properties of the mimo radar ambiguity function
DESCRIPTION
Properties of the MIMO Radar Ambiguity Function. Chun-Yang Chen and P. P. Vaidyanathan. California Institute of Technology Electrical Engineering/DSP Lab. ICASSP 2008. Outline. Review of the background Radar ambiguity function and its properties MIMO radar MIMO radar ambiguity function - PowerPoint PPT PresentationTRANSCRIPT
Properties of the MIMO Radar Ambiguity Function
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
ICASSP 2008
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
2Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
1Review: Ambiguity function and MIMO radar
3
Radar Ambiguity Function
4Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
u(t) u(t-)ej2t
: delay: Doppler
Radar Ambiguity Function
5Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
u(t) u(t-)ej2t
Matched filter output dtetuetu tjtj *'22 ))'()()((
: delay: Doppler
Radar Ambiguity Function
6Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
u(t) u(t-)ej2t
Matched filter output dtetuetu tjtj *'22 ))'()()((
dtetutu tj )'(2* ))'(()(
: delay: Doppler
Radar Ambiguity Function
7Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
u(t) u(t-)ej2t
Matched filter output dtetuetu tjtj *'22 ))'()()((
dtetutu tj )'(2* ))'(()( Radar ambiguityfunction dtetutu tj 2* )()(),(
: delay: Doppler
Radar Ambiguity Function
8Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
u(t) u(t-)ej2t
Matched filter output dtetuetu tjtj *'22 ))'()()((
dtetutu tj )'(2* ))'(()( Radar ambiguityfunction dtetutu tj 2* )()(),(
Ambiguity function characterizes the Doppler and range resolution.
: delay: Doppler
Radar Ambiguity Function
9Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets (k,k)
)(tu
Radar Ambiguity Function
10Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets (k,k)
K
k
tjk
ketu1
2)( )(tu
Radar Ambiguity Function
11Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets (k,k)
Matched filter output
K
kkkk
1
),(
K
k
tjk
ketu1
2)( )(tu
Radar Ambiguity Function
12Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Matched filter output
K
kkkk
1
),(
target 2 (,)target 1 (,)
Multiple targets (k,k)
K
k
tjk
ketu1
2)( )(tu
Radar Ambiguity Function
13Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Matched filter output
K
kkkk
1
),(
target 2 (,)target 1 (,)
),( 11
Multiple targets (k,k)
K
k
tjk
ketu1
2)( )(tu
Ambiguity function characterizes the Doppler and range resolution.
Radar Ambiguity Function
14Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
target 2 (,)target 1 (,)
),( 11
dtetutu tj 2)()(),(
Ambiguity function
Ambiguity function characterizes the Doppler and range resolution.
Radar Ambiguity Function
15Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
target 2 (,)target 1 (,)
),( 11
dtetutu tj 2)()(),(
Ambiguity function
Properties of Radar Ambiguity Function
Signal component
16Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
),(1)0,0(
Properties of Radar Ambiguity Function
Signal component
Energy
17Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
1),(2
dd
),(1)0,0(
Properties of Radar Ambiguity Function
Signal component
Energy
Symmetry
18Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
1),(2
dd
),(),(
),(1)0,0(
Properties of Radar Ambiguity Function
Signal component
Energy
Symmetry
Linear frequency modulation (LFM)
19Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
1),(2
dd
),(),(
),(),(LFM k
2
)()(LFM ktjetutu
),(1)0,0(
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar
MIMO radar
SIMO radar (Traditional)
The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.
w2w1
w0
Advantages– Better spatial resolution [Bliss & Forsythe 03]– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]– Improved parameter identifiability [Li et al. 07]
Ambiguity Function in MIMO Radar
21Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
u0(t)u1(t) uM-1(t)
…
(,f)
TX
delayDopplerfSpatial freq.
dT
Ambiguity Function in MIMO Radar
22Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
… …
MF…
MF…
MF…
(,f) (,f)
TX RX
delayDopplerfSpatial freq.
u0(t)u1(t) uM-1(t)
dT dR
)(),,( tfy
Ambiguity Function in MIMO Radar
23Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
… …
MF…
MF…
MF…
(,f) (,f)
TX RX
delayDopplerfSpatial freq.
u0(t)u1(t) uM-1(t)
dT dR
)(),,( tfy
Ambiguity Function in MIMO Radar
24Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
… …
MF…
MF…
MF…
(,f) (,f)
dttt fHf )()( ),,()',','( yy
Matched filter output
TX RX
delayDopplerfSpatial freq.
