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Radar ambiguity function
1. Properties of the MIMO Radar Ambiguity Function
Chun-Yang Chen and P. P. VaidyanathanCalifornia Institute of Technology
Electrical Engineering/DSP Lab
ICASSP 2008
2. Outline
Review of the background– Radar ambiguity function and its properties
– MIMO radar
– MIMO radar ambiguity function
Properties of the MIMO ambiguity function
–
–
–
–
Signal component
Energy
Symmetry
Linear frequency modulation (LFM)
Conclusion
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
3. Review: Ambiguity function and MIMO radar
34. Radar Ambiguity Function
u(t)u(t-t)ej2pnt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t: delay
n: Doppler
4
5. Radar Ambiguity Function
u(t)Matched filter
output
u(t-t)ej2pnt
t: delay
n: Doppler
(u (t t )e j 2pnt )(u (t t ' )e j 2pn 't )* dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
5
6. Radar Ambiguity Function
u(t)Matched filter
output
u(t-t)ej2pnt
t: delay
n: Doppler
(u (t t )e j 2pnt )(u (t t ' )e j 2pn 't )* dt
u (t )u * (t (t t ' ))e j 2p (n n ')t dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
6
7. Radar Ambiguity Function
u(t)Matched filter
output
u(t-t)ej2pnt
t: delay
n: Doppler
(u (t t )e j 2pnt )(u (t t ' )e j 2pn 't )* dt
u (t )u * (t (t t ' ))e j 2p (n n ')t dt
Radar ambiguity
function
(t ,n ) u (t )u * (t t )e j 2pnt dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
7
8. Radar Ambiguity Function
u(t)Matched filter
output
u(t-t)ej2pnt
t: delay
n: Doppler
(u (t t )e j 2pnt )(u (t t ' )e j 2pn 't )* dt
u (t )u * (t (t t ' ))e j 2p (n n ')t dt
Radar ambiguity
function
(t ,n ) u (t )u * (t t )e j 2pnt dt
Ambiguity function characterizes the Doppler and range
resolution.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
8
9. Radar Ambiguity Function
u (t )Multiple targets
(tk,nk)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
9
10. Radar Ambiguity Function
u (t )K
j 2pn k t
u
(
t
t
)
e
k
k 1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets
(tk,nk)
10
11. Radar Ambiguity Function
u (t )K
j 2pn k t
u
(
t
t
)
e
k
k 1
Matched filter
output
K
k 1
k
Multiple targets
(tk,nk)
(t t k ,n n k )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
11
12. Radar Ambiguity Function
u (t )K
j 2pn k t
u
(
t
t
)
e
k
k 1
Matched filter
output
K
k 1
k
Multiple targets
(tk,nk)
(t t k ,n n k )
n
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
12
13. Radar Ambiguity Function
u (t )K
j 2pn k t
u
(
t
t
)
e
k
k 1
Matched filter
output
n
K
k 1
k
Multiple targets
(tk,nk)
(t t k ,n n k )
(t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
13
14. Radar Ambiguity Function
Ambiguity function characterizes the Doppler and rangeresolution.
n
(t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
(t ,n ) u (t )u (t t ) e
j 2pnt
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
14
15. Radar Ambiguity Function
Ambiguity function characterizes the Doppler and rangeresolution.
n
(t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
(t ,n ) u (t )u (t t ) e
j 2pnt
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
15
16. Properties of Radar Ambiguity Function
Signal component(0,0) 1 (t ,n )
n
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
16
17. Properties of Radar Ambiguity Function
Signal component(0,0) 1 (t ,n )
Energy
2
(t ,n ) dtdn 1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
17
18. Properties of Radar Ambiguity Function
Signal component(0,0) 1 (t ,n )
Energy
2
(t ,n ) dtdn 1
Symmetry
n
t
( t , n ) (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
18
19. Properties of Radar Ambiguity Function
Signal component(0,0) 1 (t ,n )
Energy
2
(t ,n ) dtdn 1
n
Symmetry
t
( t , n ) (t ,n )
Linear frequency modulation (LFM)
u
LFM
(t ) u(t )e
jpkt 2
LFM (t ,n ) (t ,n kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
19
20. MIMO Radar
The radar systems which emits orthogonal (or noncoherent)waveforms in each transmitting antennas are called MIMO radar.
