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PHASE AND ENVELOPE OF LINEAR FM PULSE- COMPRESSION
SIGNALS FROM HIGH-VELOCITY TARGETS
TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-64-128
NOVEMBER 1964
ESD RECORD COPY
RETURN TO M. H. Ueberschaer
SCIENTIFIC & TECHNICAL INFORMATION DIVISION
(ESTI), BUILDING 1211
COPY NR. _ OF _ COPIES
Prepared for
DIRECTORATE OF RADAR AND OPTICS
ELECTRONIC SYSTEMS DIVISION
AIR FORCE SYSTEMS COMMAND
UNITED STATES AIR FORCE
L. G. Hanscom Field, Bedford, Massachusetts
Project 750
Prepared by
THE MITRE CORPORATION
Bedford, Massachusetts
Contract AF 19 (628)-2390
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ESD-TDR- 64-128
TM-03916
PHASE AND ENVELOPE OF LINEAR FM PULSE- COMPRESSION
SIGNALS FROM HIGH-VELOCITY TARGETS
TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-64-128
NOVEMBER 1964
M. H. Ueberschaer
Prepared for
DIRECTORATE OF RADAR AND OPTICS
ELECTRONIC SYSTEMS DIVISION
AIR FORCE SYSTEMS COMMAND
UNITED STATES AIR FORCE
L. G. Hanscom Field, Bedford, Massachusetts
Project 750
Prepared by
THE MITRE CORPORATION
Bedford, Massachusetts
Contract AF 19 (628)-2390
FOREWORD
The author wishes to acknowledge the help of Mr. R. D.
Haggarty, Dr. R. Manasse, Dr. J. F. A. Ormsby, and
Mr. R. W. Jacobus who also suggested the investigation
and reviewed the manuscript.
PHASE AND ENVELOPE OF LINEAR FM PULSE-COMPRESSION
SIGNALS FROM HIGH-VELOCITY TARGETS
ABSTRACT
Equations for the phase and envelope of the output signal from a linear filter,
matched to the transmitted signal, are derived. The transmitted signal is assumed
to have a flat band-limited amplitude spectrum and a linear group delay. The
input to the "matched filter" is the radar echo returned from a moving target
whose velocity is essentially constant during the illumination time. It is shown
that the returned signal is related to the transmitted signal by a time dilation.
The resulting expressions for the phase and envelope are functions which involve
Fresnel integrals. Approximations for these expressions are worked out. They
are shown to be similar in form to those which are obtained when the returned
signal is assumed to be related to the transmitted signal by a Doppler shift.
REVIEW AND APPROVAL
This technical documentary report has been reviewed and is approved.
Acting Chief, Radar Division
Directorate of Radar and Optics
iii
PHASE AND ENVELOPE OF LINEAR FM PULSE-COMPRESSION SIGNALS
FROM HIGH-VELOCITY TARGETS
SECTION 1
INTRODUCTION
This document is a first step in exploring the possibilities of using an
"all pulse-compression" (linear FM) radar system to obtain accurate estimates
of target range, radial velocity, and radial acceleration. The targets of pri¬
mary interest here are high-velocity targets, such as artificial satellites.
It is well known that linear FM signals have an inherent coupling between
range and velocity. However, it is also known that for targets whose range is
varying slowly, and for small pulse-compression ratios, this can be overcome
by transmitting alternately FM up and FM down, and taking the sum and differ¬
ence of the resulting time-delays to obtain unambiguous estimates of target
range and velocity, respectively. When the target range changes rapidly and
the pulse-compression ratio is high, certain simplifying assumptions are no
longer applicable, and the problem may become considerably more difficult.
This document investigates the problems from a fundamental point of
view. The accuracies required in the ultimate estimation of target parameters
dictate the necessity of measuring the phase of the target echo. We shall derive
several expressions (with different degrees of exactness) for the phase and
envelope of the output signal from a pulse-compression system. These expres¬
sions will be carefully compared and interpreted in a subsequent document.
1
We shall first review briefly the concepts of autocorrelation, matched
filters, and correlation functions. Given an aperiodic time function f(t), such
that
f (t) dt < 00
(1)
we define the autocorrelation function of f(t) to be
0 ( t)
f (t) f (t + r) dt
(2)
By a matched filter we mean a linear filter whose impulse response h(t) is a
reflection of the time function to which it is matched, as illustrated in Fig. 1.
Input
h(t) = f (T-t)
y(t)
Output
Fig. 1. Matched Filter
The quantity, T, is a constant which makes the filter realizable [i. e. , we
require that h(t) = 0 for t < 0] . The output is the convolution between the input
and the impulse response, i. e. ,
= 1
y(t) = \ f (x) h (t-x) dx .
Thus,
y(t) = \ f (X) f (x+T-t) dx
*^—00
Letting r = T-t, we find that
y(t) = \ f(x) f (x+t) dx ,
—OO
which is equivalent to 0 (r) defined by Eq. (1).
(3)
(4)
(5)
2
Thus, it is seen that matched filter reception and autocorrelation detection
are identical processes, provided, of course, that the filter is truly "matched"
to the input waveform.
In radar applications, the filter is often matched to the transmitted wave¬
form. The target echo may be quite different from the transmitted signal. If
this is the case, the output of the "matched filter" is no longer equal to the auto¬
correlation function but is now a cross-correlation function between the trans¬
mitted and received signals. It is often referred to loosely as an autocorrelation
function. Regardless of its name, it is this function which we are interested in
examining. We shall denote it by y(t) throughout the rest of this report.
3
SECTION 2
PHYSICAL MODEL
Consider the model shown in Fig. 2 t
h(t)
H(o>)
Fig. 2. Pulse-Compression Model
With the receiver matched to the transmitted signal, we have,
h(t) = s(T-t) , H(o>) = S*(co) exp [-jcuT] , (6)
where T makes the filter realizable, and the star denotes the complex conjugate.
A linear-FM rectangular pulse, lasting from time t = 0 to t = T, has the
form
cos
f(t) =
2* fi + K
; 0 < t < T
(7)
0 elsewhere
The Fourier transform of f(t) is a rather complicated function. How¬
ever, it has been found* that the amplitude spectrum of f(t) becomes nearly
2
rectangular as KT (the time-bandwidth product) becomes large (>100). The
present D-82 experimental radar facility employs a pulse-compression system
with a time-bandwidth product of 1, 000. A future system is proposed which
*cf. Klauder, et al. , "The Theory and Design of Chirp Radars, " B. S. T. J. ,
July 1960.