u0(t)u1(t) uM-1(t)
dT dR
1
0
1
0'
)''(2)'(2*1
0
)'(2 )'()(M
m
M
m
mffmjtvjmm
N
n
nffj edtetutue
Ambiguity Function in MIMO Radar
25Chun-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
Ambiguity Function in MIMO Radar
26Chun-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
1
0
1
0'
)''(2)'(2*1
0
)'(2 )'()(M
m
M
m
mffmjtvjmm
N
n
nffj edtetutue
1
0
1
0'
)''(2', ),()',,,(
M
m
M
m
mffmjmm eff
Ambiguity Function in MIMO Radar
27Chun-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
1
0
1
0'
)''(2)'(2*1
0
)'(2 )'()(M
m
M
m
mffmjtvjmm
N
n
nffj edtetutue
2Properties of the MIMO ambiguity function
28
Properties of the signal component
29Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,0,0( ff
),,0,0( ff
Ambiguity function:
Signal component:
)',,,( ff),,0,0( ff
'f
f
ff '
Properties of the signal component
30Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,0,0( ff
),,0,0( ff
Ambiguity function:
Signal component:
)',,,( ff),,0,0( ff
'*
' )()( mmmm dttutu For orthogonal waveforms,
'f
f
ff '
Properties of the signal component
31Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,0,0( ff
),,0,0( ff
Ambiguity function:
Signal component:
)',,,( ff),,0,0( ff
'*
' )()( mmmm dttutu For orthogonal waveforms,
If the waveforms are orthogonal, the signal component will be a
constant for all angle.
If the waveforms are orthogonal, the signal component will be a
constant for all angle.
fMff ,),,0,0('f
f
ff '
Properties of the signal component
Ambiguity function:
Signal component:
32Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff),,0,0( ff
'*
' )()( mmmm dttutu fMff ,),,0,0(
For orthogonal waveforms,
1)(2 dttum
For general waveforms,
Properties of the signal component
Ambiguity function:
Signal component:
33Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff),,0,0( ff
'*
' )()( mmmm dttutu fMff ,),,0,0(
For orthogonal waveforms,
If is integer,
1)(2 dttum
Td
For general waveforms,
dT is the spacing between the transmitting antennas
The integration of the signal component is a constant if dT is
a multiple of the wavelength.
The integration of the signal component is a constant if dT is
a multiple of the wavelength.
fMdfff ,),,0,0(
Properties of the signal component
Ambiguity function:
Signal component:
34Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff),,0,0( ff
'*
' )()( mmmm dttutu fMff ,),,0,0(
For orthogonal waveforms,
If is integer,
For the general case,
1)(2 dttum
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
fMdfff ,),,0,0(
For general waveforms,
Td
dT is the spacing between the transmitting antennas
In general, the integration of the
signal component is confined.
In general, the integration of the
signal component is confined.
Energy of the cross ambiguity function
Cross ambiguity function:
Energy of the cross ambiguity function:
35Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dtetutu tjmmmm 2*
'' )()(),(
ddmm2
' ),(
Energy of the cross ambiguity function
Cross ambiguity function:
Energy of the cross ambiguity function:
36Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dtetutu tjmmmm 2*
'' )()(),(
ddmm2
' ),(
dddtetutu tjmm
22*
' )()(
Energy of the cross ambiguity function
Cross ambiguity function:
Energy of the cross ambiguity function:
37Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dtetutu tjmmmm 2*
'' )()(),(
ddmm2
' ),(
dddtetutu tjmm
22*
' )()(
1)(
)()(
22
2*'
dttu
dtdtutu
m
mm Parserval relation
Energy of the cross ambiguity function
Cross ambiguity function:
Energy of the cross ambiguity function:
38Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dtetutu tjmmmm 2*
'' )()(),(
1),(2
' ddmm
The energy of the cross ambiguity function is a
constant.
The energy of the cross ambiguity function is a
constant.
Energy of the MIMO ambiguity function
39Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MIMO ambiguity function:
Energy of the ambiguity function
1
0
1
0'
)''(/4' ),()',,,(
M
m
M
m
mffmdjmm
Teff
')',,,(2
dfdfddff
Energy of the MIMO ambiguity function
40Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MIMO ambiguity function:
Energy of the ambiguity function
1
0
1
0'
)''(/4' ),()',,,(
M
m
M
m
mffmdjmm
Teff
')',,,(2
dfdfddff dddfdfe
M
m
M
m
mffmdjmm
T '),(21
0
1
0'
)''(/4'
dT is the spacing between the transmitting antennas
ddM
m
M
mmm
1
0
1
0'
2
' ),(
Energy of the MIMO ambiguity function
41Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MIMO ambiguity function:
Energy of the ambiguity function
1
0
1
0'
)''(/4' ),()',,,(
M
m
M
m
mffmdjmm
Teff
')',,,(2
dfdfddff dddfdfe
M
m
M
m
mffmdjmm
T '),(21
0
1
0'
)''(/4'
If dT is a multiple of the wavelength, we can apply Parserval relation for 2D DFT.
If dT is a multiple of the wavelength, we can apply Parserval relation for 2D DFT.
dT is the spacing between the transmitting antennas
Energy of the MIMO ambiguity function
42Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MIMO ambiguity function:
Energy of the ambiguity function
1
0
1
0'
)''(/4' ),()',,,(
M
m
M
m
mffmdjmm
Teff
')',,,(2
dfdfddff dddfdfe
M
m
M
m
mffmdjmm
T '),(21
0
1
0'
)''(/4'
ddM
m
M
mmm
1
0
1
0'
2
' ),(
21
0
1
0'
1 MM
m
M
m
Cross ambiguity function has
constant energy
Cross ambiguity function has
constant energy
dT is the spacing between the transmitting antennas
Energy of the MIMO ambiguity function
43Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
If dT is a multiple of the wavelength,
If dT is a multiple of the wavelength, the energy of
the MIMO ambiguity function is a constant.
