a silo mar 07

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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

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Review: Ambiguity function in MIMO radar

3

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Ambiguity Function in SIMO Radar

Ambiguity function characterizes the Doppler and range resolution.


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u(t)

(t,n) target

TX

t:delayn:Doppler

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u(t) y (t,n) (t)

(t,n) target

TX RX

t:delayn:Doppler

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dt t yt y )()( )','(),( n t n t




u(t) y (t,n) (t)

(t,n) target

TX RX

t:delayn:Doppler

Matched filter output

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dt t yt y )()( )','(),( n t n t




u(t) y (t,n) (t)

(t,n) target

TX RX

t:delayn:Doppler

dt et uet u t jt j

'22 )'()( n n t t

dt et ut ut j

)'(2

))'(()(n n t t


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dt t yt y )()( )','(),( n t n t




u(t) y (t,n) (t)

(t,n) target

TX RX

t:delayn:Doppler

dt et ut u t j

n t n t 2)()(),(

dt et uet u t jt j

'22 )'()( n n t t

dt et ut ut j

)'(2

))'(()(n n t t


Ambiguity function

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t

n

target 2 (t2,n

2)

target 1 (t1,n1)

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t

n

target 2 (t2,n

2)

target 1 (t1,n1) ),( 11 n n t t

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-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1




t

n

target 2 (t2,n

2)

target 1 (t1,n1) ),( 11 n n t t

dt et ut u t j

n t n t 2)()(),(

Ambiguity function

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MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

…

Transmitter emits

incoherent waveforms.

Transmitter: M antenna elements

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MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

Transmitter emits

incoherent waveforms.

Matched filters extractthe M orthogonal waveforms.Overall number of signals:

NM

Receiver: N antenna elementsTransmitter: M antenna elements

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Ambiguity Function in MIMO Radar


u0(t)

xT0

u1(t)

xT1

uM-1(t)

xT,M-1

…

(t,nf) TX

t:delayn:Dopplerf: Spatial freq.

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xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(t,nf) (t,nf) TX RX


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

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)(

),,(

t

f n t

y



xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(t,nf) (t,nf) TX RX


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

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)(

),,(

t

f n t

y



xT0 xT1 xT,M-1

… …

xR0 xR1 xR,M-1

MF…

MF…

MF…

(t,nf) (t,nf)

dt t t f H f )()( ),,()',','( n t n t yy


TX RX


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

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1

0

1

0'

)'(2)'(2*

1

0

)'(2 ')'()(

M

m

M

m

x f fx jt v jmm

N

n

n f f j mmedt et ut ue n t t




Receiver beamforming

dt t t f H f )()( ),,()',','( n t n t yy

t:delay

n:Dopplerf: Spatial freq.um(t): m-th waveformxm: m-th antenna locationn: receiving antenna index

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1

0

1

0'

)'(2)'(2*

1

0

)'(2 ')'()(

M

m

M

m

x f fx jt v jmm

N

n






dt t t f H f )()( ),,()',','( n t n t yy

t:delay

n:Dopplerf: Spatial freq.um(t): m-th waveformxm: m-th antenna locationn: receiving antenna index

Cross ambiguity function

* 2

, ' '( , ) ( ) ( ) j t

m m m mu t u t e dt n t n t

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1

0

1

0'

)'(2

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

M

m

M

m

x f fx j

mmmme f f

n t n t

1

0

1

0'

)'(2)'(2*

1

0

)'(2 ')'()(

M

m

M

m

x f fx jt v jmm

N

n






dt t t f H f )()( ),,()',','( n t n t yy

* 2

, ' '( , ) ( ) ( ) j t

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

t:delayn:

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

MIMO ambiguity function

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t

n

target 2 (t2,n

2,f 2) target 1 (t1,n1,f 1)

f

Ambiguity function characterizes the Doppler, range, andangular resolution.

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1

0

1

0'

)'(2

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

M

m

M

m

x f fx j

mmmme f f

n t n t



t

n

target 2 (t2,n

2,f 2) target 1 (t1,n1,f 1)

),,,( 111 f f n n t t

Ambiguityfunction

f dt et ut u t j

mmmm

n t n t 2*

', )()(),(

Ambiguity function characterizes the Doppler, range, andangular resolution.

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Proposed Waveform Design Method

25

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MIMO Radar Waveform Design Problem

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


t

n

target 1 (t1,n1,f 1) ),,,( 111 f f n n t t

f

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MIMO Radar Waveform Design Problem

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


t

n

target 1 (t1,n1,f 1) ),,,( 111 f f n n t t

f Ambiguityfunction

dt et ut u t j

mmmm

n t n t 2*

', )()(),(

1

0

1

0'

)'(2

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

M

m

M

m

x f fx j

mmmme f f

n t n t

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Imposing Waveform Structures

Pulse radar – MTI (Moving Target Indicator)

– Doppler pulse radar


)()(

1

0

L

llmm T t t u m-th waveform

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Frequency hoppingsignals

– Constant modulus

– Can be viewed asgeneralized LFM (Linear

Frequency Modulation)


)()(

1

0

L

llmm T t t u

1

0

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

q

t mqm t qt t f c jt

m-th waveform

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Frequency hoppingsignals

– Constant modulus

– Can be viewed asgeneralized LFM (Linear

Frequency Modulation)

Orthogonal waveforms

– Virtual array


)()(

1

0

L

llmm T t t u

1

0

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

q

t mqm t qt t f c jt

1

','

t f

mmqcc qmmq

m-th waveform

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Ambiguity Function of Pulse MIMO Radar