5
will have a time-bandwidth product of 10, 000. Instead of synthesizing a simple
function of time, the procedure here is to synthesize a simple function of
frequency, denoted by S(u>).
The function S(u>) has approximately a rectangular amplitude spectrum
A(oj), and a linear group delay* T(u>). For large time-bandwidth products, this
corresponds approximately to a linear FM pulse.
For our analysis, we shall use the model shown in Fig. 3.
Fig. 3. Model of A(co) and T(u>)
* Group delay is defined as
T(u>)
d0(g?)
do;
where 0(u>) is the phase of the spectrum.
6
The respective equations are:
„ W - , W
H ;“o- T s M £“o + —
A(u)) =
(8)
t)
elsewhere
TM - - 57 I
2 I ,1 t / w\
W ' ^ W yo ~ 2 ) ’
(-.*?)■
U). — — —
w
2
T , , T
w + W
co - “ —
0
W
2
i i W
| co | + ~2 for FM up
i i W
| co I ^oj + — for FM down
U L*
elsewhere (for both FM up and FM down)
(9)
The spectrum S(u:) can be written as
S(o;) = A(oj)e
j0(<^)
(10)
where 0(co) = - C T(u>) do.- plus a constant of integration. Without loss of
generality, we can let H = 1 in Fig. 3 and Eq. (8). Doing this, we obtain
J0(w)
S(w) =
w , , w
co — — — Cl) — co + —
0 2 11 0 2
elsewhere
(11)
It is clear that S(u>) is an even function, since A(u:) and T(a;) are both
even functions. For simplicity, we shall work only with the positive portion
of the spectrum throughout the rest of this paper. It must be remembered that
the actual spectra contain negative frequencies.
7
Integrating T(u>) and restricting ourselves to ui ^ 0, we obtain
S(w) =
where
j[c0 + u + c2 w2]
W ^ ^
; CO - — — CO — CO +
’ 0 2 0
elsewhere
c =
c =
!(-o- 1) jtorFMup l
~ff (“o + t) ^forFMdowni
w
2
C = =F
2 2W
\ - for FM up
1 + for FM down
(12)
(13)
(14)
and Cq is a constant of integration which will cancel out later. Equation (12)
and Fig. 2 shall be applicable throughout this paper.
The following two pulse-compression systems are of particular interest:
Pulse-Compression Systems
T
W
2ir
Cl)
_0
27T
No. 1
1 msec.
1 me.
1280 me.
3
(10 system)
No. 2
2 msec.
5 me.
1280 me.
4
(10 system)
The length of the transmitted pulse is approximately equal to T.
We shall regard the targets under discussion here as being essentially
point targets. Let the maximum radial acceleration of our targets be 200
2
m/sec . Suppose a pulse of duration T is emitted at time t = 0. Let a given
target be at range R when the leading edge of the pulse strikes it. Assuming
8
free-space propagation, this occurs at the time t = — , c being the velocity
of light. Let the radial velocity and radial acceleration of the target at that
time be ef^ual to V and A, respectively.
During the time interval ( «T) that the target is being illuminated, the
change in radial velocity is
AV = AT ^ 0. 2 m/sec for system No. 1,
^0.4 m/sec for system No. 2,
2
(since A ^200 m/sec ). The corresponding change in range due to the accelera¬
tion term alone is
1 2
AR = — AT ^ 0. 0001 m for system No. 1,
0.0004 m for system No. 2,
This is only a few degrees of phase shift for the 1280-megacycle radar, which
is very small indeed. We are primarily interested in large radial velocities
3 4
(of the order 10 and 10 m/sec, producing a AR = VT of the order of 1 to 20
meters).
Throughout this paper we shall regard the target velocity as being constant
during the illumination time.
9
SECTION 3
DOPPLER-SHIFTED SIGNAL
One assumption which is often made is that the radar signal bounced off
a moving target is a delayed and attenuated replica of the transmitted signal,
except for a Doppler shift. * This is only an approximation. It is a fairly good
one for small velocities. In Appendix I it is shown that the correlation function
(under this assumption) is equal to
The envelope has the familiar
sin x
x
form.
In Eq. (15)
k
W'
t’
P
some constant which depends on the radar cross section of
the target,
T
/2V \
- for FM up
W
{- V
+ for FM down
*By this we mean that the spectrum of the returned signal X(o j) is related to the
spectrum of the transmitted signal S(oj) by
2R
” W c / 2V \
X(w) = ke c S fw + a>0) ,
2V
where — cj„ is the "Doppler shift."
c 0
11
2V
Gl),
6
R
V
The above
it with the more
W
2
+ for FM up
- for FM down
= the target range when the pulse strikes it, and
= the target radial velocity when the pulse strikes it.
expression was derived primarily for the purpose of comparing
exact expressions to be developed in the next section.
12
SECTION 4
TIME-DILATED SIGNAL
In Appendix II it is shown that (under the assumption of essentially constant
velocity during the illumination period) the relationship between the transmitted
signal s(t) and the returned signal x(t) is given by
x(t) = k s
This corresponds to a "time dilation. " Here
c+V . 2R
a = — — ; b = -
c-V c
(18)
and k, R, V, and c have the same meanings as before. In Appendix II we also
obtain the Fourier transform of x(t):
X(w) = k a e"jal) S (aw) .
