3 I F CORRELATOR SYSTEM





Similarly, the output of the second mixer is expressed

eR(t,TR) = cos[B(t)TR] (3.2-2)

The output of the third mixer is

YO(t,T,TR) = cos[w(t)(T+TR)] + cos[m (t)(T-TR)] (3.2-3)

and after time-averaging (effected with the last filter) we obtain

1 1
Z2(T,TR) = R(T+TR) + 1 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,T), is perfectly correlated with the ref-

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, 2 R(T+TR) which affects for T > 0 is
1
the response of 2 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

zl(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 E,


f Izl(,TR)-222(T,TR) 12dT < e(TR)
0
- provided R(T) is square integrable.


(3.2-5)


Figure 3.4 shows the effect of interfering "positive" and "negative" re-

sponses for large and small TR. Of course, when TR = 0,


z2(T,0) = R(T) = ZI(T,O) = z0(T)


(3.2-6)


and the reference beat waveform is eR(t,0) = 1.


z2


T (A) LARGE TR


T (B) SMALL TR


-TR


z2


T (C) TR = 0


FIGURE 3.4 PLOTS OF z2(T,TR)


I _









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 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(t-TR), and thus also eR(t,TR), will be deterministic. In fact, from

(3.2-2) and (1.4.1-6)

eR(t,TR) = cos[s0TR+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
.2rnt
e(t,T) = an()e T (3.2-8)
n=-oo

where T is the modulation period and

an could have been written an[(T,9,Kf,m(t)] .

Similarly, let
.2Trmt
eR(t,TR) m(TR)e T .(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(T,TR) =


A/2
lim Y yo(t,T,TR)dt
A-) -A/2


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


A/2
=lim- f I
A-)- -A/2 n=-- m=-


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


00
= I
n=-00


=
n=-oo


The system

we have


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


z2(TR,TR) =
n=-mo


Ian(TR) 2


(3.2-11)


As expected, the summation above is the total power of the line spec-

trum of e. The coefficients are computed:

ST/2 2-3
an(T) = e(t,T)e dt (3.2-12)
-T/2 (A)


2nnt
T/2 -2j n
T/2 [ei[OT+KfTm(t)] + e-j[o0T+KfTm(t)]] T dt.
2T /
-T/2


For sawtooth modulation, m(t) = t for Itl ,2'


(3.2-10)
(A)


an(T)-n (TR)



an(T) Bn( R)











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


j[wgT+(KfTt 2nt]


-j [0 oT+(KfTt


2Tn
+ 2--- t
dt (3.2-13)
(A)


KfTT
sin(-- ?rn)
2
KfTT
(- T n)


ejwOt
e-J2
2-)


KfTT
sin (-- + rn)

KfTT
(--- + Tn)


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


an(TR) = Pn(Tf) =


eJWO'R sin ( 2 -rn)

2
2 KfTRT
( -2 -'un)


KfTRT
e-jW0TR sin (-2 -+ Tn)
2 KfTRT
( + in)

(3.2-14)


For


KfTRT = 27rk


all an vanish except for


ejmoTR(sgnk)
ak = 2


sgn k =


k = 0


+1 k > 0


-1 k< 0


ak = a" = cos wOTR


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


in a single frequency:


(3.2-15)


where


(3.2-16)
(A)


For


(3.2-16)
(B)


j OT
(-)


, k = 0, 1, 2, ***











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

2nkt
= cos (--- + R) (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 [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 = KT (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 in-

stantaneous difference frequency in transmitted and received signals is

c(t)-c(t-T) n
2 r -= T TR (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=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 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(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(T) = F[Wx]:


Wautocorr. () = x() (3.3-3)
(A)

IF corr. ) = /Wx() (B)


Wn-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].









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


x(t) = Re{u(t)ejOt} (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 [ f eE(t,T)eR(t,rR)dt ejtOT (3.3-6)
-T/2 (A)

= Re{z3E(T)ejW00} (B)

where
ST/2
Z3(T) = eE(t,T)eR(t,R)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 de-

scribe
.2Tnt
eE(t,T) = yn(T)e T (3.3-7)
n=-oo

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

ambiguity function as

TrnT
Yn(T) = e T X2(-T, ) (3.3-8)
(A)

and


IYnl = IX2(-r, T1 I









Thus the range response of the n-system using coherent demodulation when

the reference is defined as

27Tnt
eR(t) = cos ( + 6) (3.3-9)

with reference phase 0

becomes
27unt
T/2 m j2
E T/2 T 2nt
z3 = I Yn e cos (2T + 6)dt (3.3-10)
-T/2 n=- (A)

T/2
1 T2 2rrnt 2Tnt
= 2 Yn cos ( + Yn) cos ( t- + 6)dt (B)
-T/2


= IYnl cos (LYn 0) (C)


= X2(-T, T) cos (LX2 T ) (D)

If 8 = 0 and T << T, then


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

and any departure from the terms of the small T assumption represents a

phase error of irn/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) = Re X2(-T, R) (3.3-12)
(A)
1 2nn 1 +i 2n
= X2(T + 0) + X2(T- 0) (B)
2 KfT 2 KfT

One sees that the ambiguity function at = n/T is the sum of displaced

S= 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 pre-

dict the displaced range response, regardless of modulation non-linear-

ities, provided they are known.










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 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) = 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) = 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 X( (4-6)










Naturally, signal-to-noise may be expected to suffer because we employ

a filter with transfer function 1/X(m) 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-side-

lobe type function. Since

H() = w (4-9)
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 band-

limited, 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 re-

solution, with a (standard deviation of the output pulse) and minimum

epoch times, T (the separation between return signals), as parameters.








