IMPROVEMENT OF THE RANGE RESPONSE OF SHORTRANGE 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 Nonlinear 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 QuaziStationary 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. Delayline 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 Delayline 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 SingleChannel 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 CLOSEDFORM 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 Multidimensional 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 nDomain 117
8.3.2. The Form of the General TwoChannel
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 CLOSEDFORM 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 NONPERIODIC 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 NONMONOTONIC MODULATION 110
8.1 PHASINGTYPE SSB PROCESSING AT RF 114
8.2 TWOCHANNEL 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 SINGLECHANNEL SYSTEM USING TRIANGLE MODULATION
DC DEMODULATION 154
9.11 RESPONSE OF SINGLECHANNEL SYSTEM USING SAWTOOTH MODULATION,
THIRDHARMONIC COSINE DEMODULATION 155
9.12 RESPONSE OF TWOCHANNEL SYSTEM USING SAWTOOTH MODULATION,
THIRDHARMONIC COSINE AND SINE DEMODULATION 156
9.13 RESPONSE OF SINGLECHANNEL SYSTEM USING SAWTOOTH MODULATION,
HALFCOSINE DEMODULATION 157
9.14 RESPONSE OF TWOCHANNEL SYSTEM USING SAWTOOTH MODULATION,
HALFCOSINE AND HALFSINE DEMODULATION 158
FIGURE Page
9.15 RESPONSE OF SINGLECHANNEL SYSTEM USING TRIANGLE MODULATION,
HALFCOSINE DEMODULATION 159
9.16 RESPONSE OF SINGLECHANNEL SYSTEM USING SAWTOOTH MODULATION
AND DEMODULATION 160
9.17 RESPONSE OF SINGLECHANNEL SYSTEM USING TRIANGLE MODULATION
AND DEMODULATION 161
9.18 SPECIFIED SHORTPULSE RESPONSE 162
9.19 RESPONSE OBTAINABLE BY BANDLIMITING SHORTPULSE RESPONSE
USING RECTANGULAR WINDOW 163
9.20 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.19
FOR SAWTOOTH MODULATION 164
9.21 RESPONSE OBTAINABLE BY BANDLIMITING SHORTPULSE RESPONSE
USING HANNING WINDOW 165
9.22 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.21
FOR WASTOOTH MODULATION 166
9.23 RESPONSE OBTAINABLE BY BANDLIMITING SHORTPULSE RESPONSE
USING HAMMING WINDOW 167
9.24 FIRSTCHANNEL 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 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.26
FOR SAWTOOTH MODULATION 171
9.28 RESPONSE OBTAINABLE BY BANDLIMITING SHORTPULSE AT BT=6.4
USING RECTANGULAR WINDOW 172
9.29 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28
FOR SAWTOOTH MODULATION 173
9.30 SECONDCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SAWTOOTH MODULATION 174
9.31 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.28
FOR SINE MODULATION 175
9.32 SECONDCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE
9.28 FOR SINE MODULATION 176
FIGURE
9.33 SPECIFIED THREEUNIT EVEN PULSE RESPONSE'
9.34 RESPONSE OBTAINABLE BY BANDLIMITING THREEUNIT EVEN PULSE
RESPONSE USING RECTANGULAR WINDOW
9.35 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.34
FOR SAWTOOTH MODULATION
9.36 RESPONSE OBTAINABLE BY BANDLIMITING THREEUNIT EVEN PULSE
RESPONSE USING HAMMING WINDOW
9.37 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.36
FOR SAWTOOTH MODULATION
9.38 SPECIFIED THREEUNIT ONESIDED PULSE RESPONSE
9.39 RESPONSE OBTAINABLE BY BANDLIMITING THREEUNIT ONESIDED
PULSE RESPONSE USING HAMMING WINDOW
9.40 FIRSTCHANNEL DEMODULATION TO OBTAIN RESPONSE OF FIGURE 9.39
FOR SAWTOOTH MODULATION
9.41 SECONDCHANNEL 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 peaktopeak 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[].,F1 [] Fourier and inverseFourier 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)ejwtdt
CO
f(t) = F1[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 frequencydomain functions will be consistent;
i.e. C(*) will not be expressed alternately as C(f) and C(w).