The output of the receiver is then
Y(w) = X(w) S*(w) e"jcJr = k a e_jc0<b+T) S(aw) S* (w)
From Eq. (10) it is clear that
V *
S*(cu) =
Similarly, we have
10 (aw)
w
CO - -7T —CO — CO +
0 2 o
elsewhere
W _
w „ - — — aw ^ w +
0 2 0
W
2
W
2
S(aw) =
(19)
(20)
(21)
(22)
elsewhere
13
Since a > 0, this can be written as
W W
; U) - — U.' + —
j0(aw) ’0 2 0 2
e ' ; - -
S(aw) =
0 elsewhere
Combining (21) and (23), we obtain
i[0(ao.’) - 0(a’)]
S(aw) S*(u>) =
Cl) +
w o
uj - — ^ cc — -
0 2 a
W
0 elsewhere
Thus, over the non-zero interval, we have
0(au>) - 0(cj) = cQ + c^^ au> + c2 (aa)2 - [cQ + o> + c2 a-2]
2 2
= c^ u> (a-1) + co a; (a - 1)
Let
a - 1 = y =
2V
c-V
(23)
(24)
(25)
a - 1 = (a-1) (a+1) = y \
y =
2c
c-V
Substituting (24), (25) and (26) into (20), we obtain
2
(26)
W
Y(w) =
Cl) + —
\V ^ 0 2
Cl) - ^ Cl) — -
0 2 a
elsewhere
(27)
14
Let go" be the center of the non-zero region in the go domain, and let W" be the
bandwidth of Y(oj); i. e. , let
cj" +
0
W”
2
W"
= GO -
0
w
2
(28)
(29)
Combining (28) and (29) with (18), and solving alternately for go" and then for
W" W
in terms of and — , we obtain
"0
r
2
c
"o \cTv
w
w
T
c+v
-or
V
c+V
V
0 Vc+V
(30)
(31)
These expressions are valid for a receding target. For an approaching
target, we would obtain
0
W"
2
0 \c-V
W / c
2 (c-V
W
V
c-V
-
V
0 Vc-V
The output spectrum can now be written as
2
1j[u(-b-T+Cj y] + a;
ka e
0
We must remember that Y(co) in Eq. (32) is only the positive half of the
actual output spectrum. But since we have an even-frequency function, the
output-time function y(t) can be obtained by taking twice the real part of the
Fourier transform of the positive half of the spectrum; i. e. ,
W"
2
GO SCO,
W"
2
(32)
elsewhere
15
y(t) = 2 Re
ka P
2t r J,
a):' +
W'
2/ jco[t - b - T + C1 y] jcu y X
dco . (33)
W"
2
Here Re f(t) means "the real part of f(t). " Let
t’ = t - b - T
= t - — - T ,
c
(34)
where t' = 0 corresponds to the peak of the autocorrelation function for the
case of zero velocity. We then have
y(t) = 2 Re
ka f
2t r J
“o'+ ?
2/ jw[t' + c -y] j co c2 y\
dco .
(35)
W
0 2
This expression is exact for a constant-velocity target, and very nearly
exact if the percentage change in velocity over the pulse duration is small.
However, it cannot be integrated in closed form. It is convenient to introduce
the variable
S2 = a; - cii^' ,
(36)
so that y(t) becomes
where
y(t) = 2 Re
ka jcr P
27r 6 J
W"
2
j[c Sl+c if]
e dS2
(37)
W"
2
a = (t' + y c1) cu” + y\c2 (cd^') ,
cg = t' +yc1 + 2y\c2 to” , and
C4 = YXC2 •
(38)
16
Equation (37) can be manipulated into the form
where
y(t) = 2 Re
u . a
i j» —
e
= 2 Re kx [ Z (ux) - Z (u2)]
dx
(39)
Z(x)
■I
X j7T
e
0
a
dx
(40)
is the complex Fresnel integral. The resulting expression can then be put in
terms of the simple Fresnel integrals,
C(x)
-f
J0
cos f 7T ^-jdx ; S(x)
■r
Jo
sin ur — J dx , (40a)
which are tabulated functions. This was done (the details are given in Appendix
III), yielding
X"i /.
(41)
y(t) =
0
^[C(x3) - C(x4)] cos xi + [S(x3) - S(x4)] sin x^
for FM up
x
y(t) = — ^c(x3) - c(x4)] cos Xx ~ ts(x3) " s(x4)] sin x^
2 /
(42)
for FM down
where
ka
•>y 27r ■ ’
(t’)2 + 2y\Ci t'
4y\c2
17
X
3
x
4
+
W"
~2
, and
W"
“
(43)
The other symbols are the same as before. We can also express y(t) in the
following form:
y(t)
R cos (x^ - 0) ,
(44)
where
R = -]/ [C(x3) - C(x4)] 2 + [S(x3) - S(x4)]
(45)
for both FM up and FM down, and
-1
+ tan
S(xg) - S(x4)
0 =
- tan
-1
C(X3) " C(X4)
S(x3) - S(x4)
C(x3) - C(x4)
for FM up
for FM down
(46)
These expressions are exact for a target whose velocity is constant over the
pulse duration. Tables of the Fresnel integrals are available so that numerical
answers of the correlation function y(t) can be obtained directly.
It would, however, be desirable to obtain approximate "closed-form"
expressions for both the envelope R and the carrier term cos (x^ - 0). We
are primarily interested in the representations which are valid in the vicinity
of the peak of the envelope.
18
In Appendix III we show that the peak occurs when c = 0. This corresponds
u
to the time
(Note that y = — — = 0 when V = 0). Thus, we let t" = c in order to empha-
C — V o
size that the peak occurs at t" = 0; i. e. , let
t" = t - — - T + vc + 2 v X c o>"
(48)
In Appendix III it is also shown that (for the two pulse-compression sys¬
tems we consider here) the arguments x^ and x^ of the Fresnel integrals are
close to unity at the peak of the envelope (i. e. , at t" = 0) for large velocities.
Thus, asymptotic approximations for R and 0 which are valid in the immediate
vicinity of the peak are unsuitable except for very low target velocities. How¬
ever, the analysis of a simple Doppler-shifted signal [with Eq. (15) as the
resulting ambiguity function] is expected to be adequate for low velocities. We
shall, therefore, not attempt to get "closed-form" approximations for (45) and
(46), which are valid in this region.
As we move away from the peak, the absolute values of x and x increase
u 4
quite rapidly, allowing us to use asymptotic approximations of C(x) and S(x), for
large x, yielding approximate expressions for R and 0 which might provide
some insight. This was done in Appendix III with the result
(49)
where
(50)
19
(51)
In Eq. (44) we expressed the correlation function by
y(t) =
o
R cos (x^ - 0) .
This can be written as
y(t) = E cos 6 . (52)
In Appendix III the following approximate expressions for the envelope E and the
phase 6 were obtained:
e « oo” t" ±
/ 2V\ / 2c \ / T
\c-V/ \c-V/\2W
sin
W"
~
(■" ?)
for FM up
for FM down
(53)
(54)
(55)
These approximations are good for
.-I >“•- (sxa,) (»)-“(?)’■
It is interesting to note that the (t")^ term in the expression for 0 cancelled
with (t1 ) term in the expression for x^ [see Eq. (43)], so that 6 = x^ - 0 has
only linear time dependence.
