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Measurement Matrix Design for Compressive Sensing Based MIMO Radar

Y. Yu, A. P. Petropulu, H. V. Poor

arXiv:1102.5079v1cs.IT

TL;DR

CS-based colocated MIMO radar needs measurement matrices that preserve sparse-recovery performance while coping with interference and targets at different delays. The paper proposes two CSM/SIR-oriented designs and reports improved detection accuracy over a Gaussian random measurement matrix, with the structured design retaining comparable CSM under suitable waveforms.

  • Problem

    Measurement-matrix design must support CS recovery while balancing sensing-matrix coherence and signal-to-interference ratio in a general multi-range-bin radar setting.

  • Method

    The paper proposes one design minimizing a linear combination of CSM and inverse SIR, and another structured design whose parameters are selected for SIR enhancement across discretized target delays.

  • Results

    The structured measurement matrix can significantly improve SIR while maintaining CSM comparable to a Gaussian random measurement matrix, and the proposed matrices improve detection accuracy over that baseline.

  • Takeaways & Limitations

    Measurement-matrix design can improve CS-based MIMO-radar detection, with the effective structured design depending on the transmit waveforms.

Abstract

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In colocated multiple-input multiple-output (MIMO) radar using compressive sensing (CS), a receive node compresses its received signal via a linear transformation, referred to as measurement matrix. The samples are subsequently forwarded to a fusion center, where an L1-optimization problem is formulated and solved for target information. CS-based MIMO radar exploits the target sparsity in the angle-Doppler-range space and thus achieves the high localization performance of traditional MIMO radar but with many fewer measurements. The measurement matrix is vital for CS recovery performance. This paper considers the design of measurement matrices that achieve an optimality criterion that depends on the coherence of the sensing matrix (CSM) and/or signal-to-interference ratio (SIR). The first approach minimizes a performance penalty that is a linear combination of CSM and the inverse SIR. The second one imposes a structure on the measurement matrix and determines the parameters involved so that the SIR is enhanced. Depending on the transmit waveforms, the second approach can significantly improve SIR, while maintaining CSM comparable to that of the Gaussian random measurement matrix (GRMM). Simulations indicate that the proposed measurement matrices can improve detection accuracy as compared to a GRMM.

I. Introduction

CS-based MIMO radar exploits sparse target representations to recover angle-range-Doppler information from substantially fewer measurements. This paper designs measurement matrices to reduce sensing-matrix coherence and/or improve SIR, including a general multi-range-bin setting without sampling synchronization.

  • MIMO radar: Colocated MIMO antennas form a long virtual array, supporting high DOA resolution and parameter identification.The virtual array has as many elements as the product of transmit and receive antenna counts.
  • Compressive sensing: CS recovery uses a measurement matrix Φ incoherent with basis Ψ, forms sensing matrix ΦΨ, and estimates sparse coefficients through ℓ1 optimization.Random Gaussian matrices are a conventional choice because they satisfy the relevant incoherence condition with high probability for orthonormal bases.
  • Compressive sensing: Sparse target returns in angle-range-Doppler space enable detection and localization using few compressive samples and/or few receive nodes.
  • Problem setting: The paper addresses targets in different range bins without requiring sampling synchronization, where earlier formulations no longer apply because returns have different delays.
  • Contributions: The proposed designs optimize coherence and/or SIR; the structured design can significantly improve SIR while retaining CSM comparable to a Gaussian random measurement matrix.The first design minimizes a linear combination of CSM and inverse SIR, while the second accounts for discretized delays and transmit waveforms.
  • Contributions: Depending on transmit waveforms, the proposed measurement matrices improve detection accuracy over a Gaussian random measurement matrix.

II. Signal Model for CS-based MIMO Radar

The signal model represents compressively sampled returns from colocated MIMO radar targets on a discretized angle-speed-range grid. Fusion-center processing combines node and pulse measurements and estimates a sparse target vector.

  • System model: The system uses Mt transmit antennas and Nr colocated receive antennas transmitting periodic narrowband pulses.
  • Target model: Targets are modeled with azimuth angle θk, constant radial speed vk, and range dk(t), under far-field and slow-motion assumptions.The slow-target assumption makes Doppler shift negligible in the stated baseband model.
  • Compressive sampling: Each receive node compressively samples each received pulse by premultiplying its sampled waveform with an M × (L + ˜L) measurement matrix Φl, where M << L.The delay extension ˜L represents the maximum normalized return delay known in advance.
  • Signal model: The received data follow r_lm = Θ_lm s + Φ_l n_lm, where Θ_lm is the sensing matrix and n_lm contains jammer and thermal-noise interference.
  • Sparse representation: The angle-speed-range domain is discretized finely enough that each target lies on a grid point, producing a sparse coefficient vector s.The nonzero entries of s encode target angle, speed, and range.
  • Fusion and recovery: Receive nodes forward compressed measurements to a fusion center, which combines data across Np pulses and Nr nodes and estimates s with the Dantzig selector.

III. Measurement matrix design

The paper designs measurement matrices for CS-based MIMO radar by balancing sensing-matrix coherence and SIR. It proposes optimization-based and structured designs, with trade-offs among detection performance, interference suppression, coherence, and computational complexity.