X (W)







(A) SIGNAL SPECTRUM


S oI I I I I. i I I

u u







(B) INVERSE FILTER FOR X ())

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 anal-

ogy 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) = h(T-X)z(X)dX (4.1-1)
ou (A)
00
= f h(X)z(Tr-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 exam-

ine 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 sys-

tem, obtain a new variable Zout(T) conforming to (4.1-1).






66



The modified delay-line autocorrelation system yields a response


zI(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)
-00 (A)

00
zl I(,TR+X) h(X)dX (B)
-00


Z z Z(Tr,TR+X) h((;,dA (C)


where the main energy contribution of h(T) is between -To and TO,

T0
AX
Szl1(T,nAX+TR)h(nAX) AX (D)
n = 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,

the delay will be TR. Notice that care must be taken so that


nAX+TR > 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 .


(4.1.1-4)




































FIGURE 4.2 PARALLEL IMPLEMENTATION OF CONVOLUTION


T = T + at
x 0


sample and hold at

TR- o+td
t=
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 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 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 prob-

lem 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

1 1
z2(T,TR) = 1 R(T+TR) + R(T-TR) (4.1.2-1)
(A)


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

1(C
= Z z(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 suffi-

ciently 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(P). A probable

choice might be one of our windowing functions Wi(n). 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(0) = F-[h(T)]= -Wi() (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(n) has been represented by

an exponential Fourier series for wg-nB < 0 & wM+nB:


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

= F-I n an e Z( 0) (B)
n=-o

= an z(T- ) (C)
n=-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

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 = (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+l.

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(T) 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-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(T)

for varying values of T 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 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 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(T) and the ordered collective of all outputs as

T
Z(T) = [zi(T),z2(T),.*-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 {i)}. If an inverse, H, exists

for G such that












1 0 ..*** 0
HG = GH = I= 0 1 0 (5-3)


0 0 *** 1


we now have a linear transformation of G which contains new N-dimensional

information vectors at N points in T:


I = GH = [.out(Tl)Zout(T2)"..Out( N)] (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+l

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




I '
S Z out3T)
I
I

I \
I \
I i







put (row) comes from the transformation
i / % I \
I I


/ T 2 T I T T 3 ,- T 4 ",.
I %

FIGURE 5.1 HYPOTHETICAL RANGE RESPONSE
INDICATING CONSTRAINT POINTS



in the response Zout3(S) as defined at points {Ti}. But each system out-

put (row) comes from the transformation
N
Zoutm(T) = I hmnn(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 lin-

early 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 (k) gives additional information not contained

collectively in {zi(Tk)} 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 {zi(T)} 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 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 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.











INFORMATION DEMODULATION
ELEMENTS WAVEFORM, *n(t) ORTHOGONAL?

1. Time Samples 6(t-tn) Yes

2nnt 2Rnt
2. Trig. Coeff. sin 2 or cos2r Yes
T T

3. Walsh Coeff. wal (n -) see [26] Yes

T
4. Squarewave Demod. Squarewave of period No
n

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


TABLE 5.1. SAMPLE DEMODULATION FUNCTIONS


Then
ST/2
Zn(T) = T 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 {1n} 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(T)} in T. 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 {in(t)} by


(5.1-2)


& (t) = o *(t) .
-orth.









In the next section, we will show that this represents a linear trans-

formation of

zrth. = 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(TI)..*z(TN)I-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
Zout = hmnzn (5.2-1)
n=l (A)

N T/2
= hmn y f e(t,T) n(t)dt (B)
n=l -T/2

1 T/2 c
= f/ e(t,T)m(t)dt (C)
-T/2

where
N
c(t) hmnAn(t) (D)
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 {(n(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 in-

dependence of all vectors zi(T) ) [zi(T1)Zi(T2)...Zi(TN)] (or 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 Tmax such that we are interested

only in the case when T s Tmax. Then the instantaneous difference (IF)
BTmax
frequency will vary between 0 and ax. 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 BTmax--the maxi-

mum number of cycles over a period T--is more than a few cycles, this

spreading effect is negligible compared with the frequency range or band-
BTmax
width By the sampling theorem we know that e(t,T) is defined by

its sample points uniformly spaced by T there being a total of
2BTmax
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 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) = f e(t,T)lm(t)dt (5.3.2-1)
-T/2 (A)

ST/2
= f cos[wc(t)T]c(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 be-

cause 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:











Zoutm(n) = zoutm(T)e-_ dT (5.3.2-2)
-00 (A)

o T/2
= I/ cos [Wc(t)T~ (t)dte d- r (B)
-0T -T/2

T/2
f dt 'm(t) f cos ac(t)T e- T dT (C)
-T/2 -0

T/2

-T/2

The delta functions are non-zero only when 0 = wc(t). Thus if wc(t) is

limited to (01,02), then Zoutm(Q) is bandlimited to (Q1,Q2). For the es-

sentially bandlimited signal such that 02-~i = 2rB, zoutm(r) is band-

limited 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 trans-
-i
formation 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 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) Z3( ) Z4 )






Z4 )
B~


= I from Figure 5,2,


1
1
Z2)

z 3)


(5.4-1)
(A)


=


0 0
0 0









and
-i
H = G1 = I (5.4-2)


z = Hz = z (5.4-3)
-out -

Here, {Ti} were chosen so that the transformation H is completely inef-

fective, and the range response is not improved. Other sets {Ti} would

constrain other points to be zero in each response, and at least a dif-

ferent 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 sat-

isfying, 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 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 {Ti}.

Finally, by equation (5.3.2-2D) we know that the range response,

zoutm(T) must 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


(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 develop relationships which are easier

to use and are thus of more value and importance.




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