Unless stated otherwise, uppercase functions are the Fourier
transforms of corresponding lowercase 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/voltsec.)
KR overall target reflectivity
LP lowpass
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 roundtrip delay time to target
TO a specified delay time
Tn roundtrip 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 = wcwo
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 singlesideband processor
the output of the third mixer in a delayline 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 upperhalfplaneanalytic 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 SHORTRANGE 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 quazistatic 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 delayline intermediatefrequency (IF) corre
lator. These systems are then related to the harmonic systems or nsys
tems, for "nthharmonic" 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 nsystems 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
closedform 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 rederivation 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 twochannel processor which allows synthesis
of any desirable BL range response. Finally, computer solutions are de
veloped to solve those problems for which closedform 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 prespecified 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 costeffective.
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 continuousduty 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
quencymodulated using highindex 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 lowheight 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 continuousduty oscillator is usually simpler to
design than one that is amplitudemodulated or pulsed. The energy of a
signal is the time integral of the signal magnitude squared:
E = f Ix(t)dt (1.21)
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 (1.22)
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 = (1.23)
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. (1.24)
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 highindex modulation (B4). When the modulating waveshape is
sinusoidal, the modulation index is defined as
B/2 (1.25)
fm
where B is the instantaneous peaktopeak 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 (1.26)
Furthermore, for "highindex" 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 (1.25). A more general parameter than modulation index
is the dispersion factor, commonly defined for chirp radars [6]:
D = B *T (1.27)
For sinusoidal modulation of frequency fm = 1/T
D = 2p (1.28)
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 signaltonoise 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 quazistationary,
i.e., almost static over a modulation period T.
The assumptions are, of course, supported byand are restatements 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 =  (1.41)
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 (1.42)
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) (1.4.11)
00
where Kf is the FM modulation constant in rad./secvolt.
Then the voltage at the antenna is the sum of the signal and the delayed,
attenuated return:
t t tT
x(t)+A(T)x(tT) = cos(mot+Kf fm(X)dX)+A(T)cos(mot+Kf fm(X)dXo0T+Kf / m(A)dX)
0 0 t
(1.4.12)
(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 tT
t t
sin(0ot+Kf m(X)dX)sin(wT+Kf f m(X)dX)}
o tT
t t 1
= {[l+A(T)cos(WoT+Kf f m(X)dX)]2+A2(T)sin2(&ot+ f m(X)dX)}
tT tT
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)
tT 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)
tT
An envelope detector yields the intermediatefrequency (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 (1.4.12D):
t
el(t,T) = 1+A(T)cos(w0T+Kf / m(X)dX) .(1.4.13)
tT
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) (1.4.14)
tT
Under the assumption of a slow modulation with respect to return times,
we may consider m constant over (tT,t) such that, for the integrand of
(1.4.14),
m(A) : m(tT) z m(t) for X over (tT,t) (1.4.15)
Then the convenient mathematical approximation e3 is derived as
t
e3(t,T) = A(T)cos(0WT+Kf f m(t)dX)
tT
t
= A(T)cos(wgo+Kfm(t) / dA)
tT
= 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(tT) =2A(T)cos (wt+Kff m(X)dX)
00
tT
Scos(mo(tT)+Kf / m(X)dX) (1.4.21
0 (A)
t
= A(T)cos(woT+Kf / m(X)dX)
tT