T

)()(1

0

L

l

lmm T t t u

0 n T ll T T T 1

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T

)()(1

0

L

l

lmm T t t u

0 n T ll T T T 1

dt t t

dt et t

mm

t jmmmm

1)()(

)()(),(

*

2*',

)(

t

t n t n

)()(

', t

mmr

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1

0

2

1

0

1

0'

)'(2)( '

',)()',,,(

L

l

T j

M

m

M

m

x f fx j lmm

mmeer f f n t n t



T

)()(1

0

L

l

lmm T t t u

0 n T ll T T T 1

dt t t

dt et t

mm

t jmmmm

1)()(

)()(),(

*

2*',

)(

t

t n t n

)()(

', t

mmr

Doppler processingis separable

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)',,( f f t

1

0

2

1

0

1

0'

)'(2)( '

',)()',,,(

L

l

T j

M

m

M

m

x f fx j lmm

mmeer f f n t n t



T

)()(1

0

L

l

lmm T t t u

0 n T ll T T T 1

dt t t

dt et t

mm

t jmmmm

1)()(

)()(),(

*

2*',

)(

t

t n t n

)()(

', t

mmr

Define as

Doppler processingis separable

W f D i P bl i P l

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dt t t r mmmm )()()( *)(

', t t

Waveform Design Problem in PulseMIMO Radar


1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

x f fx j mm

mmer f f

t t

W f D i P bl i P l

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1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

x f fx j mm

mmer f f

t t

dt t t r mmmm )()()( *)(

', t t

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

sharp around {(0,f,f)}.

W f D i P bl i P l

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1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

x f fx j mm

mmer f f

t t

dt t t r mmmm )()()( *)(

', t t

)()',,( )(

0,0t t r f f



Ex: SIMO case: M=1

Wa eform Design Problem in P lse

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1

0

1

0'

)'(2)( '

',)()',,(

M

m

M

m

x f fx j mm

mmer f f

t t

dt t t r mmmm )()()( *)(

', t t

)()',,( )(

0,0t t r f f

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



Ex: SIMO case: M=1

Orthogonality of the Frequency Hopping

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Orthogonality of the Frequency HoppingSignals


1

[0, )

0

( ) exp( 2 ) 1 ( )Q

m mq t

q

t j fc t t q t

1

','

t f

mmqcc qmmq

m

m'

Frequency

Time

f

t


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1

','

t f

mmqcc qmmq

m

m'

',

* )()( mmmm dt t t

1

[0, )

0

( ) exp( 2 ) 1 ( )Q

m mq t

q

t j fc t t q t


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M er f f M

m

M

m

x x f j mm

mm

1

0

1

0'

)(2)( '

',)0(),,0(



1

','

t f

mmqcc qmmq

m

m'

',

* )()( mmmm dt t t

1

[0, )

0

( ) exp( 2 ) 1 ( )Q

m mq t

q

t j fc t t q t


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1

','

t f

mmqcc qmmq

m

m'

',

* )()( mmmm dt t t

M er f f M

m

M

m

x x f j mm

mm

1

0

1

0'

)(2)( '

',)0(),,0(

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

1

[0, )

0

( ) exp( 2 ) 1 ( )Q

m mq t

q

t j fc t t q t

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Define a vector

Optimization of the Codes


),,(vec ' f f nnnt CCω

'CC ωω wCode C is better thancode C’.

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Define a vector

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

such that



),,(vec ' f f nnnt CCω 'CC ωω w Code C is better than code C’.

CCωω

w

'

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Define a vector


such that

For any where gi are increasing convexfunctions




CC ωω w'

)()( '' CCCC ωωωω f f w

i ig f

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Define a vector


such that


So a code C is efficient if




CC ωω w'


)()( 'CC ωω f f for all C’.

i ig f

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Define a vector


such that


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

Example:




CC ωω w'


)()( 'CC ωω f f

i ig f

p

p f cc ωω )(

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1

0

1

0

')',,(min df df d f f p

t t 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

q

t mqm t qt t c jt

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Simulated Annealing Algorithm

Simulated annealing

– Create a Markov chain on the set A


)(min CCp f Csubject to

…

CC’

…

[S. Kirkpatrick et al. 85]

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Simulated annealing

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



C

C

CC

T

f Z

T

f

Z

p

T

p

T

T

)(exp

)(exp

1)(

…

CC’

…


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Simulated annealing


– Run the Markov chain Monte Carlo (MCMC)



C

C

CC

T

f Z

T

f

Z

p

T

p

T

T

)(exp

)(exp

1)(

…

CC’

…


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Simulated annealing


– Run the Markov chain Monte Carlo (MCMC)

– Decrease the temperature T from time to time

C

C

CC

T

f Z

T

f

Z

p

T

p

T

T

)(exp

)(exp

1)(




…

CC’

…


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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

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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 andbandwidth

– Initial frequencies

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Examples – Ambiguity Function


|(t,f,f’)|

Orthogonal LFMProposed Freq. Hopping Signal

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Examples – Ambiguity Function


10log10|(t,f,f’)|


Examples – Sorted Samples of Ambiguity

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0 2 4 6 8 10 -15

-10

-5

0

Sorted samples (%)

Examples Sorted Samples of AmbiguityFunctions

10log10(|(t,f,f’)|)

LFM

Randomly selected code

Proposed method

0 20 40 60 80 100 -20 -15 -10 -5 0

Sorted samples (%)

10log10(|(t,f,f’)|)

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Examples – Correlation Function Matrix



dt t t r mmmm )()()( *)(

', t t

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

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Conclusion

MIMO radar frequency hopping waveform designmethod

– Sharper ambiguity function (Better resolution)

– Applicable in the case of

• pulse radar

• orthogonal waveforms

Future work – Other optimization tools

– Phase coded signals


8/4/2019 A Silo Mar 07


Q&AThank You!

Any questions?

a silo mar 07

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