This is a direct result of the approximations we have made. The conclu-
2
sion to be drawn is that the (tM) term is negligibly small in the region indicated
by Eq. (55). This may or may not be the case in the immediate vicinity of the
peak.
20
Let us consider a different method for approximating the output function.
Going back to (37) we have
Y(t) = 2 Re
ka j a C
27 r 6 J
W"
~2~
j [c ft + c, ft ]
e 3 4 dn .
W"
~2~
It would be nice if we could simply ignore the ft term. Suppose we say that the
answer we get by doing so is a good approximation if
|c3 S2| > 100 |c4 ft I .
Letting t" = c and substituting for c , we find that (56) becomes
O rr
(56)
|tM n| > ioo
Clearly, for any particular value of t" this inequality is most difficult to satisfy
ii W"
when | S2| is largest; i.e. , when ft = ± .
Thus, (56) is satisfied if
:*•!>*" (5) (5) (#? -"(?)*■
(57)
When the ft term is ignored, (37) is integrated readily to yield, after taking
the real part,
y(t) w a
sin |
wff
L‘ — J
W"-
~2~
cos a
(58)
We note that the envelope above is identical with Eq. (53). Let us compare the
phase with Eq. (54). We have, from (38),
.2
a = (t' + yc1) wj' + y Xc2 (w”)
(38)
21
When we< express t' in terms of t", with the aid of (48), we have
co’’ V = co’’ t" -ycx co’’ - 27\c2 (co’’)
Substituting (59) mto (38), we obtain
co^' t” -yXc2 (co^')
(59)
(60)
This becomes
(y - (jj
o' * £r) (£) ©
, 2 + for FM up
0 - for FM down
(61)
Comparing this expression with the phase 9 in Eq. (54) we find that the two
expressions are the same except fbr the last factor which, in (54)
2
■ W
, is j(co”)
instead of just (oo^' ) . However, since co^' « co^ (very closely) and
W" < W, and since co^ is about three orders of magnitude greater than W, we
see that cr and 0 are identical for all practical purposes.
Note that the region over which (58) is valid is, from (57),
i"i
In (55) we required |t"| to be twice as large as that, which was a little more
conservative.
It is quite remarkable that the simple way of approximating y(t) shown
above yields virtually the same result as the rather involved procedure of
approximating the Fresnel integrals.
22
SECTION 5
SUMMARY
We have derived three different expressions* for the output function y(t).
Each expression is of the form
y(t) = E cos 0 .
(1) Simple Doppler Shift: X(c o) = k e
2R
-jcu -
where
2k W' Sm
(t* + P )
W’l
2~J
7 r 2
W' 1
( + P ) 2
= <t' +/3)«J
+ <5
[» f * ? 4
W'
w
2V
2
2
— CO
c 0
>
t'
= t -
2R
c
- T ,
P
= T
2V T
“ ^0 w
j - for
| + for
6
2V
c
[•5
T
± —
w
<x> '
0
= CO
0
?)■
(62)
*Note: The expressions listed here may differ slightly in appearance from the
corresponding ones in the test. They are identical, however.
23
c = the speed of light,
R = the target's range when the pulse strikes it, and
V = the target's radial velocity when the pulse strikes it.
-jw
(2) Time Dilation: X(u>) = k e
E =
The exact expressions (within the limits of our assumptions) are:
(t) ‘ •(»";) -i'vl*! ■
© -d
#
t" ± y X
± L -
± tan
S(x3) - S(x4)
C(x3) - C(x4)
+ for FM up
- for FM down
(63)
The terms C(x) and S(x) are the Fresnel integrals, defined by Eq. (40a), and
c+V 2V
oo
0
W"
2
c-V
y =
c-V
X =
2c
c-V
(£r) - ? (ii) ■
(c~7v) ■ “o (jTv) ’
(w)[“o- ¥ -
w
2
2R
= t - - - T ± y
c
X oj"
0
? © ’RJ)
1 w L- (£)
+
W^n j- for FM up I
2 I |+ for FM down j
+ for FM up |
- for FM down ) ’
W
~2
]
?]
and
| - for FM up |
|+ for FM down ^
24
(3) Approximations for E and 6 in Eq. (63).
fa\
WT,1
r tJ
i)
l ft-
W"1 J
9 « t" ± y X
tf) [<-;•)• -('f
|+ for FM up |
| - for FM down ^
These approximations are expected to be quite good for
(64)
|t"| >50W"y\ ^L\ M 100 T .
They may be adequate for considerably smaller values of |t"| .
The "exact" expressions in (63) are not easily interpreted. This will
have to be done numerically. We did, however, determine that the peak of the
envelope occurs at the time when t" = 0. The peak of the approximate expres¬
sion for the envelope in (64) occurs at exactly the same time. It is interesting,
also, to note that the expressions in (64), for both the phase and the envelope,
are virtually identical to those in (62) when the target velocity is small.
Due to the length of this document, we shall reserve the detailed interpre¬
tation, comparison, and application of the above expressions for a subsequent
document, ESD-TDR-64-129.
%4 Ktfa
L ¥ (1*1 <
M.H. Ueberschaer
25
APPENDIX I
ANALYSIS OF A DOPPLER-SHIFTED SIGNAL
We assume that the relationship between the transmitted and received
Fourier spectra is
2R
-j — c o
X(co) = k e C S(co — co ) , (I— 1)
d
where
k is the attenuation constant,
R is the range of the target when the pulse strikes it, and
co, is the "Doppler shift", given by
2V
“d - - - "» • ‘!-2>
where V is the radial velocity of the target when the pulse strikes it. (We are
here using the convention that V is positive when the target is receding from
the radar. ) Consider the case of a receding target, and let
2V
a = — co. = — or .
d c 0
If Eq. (11) is substituted for the transmitted spectrum, (1-1) becomes:
2R
-jco
k e
c jp(co + a)
e
X(u>) =
W W
CO - — — CO + Q! — CO + —
0 2 0 2
elsewhere
(1-3)
(1-4)
27
Again, we must remember that this is only the positive half of the frequency
spectrum. Equation (1-1) can be written as
2R
k e
^ c j 0 {oo + a)
W
Wj
X(w) =
We have
Y(u>) = X{oo) S* (uj) e
By use of Eq. (1-5), this becomes
[k (' c“ + T) ej[0(w+a) -0(w)]
Y(W) =
; a,0-a- T + T
elsewhere
-jcoT
(1-5)
(1-6)
W
W
; oo - — — U) — oo - a + — J
’0 2 0 2 1
elsewhere
Now,
0 (oj) = Cq + oo + c^ oo ,
with c and c given by (13) and (14). Thus,
X Li
(1-7)
(1-8)
{oo + Q! ) — <f>{oo) = C + C (ii) + a) + C (ft)^ + 2 U Q + a^) - [ c + c w +c.w^]
V 1 Li U 1 Li
= oo ^2 c^ of^ + (c^ a + c a2)
(1-9)
Let
2 c a = p , c a + c a = <5 .
