![Page 1: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/1.jpg)
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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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
![Page 3: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/3.jpg)
1Review: Ambiguity function in MIMO radar
3
![Page 4: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/4.jpg)
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range resolution.
4Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
![Page 5: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/5.jpg)
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
![Page 6: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/6.jpg)
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
![Page 7: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/7.jpg)
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
![Page 8: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/8.jpg)
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
![Page 9: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/9.jpg)
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
![Page 10: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/10.jpg)
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 (,)
![Page 11: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/11.jpg)
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
![Page 12: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/12.jpg)
-1-0.5
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1
-1
-0.5
0
0.5
10
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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
![Page 13: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/13.jpg)
-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
![Page 14: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/14.jpg)
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
![Page 15: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/15.jpg)
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
![Page 16: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/16.jpg)
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.
![Page 17: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/17.jpg)
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)
![Page 18: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/18.jpg)
)(),,( 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)
![Page 19: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/19.jpg)
)(),,( 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)
![Page 20: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/20.jpg)
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
![Page 21: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/21.jpg)
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
![Page 22: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/22.jpg)
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
![Page 23: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/23.jpg)
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.
![Page 24: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/24.jpg)
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.
![Page 25: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/25.jpg)
2Proposed Waveform Design Method
25
![Page 26: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/26.jpg)
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
![Page 27: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/27.jpg)
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
![Page 28: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/28.jpg)
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
![Page 29: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/29.jpg)
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
![Page 30: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/30.jpg)
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
![Page 31: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/31.jpg)
Ambiguity Function of Pulse MIMO Radar
31Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
T
)()(1
0
L
llmm Tttu
0Tll TTT 1
![Page 32: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/32.jpg)
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
![Page 33: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/33.jpg)
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
![Page 34: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/34.jpg)
)',,( 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
![Page 35: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/35.jpg)
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
![Page 36: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/36.jpg)
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)}.
![Page 37: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/37.jpg)
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
![Page 38: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/38.jpg)
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
![Page 39: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/39.jpg)
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
![Page 40: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/40.jpg)
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
![Page 41: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/41.jpg)
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
![Page 42: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/42.jpg)
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
![Page 43: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/43.jpg)
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’.
![Page 44: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/44.jpg)
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'
![Page 45: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/45.jpg)
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
![Page 46: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/46.jpg)
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
![Page 47: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/47.jpg)
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 )(
![Page 48: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/48.jpg)
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
![Page 49: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/49.jpg)
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]
![Page 50: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/50.jpg)
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]
![Page 51: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/51.jpg)
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]
![Page 52: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/52.jpg)
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]
![Page 53: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/53.jpg)
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
![Page 54: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/54.jpg)
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
![Page 55: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/55.jpg)
Examples – Ambiguity Function
55Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
|(,f,f’)|
Orthogonal LFMProposed Freq. Hopping Signal
![Page 56: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/56.jpg)
Examples – Ambiguity Function
56Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
10log10|(,f,f’)|
Orthogonal LFMProposed Freq. Hopping Signal
![Page 57: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/57.jpg)
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’)|)
![Page 58: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/58.jpg)
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
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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
![Page 60: A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP](https://reader035.vdocument.in/reader035/viewer/2022070412/56649d6f5503460f94a51198/html5/thumbnails/60.jpg)
Q&AThank You!
Any questions?
60Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007