t tT
+ A(T)cos(2wotmoT+Kf fm(X)dX+Kf / m(X)dX) (B)
CO CO
The second term is centered at 2w0 in frequency. The lowpass (LP) fil
ter is designed to pass
t
e2(t,T) = A(T)cos(w0T+Kf / m(X)dX) .(1.4.22
tT
(1.4.16)
(A)
(B)
(C)
(1.4.17)
)
)
Again, we denote the simplified math form as e3 using (1.4.15):
e2(t,T) z e3(t,T) = A(T)cos[oc(t)T] (1.4.23)
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(tTn) (1.4.31)
n=l (A)
N tTn
= An(Tn)cos(wotiwO+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 tTn
2x(t)*xreturn(t) =2cos(wot+Kf fm(X)dX). An(Tn)cos(wo(tTn)+Kf f m(X)dX)
00 n=l CO
(1.4.32)
N t (A)
= An(Tn)cos(o rn+Kf / m(X)dA)
n=l tTn
+ double frequency terms (B)
Then
N t
e2(t,t) = Y An(Tn)cos(0oTn+Kf f m(X)dX) (1.4.33)
n=1 tTn
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] (1.4.34)
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 nonlinear (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 (1.4.41)
T/2 (A)
T/2
= f A(T)cos[wc(t)r]eJtdt (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 changeor modulationof 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. SSBAM 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 (1.4.42)
T/2 (A)
27BT 27rBT
T/2 [j ( )t j( + W)t
A(T) T T
= f2 e + e dt (B)
ST/2
A(T)T sin(rBTwT/2) sin(7BT+mT/2) (C)
4 L (7BTT/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 (nonstationary, 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 bandpass or a lowpass filter or integrator. When the target is not
moving, z is a function of T only; but if the target is slowlymoving,
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 Nonlinear Section
The last part of the system will generally be nonlinear 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 nonlinear).
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) (1.5.11)
00
It is somewhat instructive to observe the instantaneous frequency,
written (as before)
Wc(t) = og+Kfm(t) (1.5.12)
Similarly, the return signal A(T)x(tT) has an instantaneous frequency of
Wreturn(t) = 0+Kfm(tT) (1.5.13)
Plots of wc and return are given in Figure 1.5(A) for the linear modu
lation case:
t T T
m(t) < t < (1.5.14)
T2
m(t) periodic such that m(t+T)=m(t), all t.
It is easy to see that "turnaround 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
linecoherently detectedto 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
Wwreturn
__ 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
drawnout and Tozzi offers no easy or closedform solution for the gen
eralcase 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 (1.5.21)
where B is the signal bandwidth
and T is the modulation period.
Resolution is determined, byandlarge, by B. Woodward gives good intu
itive reasoning in the choice of the timeresolution 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 (1.5.22)
.10 Tmax Tmax (A)
As stated previously, the period of modulation must be largeat least
ten times,saywith respect to the largest return times expected:
T > 10 Tmax (1.5.22)
(B)
Then
10
D = B *T >  10 Tmax = 100 (1.5.23)
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 (1.5.24)
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
tnTT
Xreturn(t) = A(nT+T)cos[mW(tnTT)+Kf f m(X)dX] (1.5.25)
o (A)
tT
= A(nT+T)cos[WO(tT)+Kf / m(X)dA+6] (1.5.25)
00 (B)
where
tTnT
6 = m0nT+Kf / m(X)dX = wonT (1.5.26)
tT
since the integral of zeromean 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 QuaziStationary 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
signaltonoise 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." Thischapter 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
signaltonoise 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 continuouswave (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 quazistationary model precludes the
necessity of high resolution in velocity.
(A) WIDE MAINLOBE
T
(B) HIGH SECONDARY LOBES
(C) SPREAD, LOWLEVEL
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 (2.1.3.11)
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) (2.1.3.12)
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 (2.1.3.13)
00
Equation (2.1.3.11B) 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 (2.1.3.11B). 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)
(2.1.3.14)
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
(2.1.3.15)
+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
(2.1.3.17)
where D is the denominator of (2.1.3.16).