Li X Li
(1-10)
28
Further, let co^ be the center of the non-zero region of Y(co), and let W1 be the
bandwidth of Y(co); i. e. , let
a W' W-a
“o u0 ~2 ’ T ~2~
(1-11)
Then we obtain
Ik ej6 e
--PM.,
Y(co) =
jco/3 , W _ W'
e ; co 1 — — — co — co 1 + —
’0 2 0 2
(1-12)
0 elsewhere
Our desired time function y(t) (i.e. , the correlation function) is twice the real
part of the inverse Fourier transform of Y(co); i. e. ,
yd) = 2 Re L J
Y(co) dco
Substituting (1-12), we have
±r°
, W’
+ 2 id ^
y(t) = 2Re ^ J w, k e e
"o* T
t- HE
c
- T + P
Let
f = t - — - T
c
We can integrate y(t) directly, obtaining
k id
y(t) = - Re e
is
j [t + p] J
CO
W'
o " T
W'
co' - —
0 2
(1-13)
dco (1-14)
(1-15)
W'
W'
k id j[t’ + /3]w(; ej[t'+/3] 2 _e-j[t' + /?1 2
— Re eJ e \ - rrr, - vi -
t r \ j[t' +p]
(1-16)
29
from which we finally obtain the desired expression
y(t)
/ sin
.(f + (3 ) 2
W'~
<t' +0) ^
|cos [(t' + /3) + 6]
(1-17)
30
APPENDIX H
RELATIONSHIP BETWEEN TRANSMITTED AND RECEIVED RADAR SIGNALS
For simplicity, we shall consider a point target with constant cross sec¬
tion. Suppose the transmitter sends an impulse in the direction of the target
at time t = 0. The transmitted signal is given by
(t) = 5(t) , (II-l)
where 6(t) is the delta function. Assume free-space propagation. Let the
target be at a distance
Rj = (II— 2)
when the impulse hits it, c being the velocity of light. The returned echo will
then also be an impulse which will strike the antenna at time 2t^ , i. e. ,
xi (t) = k 8 (t - 2^) = k 6 ^t - 20 , (II— 3)
where the constant k depends on the radar cross section of the target. Let
another impulse be transmitted a short time t later, so that
s2 (t) = 8 (t - r) . (II— 4)
For t^ > t, this impulse will be at a position in space equal to R^ - cr when the
first impulse strikes the target. Let the target have radial velocity V and
radial acceleration A at time t . For V « c, this second impulse will strike
the target at approximately an interval t after the first impulse did. We have
shown in the text that for the targets and the radar parameters of interest to us,
the acceleration has negligible effect (over the illumination time) on the velocity
and range. Here we assume t to be shorter than the total illumination time.
31
Thus, the relative velocity between the second impulse and the target at
time t
AR =
is AV = c - V. The distance between the target and this impulse is
cr. The time required for this impulse to "catch up" with the target is
At =
AR
SV
CT
C-V
(II-5)
The target will be at range
R„ = R, + VAt
2 1
R.
+
Vc
c-V
T
(II-6)
when this impulse strikes it. The echo returned from this impulse is then
given by
x2<*> - k 5 (»-r- ^r)= - 0* T) • <«-7>
/ 2V-\
Thus, the two returned pulses are a time t 1 1 + — -^1 apart, while the two
transmitted pulses are only a time t apart. A time-dilation has taken place.
Since we can think of our actual transmitted signal as consisting of a
sequence of impulses, we see that the target may be regarded as a time-
varying ideal delay line in cascade with a time-invariant attenuator. This is
illustrated in Fig. II— 1 .
W(t, t) k
Transmitted
s(t) -
Time-
varying
Attenuator
Received
Signal
Delay
Line
Signal
Fig. 11-1. Ideal Delay Line
The delay line is a linear, time-varying filter. The output y(t) of such a filter
is related to the input x(t) by*
*cf. Laning and Battin "Random Processes in Automatic Control," McGraw-
Hill, 1956, p. 226.
32
s( t) W (t, t) c!t ,
(H— 8)
f(t)
where W (t, t) is the time-varying impulse response. We may think of it as a
curve in the t - t plane which, for the case of constant velocity, is a straight
line, as indicated in Fig. EL— 2.
Fig. 11-2. Constant Velocity Case
t
5
Here t is the " input time" and t is the "response time. " From Eq. (II— 7) , it
is apparent that the equation of this line is given by
W(t, t)
For convenience let R =
that (H-9) becomes
Recognizing that W(t, t) =
(II— 9)
0 for t < r, we find
33
But, 1 +
2V _ c+V
c-V “ c-V
Letting
. 2R . c+V
b = — and a = — —
c c-V
we obtain
f(t) = s
The Fourier transform of f(t) is given by
(¥)
F(w)
■ s r •-m «
— oo
- i. f" 8 tS e‘iwt
27 r J a
dt
. ± r
2tt J
8(u) e'ia,b e-lu(au)
du
= a e
-jcob
- r
27 r J
s(u) e
-j(acu)u
du
(H-ll)
(11-12)
(11-13)
We recognize the last line of (11-13) as equal to
*
F(to) = e_;iajb a S(aco) . (H-14)
Thus, the signal returned from the target is related to the transmitted signal
s(t) by
x(t) = k s
(v).
(11-15)
and its Fourier transform X(co) is related to S(a>) by
X(w) = k a e~^wb S(aw) . (11-16)
Equations (11-15) and (11-16) are physically plausible, as we shall now show.
Suppose we send two impulses, one at t = 0 and one at t = r; i. e. , let
s(t) = 6(t) + 6(t - t) . (H-17)
34
Then
x(t) = k
[•(?)•*(?--)]
-■f(- ?)•*(■-?-
c+V
c-V
(11-18)
Thus, the received signal is attenuated, delayed and stretched (since the target
is receding).
