A Design Method for MIMO Radar Frequency Hopping Codes

1
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
2
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

3
Review Ambiguity function in MIMO radar
1
4
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

5
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

(t,n)
target
u(t)
TX
tdelay nDoppler
6
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

(t,n)
target
u(t)
y(t,n) (t)
TX
RX
tdelay nDoppler
7
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

Matched filter output
(t,n)
target
u(t)
y(t,n) (t)
TX
RX
tdelay nDoppler
8
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

Matched filter output
(t,n)
target
u(t)
y(t,n) (t)
TX
RX
tdelay nDoppler
9
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

Matched filter output
(t,n)
target
u(t)
y(t,n) (t)
TX
RX
tdelay nDoppler
Ambiguity function
10
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
11
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
12
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
Ambiguity function
13
Ambiguity Function in SIMO Radar

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
Ambiguity function
14
MIMO Radar
Transmitter M antenna elements
xT0
xT1
xT,M-1

u0(t)
u1(t)
uM-1(t)
Transmitter emits incoherent waveforms.
15
MIMO Radar
Receiver N antenna elements
Transmitter M antenna elements
xR0
xR1
xR,M-1
xT0
xT1
xT,M-1


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



Transmitter emits incoherent waveforms.
Matched filters extract the M orthogonal
waveforms. Overall number of signals NM
15
Chun-Yang Chen, Caltech DSP Lab Asilomar
Conference 2007
16
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
xT0
xT1
xT,M-1
TX

u0(t)
u1(t)
uM-1(t)
17
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
xT0
xT1
xT,M-1
xR0
xR1
xR,M-1
TX
RX


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



18
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
xT0
xT1
xT,M-1
xR0
xR1
xR,M-1
TX
RX


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



19
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
xT0
xT1
xT,M-1
xR0
xR1
xR,M-1
TX
RX


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



Matched filter output
20
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq. um(t) m-th
waveform xm m-th antenna location n receiving
antenna index
Matched filter output
Receiver beamforming
21
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq. um(t) m-th
waveform xm m-th antenna location n receiving
antenna index
Matched filter output
Receiver beamforming
Cross ambiguity function
22
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq. um(t) m-th
waveform xm m-th antenna location n receiving
antenna index
Matched filter output
Receiver beamforming
San Antonio et al. 07
MIMO ambiguity function
23
Ambiguity Function in MIMO Radar

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

n
target 1 (t1,n1,f1)
target 2 (t2,n2,f 2)
t
f
24
Ambiguity Function in MIMO Radar

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

n
target 1 (t1,n1,f1)
target 2 (t2,n2,f 2)
t
f
Ambiguity function
25
Proposed Waveform Design Method
2
26
MIMO Radar Waveform Design Problem

• Choose a set of waveforms um(t) so that the
  ambiguity function c(t,n,f,f) can be sharp
  around (0,0,f,f).

n
target 1 (t1,n1,f1)
t
f
27
MIMO Radar Waveform Design Problem

• Choose a set of waveforms um(t) so that the
  ambiguity function c(t,n,f,f) can be sharp
  around (0,0,f,f).

n
target 1 (t1,n1,f1)
t
f
Ambiguity function
28
Imposing Waveform Structures

• Pulse radar
• MTI (Moving Target Indicator)
• Doppler pulse radar

m-th waveform
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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)

m-th waveform
30
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

m-th waveform
31
Ambiguity Function of Pulse MIMO Radar
Tf
32
Ambiguity Function of Pulse MIMO Radar
Tf
33
Ambiguity Function of Pulse MIMO Radar
Tf
Doppler processing is separable
34
Ambiguity Function of Pulse MIMO Radar
Tf
Doppler processing is separable
Define as
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Waveform Design Problem in Pulse MIMO Radar
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Waveform Design Problem in Pulse MIMO Radar

• Choose a set of pulses fm(t) such that
  W(t,f,f) can be sharp around (0,f,f).

37
Waveform Design Problem in Pulse MIMO Radar

• Choose a set of pulses fm(t) such that
  W(t,f,f) can be sharp around (0,f,f).
• Ex SIMO case M1

38
Waveform Design Problem in Pulse MIMO Radar

• Choose a set of pulses fm(t) such that
  W(t,f,f) can be sharp around (0,f,f).
• Ex SIMO case M1

Choose a pulse with a sharp correlation function
(e.g. LFM)
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Orthogonality of the Frequency Hopping Signals
m
m'





Frequency
Time
40
Orthogonality of the Frequency Hopping Signals
m
m'
41
Orthogonality of the Frequency Hopping Signals
m
m'
42
Orthogonality of the Frequency Hopping Signals
m
m'

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

43
Optimization of the Codes

• Define a vector

Code C is better than code C.
44
Optimization of the Codes

• Define a vector
• Def a code C is efficient if there exists no
  other code C such that

Code C is better than code C.
45
Optimization of the Codes

• 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

Code C is better than code C.
46
Optimization of the Codes

• 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

Code C is better than code C.
for all C.
47
Optimization of the Codes

• 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

Code C is better than code C.
48
Optimization of the Codes
M of waveforms Q of freq. hops K of
freq.Time-bandwidth product KDfQDt
49
Simulated Annealing Algorithm
subject to

• Simulated annealing
• Create a Markov chain on the set A

S. Kirkpatrick et al. 85
C
C


50
Simulated Annealing Algorithm
subject to

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

S. Kirkpatrick et al. 85
C
C


51
Simulated Annealing Algorithm
subject to

• Simulated annealing
• Create a Markov chain on the set A with the
  equilibrium distribution
• Run the Markov chain Monte Carlo (MCMC)

S. Kirkpatrick et al. 85
C
C


52
Simulated Annealing Algorithm
subject to

• 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

S. Kirkpatrick et al. 85
C
C


53
Examples
Proposed Freq. Hopping Signals
54
Examples
Proposed Freq. Hopping Signals
Orthogonal LFM

• The same array
• The same duration and bandwidth
• Initial frequencies

55
Examples Ambiguity Function
Orthogonal LFM
Proposed Freq. Hopping Signal
W(t,f,f)
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Examples Ambiguity Function
Orthogonal LFM
Proposed Freq. Hopping Signal
10log10W(t,f,f)
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Examples Sorted Samples of Ambiguity Functions
10log10(W(t,f,f))
0

LFM
Randomly selected code
-5
Proposed method
10log10(W(t,f,f))
-10
-15

0
2
4
6
8
10
Sorted samples ()
Sorted samples ()
58
Examples Correlation Function Matrix
Orthogonal LFM
Proposed Freq. Hopping Signal
58
Chun-Yang Chen, Caltech DSP Lab Asilomar
Conference 2007
59
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

60
Thank You!
QA
Any questions?
60
Chun-Yang Chen, Caltech DSP Lab Asilomar
Conference 2007