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MIMO Radar Using Compressive Sampling

Yao Yu, Athina P. Petropulu, H. Vincent Poor

arXiv:0911.4752v1cs.IT

TL;DR

Distributed MIMO radar needs a flexible way to estimate target angle and Doppler information with limited measurements from wireless-network nodes. The paper combines compressive sampling with l1 optimization and reports performance at least as good as several conventional techniques while using significantly fewer samples.

  • Problem

    The paper addresses angle-Doppler estimation in flexible distributed MIMO radar implemented by a small wireless network rather than a fixed antenna array.

  • Method

    Receive nodes in a distributed MIMO radar apply compressive sampling, then a fusion center uses l1 optimization to estimate sparse target locations in angle-Doppler space.

  • Results

    The proposed method is at least as good as Capon, APES, GLRT, and MUSIC while using a significantly smaller number of samples.

  • Takeaways & Limitations

    The approach provides distributed MIMO radar angle-Doppler estimation with substantially fewer forwarded samples than the compared techniques.

Abstract

from arXiv · show

A MIMO radar system is proposed for obtaining angle and Doppler information on potential targets. Transmitters and receivers are nodes of a small scale wireless network and are assumed to be randomly scattered on a disk. The transmit nodes transmit uncorrelated waveforms. Each receive node applies compressive sampling to the received signal to obtain a small number of samples, which the node subsequently forwards to a fusion center. Assuming that the targets are sparsely located in the angle- Doppler space, based on the samples forwarded by the receive nodes the fusion center formulates an l1-optimization problem, the solution of which yields target angle and Doppler information. The proposed approach achieves the superior resolution of MIMO radar with far fewer samples than required by other approaches. This implies power savings during the communication phase between the receive nodes and the fusion center. Performance in the presence of a jammer is analyzed for the case of slowly moving targets. Issues related to forming the basis matrix that spans the angle-Doppler space, and for selecting a grid for that space are discussed. Extensive simulation results are provided to demonstrate the performance of the proposed approach at difference jammer and noise levels.

I. INTRODUCTION

This paper develops an infrastructure-less distributed MIMO radar using randomly placed single-antenna nodes and compressive sampling for sparse angle-Doppler estimation. It also analyzes jammer effects, derives signal-to-jammer expressions, and proposes a measurement-matrix modification to improve SJR.

  • System concept: Distributed MIMO radar uses single-antenna nodes randomly distributed on a disk, enabling flexible antenna placement without fixed infrastructure.The nodes can be freely chosen within the small-scale network.
  • Proposed estimation approach: The nodes transmit independent waveforms, and the paper extends compressive-sampling DOA estimation to sparse angle-Doppler estimation for MIMO radar.Because targets are typically fewer than available snapshots, the estimation problem is formulated as recovery of a sparse vector.
  • Proposed estimation approach: Known transmitted waveforms and transmitter locations let receive nodes construct the basis matrix locally, while a fusion center handles cases involving missing location information, limited computation, or interference.The fusion center formulates an augmented ℓ1-optimization problem whose solution provides target angle and Doppler information.
  • Jammer analysis: The paper derives average signal-to-jammer ratio expressions and proposes a modified measurement matrix that improves SJR.A jammer acts as noise and degrades ℓ1-optimization performance through its effect on the noise level.
  • Evaluation scope: The study focuses on randomly placed transmit and receive antennas rather than a uniform linear array and examines jammer effects on estimation performance.Stationary-target comparisons include Capon, APES, GLRT, and MUSIC; moving-target comparisons use the matched-filter method.

II. SIGNAL MODEL FOR MIMO RADAR

The signal model describes a clutter-free, synchronized planar MIMO radar with uniformly distributed transmit and receive nodes, point targets, narrowband periodic waveforms, and sampled noisy returns. Slowly moving targets produce Doppler shifts modeled under far-field and small-shift assumptions.

  • Network and propagation assumptions: Transmit and receive nodes are uniformly distributed on a small-radius disk, with targets and nodes constrained to a common plane.The model also assumes a clutter-free environment, perfect synchronization, and perfect node localization.
  • Target model: The scene contains K point targets at azimuth angles θ_k, each moving with constant radial speed v_k and range d_k(t) = d_k(0) − v_kt.Propagation distances are approximated under the far-field condition d_k(t) ≫ r_t/r_i.
  • Waveform and received-signal model: Each transmit antenna sends a narrowband periodic waveform x_i(t)e^j2πft, with all transmit nodes sharing carrier frequency f and period T.The narrowband assumption permits delay to be retained only in the phase term.
  • Noise and target amplitudes: The received signal includes complex target amplitudes β_k proportional to RCS and additive i.i.d. Gaussian noise with zero mean and variance σ^2.The amplitudes are assumed identical across receivers because the receive nodes see the same target aspect in a small network.
  • Sampling and Doppler: Target motion induces Doppler shift f_k = 2v_kf/c, and received samples are collected over L snapshots with sampling period T_s during each pulse.The model assumes a small Doppler shift, f_kT_s << 1, because targets move slowly.

