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

0

0.5

1

0

0.5

10

2

4

6

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

3

Ambiguity Function in SIMO Radar

Ambiguity function characterizes the Doppler and range resolution.





u(t)

(,)target

TX

delayDoppler




u(t) y(,) (t)

(,)target

TX RX

delayDoppler

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




u(t) y(,) (t)

(,)target

TX RX

delayDoppler

Matched filter output

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




u(t) y(,) (t)

(,)target

TX RX

delayDoppler

dtetuetu tjtj '22 )'()(

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


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




u(t) y(,) (t)

(,)target

TX RX

delayDoppler

dtetutu tj 2)()(),(

dtetuetu tjtj '22 )'()(

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


Ambiguity function




target 2 (,)target 1 (,)





),( 11

-1-0.5

00.5

1

-1

-0.5

0

0.5

10

0.2

0.4

0.6

0.8

1





),( 11

dtetutu tj 2)()(),(

Ambiguity function

MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

…

Transmitter emits

incoherent waveforms.

Transmitter emits


Transmitter: M antenna elements

MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

Transmitter emits


Transmitter emits


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

NM

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

NM

Receiver: N antenna elementsTransmitter: M antenna elements

Ambiguity Function in MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

…

(,f)

TX

delayDopplerfSpatial freq.



xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(,f) (,f)

TX RX


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

)(),,( tfy



xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(,f) (,f)

TX RX


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

)(),,( tfy



xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(,f) (,f)

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


TX RX


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

1

0

1

0'

)'(2)'(2*1

0

)'(2 ')'()(M

m

M

m

xffxjtvjmm

N

n

nffj mmedtetutue




Receiver beamforming

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

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

1

0

1

0'

)'(2)'(2*1

0

)'(2 ')'()(M

m

M

m

xffxjtvjmm

N

n

nffj mmedtetutue





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


Cross ambiguity function

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

m m m mu t u t e dt

1

0

1

0'

)'(2',

'),()',,,(M

m

M

m

xffxjmm

mmeff

1

0

1

0'

)'(2)'(2*1

0

)'(2 ')'()(M

m

M

m

xffxjtvjmm

N

n

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)

f

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

1

0

1

0'

)'(2',

'),()',,,(M

m

M

m

xffxjmm

mmeff



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

25

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

f

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*', )()(),(

1

0

1

0'

)'(2',

'),()',,,(M

m

M

m

xffxjmm

mmeff

Imposing Waveform Structures Pulse radar

– MTI (Moving Target Indicator)

– Doppler pulse radar


)()(1

0

L

llmm Tttu m-th waveform




Frequency hopping signals– Constant modulus– Can be viewed as

generalized LFM (Linear

Frequency Modulation)


)()(1

0

L

llmm Tttu

1

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

Q

qtmqm tqttfcjt

m-th waveform




Frequency hopping signals– Constant modulus– Can be viewed as

generalized LFM (Linear

Frequency Modulation)

Orthogonal waveforms– Virtual array


)()(1

0

L

llmm Tttu

1

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

Q

qtmqm tqttfcjt

1

','

tf

mmqcc qmmq

m-th waveform

Ambiguity Function of Pulse MIMO Radar


T

)()(1

0

L

llmm Tttu

0Tll TTT 1



T

)()(1

0

L

llmm Tttu

0Tll TTT 1

dttt

dtett

mm

tjmmmm

1)()(

)()(),(

*

2*',

)(

)()(',

mmr

1

0

21

0

1

0'

)'(2)( '

',)()',,,(

L

l

TjM

m

M

m

xffxj lmm

mmeerff



T

)()(1

0

L

llmm Tttu

0Tll TTT 1

dttt

dtett

mm

tjmmmm

1)()(

)()(),(

*

2*',

)(

)()(',

mmr

Doppler processingis separable

)',,( ff

1

0

21

0

1

0'

)'(2)( '

',)()',,,(

L

l

TjM

m

M

m

xffxj lmm

mmeerff



T

)()(1

0

L

llmm Tttu

0Tll TTT 1

dttt

dtett

mm

tjmmmm

1)()(

)()(),(

*

2*',

)(

)()(',

mmr

Define asDoppler processingis separable

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

Waveform Design Problem in Pulse MIMO Radar


1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

xffxj mm

mmerff



1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

xffxj mm

mmerff

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

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



1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

xffxj mm

mmerff

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

)()',,( )(

0,0 rff


Ex: SIMO case: M=1



1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

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


1

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

1

','

tf

mmqcc qmmq

m

m'

Frequency

Time

f

t



1

','

tf

mmqcc qmmq

m

m'

',* )()( mmmm dttt

1

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t

MerffM

m

M

m

xxfj mm

mm

1

0

1

0'

)(2)( '

',)0(),,0(



1

','

tf

mmqcc qmmq

m

m'

',* )()( mmmm dttt

1

[0, )0

( ) exp( 2 ) 1 ( )Q

m mq tq

t j fc t t q t



1

','

tf

mmqcc qmmq

m

m'

',* )()( mmmm dttt

MerffM

m

M

m

xxfj mm

mm

1

0

1

0'

)(2)( '

',)0(),,0(

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

1

[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

i igf

Define a vector



So a code C is efficient if





CC ωω w'


)()( 'CC ωω ff for all C’.

i igf

Define a vector



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





CC ωω w'


)()( 'CC ωω ff

i igf

p

pf cc ωω )(



1

0

1

0

')',,(min dfdfdffp

C

',

}1,1,0{

' mmqcc

K

qmmq

MQC

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

1

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

Q

qtmqm tqttcjt

Simulated Annealing Algorithm

Simulated annealing– Create a Markov chain on the set A


)(min CC

pf Csubject to

…

CC’

…[S. Kirkpatrick et al. 85]


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


)(min CC

pf Csubject to

C

C

CC

T

fZ

T

f

Z

pT

p

TT

)(exp

)(exp

1)(

…

CC’




– Run the Markov chain Monte Carlo (MCMC)


)(min CC

pf Csubject to

C

C

CC

T

fZ

T

f

Z

pT

p

TT

)(exp

)(exp

1)(

…

CC’



– Run the Markov chain Monte Carlo (MCMC)

– Decrease the temperature T from time to time

C

C

CC

T

fZ

T

f

Z

pT

p

TT

)(exp

)(exp

1)(



)(min CC

pf Csubject to

…

CC’


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

0

1

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

0

1

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

0

1

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

0

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

0

1

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

0

1

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

0

1

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

0

1

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

0

1

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

0

1

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

0

1

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

0

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

-10

-5

0

Sorted samples (%)

Examples – Sorted Samples of Ambiguity Functions

10log10(|(,f,f’)|)

LFM

Randomly selected code

Proposed method

0 20 40 60 80 100-20

-15

-10

-5

0

Sorted samples (%)

10log10(|(,f,f’)|)

Examples – Correlation Function Matrix



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

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

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?


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

Documents

dopplerchunyang chen

n1chunyang chen

simo radarambiguity

range resolution

mimo radarthe

mimo pulse radarfrequency

future work