MIMO radar
f2(t)
f1(t)
f0(t)
SIMO radar (Traditional)
w2f(t)
w1f(t)
w0f(t)
Advantages
– Better spatial resolution [Bliss & Forsythe 03]
– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
– Improved parameter identifiability [Li et al. 07]
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
21. Ambiguity Function in MIMO Radar
(t,n,f) t:delayn:Doppler
f: Spatial freq.
TX
dT
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
21
22. Ambiguity Function in MIMO Radar
(t,n,f)TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
22
23. Ambiguity Function in MIMO Radar
(t,n,f)TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
(t )
23
24. Ambiguity Function in MIMO Radar
(t,n,f)TX
RX
…
dT
u0(t) u1(t)
uM-1(t)
Matched filter output
(y
(t ',n ', f ')
t:delay
n:Doppler
f: Spatial freq.
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
MF
…
(t )
(t ) ) y (t ,n , f ) (t )dt
H
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
24
25. Ambiguity Function in MIMO Radar
Matched filter output(y
(t ',n ', f ')
(t ) ) y (t ,n , f ) (t )dt
H
( u
N 1
M 1 M 1
n 0
m 0 m ' 0
j 2p ( f f ') n
e
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t
t
)
u
(
t
t
'
)
e
dt
e
m
m
Receiver beamforming
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
25
26. Ambiguity Function in MIMO Radar
Matched filter output(y
(t ',n ', f ')
(t ) ) y (t ,n , f ) (t )dt
H
( u
N 1
M 1 M 1
n 0
m 0 m ' 0
j 2p ( f f ') n
e
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t
t
)
u
(
t
t
'
)
e
dt
e
m
m
Receiver beamforming
m,m ' (t ,n ) um (t )um* ' (t t )e j 2pn t dt
Cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
26
27. Ambiguity Function in MIMO Radar
Matched filter output(y
(t ',n ', f ')
(t ) ) y (t ,n , f ) (t )dt
H
( u
N 1
M 1 M 1
n 0
m 0 m ' 0
j 2p ( f f ') n
e
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t
t
)
u
(
t
t
'
)
e
dt
e
m
m
Receiver beamforming
m,m ' (t ,n ) um (t )um* ' (t t )e j 2pn t dt
[San Antonio et al. 07]
M 1 M 1
(t ,n , f , f ' ) m,m ' (t ,n )e j 2p ( fm f 'm ')
m 0 m ' 0
MIMO ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
27
28. Properties of the MIMO ambiguity function
2829. Properties of the signal component
Ambiguity function:Signal component:
(t ,n , f , f ' )
(0,0, f , f )
(0,0, f , f ' )
f ' f
f'
(0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
29
30. Properties of the signal component
Ambiguity function:Signal component:
(0,0, f , f ' )
(t ,n , f , f ' )
(0,0, f , f )
For orthogonal waveforms,
um (t )um* ' (t )dt mm'
f ' f
f'
(0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
30
31. Properties of the signal component
Ambiguity function:Signal component:
(0,0, f , f ' )
(t ,n , f , f ' )
(0,0, f , f )
For orthogonal waveforms,
f ' f
f'
(0,0, f , f )
um (t )um* ' (t )dt mm'
(0,0, f , f ) M , f
If the waveforms are orthogonal,
the signal component will be a
constant for all angle.
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
31
32. Properties of the signal component
Ambiguity function:Signal component:
(t ,n , f , f ' )
(0,0, f , f )
For general waveforms,
For orthogonal waveforms,
um (t ) dt 1
2
um (t )um* ' (t )dt mm'
(0,0, f , f ) M , f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
32
33. Properties of the signal component
Ambiguity function:Signal component:
(t ,n , f , f ' )
(0,0, f , f )
For general waveforms,
If
dT
For orthogonal waveforms,
um (t ) dt 1
2
is integer,
um (t )um* ' (t )dt mm'
(0,0, f , f ) M , f
(0,0, f , f ) df M , f
The integration of the signal
component is a constant if dT is
a multiple of the wavelength.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
33
34. Properties of the signal component
Ambiguity function:Signal component:
(t ,n , f , f ' )
(0,0, f , f )
For general waveforms,
If
For orthogonal waveforms,
um (t ) dt 1
dT
dT is the spacing
between the
transmitting antennas
2
is integer,
um (t )um* ' (t )dt mm'
(0,0, f , f ) M , f
(0,0, f , f ) df M , f
For the general case,
2dT / M (0,0, f , f ) df 2dT / M
2d T /
2d T /
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
In general, the
integration of the
signal component is
confined.
