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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

3. Review: Ambiguity function and MIMO radar1
Review: Ambiguity function and MIMO radar

4. Radar Ambiguity Function
t: delay
n: Doppler
u(t-t)ej2pnt
u(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

5. Radar Ambiguity Function
t: delay
n: Doppler
u(t-t)ej2pnt
u(t)
Matched filter
output
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

6. Radar Ambiguity Function
t: delay
n: Doppler
u(t-t)ej2pnt
u(t)
Matched filter
output
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

7. Radar Ambiguity Function
t: delay
n: Doppler
u(t-t)ej2pnt
u(t)
Matched filter
output
Radar ambiguity
function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

8. Radar Ambiguity Function
t: delay
n: Doppler
u(t-t)ej2pnt
u(t)
Matched filter
output
Radar ambiguity
function
Ambiguity function characterizes the Doppler and range resolution.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

9. Radar Ambiguity Function
Multiple targets
(tk,nk)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

10. Radar Ambiguity Function
Multiple targets
(tk,nk)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

11. Radar Ambiguity Function
Multiple targets
(tk,nk)
Matched filter
output
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

12. Radar Ambiguity Function
Multiple targets
(tk,nk)
Matched filter
output n
target 1 (t1,n1)
target 2 (t2,n2) t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

13. Radar Ambiguity Function
Multiple targets
(tk,nk)
Matched filter
output n
target 1 (t1,n1)
target 2 (t2,n2) t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

14. Radar Ambiguity Function
Ambiguity function characterizes the Doppler and range resolution. n
target 1 (t1,n1)
target 2 (t2,n2) t
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

15. Radar Ambiguity Function
Ambiguity function characterizes the Doppler and range resolution. n
target 1 (t1,n1)
target 2 (t2,n2) t
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

16. Properties of Radar Ambiguity Function
Signal component n t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

17. Properties of Radar Ambiguity Function
Signal component
Energy n t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

18. Properties of Radar Ambiguity Function
Signal component
Energy
Symmetry n t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

19. Properties of Radar Ambiguity Function
Signal component
Energy
Symmetry
Linear frequency modulation (LFM) n t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

20. MIMO Radar f2(t) w2f(t) f1(t) w1f(t) f0(t) w0f(t) Advantages
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:delay
n:Doppler
f: Spatial freq.
(t,n,f) TX … dT
u0(t)
u1(t)
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

22. Ambiguity Function in MIMO Radar
(t,n,f)
t:delay
n:Doppler
f: Spatial freq.
(t,n,f) TX RX … … dT dR
u0(t)
u1(t)
uM-1(t) MF MF MF … … …
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

23. Ambiguity Function in MIMO Radar
(t,n,f)
t:delay
n:Doppler
f: Spatial freq.
(t,n,f) TX RX … … dT dR
u0(t)
u1(t)
uM-1(t) MF MF MF … … …
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

24. Ambiguity Function in MIMO Radar
(t,n,f)
t:delay
n:Doppler
f: Spatial freq.
(t,n,f) TX RX … … dT dR
u0(t)
u1(t)
uM-1(t) MF MF MF … … …
Matched filter output
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

25. Ambiguity Function in MIMO Radar
t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
Matched filter output
Receiver beamforming
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

26. Ambiguity Function in MIMO Radar
t:delay
n:Doppler
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

27. Ambiguity Function in MIMO Radar
t:delay
n:Doppler
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

28. Properties of the MIMO ambiguity function2
Properties of the MIMO ambiguity function

29. Properties of the signal component
Ambiguity function:
Signal component:
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

30. Properties of the signal component
Ambiguity function:
Signal component:
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

32. Properties of the signal component
Ambiguity function:
Signal component:
For general waveforms,
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

33. Properties of the signal component
Ambiguity function:
Signal component:
For general waveforms,
For orthogonal 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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

34. Properties of the signal component
dT is the spacing between the transmitting antennas
Ambiguity function:
Signal component:
For general waveforms,
For orthogonal waveforms,
If is integer,
For the general case,
In general, the integration of the signal component is confined.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

35. Energy of the cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

36. Energy of the cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

37. Energy of the cross ambiguity function
Parserval relation
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

38. Energy of the cross ambiguity function
The energy of the cross ambiguity function is a constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

39. Energy of the MIMO ambiguity function
Energy of the ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

40. Energy of the MIMO ambiguity function
Energy of the ambiguity function
dT is the spacing between the transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

41. Energy of the 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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

42. Energy of the MIMO ambiguity function
Energy of the ambiguity function
Cross ambiguity function has constant energy
dT is the spacing between the transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

47. Symmetry properties Symmetry of the cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

48. It suffices to show only half of the ambiguity function (t>0).
Symmetry properties
Symmetry of the cross ambiguity function
Symmetry of the MIMO ambiguity function
It suffices to show only half of the ambiguity function (t>0).
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

49. Linear frequency modulation (LFM)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

50. Linear frequency modulation (LFM)
Cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

51. Linear frequency modulation (LFM)
Cross ambiguity function
MIMO ambiguity function
Shear off
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

52. Linear frequency modulation (LFM)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

53. Linear frequency modulation (LFM)
Shear off n t n t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

54. Linear frequency modulation (LFM)
Shear off n t n t
The range resolution is improved by LFM. n n t t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

55. Conclusion Properties of the MIMO ambiguity function Signal component
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

56. Conclusion Properties of the MIMO ambiguity function Signal component
Energy
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

57. Conclusion Properties of the MIMO ambiguity function Signal component
Energy
Symmetry
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

58. Conclusion Properties of the MIMO ambiguity function Signal component
Energy
Symmetry
LFM
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

59. Q&A Thank You! Any questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008 59

60. Properties of the signal component
If the waveforms are orthogonal, the signal component will be a constant for all angle.
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

63. MIMO Radar SIMO Radar MIMO Radar TX RX TX RX u (t) u0(t) u1(t) uM-1(t)… … MF MF MF
u (t) TX RX
MIMO
Radar … …
u0(t)
u1(t)
uM-1(t) MF MF MF … … … … … …
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

64. MIMO Radar MIMO Radar Advantages RX TX
Better spatial resolution [Bliss & Forsythe 03]
Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
Improved parameter identifiability [Li et al. 07] TX RX
MIMO
Radar … …
u0(t)
u1(t)
uM-1(t) MF MF MF … … … … … …
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008