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### Transcript of Radar Signal Processing - UC3Mtsc.uc3m.es/~fernando/Lecture5.pdf · Radar Signal Processing...

Ambiguity Function and Waveform DesignGolay Complementary Sequences (Golay Pairs)

Golay Pairs for Radar: Zero Doppler

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 6= 0) andtarget absent (h = 0) from the radar measurement r(t).

Estimate target range d0.

Estimate target range rate (velocity) v0.

• Ambiguity Function

Correlate the radar return r(t) with the transmit waveforms(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).

• 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()

• 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

• 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.

• Waveform Design

Waveform Design Problem: Design a waveform with a goodambiguity function.

A point target with delay o and Doppler shift o manifests asthe ambiguity function A(, o) centered at o.For multiple point targets we have a superposition ofambiguity functions.

A weak target located near astrong target can be maskedby the sidelobes of theambiguity function centeredaround the strong target.

Picture: Skolnik, ch. 11

• Waveform Design

Phase coded waveform:

s(t) =L1=0

x(`)u(t `T )

The pulse shape u(t) and the chip rate T are dictated bythe radar hardware.

x(`) is a length-L discrete sequence (or code) that we design.

Control the waveform ambiguity function by controlling theautocorrelation function of x(`).

Waveform design: Design of discrete sequences with goodautocorrelation properties.

• Phase Codes with Good Autocorrelations

Frank Code Barker Code

Golay Complementary Codes

• 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( kT )

=

2(L1)k=2(L1)

Cx(k)Ru( kT )

• Impulse-like Autocorrelation

Ideal waveform for resolving targets in range (no rangesidelobes):

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 pair of sequences x(`) and y(`) so that

Cx(k) + Cy(k) = 2Lk,0.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• Golay Pairs: Examplex

x y

y

Time reversal:

x : 1 1 1 1 1 1 1 1

x : 1 1 1 1 1 1 1 1

If (x, y) is a Golay pair then (x,y), (x,y), and(x,y) are also Golay pairs.

• Golay Pairs: Construction

1 11 1

)and apply the

construction (AB

)

A BA BB AB A

Example:

(1 11 1

)

1 1 1 11 1 1 11 1 1 11 1 1 1

1 1 1 1 1 1 1 11 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 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

The waveforms coded by Golay pairs xand y are transmitted over two PulseRepetition 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

Frank coded waveforms Golay complementary waveforms

Weaker target is masked Weaker target is resolved

• Sensitivity to Doppler

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 6= ()k,0

• Sensitivity to Doppler

Range Sidelobes Problem: A weak target located near a strongtarget can be masked by the range sidelobes of the ambiguityfunction centered around the strong target.

Range-Doppler imageobtained with conventionalpulse trainx y x y

• References

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.