III. COMPRESSIVE SENSING FOR MIMO RADAR

This section discretizes the angle-Doppler plane and models target locations as sparse, enabling compressed measurements and l1-based recovery at individual nodes or a fusion center. Recovery uses sensing matrices with fewer measurements than grid dimensions and a Dantzig-selector formulation under a probability condition.

  • Sparse angle-Doppler representation: The angle-Doppler plane is discretized on a fine grid, and a small number of targets makes the coefficient vector s sparse.This sparse representation supports compressive sensing over the angle-Doppler plane.
  • Compressive measurements: Each receive node measures linear projections using an M × L Gaussian sensing matrix with M < L and small correlation with the basis matrix.The number of projections M must exceed the number of targets.
  • Local recovery: With transmitter identities and coordinates, a node constructs Ψlm and recovers s from its received data through l1-optimization.Location information for other nodes may be supplied by higher network layers.
  • Fusion-center recovery: When location information is unavailable or interference is strong, receive nodes forward linear projections to a fusion center for joint recovery.The fusion center combines outputs from Np pulses and Nr receive antennas using global and local information.
  • Recovery condition: The fusion center recovers s by applying the Dantzig selector, with very high probability when the parameter µ satisfies the stated sensing-matrix and noise condition.The recommended µ range also depends on experimentation, a lower bound, and avoiding the trivial solution s = 0.

A. Resolution · B. Maximum grid size for the angle-Doppler space

Resolution improves as the numbers of pulses and receive or transmit nodes increase, while simulations show comparable resolution with far fewer received samples. Grid design must balance target capture against sensing-matrix coherence and the UUP, with small pulse and receive-node counts still producing strong performance.

  • A. Resolution: Increasing Np, Nr, or Mt reduces sensing-matrix column correlation and improves the conditions for sparse recovery.Larger Np and Nr increase the sensing-matrix column dimension, making columns less similar; Mt has the same correlation-reduction effect.
  • A. Resolution: A finer angle-Doppler grid improves resolution but, for fixed Np, Nr, and Mt, creates more correlated sensing-matrix columns.The resulting correlation can be mitigated by increasing the number of measurements M.
  • A. Resolution: The proposed approach can match conventional resolution while using far fewer received samples.This conclusion is based on extensive simulations.
  • B. Maximum grid size for the angle-Doppler space: Grid points must be close enough to targets that off-grid reflections remain sufficiently correlated with nearby basis-matrix columns.This requirement conflicts with the UUP’s approximate-orthogonality condition, creating a tradeoff between column correlation and grid size.
  • B. Maximum grid size for the angle-Doppler space: Without prior target information, the maximum grid spacing can be selected from a worst-case midpoint analysis using a uniform angle-Doppler discretization.The worst case places targets midway between adjacent grid points.
  • B. Maximum grid size for the angle-Doppler space: Varying (∆a, ∆b) until average zero-lag correlation reaches a threshold provides a practical grid-selection procedure.The threshold should capture off-grid targets while satisfying the UUP, ensuring estimates fall on the constructed basis-matrix grid.
  • B. Maximum grid size for the angle-Doppler space: When targets lie between grid points, increasing Np or Nr does not necessarily improve performance, but simulations show very good performance with very small Np and Nr.For similar performance, the conventional matched-filter method requires much greater Np and Nr.

C. Range of unambiguous speed … A. Analysis of Signal-to-Jammer Ratio

The paper characterizes unambiguous speed, computational complexity, jammer-induced degradation, and signal-to-jammer ratio for the compressive-sampling approach. It balances pulse duration, grid refinement, optimization cost, and jammer suppression in its analysis.

