Properties of the MIMO Radar Ambiguity Function

1
Properties of the MIMO Radar Ambiguity Function

• Chun-Yang Chen and P. P. Vaidyanathan

California 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

3
Review Ambiguity function and MIMO radar
1
4
Radar Ambiguity Function
t delay n Doppler
u(t)
5
Radar Ambiguity Function
t delay n Doppler
u(t)
Matched filter output
6
Radar Ambiguity Function
t delay n Doppler
u(t)
Matched filter output
7
Radar Ambiguity Function
t delay n Doppler
u(t)
Matched filter output
Radar ambiguity function
8
Radar Ambiguity Function
t delay n Doppler
u(t)
Matched filter output
Radar ambiguity function

• Ambiguity function characterizes the Doppler and
  range resolution.

9
Radar Ambiguity Function
Multiple targets (tk,nk)
10
Radar Ambiguity Function
Multiple targets (tk,nk)
11
Radar Ambiguity Function
Multiple targets (tk,nk)
Matched filter output
12
Radar Ambiguity Function
Multiple targets (tk,nk)
Matched filter output
n
target 1 (t1,n1)
target 2 (t2,n2)
t
13
Radar Ambiguity Function
Multiple targets (tk,nk)
Matched filter output
n
target 1 (t1,n1)
target 2 (t2,n2)
t
14
Radar Ambiguity Function

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
Ambiguity function
15
Radar Ambiguity Function

• Ambiguity function characterizes the Doppler and
  range resolution.

n
target 1 (t1,n1)
target 2 (t2,n2)
t
Ambiguity function
16
Properties of Radar Ambiguity Function

• Signal component

n
t
17
Properties of Radar Ambiguity Function

• Signal component
• Energy

n
t
18
Properties of Radar Ambiguity Function

• Signal component
• Energy
• Symmetry

n
t
19
Properties of Radar Ambiguity Function

• Signal component
• Energy
• Symmetry
• Linear frequency modulation (LFM)

n
t
20
MIMO Radar
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

21
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
TX

dT
u0(t)
u1(t)
uM-1(t)
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Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
TX
RX


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



23
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
TX
RX


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



24
Ambiguity Function in MIMO Radar
tdelay nDoppler f Spatial freq.
(t,n,f)
(t,n,f)
TX
RX


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



Matched filter output
25
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
26
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
27
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
28
Properties of the MIMO ambiguity function
2
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Properties of the signal component

• Ambiguity function
• Signal component

30
Properties of the signal component

• Ambiguity function
• Signal component

For orthogonal waveforms,
31
Properties of the signal component

• Ambiguity function
• Signal component

For orthogonal waveforms,
If the waveforms are orthogonal, the signal
component will be a constant for all angle.
32
Properties of the signal component

• Ambiguity function
• Signal component

For orthogonal waveforms,
For general waveforms,
33
Properties of the signal component

• Ambiguity function
• Signal component

For orthogonal waveforms,
For general waveforms,

• If is integer,

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
34
Properties of the signal component
dT is the spacing between the transmitting
antennas

• Ambiguity function
• Signal component

For orthogonal waveforms,
For general waveforms,

• If is integer,
• For the general case,

In general, the integration of the signal
component is confined.
35
Energy of the cross ambiguity function

• Cross ambiguity function
• Energy of the cross ambiguity function

36
Energy of the cross ambiguity function

• Cross ambiguity function
• Energy of the cross ambiguity function

37
Energy of the cross ambiguity function

• Cross ambiguity function
• Energy of the cross ambiguity function

Parserval relation
38
Energy of the cross ambiguity function

• Cross ambiguity function
• Energy of the cross ambiguity function

The energy of the cross ambiguity function is a
constant.
39
Energy of the MIMO ambiguity function

• MIMO ambiguity function
• Energy of the ambiguity function

40
Energy of the MIMO ambiguity function

• MIMO ambiguity function
• Energy of the ambiguity function

dT is the spacing between the transmitting
antennas
41
Energy of the MIMO ambiguity function

• MIMO ambiguity function
• Energy of the ambiguity function

dT is the spacing between the transmitting
antennas
If dT is a multiple of the wavelength, we can
apply Parserval relation for 2D DFT.
42
Energy of the MIMO ambiguity function

• MIMO ambiguity function
• Energy of the ambiguity function

Cross ambiguity function has constant energy
dT is the spacing between the transmitting
antennas
43
Energy of the MIMO ambiguity function

• If dT is a multiple of the wavelength,

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.
44
Energy of the MIMO ambiguity function

• If dT is a multiple of the wavelength,
• Recall that the signal component satisfies,
• Because energy and the signal component are both
  constants, we can only spread the energy to
  minimize the peak.

dT is the spacing between the transmitting
antennas
45
Energy of the MIMO ambiguity function

• If dT is a multiple of the wavelength,
• In general, the energy satisfies,

dT is the spacing between the transmitting
antennas
In general, the energy of the MIMO ambiguity
function is confined in a certain range.
46
Energy of the MIMO ambiguity function

• If dT is a multiple of the wavelength,
• In general, the energy satisfies,
• In general, the signal component satisfies,

dT is the spacing between the transmitting
antennas
47
Symmetry properties

• Symmetry of the cross ambiguity function

48
Symmetry properties

• Symmetry of the cross ambiguity function
• Symmetry of the MIMO ambiguity function

It suffices to show only half of the ambiguity
function (tgt0).
49
Linear frequency modulation (LFM)

• Linear frequency modulation

50
Linear frequency modulation (LFM)

• Linear frequency modulation
• Cross ambiguity function

51
Linear frequency modulation (LFM)

• Linear frequency modulation
• Cross ambiguity function
• MIMO ambiguity function

Shear off
52
Linear frequency modulation (LFM)
n
t
53
Linear frequency modulation (LFM)
LFM
Shear off
n
t
n
t
54
Linear frequency modulation (LFM)
LFM
Shear off
n
t
n
t
The range resolution is improved by LFM.
n
n
t
t
55
Conclusion

• Properties of the MIMO ambiguity function
• Signal component

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Conclusion

• Properties of the MIMO ambiguity function
• Signal component
• Energy

57
Conclusion

• Properties of the MIMO ambiguity function
• Signal component
• Energy
• Symmetry

58
Conclusion

• Properties of the MIMO ambiguity function
• Signal component
• Energy
• Symmetry
• LFM

59
Thank You!
QA
Any questions?
59
Chun-Yang Chen, Caltech DSP Lab ICASSP 2008
60
Properties of the signal component
If the waveforms are orthogonal, the signal
component will be a constant for all angle.
For orthogonal waveforms,
61
Properties of the signal component
The integration of the signal component is a
constant if dT is a multiple of the wavelength.
For general waveforms,

• If is integer,

dT is the spacing between the transmitting
antennas
62
Properties of the signal component
In general, the integration of the signal
component is confined in a certain range.

• For the general case,

dT is the spacing between the transmitting
antennas
63
MIMO Radar
TX
RX
SIMO Radar


MF
MF
MF
u (t)
RX
TX
MIMO Radar


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






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)
MF
MF
MF