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A Design Method for MIMO Radar
Frequency Hopping Codes
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Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
Asilomar Conference 2007
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 2
Review: Ambiguity function in MIMO radar
3
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 4
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
target (t,n)
u(t)
TX
t:delay
n:Doppler
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 5
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
target (t,n)
u(t) y(t,n) (t)
TX RX
t:delay
n:Doppler
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 6
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
Matched filter output
(t,n)
y
target (t ,n ) (t ',n ')
(t ) y (t ) dt
u(t) y(t,n) (t)
TX RX
t:delay
n:Doppler
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 7
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
Matched filter output
(t,n)
y
target (t ,n ) (t ',n ')
(t ) y (t ) dt
u (t t )e j 2n t u (t t ' ) e j 2n 't dt
u (t )u (t (t t ' )) e j 2 (n n ')t dt
u(t) y(t,n) (t)
TX RX
t:delay
n:Doppler
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 8
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
Matched filter output
(t,n)
y
target (t ,n ) (t ',n ')
(t ) y (t ) dt
u (t t )e j 2n t u (t t ' ) e j 2n 't dt
u (t )u (t (t t ' )) e j 2 (n n ')t dt
u(t) y(t,n) (t)
(t ,n ) u(t )u(t t ) e
TX RX j 2n t
dt
t:delay
n:Doppler Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 9
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
n
target 1 (t1,n1)
target 2 (t2,n2)
t
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 10
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
n (t t 1 ,n n 1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 11
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
n (t t 1 ,n n 1 )
1
0.8
0.6
target 1 (t1,n1) 0.4
target 2 (t2,n2)
0.2
0
1
t
0.5 1
0 0.5
0
-0.5 -0.5
-1 -1
(t ,n ) u(t )u(t t ) e j 2n t
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 12
Ambiguity Function in SIMO Radar
Ambiguity function characterizes the Doppler and range
resolution.
n (t t 1 ,n n 1 )
1
0.8
0.6
target 1 (t1,n1) 0.4
target 2 (t2,n2)
0.2
0
1
t
0.5 1
0 0.5
0
-0.5 -0.5
-1 -1
(t ,n ) u(t )u(t t ) e j 2n t
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 13
MIMO Radar
Transmitter: M antenna elements
xT0 xT1 xT,M-1
…
u0(t) u1(t) uM-1(t)
Transmitter emits
incoherent waveforms.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 14
MIMO Radar
Transmitter: M antenna elements Receiver: N antenna elements
xT0 xT1 xT,M-1 xR0 xR1 xR,M-1
… …
u0(t) u1(t) uM-1(t) MF MF MF
… … …
Matched filters extract
Transmitter emits
the M orthogonal waveforms.
incoherent waveforms. Overall number of signals:
NM
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 15
Ambiguity Function in MIMO Radar
(t,nf) t:delay
n:Doppler
xT0 xT1 xT,M-1 f: Spatial freq.
TX …
u0(t) u1(t) uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 16
Ambiguity Function in MIMO Radar
t:delay
(t,nf) n:Doppler (t,nf)
f: Spatial freq.
xT0 xT1 xT,M-1 xR0 xR1 xR,M-1
TX … RX …
u0(t) u1(t) uM-1(t) MF MF MF
… … …
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 17
Ambiguity Function in MIMO Radar
t:delay
(t,nf) n:Doppler (t,nf)
f: Spatial freq.
xT0 xT1 xT,M-1 xR0 xR1 xR,M-1
TX … RX …
u0(t) u1(t) uM-1(t) MF MF MF
… … …
(t ,n , f )
y (t )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 18
Ambiguity Function in MIMO Radar
t:delay
(t,nf) n:Doppler (t,nf)
f: Spatial freq.
xT0 xT1 xT,M-1 xR0 xR1 xR,M-1
TX … RX …
u0(t) u1(t) uM-1(t) MF MF MF
… … …
(t ,n , f )
y (t )
Matched filter output
y
(t ',n ', f ')
y (t ,n , f ) (t )dt
H
(t )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 19
Ambiguity Function in MIMO Radar
t:delay
Matched filter output n:Doppler
f: Spatial freq.
y
(t ',n ', f ') H (t ,n , f )
y
um(t): m-th waveform
(t ) (t )dt xm: m-th antenna location
n: receiving antenna index
u (t t )u (t t ' )e
N 1 M 1 M 1
e j 2 ( f f ') n
n 0 m 0 m '0
m
*
m
j 2 (n v ') t
dt e j 2 ( fxm f ' xm ' )
Receiver beamforming
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 20
Ambiguity Function in MIMO Radar
t:delay
Matched filter output n:Doppler
f: Spatial freq.