  • C. Range of unambiguous speed: Distinct Doppler grid points at the same angle produce different basis-matrix columns when their pulse-to-pulse phase factors differ.This establishes the condition used to characterize unambiguous relative speed.
  • C. Range of unambiguous speed: The pulse duration T must balance competing requirements: smaller T enlarges unambiguous speed range, while larger T supports the narrowband waveform assumption.T must also keep Doppler shift nearly constant during each pulse.
  • D. Complexity: The convex program becomes more complex as the number of detected targets increases, and its complex-valued formulation can be recast as an SOCP solvable in polynomial time.The relevant polynomial-time dimension is that of the unknown vector.
  • D. Complexity: Fine grids increase computational complexity; coarse-grid initialization, local refinement, and restricting candidate angle-Doppler space reduce basis-matrix size and l1-optimization cost.The receiver required to obtain the compressive samples is also more complex.
  • IV. PERFORMANCE ANALYSIS IN THE PRESENCE OF A JAMMER SIGNAL: Under an uncorrelated jammer, increased interference power degrades the Dantzig selector and therefore deteriorates the proposed compressive-sensing method.The analysis provides signal-to-jammer expressions and proposes a modified measurement matrix for jammer suppression.
  • A. Analysis of Signal-to-Jammer Ratio: The jammer contribution is modeled similarly to additive noise when jammer waveforms are uncorrelated with the transmitted radar waveforms, with jammer power assumed dominant in the analysis.Simulations later consider additive noise together with the jammer.
  • A. Analysis of Signal-to-Jammer Ratio: Average desirable-signal and jammer powers over node locations yield the average SJR, defined as SJR= Ps(l)/Pj(l).The denominator is independent of node locations, so averaging preserves the power ratio.

B. SJR based on a modified measurement matrix

The section proposes a modified measurement matrix that correlates with the transmitted signal to improve SJR while preserving compressive-sensing stability. For stationary targets, it improves SJR by a factor of L/M_t when L ≫ M_t, although increasing L raises the required sampling rate.

  • B. SJR based on a modified measurement matrix: The modified measurement matrix correlates the jammer signal with the transmitted signal to improve SJR.The analysis compares the modified matrix with the original random matrix for stationary and moving targets.
  • B. SJR based on a modified measurement matrix: The proposed measurement matrix has low coherence with the basis matrix, guaranteeing a stable solution to the l1-optimization problem.Its Gaussian construction follows the near-incoherence of i.i.d. random measurement matrices with fixed basis matrices.
  • B. SJR based on a modified measurement matrix: Using the modified matrix does not affect the jammer signal’s average power.The jammer-power expression is obtained from the original formulation by replacing Φ_l with ˜Φ_l.
  • Stationary Targets: L/M_t: the modified matrix improves SJR by a factor of L/M_t when L ≫ M_t for stationary targets.Stationary targets have zero Doppler shift in the derivation.
  • Stationary Targets: Increasing L improves SJR but requires a higher sampling rate when the pulse duration is fixed.The sampling-rate cost accompanies the SJR improvement from increasing L.

2) Slowly Moving Targets:

For slowly moving targets with f_sT << 1, the sensing and interference characteristics are approximated using small-Doppler-shift assumptions. Under these conditions, target performance is approximately the same as for stationary targets for both random measurement matrices.

  • Slowly Moving Targets: For moving targets with f_sT << 1, P_s(l) is approximately the same as for stationary targets.This result follows from incorporating the Doppler shift into the measurement matrix Φ_l.
  • Slowly Moving Targets: When f_kT_s << 1 and L is relatively large, approximations for the relevant matrix terms are derived.The derivation uses entries involving XH_DH(f_k)X and convolution terms.
  • Slowly Moving Targets: Off-diagonal elements are ignored because they are small compared with diagonal elements.This simplification leads to the stated approximation for the matrix structure.
  • Slowly Moving Targets: For f_sT << 1, the SJR of moving targets is approximately equal to that of stationary targets for both random measurement matrices.The conclusion applies to the analyzed slowly moving-target scenarios.

V. SIMULATION RESULTS · A. Stationary Targets · 1) Targets falling on the grid:

Simulations evaluate compressive-sampling (CS) MIMO radar against conventional methods for stationary targets, including noise, jamming, parameter effects, and angular separation. CS generally provides clean target detection and competitive performance with substantially fewer samples, while threshold, jammer strength, and receive-node count affect results.

