Golay codes.pdf

8/14/2019 Golay codes.pdf

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Radar Signal Processing

Ambiguity Function and Waveform Design

Golay Complementary Sequences (Golay Pairs)Golay Pairs for Radar: Zero Doppler

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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.

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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).

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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()

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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.


C G
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Phase Codes with Good Autocorrelations

Frank Code Barker Code

Golay Complementary Codes


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)


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.


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.


G l C l t S (G l P i )
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G l C l t S s (G l P i s)


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Definition: Two length L unimodular sequences x() and y()are Golay complementary if the sum of theirautocorrelation functions satisfies




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G y C p y S q (G y )





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y p y q ( y )





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y p y q ( y )





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


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:


x

x y

y

Discrete Sequence Coded Waveform


Golay Pairs for Radar: Advantage
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Frank coded waveforms Golay complementary waveforms

Weaker target is masked Weaker target is resolved


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


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


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.

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