From multiplying both sides by D2 we obtain
2/fUoo*df{/UoUo*dff*/UoI2(AUo*+A*Uo)dff(UoUo*)2dff(AUo*+A*Uo)df} = 0
f(AU+A*U){IU0 12flUo(a) 2daIUg (a)I 4da}df = 0
(2.1.3.18)
(2.1.3.16)
Since A must be arbitrary
f{IUo(f)12.IUo(a)12Uo(a)4}da = 0
B B
for  f 6 B (2.1.3.19)
2 2
or
B/2
/ Uo(ca)2[jIU(ca)2I U(f)12]da = 0 f (2.1.3.110)
B/2
which has a solution at
B B
2 2
IUo(a) = (2.1.3.111)
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 modulationand thus a flat spectrum under the high
dispersion assumptionprovides best resolution and the least spread dis
tribution of power in the various other harmonic lines of the IF spec
trum [7].
Intuitively speaking, the wider the bandwidth or occupied sec
tions of the spectrum, the better the target resolution. Equal weighting
of each section of spectrum occupied has been shown to be in the best
interest of total ambiguity (in terms of 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 (2.1.3.21)
K(0)2 (A)
0 lu(t)14dt
=  (B)
m 2
f Ju(t)j2dt
where
K(<) = / U*(f)U(f+O)df (2.1.3.22)
00
and
U(f) = F[u(t)] (2.1.3.23)
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 quazistatic 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 (2.1.3.31)
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} (2.1.3.31)
(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 (2.1.3.31A) is simply doubled if the analytic forms
+ +
xl and x2 are substituted for xl and x2. Then
j27T(fo )t j27T(fo + )t 2
2ISE = / lu(t )e u(t + )e I dt
(2.1.3.32)
(A)
00o O
= f u(t 1)]2dt + f Ju(t + ) 12dt
I u(t )u*(t + )ej27tdt I u*(t )u(t + j)ej2 Ttdt
co o00
(B)
Notice that the first two terms of (2.1.3.32B) 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
(2.1.3.33)
(A)
The function is sometimes defined
00
X2(T,<) = f u(t)u*(t + T)eJ2 dt
(2.1.3.33)
(B)
which varies from X only in phase. As indicated above, we are gener
ally interested in the magnitude only; clearly
(2.1.3.34)
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)
(2.1.3.36)
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
(2.1.3.37)
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. JU
~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. 3.35)
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 (2.1.3.39)
00 W0
for signals of unit energy. Thus, the twodimensional analogy to total
ambiguity along the Taxis 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 rangedoppler 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(tT)ej27ft x*(t) = f x(AT)ej27ax*(Xt)dA (2.1.3.310)
0 (A)
= /0eJ 2f0Tu(XT)ej 27r(f 0+) Xu*(t)e j27O (tX)dX
(B)
= u(Xr)u*(Xt)eJ 27r eJ2 fO (tT)dX (C)
00
= ej2fg(Tt) X2(Tt,4) (D)
which is the ambiguity function times an RF phase function. Note that
the matched receiver is designed for a maximum peaksignal to average
noisepower 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
ST 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 redis
tribute effective ambiguity in the rangedoppler 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
finestructure 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 twodimensional Fourier transform of IX(T,) 12 produces
IX(t,f) 12.
of IX(T,4) 12ej2nfT2j2TtdTd1 = IX(t,f)12
00 00
8. f IX(T,) 12dT = I IX(T,0) 2ej27OTd
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(TpO,,)
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 quazistatic 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 Taxis,
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) (2.2.11)
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, ..
(2.2.12)
and the ambiguity function may be redefined as
T/2
Xp(T,) = f up(t)up(t+T)e 21tdt .(2.2.13)
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 ,(2.2.14)
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 (2.1.3.39). 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 normalizedlet 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 pseudorandom 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
"singleperiod" finiteenergy signal u(t):
(2.2.15)
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 nonperiodic 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 welldesigned system this will be the case.
Our range responseA(T) neglectedcan be considered
{Rp(r) MI < I
z(T) = 2 (2.2.19)
o Iri > I
because of the spaceloss weighting that will eventually be imposed; A(r)
will be very small for IT > T/2.