  • The design goal is to reduce sensing-matrix coherence while increasing SIR for CS-based MIMO radar.
  • A. Measurement matrix design #1: The first approach optimizes a linear combination of CSM and reciprocal SIR, with a positive weight controlling their tradeoff.
  • A. Measurement matrix design #1: The alternative SCSM formulation increases the number of low-coherence column pairs compared with the maximum-coherence formulation.
  • The proposed optimization methods improve detection performance without amplifying interference, but require more computation than the conventional measurement matrix.
  • The optimization can become prohibitively expensive for large delay dimensions or many grid points, and updates may be needed when the search basis changes.
  • B. Measurement matrix design #2: The structured second design controls the number of optimization variables through a parameter and enables two-step processing that simplifies receive-node hardware.
  • B. Measurement matrix design #2: The second design targets SIR improvement only and can suppress interference uncorrelated with transmit waveforms while retaining coherence comparable to a Gaussian random matrix.

3) The SIR gain:

The proposed structured measurement matrix provides SIR gains whose magnitude depends on the transmit waveform and system dimensions. The gain is guaranteed above one for QPSK under stated dimensional conditions, while rectangular pulses can provide a larger gain than Hadamard codes.

  • For QPSK waveforms, the SIR gain is greater than 1 when ˜L < L and Mt < L.
  • For sufficiently large L and moderate Mt, the SIR gain for Hadamard codes exceeds that for QPSK waveforms.
  • The SIR gain from rectangular pulses is approximately Mt times greater than the gain from Hadamard codes.

4) The CSM

The structured measurement matrix can increase maximum CSM relative to a Gaussian random matrix, but this increase is negligible when its design matrix is well conditioned. Its principal trade-off is stronger SIR with potentially higher coherence.

  • The coherence of the sensing matrix is approximately determined by the coherence of WHWV under the structured design.
  • The structured matrix increases maximum CSM relative to a Gaussian random measurement matrix with high probability.
  • When W is well conditioned, the resulting increase in maximum CSM is negligible.
  • The first design jointly decreases coherence and enhances SIR, whereas the second design requires knowledge only of discretized time delays.
  • The first design is expected to perform better under low interference, while the second design is favored under strong interference because of its higher SIR.

IV. Simulation Results

The simulations evaluate proposed measurement matrices in a colocated MIMO radar under Gaussian noise, jammer interference, and sparse, grid-aligned targets. Each receive node forwards compressed measurements to the fusion center for CS-based processing.

  • The experiments demonstrate CS-based MIMO radar performance using the proposed measurement matrices Φ#1 and Φ#2.
  • The evaluation uses a MIMO radar with nodes uniformly distributed on a 10 m-radius disk and a 5 GHz carrier.
  • Received signals contain zero-mean Gaussian noise, while a jammer transmits an unknown Gaussian waveform from angle 7o.
  • Targets are assumed to lie on grid points, and each receive node forwards M = 30 compressed measurements to the fusion center.

1) SIR improvement:

The simulations compare the proposed measurement matrices with the Gaussian random measurement matrix and matched-filter radar across SIR, coherence, and detection performance. Results show waveform- and interference-dependent gains, with trade-offs between SIR and CSM.

  • SIR improvement: A decrease in maximum delay can significantly improve SIR for QPSK waveforms when ˜L is less than 10.
  • SIR improvement: Hadamard and rectangular-pulse waveforms produce nearly unchanged SIR across different maximum-delay values, unlike QPSK waveforms.
  • The CSM: Φ#2 increases maximum CSM relative to GRMM with high probability, although its adjacent-column correlation distribution changes little from the conventional matrix.
  • The CSM: Rectangular pulses yield the worst CSM distribution because their high autocorrelation produces high CSM independently of the measurement matrix.
  • SIR performance: Φ#2 outperforms GRMM and Φ#1 in SIR, while Φ#1 with ˜λ = 0.6 yields slightly better SIR than GRMM.
  • Detection performance: Under strong interference, Φ#1 with ˜λ = 1.5 performs better than with ˜λ = 0.6, whereas all matrices perform similarly when SNR = 10dB and β = 0.
  • Detection performance: The CS approach outperforms matched-filter radar despite using fewer measurements.

V. Conclusions

The paper proposes two measurement matrices for CS-based MIMO radar: one jointly targets SIR and CSM, while the other targets SIR through waveform- and delay-aware structure. Simulations show improved detection with suitable waveforms, but also identify computational and waveform-selection constraints.

  • The paper proposes Φ#1 and Φ#2 to improve target detection performance in CS-based MIMO radar.
  • Φ#1 jointly enhances SIR and reduces CSM through a convex optimization problem.
  • Φ#1 has heavier computational demands than the conventional measurement matrix and must adapt to a particular basis matrix.
  • Φ#2 targets SIR improvement, is constructed from transmit waveforms and possible discretized delays, and depends only on the range grid.
  • Φ#2 can improve SIR with reduced-bandwidth waveforms, but excessively narrowband waveforms increase CSM and can invalidate CS conditions.
  • With suitable waveforms such as Hadamard codes, Φ#1 and Φ#2 improve detection accuracy over GRMM under small and strong interference, respectively.
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