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DSP Group, EE, Caltech, Pasadena CA1 Beamforming Issues in Modern MIMO Radars with Doppler Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology

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DSP Group, EE, Caltech, Pasadena CA2 Outline Review of the MIMO radar Spatial resolution. [D. W. Bliss and K. W. Forsythe, 03] MIMO space-time adaptive processing (STAP) Problem formulation Clutter rank in MIMO STAP Clutter subspace in MIMO STAP Numerical example

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DSP Group, EE, Caltech, Pasadena CA3 SIMO Radar Transmitter: M elementsReceiver: N elements dTdT e j2 (ft-x/ ) w 2 w 1 w 0 dRdR e j2 (ft-x/ ) Transmitter emits coherent waveforms. Number of received signals: N

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DSP Group, EE, Caltech, Pasadena CA4 Transmitter: M elementsReceiver: N elements dTdT e j2 (ft-x/ ) dRdR e j2 (ft-x/ ) MF … … Transmitter emits orthogonal waveforms. Matched filters extract the M orthogonal waveforms. Overall number of signals: NM MIMO Radar

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DSP Group, EE, Caltech, Pasadena CA5 Transmitter: M elementsReceiver: N elements Virtual array: NM elements d T =Nd R dRdR e j2 (ft-x/ ) MF … … MIMO Radar (2) The spacing d T is chosen as Nd R, such that the virtual array is uniformly spaced.

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DSP Group, EE, Caltech, Pasadena CA6 The clutter resolution is the same as a receiving array with NM physical array elements. A degree-of-freedom NM can be created using only N+M physical array elements. Receiver: N elements Virtual array: NM elements Transmitter : M elements += [D. W. Bliss and K. W. Forsythe, 03] MIMO Radar (3)

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DSP Group, EE, Caltech, Pasadena CA7 Space-Time Adaptive Processing (STAP) v v sin i airborne radar jammer target i-th clutter vtvt The clutter Doppler frequencies depend on looking directions. The problem is non-separable in space-time. ii The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).

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DSP Group, EE, Caltech, Pasadena CA8 Formulation of MIMO STAP d T =Nd R e j2 (ft-x/ ) dRdR e j2 (ft-x/ ) MF … … Transmitter : M elementsReceiver: N elements vsin target vtvt vtvt clutter jammer noise NML NML x NML

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DSP Group, EE, Caltech, Pasadena CA9 Clutter in MIMO Radar size: NML size: NMLxNML

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DSP Group, EE, Caltech, Pasadena CA10 Clutter Rank in MIMO STAP: Integer Case Integer case: and are both integers. This result can be viewed as the MIMO extension of Brennan’s rule. Theorem: If and are integers, The set {n+ m+ l} has at most N+ (M-1)+ (L-1) distinct elements.

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DSP Group, EE, Caltech, Pasadena CA11 Clutter Signals and Truncated Sinusoidal Functions c i is NML vector which consists of It can be viewed as a non-uniformly sampled version of truncated sinusoidal signals. The “time-and-band limited” signals can be approximated by linear combination of prolate spheroidal wave functions. X 2W

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DSP Group, EE, Caltech, Pasadena CA12 Prolate Spheroidal Wave Function (PSWF) Time window Frequency window X-WW0 Prolate spheroidal wave functions (PSWF) are the solutions to the integral equation [van tree, 2001]. in [0,X] Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962]. The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.

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DSP Group, EE, Caltech, Pasadena CA13 PSWF Representation for Clutter Signals The “time-and-band limited” signals can be approximated by 2WX+1 PSWF basis elements. clutter rank in integer case

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DSP Group, EE, Caltech, Pasadena CA14 PSWF Representation for Clutter Signals (2) The PSWF (x) can be computed off-line The vector u k can be obtained by sampling the PSWF. non-uniformly sample U: NML x r c A: r c x r c

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DSP Group, EE, Caltech, Pasadena CA15 i-th clutter signal truncated sinusoidal PSWF Non-uniformly sample Linear combination Non-uniformly sample Sampled PSWF Linear combination Stack i-th clutter signal Stack Sampled PSWF Linear combination Clutter covariance matrix U: NML x r c A: r c x r c

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DSP Group, EE, Caltech, Pasadena CA16 Numerical Example N=10 M=5 L=16 =N=10 NML=800 N+ (M-1)+ (L-1)=72.5 Proposed method Eigenvalues The figure shows the clutter power in the orthonormalized basis elements. The proposed method captures almost all the clutter power. Parameters: k q k H R c q k

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DSP Group, EE, Caltech, Pasadena CA17 Conclusion The clutter subspace in MIMO radar is explored. Clutter rank for integer/non-integer and Data-independent estimation of the clutter subspace. Advantages of the proposed subspace estimation method. It is data-independent. It is accurate. It can be computed off-line.