Then
(2.2.16)
(A)
(2.2.17)
(A)
(2.2.18)
u(t)
0 to t
up(t)
T+t0 0 to T t
(A) SIGNALS
C(f)
(C) SPECTI
P p(f)
A A
,til
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 NONPERIODIC EXAMPLES
Equation (2.2.17E) may be rewritten
Rp(T) = C(T) I
n=aa
Transforming both sides we obtain
Pp(f) = C(f) *
n=
6(T+nT)
(2.2.110)
6(f )
(2.2.111)
We will consider a power spectrum P(f) which is proportional to the
envelope of Pp(f) and which corresponds to the nonperiodic autocorrela
tion and range response:
  ..._ .. r
i
LCC r
I
V I, hIVL ,L. Jl ., IY
P(f) = F[R(T)] = C(f) (2.2.112)
(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) (2.2.21)
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 functiontends to "smear" the windowed
function's transform through convolution. This rectangular window is de
finded as
1 7B < w( < rB
WO(W) = (2.2.31)
0 otherwise
In the transform domain this window is
sin irBt
w0(t) = B s Bt (2.2.32)
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(&) (2.2.33)
where Wi is any window.
Transforming (2.2.33) we obtain the convolution (*)
wo(t) wi(t) = wi(t) (2.2.34)
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) (2.2.35)
0 otherwise
which has transform
1 1 1 1 1
w1(t) M w0(t) + w0(t +) + w0(t )
2~ w 4t F
(2.2.36)
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
(2.2.37)
otherwise
having, of course, a transform of
w2(t) = .54wo(t) + .23wo(t + ) + .23wo(t ) (2.2.38)
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 DolphTchebycheff 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 DolphTchebycheff window trans
form assumes the form
w(t) = os Bt2A2 (2.2.39)
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) = (2.2.310)
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 finitelength 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
zeroorder 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
(2.2.41)
where
WI(w) = [Wi(m+2nnB)]
n=oo
(2.2.42)
The transform of periodic Wi(w) is, of course, a series of delta func
tions in the t domain. All of the above windowsand, indeed, all real,
even windows, if N is large enough may be expressed
N
nw
Wi(w) = M ani cos BB
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
(2.2.43)
2.2.44)
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 (2.2.44). 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 ]) (2.2.45)
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
delayline 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 nonlinear 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 dopplerpass 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 (3.11)
M0o M/2 (A)
SM/2
lim M / A(T)x(t)x(tT)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
(3.11)
(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 closein
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) (3.12)
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(ts)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 nonnegative 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(TTR)
where TR is the delay of the delay line (Figure 3.2). Such a range
e(t,r TTR)
TO NL
CIRCUITS
x(tTR)
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 KIH(2Tf)12 where H(w) is the fil
ter transfer function.
3. Use nonlinear 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 highpower systems) and be
cause of a reduced average power when maximum peak power is fixed. Fil
tering also reduces average power of the signal, and if the center fre
quency of the transmitter is altered from that for which the filter was
designed, the shaping of the spectrum will be other than desired. Thus,
filters require extra stability measures on the oscillator. If non
linear FM is used, a high average power is preserved; the power is dis
tributed according to the desired window through selection of the suit
able modulation, m(t).
Shaping may also be accomplished in the "receiver" (if the sys
tem is so separablei.e., the system must have separate antennas) by
filtering methods similar to 2. above. Obviously, the system is no
longer "matched" and the signaltonoise 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 Delayline IF Correlator Systems
The delayline 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 doublefrequency
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]
(3.21)
FIGURE 3.3 I F CORRELATOR SYSTEM
Similarly, the output of the second mixer is expressed
eR(t,TR) = cos[B(t)TR] (3.22)
The output of the third mixer is
YO(t,T,TR) = cos[w(t)(T+TR)] + cos[m (t)(TTR)] (3.23)
and after timeaveraging (effected with the last filter) we obtain
1 1
Z2(T,TR) = R(T+TR) + 1 R(TTR). See (3.1.1A) (3.24)
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 noncausal 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.
(3.25)
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)
(3.26)
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
highcapacity channels may be constructed either of distributed parame
ters (as in a transmission line) or of a series of lumped sections.