y
(t ',n ', f ') H (t ,n , f )
y
um(t): m-th waveform
(t ) (t )dt xm: m-th antenna location
n: receiving antenna index
u (t t )u (t t ' )e
N 1 M 1 M 1
e j 2 ( f f ') n
n 0 m 0 m '0
m
*
m
j 2 (n v ') t
dt e j 2 ( fxm f ' xm ' )
Receiver beamforming
m,m ' (t ,n ) um (t )um ' (t t )e j 2n t dt
*
Cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 21
Ambiguity Function in MIMO Radar
t:delay
Matched filter output n:Doppler
f: Spatial freq.
y
(t ',n ', f ') H (t ,n , f )
y
um(t): m-th waveform
(t ) (t )dt xm: m-th antenna location
n: receiving antenna index
u (t t )u (t t ' )e
N 1 M 1 M 1
e j 2 ( f f ') n
n 0 m 0 m '0
m
*
m
j 2 (n v ') t
dt e j 2 ( fxm f ' xm ' )
Receiver beamforming
m,m ' (t ,n ) um (t )um ' (t t )e j 2n t dt
*
[San Antonio et al. 07]
M 1 M 1
(t ,n , f , f ' ) m ,m ' (t ,n )e j 2 ( fx m f ' xm ' )
m 0 m ' 0
MIMO ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 22
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 23
Ambiguity Function in MIMO Radar
Ambiguity function characterizes the Doppler, range, and
angular resolution.
n (t t 1 ,n n 1 , f1 , f )
target 1 (t1,n1,f1)
target 2 (t2,n2,f 2)
t
f Ambiguity
m,m' (t ,n ) um (t )um (t t )e j 2n t dt
M 1 M 1
*
function (t ,n , f , f ' ) m ,m ' (t ,n )e j 2 ( fxm f ' xm ' )
m 0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 24
Proposed Waveform Design Method
25
MIMO Radar Waveform Design Problem
Choose a set of waveforms {um(t)} so that the ambiguity
function tnf,f’ can be sharp around {(0,0,f,f)}.
n (t t 1 ,n n 1 , f1 , f )
target 1 (t1,n1,f1)
t
f
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 26
MIMO Radar Waveform Design Problem
Choose a set of waveforms {um(t)} so that the ambiguity
function tnf,f’ can be sharp around {(0,0,f,f)}.
n (t t 1 ,n n 1 , f1 , f )
target 1 (t1,n1,f1)
t
f Ambiguity
m,m' (t ,n ) um (t )um (t t )e j 2n t dt
M 1 M 1
*
function (t ,n , f , f ' ) m ,m ' (t ,n )e j 2 ( fxm f ' xm ' )
m 0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 27
Imposing Waveform Structures
L 1
Pulse radar u m (t ) m (t Tl ) m-th waveform
– MTI (Moving Target Indicator) l 0
– Doppler pulse radar
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 28
Imposing Waveform Structures
L 1
Pulse radar u m (t ) m (t Tl ) m-th waveform
– MTI (Moving Target Indicator) l 0
– Doppler pulse radar
Frequency hopping
signals
– Constant modulus
Q 1
– Can be viewed as
generalized LFM (Linear
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt )
q 0
Frequency Modulation)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 29
Imposing Waveform Structures
L 1
Pulse radar u m (t ) m (t Tl ) m-th waveform
– MTI (Moving Target Indicator) l 0
– Doppler pulse radar
Frequency hopping
signals
– Constant modulus
Q 1
– Can be viewed as
generalized LFM (Linear
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt )
q 0
Frequency Modulation)
Orthogonal waveforms cmq cm'q q, m m'
– Virtual array ft 1
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 30
Ambiguity Function of Pulse MIMO Radar
L 1
u m (t ) m (t Tl )
l 0 T
T Tl 1 Tl nT 0
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 31
Ambiguity Function of Pulse MIMO Radar
L 1
u m (t ) m (t Tl )
l 0 T
T Tl 1 Tl nT 0
( ) m,m' (t ,n ) m (t )m (t t )e j 2n t dt
*
m (t )m (t t ) 1dt
*
rmm' (t )
( )
,
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 32
Ambiguity Function of Pulse MIMO Radar
L 1
u m (t ) m (t Tl )
l 0 T
T Tl 1 Tl nT 0
