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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
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10
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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.
4Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
5Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u(t)
(,)target
TX
delayDoppler
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
6Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u(t) y(,) (t)
(,)target
TX RX
delayDoppler
dttyty )()( )','(),(
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
7Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u(t) y(,) (t)
(,)target
TX RX
delayDoppler
Matched filter output
dttyty )()( )','(),(
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
8Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u(t) y(,) (t)
(,)target
TX RX
delayDoppler
dtetuetu tjtj '22 )'()(
dtetutu tj )'(2))'(()(
Matched filter output
dttyty )()( )','(),(
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
9Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u(t) y(,) (t)
(,)target
TX RX
delayDoppler
dtetutu tj 2)()(),(
dtetuetu tjtj '22 )'()(
dtetutu tj )'(2))'(()(
Matched filter output
Ambiguity function
Ambiguity function characterizes the Doppler and range resolution.
Ambiguity Function in SIMO Radar
10Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
target 2 (,)target 1 (,)
Ambiguity function characterizes the Doppler and range resolution.
Ambiguity Function in SIMO Radar
11Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Ambiguity function characterizes the Doppler and range resolution.
Ambiguity Function in SIMO Radar
12Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
target 2 (,)target 1 (,)
),( 11
dtetutu tj 2)()(),(
Ambiguity function
-1-0.5
00.5
1
-1
-0.5
0
0.5
10
0.2
0.4
0.6
0.8
1
Ambiguity function characterizes the Doppler and range resolution.
Ambiguity Function in SIMO Radar
13Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
target 2 (,)target 1 (,)
),( 11
dtetutu tj 2)()(),(
Ambiguity function
MIMO Radar
14Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u0(t)
xT0
u1(t)
xT1
uM-1(t)
xT,M-1
…
Transmitter emits
incoherent waveforms.
Transmitter emits
incoherent waveforms.
Transmitter: M antenna elements
MIMO Radar
15Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u0(t)
xT0
u1(t)
xT1
uM-1(t)
xT,M-1
… …
xR0 xR1 xR,M-1
MF…
MF…
MF…
Transmitter emits
incoherent waveforms.
Transmitter emits
incoherent waveforms.
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
16Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u0(t)
xT0
u1(t)
xT1
uM-1(t)
xT,M-1
…
(,f)
TX
delayDopplerfSpatial freq.
Ambiguity Function in MIMO Radar
17Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
xT0 xT1 xT,M-1
… …
xR0 xR1 xR,M-1
MF…
MF…
MF…
(,f) (,f)
TX RX
delayDopplerfSpatial freq.
u0(t)u1(t) uM-1(t)
)(),,( tfy
Ambiguity Function in MIMO Radar
18Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
xT0 xT1 xT,M-1
… …
xR0 xR1 xR,M-1
MF…
MF…
MF…
(,f) (,f)
TX RX
delayDopplerfSpatial freq.
u0(t)u1(t) uM-1(t)
)(),,( tfy
Ambiguity Function in MIMO Radar
19Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
xT0 xT1 xT,M-1
… …
xR0 xR1 xR,M-1
MF…
MF…
MF…
(,f) (,f)
dttt fHf )()( ),,()',','( yy
Matched filter output
TX RX
delayDopplerfSpatial freq.
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
Ambiguity Function in MIMO Radar
20Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Matched filter output
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
Ambiguity Function in MIMO Radar
21Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Matched filter output
Receiver beamforming
dttt fHf )()( ),,()',','( yy
delayDopplerfSpatial 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
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
Ambiguity Function in MIMO Radar
22Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Matched filter output
Receiver beamforming
dttt fHf )()( ),,()',','( yy
* 2, ' '( , ) ( ) ( ) j t
m m m mu t u t e dt [San Antonio et al. 07]
delayDopplerfSpatial freq.um(t): m-th waveformxm: m-th antenna locationn: receiving antenna index
MIMO ambiguity function
Ambiguity Function in MIMO Radar
23Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Ambiguity Function in MIMO Radar
24Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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)}.
26Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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)}.
27Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
28Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)()(1
0
L
llmm Tttu m-th waveform
Imposing Waveform Structures Pulse radar
– MTI (Moving Target Indicator)
– Doppler pulse radar
Frequency hopping signals– Constant modulus– Can be viewed as
generalized LFM (Linear
Frequency Modulation)
29Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)()(1
0
L
llmm Tttu
1
0),0[ )(1)2exp()(
Q
qtmqm tqttfcjt
m-th waveform
Imposing Waveform Structures Pulse radar
– MTI (Moving Target Indicator)
– Doppler pulse radar
Frequency hopping signals– Constant modulus– Can be viewed as
generalized LFM (Linear
Frequency Modulation)
Orthogonal waveforms– Virtual array
30Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)()(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
31Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
T
)()(1
0
L
llmm Tttu
0Tll TTT 1
Ambiguity Function of Pulse MIMO Radar
32Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Ambiguity Function of Pulse MIMO Radar
33Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Ambiguity Function of Pulse MIMO Radar
34Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
35Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
1
0
1
0'
)'(2)( '
',)()',,(
M
m
M
m
xffxj mm
mmerff
Waveform Design Problem in Pulse MIMO Radar
36Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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)}.
Waveform Design Problem in Pulse MIMO Radar
37Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
1
0
1
0'
)'(2)( '
',)()',,(
M
m
M
m
xffxj mm
mmerff
dtttr mmmm )()()( *)(',
)()',,( )(
0,0 rff
Choose a set of pulses {m(t)} such that (,f,f’) can be sharp around {(0,f,f)}.
Ex: SIMO case: M=1
Waveform Design Problem in Pulse MIMO Radar
38Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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)
Choose a set of pulses {m(t)} such that (,f,f’) can be sharp around {(0,f,f)}.
Ex: SIMO case: M=1
Orthogonality of the Frequency Hopping Signals
39Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Orthogonality of the Frequency Hopping Signals
40Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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(
Orthogonality of the Frequency Hopping Signals
41Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
Orthogonality of the Frequency Hopping Signals
42Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
43Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
),,( 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
Optimization of the Codes
44Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
),,( vec 'ff nnnCCω
'CC ωω w Code C is better than code C’.
CC ωω w'
Define a vector
Def: a code C is efficient if there exists no other code C’ such that
For any where gi are increasing convex functions
Optimization of the Codes
45Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
),,( vec 'ff nnnCCω
'CC ωω w Code C is better than code C’.
CC ωω w'
)()( '' CCCC ωωωω ffw
i igf
Define a vector
Def: a code C is efficient if there exists no other code C’ such that
For any where gi are increasing convex functions
So a code C is efficient if
Optimization of the Codes
46Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
),,( vec 'ff nnnCCω
'CC ωω w Code C is better than code C’.
CC ωω w'
)()( '' CCCC ωωωω ffw
)()( 'CC ωω ff for all C’.
i igf
Define a vector
Def: a code C is efficient if there exists no other code C’ such that
For any where gi are increasing convex functions
So a code C is efficient if for all C’. Example:
Optimization of the Codes
47Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
),,( vec 'ff nnnCCω
'CC ωω w Code C is better than code C’.
CC ωω w'
)()( '' CCCC ωωωω ffw
)()( 'CC ωω ff
i igf
p
pf cc ωω )(
Optimization of the Codes
48Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
49Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)(min CC
pf Csubject to
…
CC’
…[S. Kirkpatrick et al. 85]
Simulated Annealing Algorithm
Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution
50Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)(min CC
pf Csubject to
C
C
CC
T
fZ
T
f
Z
pT
p
TT
)(exp
)(exp
1)(
…
CC’
…[S. Kirkpatrick et al. 85]
Simulated Annealing Algorithm
Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution
– Run the Markov chain Monte Carlo (MCMC)
51Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)(min CC
pf Csubject to
C
C
CC
T
fZ
T
f
Z
pT
p
TT
)(exp
)(exp
1)(
…
CC’
…[S. Kirkpatrick et al. 85]
Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution
– 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)(
Simulated Annealing Algorithm
52Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
)(min CC
pf Csubject to
…
CC’
…[S. Kirkpatrick et al. 85]
Examples
53Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
54Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
55Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
|(,f,f’)|
Orthogonal LFMProposed Freq. Hopping Signal
Examples – Ambiguity Function
56Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
10log10|(,f,f’)|
Orthogonal LFMProposed Freq. Hopping Signal
57Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
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
58Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Orthogonal LFMProposed Freq. Hopping Signal
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
59Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
Q&AThank You!
Any questions?
60Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007