34
35. Energy of the cross ambiguity function
Cross ambiguity function:mm' (t ,n ) um (t )um* ' (t t )e j 2pnt dt
Energy of the cross ambiguity function:
mm' (t ,n ) dt dn
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
35
36. Energy of the cross ambiguity function
Cross ambiguity function:mm' (t ,n ) um (t )um* ' (t t )e j 2pnt dt
Energy of the cross ambiguity function:
mm' (t ,n ) dt dn
2
u
m
(t )u (t t )e
*
m'
j 2pnt
2
dt dndt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
36
37. Energy of the cross ambiguity function
Cross ambiguity function:mm' (t ,n ) um (t )um* ' (t t )e j 2pnt dt
Energy of the cross ambiguity function:
mm' (t ,n ) dt dn
2
u
m
(t )u (t t )e
*
m'
j 2pnt
2
dt dndt
um (t )u (t t ) dtdt
*
m'
2
Parserval
relation
( u (t ) dt ) 1
2
2
m
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
37
38. Energy of the cross ambiguity function
Cross ambiguity function:mm' (t ,n ) um (t )um* ' (t t )e j 2pnt dt
Energy of the cross ambiguity function:
mm' (t ,n ) dt dn 1
2
The energy of the cross
ambiguity function is a
constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
38
39. Energy of the MIMO ambiguity function
MIMO ambiguity function:M 1 M 1
(t ,n , f , f ' ) mm' (t ,n )e j 4pd
T
/ ( fm f 'm ')
m 0 m ' 0
Energy of the ambiguity function
(t ,n , f , f ' )
2
dtdndfdf '
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
39
40. Energy of the MIMO ambiguity function
MIMO ambiguity function:M 1 M 1
(t ,n , f , f ' ) mm' (t ,n )e j 4pd
T
/ ( fm f 'm ')
m 0 m ' 0
Energy of the ambiguity function
(t ,n , f , f ' )
M 1 M 1
m 0 m ' 0
2
dtdndfdf '
2
j 4pdT / ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
40
41. Energy of the MIMO ambiguity function
MIMO ambiguity function:M 1 M 1
(t ,n , f , f ' ) mm' (t ,n )e j 4pd
T
/ ( fm f 'm ')
m 0 m ' 0
Energy of the ambiguity function
(t ,n , f , f ' )
M 1 M 1
m 0 m ' 0
M 1 M 1
2
dT is the spacing
between the
transmitting antennas
dtdndfdf '
2
j 4pdT / ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
mm' (t ,n ) dtdn
2
m 0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
If dT is a multiple of
the wavelength, we
can apply Parserval
relation for 2D DFT.
41
42. Energy of the MIMO ambiguity function
MIMO ambiguity function:M 1 M 1
(t ,n , f , f ' ) mm' (t ,n )e j 4pd
T
/ ( fm f 'm ')
m 0 m ' 0
Energy of the ambiguity function
(t ,n , f , f ' )
M 1 M 1
m 0 m ' 0
M 1 M 1
2
dtdndfdf '
2
j 4pdT / ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
mm' (t ,n ) dtdn
2
m 0 m ' 0
M 1 M 1
1 M
m 0 m ' 0
2
Cross ambiguity
function has
constant energy
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
42
43. Energy of the MIMO ambiguity function
If dT is a multiple of the wavelength,2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
2
dT is the spacing
between the
transmitting antennas
If dT is a multiple of the
wavelength, the energy of
the MIMO ambiguity function
is a constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
43
44. Energy of the MIMO ambiguity function
If dT is a multiple of the wavelength,2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
2
dT is the spacing
between the
transmitting antennas
Recall that the signal component satisfies,
(0,0, f , f ) df
M , f
– Because energy and the signal component are both constants,
we can only spread the energy to minimize the peak.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
44
45. Energy of the MIMO ambiguity function
If dT is a multiple of the wavelength,2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
2
dT is the spacing
between the
transmitting antennas
In general, the energy satisfies,
2dT /
2
2dT / M 2
2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
(2dT / )2
(2dT / )2
2
2
In general, the energy of the
MIMO ambiguity function is
confined in a certain range.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
45
46. Energy of the MIMO ambiguity function
If dT is a multiple of the wavelength,2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
2
dT is the spacing
between the
transmitting antennas
In general, the energy satisfies,
2dT /
2
2dT / M 2
2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
M
(2dT / )2
(2dT / )2
2
2
In general, the signal component satisfies,
2dT / M (0,0, f , f ) df 2dT / M
2d T /
2d T /
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
46
47. Symmetry properties
Symmetry of the cross ambiguity functionmm' ( t , n ) m'm (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
47
48. Symmetry properties
Symmetry of the cross ambiguity functionmm' ( t , n ) m'm (t ,n )
Symmetry of the MIMO ambiguity function
( t , n , f , f ' ) (t ,n , f ' , f )
It suffices to show only half of
the ambiguity function (t>0).