( ) m,m' (t ,n ) m (t )m (t t )e j 2n t dt
*
m (t )m (t t ) 1dt
*
rmm' (t )
( )
,
M 1 M 1 ( ) j 2 ( fxm f ' xm ' )
L 1
j 2n Tl
(t ,n , f , f ' ) rm ,m ' (t )e e
m 0 m '0 l 0
Doppler processing
is separable
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 33
Ambiguity Function of Pulse MIMO Radar
L 1
u m (t ) m (t Tl )
l 0 T
T Tl 1 Tl nT 0
( ) m,m' (t ,n ) m (t )m (t t )e j 2n t dt
*
m (t )m (t t ) 1dt
*
rmm' (t )
( )
,
M 1 M 1 ( ) j 2 ( fxm f ' xm ' )
L 1
j 2n Tl
(t ,n , f , f ' ) rm ,m ' (t )e e
m 0 m '0 l 0
Doppler processing
Define as (t , f , f ' ) is separable
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 34
Waveform Design Problem in Pulse
MIMO Radar
M 1 M 1
(t , f , f ' ) rm(m)' (t )e j 2 ( fxm f ' xm ' )
,
m 0 m ' 0
rmm' (t ) m (t )m (t t )dt
( )
,
*
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 35
Waveform Design Problem in Pulse
MIMO Radar
M 1 M 1
(t , f , f ' ) rm(m)' (t )e j 2 ( fxm f ' xm ' )
,
m 0 m ' 0
rmm' (t ) m (t )m (t t )dt
( )
,
*
Choose a set of pulses {m(t)} such that (t,f,f’) can be
sharp around {(0,f,f)}.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 36
Waveform Design Problem in Pulse
MIMO Radar
M 1 M 1
(t , f , f ' ) rm(m)' (t )e j 2 ( fxm f ' xm ' )
,
m 0 m ' 0
rmm' (t ) m (t )m (t t )dt
( )
,
*
Choose a set of pulses {m(t)} such that (t,f,f’) can be
sharp around {(0,f,f)}.
Ex: SIMO case: M=1
(t , f , f ' ) r0(, ) (t )
0
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 37
Waveform Design Problem in Pulse
MIMO Radar
M 1 M 1
(t , f , f ' ) rm(m)' (t )e j 2 ( fxm f ' xm ' )
,
m 0 m ' 0
rmm' (t ) m (t )m (t t )dt
( )
,
*
Choose a set of pulses {m(t)} such that (t,f,f’) can be
sharp around {(0,f,f)}.
Ex: SIMO case: M=1
(t , f , f ' ) r0(, ) (t )
0
Choose a pulse with a sharp correlation function (e.g. LFM)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 38
Orthogonality of the Frequency Hopping
Signals
Q 1
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt ) m
q 0
cmq cm'q q, m m' m'
ft 1
Frequency
f
t
Time
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 39
Orthogonality of the Frequency Hopping
Signals
Q 1
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt ) m
q 0
cmq cm'q q, m m' m'
ft 1
m (t )m (t )dt m,m'
*
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 40
Orthogonality of the Frequency Hopping
Signals
Q 1
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt ) m
q 0
cmq cm'q q, m m' m'
ft 1
m (t )m (t )dt m,m'
*
M 1 M 1
(0, f , f ) rm(m)' (0)e j 2f ( xm xm ' ) M
,
m 0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 41
Orthogonality of the Frequency Hopping
Signals
Q 1
m (t ) exp( j 2fcmqt ) 1[0,t ) (t qt ) m
q 0
cmq cm'q q, m m' m'
ft 1
m (t )m (t )dt m,m'
*
M 1 M 1
(0, f , f ) rm(m)' (0)e j 2f ( xm xm ' ) M
,
m 0 m ' 0
is a constant along {(0,f,f)}, no matter what codes
are chosen.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 42
Optimization of the Codes
Define a vector ωC vec C (nt , n f , n f ' )
ω C w ω C' Code C is better than
code C’.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 43
Optimization of the Codes
Define a vector ωC vec C (nt , n f , n f ' )
ω C w ω C' Code C is better than code C’.
Def: a code C is efficient if there exists no other code C’
such that
ω C' w ω C
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 44
Optimization of the Codes
Define a vector ωC vec C (nt , n f , n f ' )
ω C w ω C' Code C is better than code C’.
