a design method for mimo radar frequency hopping codes chun-yang chen and p. p. vaidyanathan...

A Design Method for MIMO Radar Frequency Hopping Codes

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of Technology

Electrical Engineering/DSP Lab

Asilomar Conference 2007

Outline

Review of the background– Ambiguity function– Ambiguity function in MIMO radar

The proposed waveform design method– Ambiguity function for MIMO pulse radar– Frequency hopping signals– Optimization of the frequency hopping codes– Examples

Conclusion and future work

2Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

1Review: Ambiguity function in MIMO radar

Ambiguity Function in SIMO Radar

Ambiguity function characterizes the Doppler and range resolution.

(,)target

delayDoppler

u(t) y(,) (t)

(,)target

delayDoppler

dttyty )()( )','(),(

u(t) y(,) (t)

(,)target

delayDoppler

Matched filter output

dttyty )()( )','(),(

u(t) y(,) (t)

(,)target

delayDoppler

dtetuetu tjtj '22 )'()(

dtetutu tj )'(2))'(()(

dttyty )()( )','(),(

u(t) y(,) (t)

(,)target

delayDoppler

dtetutu tj 2)()(),(

dtetuetu tjtj '22 )'()(

dtetutu tj )'(2))'(()(

Ambiguity function

target 2 (,)target 1 (,)

),( 11

-1-0.5

),( 11

dtetutu tj 2)()(),(

Ambiguity function

-1-0.5

),( 11

dtetutu tj 2)()(),(

Ambiguity function

MIMO Radar

uM-1(t)

xT,M-1

Transmitter emits

incoherent waveforms.

Transmitter emits

Transmitter: M antenna elements

MIMO Radar

uM-1(t)

xT,M-1

… …

xR0 xR1 xR,M-1

Transmitter emits

Matched filters extract the M orthogonal waveforms.Overall number of signals:

Receiver: N antenna elementsTransmitter: M antenna elements

Ambiguity Function in MIMO Radar

uM-1(t)

xT,M-1

delayDopplerfSpatial freq.

xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

(,f) (,f)

u0(t)u1(t) uM-1(t)

)(),,( tfy

xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

(,f) (,f)

u0(t)u1(t) uM-1(t)

)(),,( tfy

xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

(,f) (,f)

dttt fHf )()( ),,()',','( yy

u0(t)u1(t) uM-1(t)

)'(2)'(2*1

)'(2 ')'()(M

xffxjtvjmm

nffj mmedtetutue

Receiver beamforming

dttt fHf )()( ),,()',','( yy

delayDopplerfSpatial freq.um(t): m-th waveformxm: m-th antenna locationn: receiving antenna index

)'(2)'(2*1

)'(2 ')'()(M

xffxjtvjmm

nffj mmedtetutue

dttt fHf )()( ),,()',','( yy

Cross ambiguity function

* 2, ' '( , ) ( ) ( ) j t

m m m mu t u t e dt

)'(2',

'),()',,,(M

xffxjmm

)'(2)'(2*1

)'(2 ')'()(M

xffxjtvjmm

nffj mmedtetutue

dttt fHf )()( ),,()',','( yy

* 2, ' '( , ) ( ) ( ) j t

m m m mu t u t e dt [San Antonio et al. 07]

MIMO ambiguity function

target 2 (,,f 2)target 1 (,,f1)

Ambiguity function characterizes the Doppler, range, and angular resolution.

)'(2',

'),()',,,(M

xffxjmm

target 2 (,,f 2)target 1 (,,f1)

),,,( 111 ff

Ambiguity function

f dtetutu tjmmmm

2*', )()(),(

Ambiguity function characterizes the Doppler, range, and angular resolution.

2Proposed Waveform Design Method

MIMO Radar Waveform Design Problem

Choose a set of waveforms {um(t)} so that the ambiguity function f,f’can be sharp around {(0,0,f,f)}.

target 1 (,,f1)

),,,( 111 ff

MIMO Radar Waveform Design Problem

Choose a set of waveforms {um(t)} so that the ambiguity function f,f’can be sharp around {(0,0,f,f)}.

target 1 (,,f1)

),,,( 111 ff

fAmbiguity function

dtetutu tjmmmm

2*', )()(),(

)'(2',

'),()',,,(M

xffxjmm

Imposing Waveform Structures Pulse radar

– MTI (Moving Target Indicator)

– Doppler pulse radar

llmm Tttu m-th waveform

Frequency hopping signals– Constant modulus– Can be viewed as

generalized LFM (Linear

Frequency Modulation)

llmm Tttu

0),0[ )(1)2exp()(

qtmqm tqttfcjt

m-th waveform

Frequency hopping signals– Constant modulus– Can be viewed as

generalized LFM (Linear

Frequency Modulation)