Next consider a C\V signal; i. e. , let
S(co) = 6(cd - cr^) + <5(co + cr^)
from (II— 1G) we obtain, for the received signal,
X(oj) = ka e [6 (acd - cd^) + 6 (acd + cd^)]
(11-19)
= ka e
-jedb
Cd .
0
Cd
0
6 I co — — I + 6 I cd + —
Cd,
The new carrier frequency is therefore equal to — , where a =
Now,
c+V
c-V
1 +
2V
c-V
(11-20)
c+V
c^V ’
(11-21)
for V < — this can be expanded in a power series, i.e. ,
tl
(11-22)
2 V
For V < < c this becomes approximately equal to 1 - — , so that the new
frequency is
"o /. 2V
T o y T
2V \
(11-23)
which we recognize as the familiar "doppler-shifted" frequency.
35
APPENDIX III
ANALYSIS OF A TIME-DILATED SIGNAL AND
APPROXIMATIONS TO THE FRESNEL INTEGRAL EXPRESSIONS
We start with Eq. (37 of the text, which reads
— 2
ka i a C 2 n + C4 n 1
y(t) = 2 Re - e] e dS2 .
2n J W"
~2
The exponent inside the integral can be written as
j[o3n*c4a2] = jc4 | n +
so that (HI-1) becomes
c.
j Iff -
kn.
y(t) = 2 Re, — e
w
4c- 1 ~2
I
W"
2
ln+ 2-r
dS2 .
Let
3 1 IT a
c, (H+ - —
4 * 2 C4
. - h
7 r I 2c
dn = JJL ' da
V2C4
(HI-1)
(III- 2)
(HI— 3)
(HI— 4)
37
Thus, we obtain
y(t)
j
ka
= — Re e
7T
2
a
This can be written as
y(t)
(m-5)
(m-6)
where Z(u ) and Z(u ) are the complex Fresnel integrals, with u and u given
1 C* X Li
by
(HI-7)
We must take the real part of the product of several complex quantities. For
convenience, let us list all the other symbols appearing in (HI-6) in terms of
the more elementary parameters; i. e. ,
38
c+V 2 V 2c
a ~ c-V ’ ^ _ c-V ’ X "" c-V ’
a = (t' +yci) co” +yXc2 (coJ') ,
C3 = v +Tci + 2 7^c2 uq >
C4 = yXC2 ’
1 L _ w\
W 1 0 2 y
T ( W\
W (^0 + 2 j
for FM up ,
_T_
2W
for FM down ,
for FM up ,
2W
for FM down ,
co"
0
W"
2
/ c \ _ W / V \
^0 l c+vJ 2 (c+v)
(cTv) - “o (cTv)
w
2
2R
t' = t - T
( III— 8)
39
We note that
is imaginary for FM up and real for FM down; i. e. ,
for FM up
for FM down
(HI- 9)
where j =
Now let
ka
2
c
(t')2 + 2 y\c V
4 yXc2
2y\c2
2y\c(
(III- 10)
Using these symbols, we have, for FM up,
{cos x + j sin x i
- - - - [Z(jx3)-Z(jx4)]J
(III-ll)
40
Similarly, we obtain, for FM down,
y(t) = x Re
cos + j sin x
0 ™ )'
- [Z(x3) - Z(x4)]| .
(Ill- 12)
All the x's are now real quantities. The complex Fresnel integral Z (x) can be
expressed (for real x) as follows:
Z (x) = C (x) + j S (x)
Z ( jx) = S (x) + j C (x)
(III— 13)
where
f\X 2 rx 2
(o' [a
C (x) = \ cos 7r — do1 ; S (x) = \ sin 7 r — do: .
^ n * 2
(m-i4)
These are tabulated functions. Using (III— 13) , we obtain
x
y(t) = — ][c(x3) " c(x4)l cos X1 + [ s(x3) " s(x4)l sin X1 [ (HI— 15)
2
for FM up, and
x
y(t) = — }tC(x3) - C(x4)l cos Xx " tS(x3) " S<x4)l sin xj } • (HI- 16 )
2
for FM down. (Note that x , x and x have different values for FM up and FM
1 u Tr
down. ) We can express y(t) in the following form:
y(t) = ( 3T J R cos (xx - p) .
(Ill— 1 7)
41
where
R =
[C(Xg) - C(X4)]2 + [S(Xg) - Stx^]2
(m-i8)
for both FM up and FM down, and
i rs<*3)-s(x4)
S(x3) - S(x4)
for FM up ,
for FM down .
(Ill- 19)
These expressions are exact (within the limits of our approximations), and
can be evaluated with the aid of available tables of Fresnel integrals. However,
we shall try to obtain approximate closed form representations for R and 0 in
the vicinity of the peak of the envelope.
Let us first consider the envelope R. We wish to determine its maximum.
From the analysis leading to Eq. (16) in the text, we expect this maximum to
occur near c^ = 0. Let us postulate that the peak actually occurs at that point.
A necessary condition for this to be so is that
when
c3 = 0 = t'+yc1 + 2y\ c2 a,"
(HI-20)
t- T -T+yc1 + 2yXc2 co-
42
Let us check whether this condition is satisfied. From (III— 18) we have,
W = -k {2[c(x3> - C(x4)1 S [c(x3> - C(X4)I
+ 2 [S(x3) - S(x4)] j (S(x3)-S(x4)]| .
Equation (III— 2 1 ) will be equal to zero if both
i [C(x3) - C(x4)] = 0, and
Tt lS(X3> - S(x4)1 =°'
(III— 2 1 )
(III— 22)
provided that R does not equal zero at the same time. The Fresnel integrals in
(III— 22 ) are all of the form
f (a) da = F [x (t)] - F (0) ,
(III— 23 )
where F is the indefinite integral of f. Differentiating (III— 23) with respect to t,
we have
f (a) da = f[x(t)] . (Ill— 24)
In our case, x(t) is equal to either x or x . Referring to (III— 10), we see that
(III— 25)
x„ =
kl C3 + k2
x. =
k, C - k
13 2
43
where k and k are constants, and c is of the form c = t + another constant.