Transmission lines are long and bulky and high bandwidth lumped parame
ter delays must contain many sections, thereby increasing their com
plexity and cost.
It is apparent that, if the signal x(t) is deterministic, then
x(tTR), and thus also eR(t,TR), will be deterministic. In fact, from
(3.22) and (1.4.16)
eR(t,TR) = cos[s0TR+KfTRm(t)] (3.27)
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 (3.28)
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 .(3.29)
= ( .29
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
Ao 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
(3.211)
As expected, the summation above is the total power of the line spec
trum of e. The coefficients are computed:
ST/2 23
an(T) = e(t,T)e dt (3.212)
T/2 (A)
2nnt
T/2 2j n
T/2 [ei[OT+KfTm(t)] + ej[o0T+KfTm(t)]] T dt.
2T /
T/2
For sawtooth modulation, m(t) = t for Itl ,2'
(3.210)
(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 (3.213)
(A)
KfTT
sin( ?rn)
2
KfTT
( T n)
ejwOt
eJ2
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
ejW0TR sin (2 + Tn)
2 KfTRT
( + in)
(3.214)
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:
(3.215)
where
(3.216)
(A)
For
(3.216)
(B)
j OT
()
, k = 0, 1, 2, ***
.27kt .2rrkt
T T
eR(t,TR) = ak e + ak e (3.217)
(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 (3.31)
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(tT) n
2 r = T TR (3.32)
for sawtooth modulation.
Such harmonic systems or "nsystems," 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 nsystem is equivalent to a delayline
IF correlator system when the modulation is
linear.
3. The nsystem becomes an autocorrelation
system or, equivalently, a matched filter
system when n=0. R(T) is sometimes re
ferred to as the zeroresponse for this
reason.
As per 2. above, the equivalence of the nsystem 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 nonlinear 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 delayline system, the equivalent window for the
nsystem 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() (3.33)
(A)
IF corr. ) = /Wx() (B)
Wnsyst. () = Wx(m) (C)
(Linear)
Note that we did not prescribe the manner in which m(t) might be designed
for nonlinear 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 nonlinearities 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 nonlinear waveshape for n > 0 [22].
Just as the RF signal may be represented in complex envelope form
x(t) = Re{u(t)ejOt} (3.34)
we may represent the IF signal as
e(t,T) = cos[(4m(t)+w0)T] = Re{eE(t,T)ejOT} (3.35)
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 (3.36)
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 (3.37)
n=oo
Couch [22] has shown that the Fourier coefficients, {yn}, relate to the
ambiguity function as
TrnT
Yn(T) = e T X2(T, ) (3.38)
(A)
and
IYnl = IX2(r, T1 I
Thus the range response of the nsystem using coherent demodulation when
the reference is defined as
27Tnt
eR(t) = cos ( + 6) (3.39)
with reference phase 0
becomes
27unt
T/2 m j2
E T/2 T 2nt
z3 = I Yn e cos (2T + 6)dt (3.310)
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, )} (3.311)
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
(2.2.15), (3.24), (3.31) and (3.311)
z3E(T) = Re X2(T, R) (3.312)
(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 linearFM (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 zeroorder Bessel
function).
In a later chapter, a relation will be developed which will pre
dict the displaced range response, regardless of modulation nonlinear
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 timevariable gain and a
timeinvarientfilter.
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(tTn) (41)
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) (42)
The object in the inverse filtering approach is to find a filter, h(t),
such that
h(t) x(t) = 6(t) (43)
Then one may filter the composite return to obtain
N
Xreturn(t) h(t) = Yn6(tTn) (44)
n=l
for which no resolution problems or ambiguities exist. In the frequency
domain
H(w)X(w) = 1 ; (45)
h(t) = F X( (46)
Naturally, signaltonoise 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) (47)
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) (48)
(A)
H(w) X(w) = G(w) (B)
where g(t) is to be specified, hopefully being a very peaked, lowside
lobe type function. Since
H() = w (49)
X(w) '
We must be very careful to provide zeros of G(w) to coincide with those
of X(w). An example of the problems associated with nulls is given by
the spectra of Figure 4.1(B) [25]. Especially if X(w) were to be 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 signaltonoise 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 realtime 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(TX)z(X)dX (4.11)
ou (A)
00
= f h(X)z(TrX)dX (B)
00
We see by (4.11) that the value of Zout(T) depends upon values of z(X)
for all arguments except those for which h(TX) = 0. We shall now exam
ine each system type for devices by which we might obtain the convolution
of (4.11) 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 (4.11).