Orthogonal waveforms– Virtual array

llmm Tttu

0),0[ )(1)2exp()(

qtmqm tqttfcjt

mmqcc qmmq

m-th waveform

Ambiguity Function of Pulse MIMO Radar

llmm Tttu

0Tll TTT 1

llmm Tttu

0Tll TTT 1

tjmmmm

)()(),(

)()(',

)'(2)( '

',)()',,,(

xffxj lmm

mmeerff

llmm Tttu

0Tll TTT 1

tjmmmm

)()(),(

)()(',

Doppler processingis separable

)',,( ff

)'(2)( '

',)()',,,(

xffxj lmm

mmeerff

llmm Tttu

0Tll TTT 1

tjmmmm

)()(),(

)()(',

Define asDoppler processingis separable

dtttr mmmm )()()( *)(',

Waveform Design Problem in Pulse MIMO Radar

)'(2)( '

',)()',,(

xffxj mm

mmerff

)'(2)( '

',)()',,(

xffxj mm

mmerff

dtttr mmmm )()()( *)(',

Choose a set of pulses {m(t)} such that (,f,f’) can be sharp around {(0,f,f)}.

)'(2)( '

',)()',,(

xffxj mm

mmerff

dtttr mmmm )()()( *)(',

)()',,( )(

0,0 rff

Ex: SIMO case: M=1

)'(2)( '

',)()',,(

xffxj mm

mmerff

dtttr mmmm )()()( *)(',

)()',,( )(

0,0 rff

Choose a pulse with a sharp correlation function (e.g. LFM)

Ex: SIMO case: M=1

Orthogonality of the Frequency Hopping Signals

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

mmqcc qmmq

Frequency

mmqcc qmmq

',* )()( mmmm dttt

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

MerffM

xxfj mm

)(2)( '

',)0(),,0(

mmqcc qmmq

',* )()( mmmm dttt

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

mmqcc qmmq

',* )()( mmmm dttt

MerffM

xxfj mm

)(2)( '

',)0(),,0(

is a constant along {(0,f,f)}, no matter what codes are chosen.

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

Define a vector

Optimization of the Codes

),,( vec 'ff nnnCCω

'CC ωω w Code C is better than code C’.

Define a vector

Def: a code C is efficient if there exists no other code C’ such that

CC ωω w'

Define a vector

For any where gi are increasing convex functions

CC ωω w'

)()( '' CCCC ωωωω ffw

Define a vector

So a code C is efficient if

CC ωω w'

)()( 'CC ωω ff for all C’.

Define a vector

So a code C is efficient if for all C’. Example:

CC ωω w'

)()( 'CC ωω ff

pf cc ωω )(

')',,(min dfdfdffp

}1,1,0{

' mmqcc

M:# of waveformsQ: # of freq. hopsK: # of freq.Time-bandwidth product:KfQt

0),0[ )(1)2exp()(

qtmqm tqttcjt

Simulated Annealing Algorithm

Simulated annealing– Create a Markov chain on the set A

)(min CC

pf Csubject to

…[S. Kirkpatrick et al. 85]

Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution

)(min CC

pf Csubject to

– Run the Markov chain Monte Carlo (MCMC)

)(min CC

pf Csubject to

– Run the Markov chain Monte Carlo (MCMC)

– Decrease the temperature T from time to time

)(min CC

pf Csubject to

Examples

Parameters:Uniform linear array# of waveforms M =4# of hops Q=10# of freq. K=15norm type p=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

Proposed Freq. Hopping Signals

Examples

Parameters:Uniform linear array# of waveforms M =4# of hops Q=10# of freq. K=15norm type p=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

Orthogonal LFMProposed Freq. Hopping Signals

Parameters:– The same array– The same duration and

bandwidth– Initial frequencies

Examples – Ambiguity Function

|(,f,f’)|

Orthogonal LFMProposed Freq. Hopping Signal

Examples – Ambiguity Function

10log10|(,f,f’)|

0 2 4 6 8 10-15

Sorted samples (%)

Examples – Sorted Samples of Ambiguity Functions

10log10(|(,f,f’)|)

Randomly selected code

Proposed method

0 20 40 60 80 100-20

Sorted samples (%)

10log10(|(,f,f’)|)

Examples – Correlation Function Matrix

dtttr mmmm )()()( *)(',

0 0.5 1-1

Conclusion

MIMO radar frequency hopping waveform design method– Sharper ambiguity function (Better resolution)– Applicable in the case of

• pulse radar • orthogonal waveforms

Future work– Other optimization tools– Phase coded signals

Q&AThank You!

Any questions?

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