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DSP Group, EE, Caltech, Pasadena CA18 Further and Future Work Further work The STAP method applying the subspace estimation is submitted to ICASSP 07. Future work In practice, some effects such as internal clutter motion (ICM) will change the clutter space. Estimating the clutter subspace by using a combination of both the geometry and the data will be explored in the future. New method

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DSP Group, EE, Caltech, Pasadena CA19 References [1] D. W. Bliss and K. W. Forsythe, “ Multiple-input multiple-output (MIMO) radar and imaging: degrees of freedom and resolution, ” Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp , Nov [2] D. Slepian, and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-III: the dimension of the space of essentially time-and-band-limited signals," Bell Syst. Tech. J., pp , July [3] D. J. Rabideau and P. Parker, "Ubiquitous MIMO Multifunction Digital Array Radar," Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp , Nov [4] N. A. Goodman and J.M. Stiles, "On Clutter Rank Observed by Arbitrary Arrays," accepted to IEEE Trans. on Signal Processing.

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DSP Group, EE, Caltech, Pasadena CA20 Thank you

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DSP Group, EE, Caltech, Pasadena CA21 Comparison of the Clutter Rank in MIMO and SIMO Radar MIMOSIMO Clutter rank N+ (M-1)+ (L-1)N+ (L-1) Total dimension NMLNL Ratio ( =N) < The clutter rank is a smaller portion of the total dimension. The MIMO radar receiver can null out the clutter subspace without affecting the SINR too much. > >

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DSP Group, EE, Caltech, Pasadena CA22 Formulation of MIMO STAP (2) dTdT e j2 (ft-x/ ) dRdR e j2 (ft-x/ ) MF … … Transmitter : M elementsReceiver: N elements vsin target vtvt vtvt T: Radar pulse period

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DSP Group, EE, Caltech, Pasadena CA23 Fully Adaptive STAP for MIMO Radar ^ Difficulty: The size of R y is NML which is often large. The convergence of the fully adaptive STAP is slow. The complexity is high. Solution:

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DSP Group, EE, Caltech, Pasadena CA24 Clutter Subspace in MIMO STAP: Non-integer Case Non-integer case: and not integers. Basis need for representation of clutter steering vector c i. Data independent basis is preferred. Less computation Faster convergence of STAP We study the use of prolate spheroidal wave function (PSWF) for this.

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DSP Group, EE, Caltech, Pasadena CA25 Extension to Arbitrary Array This result can be extended to arbitrary array. X R,n is the location of the n-th receiving antenna. X T,m is the location of the m-th transmitting antenna. u i is the location of the i-th clutter. v is the speed of the radar station.

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DSP Group, EE, Caltech, Pasadena CA26 Review of MIMO radar: Diversity approach dRdR e j2 (ft-x/ ) MF … … Receiver: If the transmitting antennas are far enough, the received signals of each orthogonal waveforms becomes independent. [E. Fishler et al. 04] This diversity can be used to improve target detection.

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DSP Group, EE, Caltech, Pasadena CA27 Prolate Spheroidal Wave Function (PSWF) (2) By the maximum principle, this basis concentrates most of its energy on the band [-W, W] while maintaining the orthogonality. Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962]. The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.

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DSP Group, EE, Caltech, Pasadena CA28 Review of MIMO Radar: Degree-of-Freedom Approach The clutter resolution is the same as a receiving array with NM physical array elements. A degree-of-freedom NM can be created using only N+M physical array elements. Receiver: N elements Virtual array: NM elements Transmitter : M elements d T =Nd R e j2 (ft-x/ ) dRdR e j2 (ft-x/ ) MF … … += [D. W. Bliss and K. W. Forsythe, 03]

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