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
48
49. Linear frequency modulation (LFM)
Linear frequency modulationu
LFM
m
(t ) um (t )e
jpkt 2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
49
50. Linear frequency modulation (LFM)
Linear frequency modulationu
LFM
m
(t ) um (t )e
jpkt 2
Cross ambiguity function
LFM
mm'
(t ,n ) mm' (t ,n kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
50
51. Linear frequency modulation (LFM)
Linear frequency modulationu
LFM
m
(t ) um (t )e
jpkt 2
Cross ambiguity function
LFM
mm'
(t ,n ) mm' (t ,n kt )
MIMO ambiguity function
LFM
Shear off
(t ,n , f , f ' ) (t ,n kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
51
52. Linear frequency modulation (LFM)
(t ,n , f , f ' )n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
52
53. Linear frequency modulation (LFM)
(t ,n kt , f , f ' )(t ,n , f , f ' )
LFM
n
Shear off
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
53
54. Linear frequency modulation (LFM)
(t ,n kt , f , f ' )(t ,n , f , f ' )
LFM
Shear off
n
The range
resolution is
improved by
LFM.
t
n
n
t
n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
54
55. Conclusion
Properties of the MIMO ambiguity function– Signal component
2dT / M (0,0, f , f ) df 2dT / M
2d T /
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2d T /
55
56. Conclusion
Properties of the MIMO ambiguity function– Signal component
2dT / M (0,0, f , f ) df 2dT / M
2d T /
– Energy
2d T /
2
2dT / M 2
2dT / M 2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
(2dT / )2
(2dT / )2
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
56
57. Conclusion
Properties of the MIMO ambiguity function– Signal component
2dT / M (0,0, f , f ) df 2dT / M
2d T /
– Energy
– Symmetry
2d T /
2
2dT / M 2
2dT / M 2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
(2dT / )2
(2dT / )2
2
2
( t , n , f , f ' ) (t ,n , f ' , f )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
57
58. Conclusion
Properties of the MIMO ambiguity function– Signal component
2dT / M (0,0, f , f ) df 2dT / M
2d T /
– Energy
– Symmetry
– LFM
2d T /
2
2dT / M 2
2dT / M 2
(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'
(2dT / )2
(2dT / )2
2
2
( t , n , f , f ' ) (t ,n , f ' , f )
LFM (t ,n , f , f ' ) (t ,n kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
58
59. Thank You!
Q&AAny questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
59
60. Properties of the signal component
If the waveforms areorthogonal, the
signal component
will be a constant
for all angle.
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
um (t )um* ' (t )dt mm'
(0,0, f , f ) M , f
60
61. Properties of the signal component
For general waveforms,If
dT
um (t ) dt 1
2
is integer,
(0,0, f , f ) df M , f
The integration of
the signal
component is a
constant if dT is a
multiple of the
wavelength.
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
61
62. Properties of the signal component
In general, theintegration of the signal
component is confined
in a certain range.
For the general case,
2dT / M (0,0, f , f ) df 2dT / M
2d T /
2d T /
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
62
63. MIMO Radar
TXRX
…
SIMO
Radar
…
MF
MF
MF
u (t)
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
63
64. MIMO Radar
Advantages– Better spatial resolution [Bliss & Forsythe 03]
– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
– Improved parameter identifiability [Li et al. 07]
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
64