Def: a code C is efficient if there exists no other code C’
such that ω C' w ω C
For any f i gi where gi are increasing convex
functions
ωC' w ωC f (ωC' ) f (ωC )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 45
Optimization of the Codes
Define a vector ωC vec C (nt , n f , n f ' )
ω C w ω C' Code C is better than code C’.
Def: a code C is efficient if there exists no other code C’
such that ω C' w ω C
For any f i gi where gi are increasing convex
functions ωC' w ωC f (ωC' ) f (ωC )
So a code C is efficient if f (ωC ) f (ωC' )
for all C’.
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 46
Optimization of the Codes
Define a vector ωC vec C (nt , n f , n f ' )
ω C w ω C' Code C is better than code C’.
Def: a code C is efficient if there exists no other code C’
such that ω C' w ω C
For any f i gi where gi are increasing convex
functions ωC' w ωC f (ωC' ) f (ωC )
So a code C is efficient if f (ωC ) f (ωC' ) for all C’.
Example: f (ωc ) ωc p
p
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 47
Optimization of the Codes
1 1
min (t , f , f ' ) dt df df '
p
C
0 0
C {0,1, K 1}MQ
cmq cm 'q q, m m'
M:# of waveforms
Q 1
Q: # of freq. hops
K: # of freq. m (t ) exp( j 2cmqt ) 1[0,t ) (t qt )
q 0
Time-bandwidth product:
KfQt
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 48
Simulated Annealing Algorithm
min f p (C)
C
subject to C
Simulated annealing [S. Kirkpatrick et al. 85]
– Create a Markov chain on the set A
C’
C
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 … 49
Simulated Annealing Algorithm
min f p (C)
C
subject to C
Simulated annealing [S. Kirkpatrick et al. 85]
– Create a Markov chain on the set A with the equilibrium distribution
1 f p (C) C’
T (C) exp
T
C
ZT
f p (C) …
ZT exp
C T
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 … 50
Simulated Annealing Algorithm
min f p (C)
C
subject to C
Simulated annealing [S. Kirkpatrick et al. 85]
– Create a Markov chain on the set A with the equilibrium distribution
1 f p (C) C’
T (C) exp
T
C
ZT
f p (C) …
ZT exp
C T
…
– Run the Markov chain Monte Carlo (MCMC)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 51
Simulated Annealing Algorithm
min f p (C)
C
subject to C
Simulated annealing [S. Kirkpatrick et al. 85]
– Create a Markov chain on the set A with the equilibrium distribution
1 f p (C) C’
T (C) exp
T
C
ZT
f p (C) …
ZT exp
C T
…
– Run the Markov chain Monte Carlo (MCMC)
– Decrease the temperature T from time to time
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 52
Examples
Proposed Freq. Hopping Signals
Parameters:
Uniform linear array
# of waveforms M =4
# of hops Q=10
# of freq. K=15
norm type p=3
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 53
Examples
Proposed Freq. Hopping Signals Orthogonal LFM
Parameters: Parameters:
Uniform linear array
– The same array
# of waveforms M =4
# of hops Q=10 – The same duration and
# of freq. K=15 bandwidth
norm type p=3 – Initial frequencies
1 1
0 0
-1 -1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1
0 0
-1 -1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1
0 0
-1 -1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1
0 0
-1 -1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 54
Examples – Ambiguity Function
Proposed Freq. Hopping Signal Orthogonal LFM
|(t,f,f’)|
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 55
Examples – Ambiguity Function
Proposed Freq. Hopping Signal Orthogonal LFM
10log10|(t,f,f’)|
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 56
Examples – Sorted Samples of Ambiguity
Functions
10log10(|(t,f,f’)|)
0
LFM
Randomly selected code
-5
Proposed method
10log10(|(t,f,f’)|)
-10 0
-5
-10
-15
0 2 4 6 8 10
-15
Sorted samples (%)
-20
0 20 40 60 80 100
Sorted samples (%)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 57
Examples – Correlation Function Matrix
Proposed Freq. Hopping Signal Orthogonal LFM
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
-1 -1 -1 -1 -1 -1 -1 -1
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
-1 -1 -1 -1 -1 -1 -1 -1
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
-1 -1 -1 -1 -1 -1 -1 -1
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
-1 -1 -1 -1 -1 -1 -1 -1
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
rmm' (t ) m (t )m (t t )dt
( )
,
*
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 58
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 59
Thank You!
Q&A Any questions?
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 60
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