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Radar Signal Processing
Ambiguity Function and Waveform Design
Golay Complementary Sequences (Golay Pairs)Golay Pairs for Radar: Zero Doppler
Radar Signal Processing
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Radar Problem
Transmit a waveform s(t) andanalyze the radar return r(t):
r(t) =hs(to)ej(to)+n(t)
h: target scattering coefficient; o= 2do/c: round-trip time;= 2fo
2voc
: Doppler frequency; n(t): noise
Target detection: decide between target present (h = 0) andtarget absent (h= 0) from the radar measurement r(t).
Estimate target range d0.
Estimate target range rate (velocity) v0.
Radar Signal Processing
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Ambiguity Function
Correlate the radar return r(t) with the transmit waveform
s(t). The correlator output is given by
m( o, ) =
hs(t o)s(t )ej(to)dt+noise term
Without loss of generality, assume o= 0. Then, the receiveroutput is
m(, ) =hA(, ) +noise term
where
A(, ) =
s(t)s(t )ejtdt
is called the ambiguity function of the waveform s(t).
Radar Signal Processing
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Ambiguity Function
Ambiguity function A(, ) is a two-dimensional function ofdelay and Doppler frequency that measures the correlationbetween a waveform and its Doppler distorted version:
A(, ) =
s(t)s(t )ejtdt
The ambiguity function along the zero-Doppler axis (= 0) isthe autocorrelation function of the waveform:
A(, 0) =
s(t)s(t )dt=Rs()
Radar Signal Processing
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Ambiguity Function
Example: Ambiguity function of a square pulse
Picture: Skolnik, ch. 11
Constant velocity (left) and constant range contours (right):
Pictures: Skolnik, ch. 11
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Ambiguity Function: Properties
Symmetry:
A(, ) =A(,)
Maximum value:
|A(, )| |A(0, 0)| =
|s(t)|2dt
Volume property(Moyals Identity):
|A(, )|2dd= |A(0, 0)|2
Pushing |A(, )|2 down in one place makes it pop outsomewhere else.
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Waveform Design
Waveform Design Problem: Design a waveform with a goodambiguity function.
A point target with delay o and Doppler shifto manifests asthe ambiguity function A(, o) centered at o.
For multiple point targets we have a superposition of
ambiguity functions.
A weak target located near astrong target can be maskedby the sidelobes of theambiguity function centeredaround the strong target.
Picture: Skolnik, ch. 11
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Waveform Design
Phase coded waveform:
s(t) =L1=0
x()u(t T)
The pulse shape u(t) and the chip rate Tare dictated bythe radar hardware.
x() is a length-L discrete sequence (or code) that we design.
Control the waveform ambiguity function by controlling the
autocorrelation function ofx().Waveform design: Design of discrete sequences with goodautocorrelation properties.
Radar Signal Processing
C G
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Phase Codes with Good Autocorrelations
Frank Code Barker Code
Golay Complementary Codes
Radar Signal Processing
W f D i Z D l
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Waveform Design: Zero Doppler
Suppose we wish to detect stationary targets in range.
The ambiguity function along the zero-Doppler axis is thewaveform autocorrelation function:
Rs() =
s(t)s(t )dt
=L1=0
L1m=0
x()x(m)
u(t T)u(t mT)dt
=
L1=0
L1m=0
x()x(m)Ru(+ (m )T)
=
2(L1)
k=2(L1)
L1
=0
x(
)
x(
k)
Ru(
k
T)
=
2(L1)k=2(L1)
Cx(k)Ru( kT)
Radar Signal Processing
I l lik A l i
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Impulse-like Autocorrelation
Ideal waveform for resolving targets in range (no range
sidelobes):
Rs() =
2(L1)k=2(L1)
Cx(k)Ru( kT) ()
We do not have control over Ru().
Question: Can we find the discrete sequence x() so thatCx(k) is a delta function?
Answer: This is not possible with a single sequence, but wecan find a pairof sequences x() and y() so that
Cx(k) +Cy(k) = 2Lk,0.
Radar Signal Processing
G l C l t S (G l P i )
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
G l C l t S (G l P i )
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
G l C l t S s (G l P i s)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()
are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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Golay Complementary Sequences (Golay Pairs)
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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G y C p y S q (G y )
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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y p y q ( y )
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Complementary Sequences (Golay Pairs)
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y p y q ( y )
Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies
Cx(k) +Cy(k) = 2Lk,0
for all (L 1) k L 1.
Radar Signal Processing
Golay Pairs: Example
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y px
x y
y
Time reversal:
x: 1 1 1 1 1 1 1 1x: 1 1 1 1 1 1 1 1If(x, y) is a Golay pair then (x,
y), (
x,y), and
(x,y) are also Golay pairs. Radar Signal Processing
Golay Pairs: Construction
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Standard construction: Start with
1 11 1
and apply the
construction AB
A B
A BB A
B A
Example:
1 11 1
1 1 1 11 1 1 11 1 1 1
1 1 1 1
1 1 1
1 1 1
1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1
1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1
1 1 1 1 1 1 1 11 1 1 1 1 1 1 1
Other constructions:Weyl-Heisenberg Construction: Howard, Calderbank, andMoran, EURASIP J. ASP 2006Davis and Jedwab: IEEE Trans. IT 1999
Radar Signal Processing
Golay Pairs for Radar: Zero Doppler
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The waveforms coded by Golay pairs xand y are transmitted over two Pulse
Repetition Intervals (PRIs) T.
Each return is correlated with its corresponding sequence:
Cx(k) +Cy(k) = 2Lk,0
x
x y
y
Discrete Sequence Coded Waveform
Radar Signal Processing
Golay Pairs for Radar: Advantage
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Frank coded waveforms Golay complementary waveforms
Weaker target is masked Weaker target is resolved
Radar Signal Processing
Sensitivity to Doppler
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Asx (, ) +ej2TAsy (, )
Although the autocorrelationsidelobe level is zero, theambiguity function exhibitsrelatively high sidelobes fornonzero Doppler. [Levanon,Radar Signals, 2004, p. 264]
Why? Roughly speaking
Cx(k) +Cy(k)ej =()k,0
Radar Signal Processing
Sensitivity to Doppler
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Range Sidelobes Problem: A weak target located near a strongtarget can be masked by the range sidelobes of the ambiguity
function centered around the strong target.
Range-Doppler image
obtained with conventionalpulse trainx y x y
Radar Signal Processing
References
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1 M. I. Skolnik, An introduction and overview of radar, in Radar Handbook, M. I. Skolnik, Ed. New York:McGraw-Hill, 2008.
2 M. R. Ducoff and B.W. Tietjen, Pulse compression radar, in Radar Handbook, M. I. Skolnik, Ed. NewYork: McGraw-Hill, 2008.
3 S. D. Howard, A. R. Calderbank, and W. Moran, The finite Heisenberg-Weyl groups in radar andcommunications,EURASIP Journal on Applied Signal Processing, Article ID 85685, 2006.
4 N. Levanon and E. Mozeson, Radar Signals, New York: Wiley, 2004.
5 M. Golay, Complementary series, IRE Trans. Inform. Theory, vol. 7, no. 2, pp. 82-87, April 1961.
Radar Signal Processing
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