66
The modified delayline autocorrelation system yields a response
zI(T,TR) = R(TTR) (4.1.11)
Then
Co
zout(T) = f R(TTRX) h(X)dX (4.1.12)
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 > TRTO > 0 (4.1.13)
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 quazifixed, TR
being allowed to vary slowly in time as illustrated in Figure 4.3. The
delay is varied as
Tx = TO+at .
(4.1.14)
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 noncausal filter response in T by a
realizable one in t, we scale and translate (delay) the response of h
to be
h(To+attd) (4.1.15)
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 quazifixed 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 Delayline IF Correlator System
Since the delayline IF correlator system yields a response of
1 1
z2(T,TR) = 1 R(T+TR) + R(TTR) (4.1.21)
(A)
7 R(TTR) 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 delayline system, provided the contribution of R(TTR) 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) (4.1.31)
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() (4.1.32)
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 wgnB < 0 & wM+nB:
Zout(T) =ut( )] = F1[H(R)Z(Z)] (4.1.31)
(A)
= FI 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 (4.1.3C) 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 = (4.1.41)
Then the parallel convolution construction of Figure 4.2 is applicable
to harmonic nsystems with the restriction that
AX = (4.1.42)
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 (3.312A), but equation (3.312B) 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 decon
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 delayline IF correla
tor system, the problem was solved by using an approximating pointby
point convolution using parallel implementation. Alternately, by using
a variable delay and a scaled timedomain filter, it is possible to ob
tain a timedomain 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 nonlinear 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 subsystem
output as Zn(T) and the ordered collective of all outputs as
T
Z(T) = [zi(T),z2(T),.*ZN(T)] (51)
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)] (52)
(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 (53)
0 0 *** 1
we now have a linear transformation of G which contains new Ndimensional
information vectors at N points in T:
I = GH = [.out(Tl)Zout(T2)"..Out( N)] (54)
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) (55)
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 subsystems, 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. Walshfunction expansion coefficients,
4. the outputs of squarewave 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) timeaverages 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(ttn) 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 (5.11)
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
(5.12)
& (t) = o *(t) .
orth.
In the next section, we will show that this represents a linear trans
formation of
zrth. = z (5.13)
Then if we transform using H' (derived from zorth.) we obtain
Zout = H' zorth. = H' 9 z = Hz (5.14)
so that we are still performing one transformation by H, determined as
H = G1 = [z(TI)..*z(TN)I1 (5.15)
Thus we perceive no generalizable advantage of orthogonal sets over non
orthogonal ones.
5.2 Equivalent SingleChannel 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 (5.21)
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 (5.21).
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 = G1, the
existence of that matrix comes into question. That existence depends
upon G having a nonzero 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 BTmaxthe maxi
mum number of cycles over a period Tis more than a few cycles, this
spreading effect is negligible compared with the frequency range or band
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 (5.3.11)
When the modulation is monotonic but not linear, we may consider a piece
wiselinear 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 (5.3.21)
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 (5.3.22)
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 nonzero 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 (5.3.23)
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, Gand thus Hdepends
upon the {Ti} by (52A)
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)
(5.41)
(A)
=
0 0
0 0
and
i
H = G1 = I (5.42)
z = Hz = z (5.43)
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 (5.3.22D) 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
(5.44)
H'G = R
so that
H' = RH = RG1 (5.45)
where R is the desired range response matrix
and H' defines the appropriate linear combinational matrix for the
system.
However, the abovementioned 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.