12 3 3
Hence,
dx dx ,
3 4
dT = kl dT
(IH-26)
Thus, the — in (III-24) is equal to k, , a constant. Now, the function f (a) in
dt
’ dt ' 7 ? v i / 2v
our case is either equal to cos I- a J or sin l - a J . Combining
all this
information. (Ill— 22) becomes
dT lC<X3>-C(X4)] - k!
dt tS(X3-S<X4)!
[cos (* X2^
!- :\1
1)
C0S (2 x3j
I - cos Ah
1
[sin (1 4)
- si" (1 <)] j
(111-27)
We wish to find out whether both these equations in (III— 2 7) are identically zero
when c = 0. Substituting (III— 25) into (III— 27), we obtain first
*3
dt 1C<X3>-C(V! = k
1 [C°S T (kl °3 + k2 + “l k2 C3)
— (k2 c2 + k^ - 2k k A]
2 \ 1 3 2 1 2 3/J
- cos
(III— 28 )
Let
y = — (2k k c )
J 2 ' 1 2 37
(III— 29)
44
Making use of the trig-identity
cos (x ± y) = cos x cos y + sin x sin y ,
Eq. (Ill— 28) becomes
s [C(V-C<X4>] =ki[-2
Now, since y contains c as a factor, we
o
Hence, sin y = 0 when c = 0, and
O
^ [C(xg) - C(x4)l = 0. when c3 = 0 . (III-31)
Similarly, using (III— 29) again and the trig-identity sin (x ± y) = sin x cos y
± cos x sin y, we obtain, for the second equation in (III— 2 7) ,
^ [S(x3) - S(x4)] = k1 [2 cos x sin y] , (HI-32)
which again contains sin y as a factor, yielding
[S (x ) - S (x )] = 0, when c = 0 . (HI-33)
dt 3 4 3
sin x sin y] . (Ill— 30)
see that y = 0 when c = 0.
Thus we see that (III— 22) is satisfied. It can readliy be checked that R is not
identically zero when c equals zero. Hence, we conclude that
O
dR ,
— — = 0 when c
dt f
= 0 ^i. e. ,
2R
when t = — + T- yCj - 2 y X c^ c u" J . (Ill— 34 )
')
Let us, therefore, think of c as a shifted time variable; i. e. , let
O
t" = c = t -
2R
C -T+yc -2yXc
(III— 35 )
45
Note that Eq. (Ill— 34) is satisfied regardless of the target velocity V. When
/ 2 V '
V = 0, y = 0 fe
<)
^eince y = — — | , which serves as a check.
Let us, therefore, accept our postulate as being true; namely, that the
envelope R has its peak when t" = 0.
When t" = c = 0, the arguments x and x of the Fresnal integrals are
(from III— 10) equal to
From (HI-8) we see that
W"
2
_ /y\T /W "\
V 7rw y 2 / *
/yXT ( w»\
V 7TW \ 2 f
which, for V < < c, reduces approximately to
Similarly, we have
so
A
= V7
W"
w
2
« - -c
Y
2V
c-V
2V
c-V
|x4l
when c =
3
/4vt
C7T W
’w
— — oo
2 0
v
0 c
2V
c
_
TW /2V\ f /~T~
7 r \ c / 7rW
(HI-36)
(III— 37)
46
Letting B = — and f
B 2tt 0
— , this becomes
27 r
(III— 38)
For the particular radar of interest, is approximately 1300 megacycles, and
there are two pulse-compression systems to be considered:
System No. 1:
T = 1 millisecond, B = 1 megacycle ;
System No. 2:
T = 2 milliseconds, B = 5 megacycles .
4
The range of radial velocities we consider here are 0 ^ V ^10 m/sec. Let us
check if x has a maximum or a minimum in this region; i. e. , using (III— 38) ,
we solve for
dx
dV
0
d_
dV
[a V1/2 - b V3/2'
— a . , _ 5 b Vv" = 0.
2 vH7 2
For V / 0, we obtain a - 3b V
V =
3b
which simplifies to
V
Be
6fo
(in-39)
47
4
For System No. 1, this corresponds to V = 3.85 x 10 m/sec, which fall6
outside our region of interest. For System No. 2, the value is even larger.
4
Thus, since there are no local maxima or minima between 0 ^ V ^ 10 m/sec,
we can compute the extreme values of x by taking the end points. For V = 0,
4
we simply get zero. When V = 10 m/sec, we obtain
x w 0. 237 for System No. 1 ;
x « 0. 745 for System No. 2 .
(IH-40)
Asymptotic series expansions of the Fresnel integrals C(x) and S(x) exist
both in ascending powers of x (for x < 1) and in descending powers of s (for x > 1).
The former are useful for x < .< 1 and the latter for x > > 1, since a few terms
of the series then give us a good approximation.
From the above estimates, we see that at the peak of the envelope the
4
argument of the Fresnel integrals is close to one when V = 10 m/sec. Thus,
asymptotic expansions appear to be unsuitable in the immediate vicinity of the
peak for large velocities.
We could, of course, consider smaller velocities. Suppose we require
that
x = 0. 01 < < 1 ,
(where x = x = |x | when c = 0).
O 4 O
Using Eq. (Ill— 10), we find that this corresponds to a radial velocity of
approximately
V » 10.5 m/sec for System No. 1, and
V « 1.5 m/sec for System No. 2 .
48
However, for such low velocities, the analysis of a simple Doppler-
shifted signal (as done in Appendix I) is expected to be quite adequate. It does
not seem worthwhile to go through the necessary approximations.
Let us consider whether we can infer anything about the nature of y (t) as
we move away from the peak of the envelope. Going back to (III-10) and substi¬
tuting t" for c , we have
O
(M. ATT
’ tM u
( — 1
|x4, V,w
2y\c t !
f 2 J
(III— 41 )
(Here we imply that + goes with x , - with x )
O 4
Let us rewrite (III— 41 ) as
(III— 42)
We have previously (III— 40) found that the product k k is of the order one when
1 O
t" = 0. We would now like to know how large t" has to be such that |x | or
O
lx I are of the order 100 (so that we might use asymptotic expansions for
i 4>
large x).
We let
t"
k„
= 100 k
(III— 43)
Substituting for k and k , we have
Z u
|t"J = 100
This is approximately
|t"| a 200
[? (*) - (*)]
l2yxcl
(III— 44)
[2 0 cJ[Vc/wJ
49
Using V = 10 m/sec, we have
1 1" | « 6psec for System No. 1;
1 1" | « 12psec for System No. 2.