If dT is a multiple of the wavelength, the energy of
the MIMO ambiguity function is a constant.
22')',,,( Mdfdfddff
dT is the spacing between the transmitting antennas
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.
Energy of the MIMO ambiguity function
44Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing between the transmitting antennas22
')',,,( Mdfdfddff
fMdfff ,),,0,0(
If dT is a multiple of the wavelength,
In general, the energy satisfies,
Energy of the MIMO ambiguity function
45Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
22')',,,( Mdfdfddff
22
222
2
2
/2
/2')',,,(
/2
/2M
d
ddfdfddffM
d
d
T
T
T
T
In general, the energy of the MIMO ambiguity function is confined in a certain range.
In general, the energy of the MIMO ambiguity function is confined in a certain range.
dT is the spacing between the transmitting antennas
If dT is a multiple of the wavelength,
In general, the energy satisfies,
In general, the signal component satisfies,
Energy of the MIMO ambiguity function
46Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
22')',,,( Mdfdfddff
22
222
2
2
/2
/2')',,,(
/2
/2M
d
ddfdfddffM
d
d
T
T
T
T
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
dT is the spacing between the transmitting antennas
Symmetry properties
Symmetry of the cross ambiguity function
47Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
),(),( '' mmmm
Symmetry of the cross ambiguity function
Symmetry of the MIMO ambiguity function
Symmetry properties
48Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
),(),( '' mmmm
),',,()',,,( ffff
It suffices to show only half of the ambiguity function (>0).It suffices to show only half of the ambiguity function (>0).
Linear frequency modulation
2
)()(LFM ktjmm etutu
Linear frequency modulation (LFM)
49Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Linear frequency modulation (LFM)
Linear frequency modulation
Cross ambiguity function
50Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
)()(LFM ktjmm etutu
),(),( 'LFM
' kmmmm
Linear frequency modulation
Cross ambiguity function
MIMO ambiguity function
),(),( 'LFM
' kmmmm
Linear frequency modulation (LFM)
51Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
)()(LFM ktjmm etutu
)',,,()',,,(LFM ffkff Shear offShear off
Linear frequency modulation (LFM)
52Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff
Linear frequency modulation (LFM)
53Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff
)',,,( ffk
LFM
Shear off
Linear frequency modulation (LFM)
54Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
)',,,( ff
)',,,( ffk
LFM
The range resolution is improved by
LFM.
The range resolution is improved by
LFM.
Shear off
Conclusion
Properties of the MIMO ambiguity function– Signal component
55Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
Conclusion
Properties of the MIMO ambiguity function– Signal component
– Energy
56Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
22
222
2
2
/2
/2')',,,(
/2
/2M
d
ddfdfddffM
d
d
T
T
T
T
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
Conclusion
Properties of the MIMO ambiguity function– Signal component
– Energy
– Symmetry
57Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
22
222
2
2
/2
/2')',,,(
/2
/2M
d
ddfdfddffM
d
d
T
T
T
T
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
),',,()',,,( ffff
Conclusion
Properties of the MIMO ambiguity function– Signal component
– Energy
– Symmetry
– LFM
58Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
22
222
2
2
/2
/2')',,,(
/2
/2M
d
ddfdfddffM
d
d
T
T
T
T
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
),',,()',,,( ffff
)',,,()',,,(LFM ffkff
Q&AThank You!
Any questions?
59Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Properties of the signal component
60Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
'*
' )()( mmmm dttutu fMff ,),,0,0(
For orthogonal waveforms,If the waveforms are orthogonal, the
signal component will be a constant
for all angle.
If the waveforms are orthogonal, the
signal component will be a constant
for all angle.
Properties of the signal component
61Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
If is integer,
1)(2 dttum
Td
fMdfff ,),,0,0(
For general waveforms,
The integration of the signal
component is a constant if dT is a
multiple of the wavelength.
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
Properties of the signal component
62Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
For the general case,
Md
ddfffM
d
d
T
T
T
T
/2
/2),,0,0(
/2
/2
In general, the integration of the signal component is confined
in a certain range.
In general, the integration of the signal component is confined
in a certain range.
dT is the spacing between the transmitting antennas
MIMO Radar
63Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
… …
MF…
MF…
MF…
TX RX
u0(t)u1(t) uM-1(t)
…
u (t)
MIMORadar
SIMORadar
TX RX
…
MFMFMF
… … …
MIMO Radar
64Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
… …
MF…
MF…
MF…
TX RX
u0(t)u1(t) uM-1(t)
MIMORadar
… … …
Advantages– Better spatial resolution [Bliss & Forsythe 03]– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]– Improved parameter identifiability [Li et al. 07]