(HI-45)
The following asymptotic approximations* are expected to be quite good for
large values of x (say, x ^ 100).
2 V
1 sin 7rx /2 I
2 + 7rx ’ r
> (m-46)
2 i
1 COS 7TX /2 |
2 7TX ' f
2 .
Letting o: = n x /2 and using the approximations in (HI-46), we have
C(x)
S(x)
C(x3) - C(x4)
S (xg) - C (x4)
(III— 47)
(Although we are using equal signs, it is understood that these are only approxi¬
mations . )
Substituting (III— 47) into (III— 1 8) , we have
R
3 4
x„ x,
3 4
(sin a sin a + cos a cos a.
3 4 3 4
>]
cos (a
O
(III— 48)
*c.f. Watson "Theory of Bessel Functions," p. 545.
50
Let us rewrite (III— 42) as
where
|A| <<|a|
Thus, we have
X3 A + A A
i ^1 - — J (ignoring higher order terms).
Similarly,
^ ^1 + (ignoring higher order terms),
(in-49)
Furthermore, we have
2(i)(-|)s (1 + f)=
2 2
- = — (again, ignoring higher order terms).
X„ X .Li
3 e A
51
Similarly,
7T / 2 2 \
a3 " "4 " 2 V 3 X4 )
= J [A2 + 2 A A + A2 - (A2 - 2A A + A2)]
Li
- 3 [4 A A]
= 277 A A .
Substituting the above approximations into (III-48), we have
1/2
B - 1 fj. ±
" ; [a a2
cos 2 7r A A
]
= [1 - COS 2 7T A A] 1/2
7T A
But 1 - cos 2x = 2 sin x, so that (III— 50) becomes
R » 2A
Referring back to (III— 1 7) , the whole expression for the envelope E is
(sin 7 r A A\
7T A A J
(ni-50)
(HI-51)
E = — 2 A — A-A
X 7T A A
Li
Substituting our elementary parameters back, we finally obtain
E = a
(3) ft)
FW)J
(HI-52)
52
This is a good approximation for
If, ,100 (sp) ,27xc2| - 100 E: (^)(^) i - ,00 (2) T
Let us now look at the phase angle 0 in the same region. We had
0 = ± tan
1
S (Xg) - S (X4)
C(x3) - C(x4)
](■*
for FM up )
for FM down /
Let
Using (III— 47) , we have
ip = ± tan 0 .
cos a, cos «
4> =
sin a sin a.
(Ill— 53)
(III— 54)
We have, as before,
*(i) * 1
With A << A, let us ignore all terms except the first. Then we get
ili =
cos cv - cos a
4 _ 3_
sin a - sin a
3 4
(III— 55)
53
But
Letting
we have
7 r A 2
“3 = 2 (A + A)
7T . 2
“4 = 2 (A ■ A)
7T 2 2
“ (A + A + 2 A A) ;
£ (A2 + A2 - 2 A A) .
X
y
7T
2
7T
2
2 2
(A + A ) ,
(2 A A) ,
cos (x - y) - cos (x + y)
sin (x + y) - sin (x - y)
(III-56)
Using some trig- identities, we obtain
2 sin x sin y
2 cos x sin y
tan x .
But if> = ± tan 0, so that 0 = ± x. Substituting for x, we finally have
0 = ±
y X T
2W
N’
+ for FM up
- for FM down
r
(in-58)
The carrier term is (from III— 1 7) equal to cos (x^ - 0). Letting 9 = x^ - 0
we obtain, after a fair amount of algebra,
e
"o »t» +
yXT
2W
f
* (c”)
T
(W”)21
“J
(III— 59)
54
where the upper signs are for FM up, and the lower signs for FM down. Now,
since W" < < u^' by about three orders of magnitude, suppose we ignore the
second term in the brackets. Substituting for y and X , we have
6 ~ (jj.
b . . (S) (S) (A <»;•'
+ for FM up |
- for FM down i
(III— 60)
Again, this is for
t" > 100
55
Security Classification
DOCUMENT CONTROL DATA • R&D
( Security c/eaa///caHon 0/ title body 0/ ebatracf end indexing ennotetion muet be entered when the overell report ie c leaeilied)
1 QRIGINATIN G ACTIVITY ( Corporal . tulhor)
MITRE Corporation
Bedford, Mass.
2*. REPORT SECURITY CLASSIFICATION
UNCLASSIFIED
26 GROUP
n/a
3 REPORT TITLE
Phase and Envelope of Linear FM Pulse-Compression Signals From High-Velocity
Targets
4 DESCRIPTIVE NOTES (Type 0/ report en<f induelve detee)
5 AUTHORS Ctaaf n«ma, ttret neme, inlttel)
Ueberschaer, M.H.
6. REPO RT DATE
Nov 64
7. TOTAL NO. OF PAGE* 7b. NO O P REF*
SB 0
8« CONTRACT OR GRANT NO.
AF19(628)2390
b. PROJECT NO.
750
C-
d
• a. ORIGINATOR'* REPORT NUMBCRfSj
TM-03916
f b. OTHER REPORT NOfSj (A ny other numbere thet mey be eeeigned
thle report)
ESD-TDR-64-128
10. A VA IL ABILITY/LIMITATION NOTICES
Qualified Requesters May Obtain From DDC.
Aval From OTS.
11 SUPPLEMENTARY NOTES
12. SPONSORING MILITARY ACTIVITY
Directorate of Radar and Optics, ESD,
L.G. Hanscom Field, Bedford, Mass.
13. ABSTRACT
Equations for the phase and envelope of the output signal from a
linear filter, matched to the transmitted signal, are derived. The trans-
mitted signal is assumed to have a flat band-limited amplitude spectrum
and a linear group delay. The input to the "matched filter" is the radar
echo returned from a moving target whose velocity is essentially constant
during the illumination time. It is shown that the returned signal is related
to the transmitted signal by a time dilation. The resulting expressions for the
phase and envelope are functions which involve Fresnel integrals. Approximations
for these expressions are worked out. They are shown to be similiar in form
to those which are obtained when the returned signal is assumed to be trans-
mitted signal by a Doppler shift.
DD .MM. 1473
Security Classification
Security Classification
14.
KEY WORDS
LINK A
LINK B
. ROLf
WT
LINK C
Raoar
Targets
Pulse Compression
Linear FM
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Security Classification