  • V. SIMULATION RESULTS: The simulations assess CS target detection under noise and jamming, comparing it with conventional MIMO radar methods to quantify relative weaknesses and advantages.For stationary targets, the comparison methods are Capon, APES, GLRT, and MUSIC; moving-target comparisons use matched filtering.
  • A. Stationary Targets: The target-information vector produces clean regions away from targets and well-distinguished peaks, supporting target detection with a small probability of false alarm.Peak location indicates target location, while peak magnitude indicates RCS magnitude.
  • 1) Targets falling on the grid:: 5.8% (= 30/512) is the sample count used by CS relative to conventional methods while achieving at least comparable jammer-dominated performance.The comparison methods use L = 512 samples, whereas CS uses M = 30 samples.
  • 1) Targets falling on the grid:: The grid-aligned experiment correctly identifies two targets, although CS shows a small error in target RCS magnitude and comparison methods exhibit severe ripples.With one receive antenna, the comparison methods yield PRR close to 1, indicative of severe ripples.
  • 1) Targets falling on the grid:: Increasing threshold µ reduces ripples but degrades RCS amplitude estimates; stronger jamming increases the probability of low PRR and PJR for CS.Increasing µ can also improve target DOA estimates and reduce the probability of missing a target, at the cost of increased ripples.
  • 1) Targets falling on the grid:: Increasing Nr or M improves CS under strong jamming, while increasing M beyond Mt does not help because of the maximal rank of Φ′.For the strong-jammer case β2 = 3600, Nr is increased to 30 to demonstrate improvement.
  • 1) Targets falling on the grid:: CS remains effective for closely spaced targets under strong jamming, whereas MUSIC fails, Capon and APES often produce PRR≈1, and GLRT performs well.CS has only a few exceptions with PRR or PJR less than 1, occurring with very small probability.
  • 1) Targets falling on the grid:: In thermal-noise-dominated conditions, CS performs well for PRR and PJR, is less threshold-sensitive in angle MSE, and remains comparable to GLRT with L = 256.Comparison methods become noisy, while all methods can achieve low angle MSE and PFA with appropriate thresholding; thresholding choices affect specific values.

2) Targets falling off the grid points:

The section evaluates angle-Doppler estimation when targets lie between discretized angle-grid points, using a 0.2o grid from −8o to 8o and four off-grid targets. With jammer powers of 400 and 3600, the proposed method captures these targets well, while GLRT shows high variance.

  • Grid selection: 0.2o increments discretize the angle space from −8o to 8o, producing a grid that supports evaluation of unknown target locations.The grid is selected using the procedures described in Section III-B because the best grid is not known in advance.
  • Experimental setup: Four targets are placed off-grid at θk = {−1.1o, −0.3o, 0.3o, 1.1o}, each with reflection coefficient βk = 1, while a jammer remains at 7o.Because the targets lie between grid points, PRR and PJR cannot be plotted as in the on-grid case.
  • Evaluation: 400 and 3600 are the jammer powers evaluated in the left and right columns of Fig. 11, respectively.The results show the mean plus and minus one standard deviation for the DOA-estimate amplitude at each grid point.
  • Results: The proposed method captures targets that do not fall on grid points well when proper grid points are used.This result is reported for the off-grid target experiment across the jammer-power cases shown in Fig. 11.
  • Results: GLRT is the next best method, capturing the targets but exhibiting high variance in the shaded region around its mean.The shaded region represents the mean plus and minus one standard deviation.

B. Moving Targets · 1) Targets falling onto the grid points:

For moving targets on angle-Doppler grid points, the compressive-sampling approach outperforms matched filtering under the stated jammer and noise conditions. It achieves desired performance with substantially fewer receive nodes and pulses, while additional receive antennas reduce pulse requirements.

  • B. Moving Targets: The experiment uses orthogonal QPSK waveforms, a jammer at 7o with power 400, SNR 0 dB, and M = 30 measurements per receive node.These conditions define the moving-target evaluation scenario.
  • B. Moving Targets: Three targets are placed at θk = −1o, 0o, 1o and move at vk = 60m/s, 70m/s, 80m/s, respectively.The target speeds correspond to the listed target angles in order.
  • 1) Targets falling onto the grid points:: The angle-Doppler space is sampled with increment (0.5o, 5m).The resulting basis vector spans angles from −8o to 8o and speeds from 50m/s to 110m/s.
  • 1) Targets falling onto the grid points:: The matched-filtering method performs worse than the CS approach even when using data from 30 pulses.The matched filter correlates received signals with transmit signals distorted by Doppler shifts and steering vectors.
  • 1) Targets falling onto the grid points:: The proposed CS approach delivers desired performance with a single receive node and as low as 5 pulses.This demonstrates reduced pulse requirements relative to the matched-filtering comparison.
  • 1) Targets falling onto the grid points:: Increasing Nr reduces the number of pulses required to produce good performance.The effect is observed by comparing the one-node and ten-node columns of Figure 12.

2) Targets falling off the grid points: · VI. CONCLUSIONS

The proposed compressive-sampling MIMO radar refines off-grid angle-Doppler estimation and captures targets outside the grid. It uses fewer forwarded samples while maintaining competitive performance, but assumes synchronization and perfect node locations.

  • 2) Targets falling off the grid points:: Column correlation is more sensitive to the angle step than the speed step, so initial grids should be denser in angle and sparser in speed.This follows from fT_s << 1; denser angle-Doppler sampling around the initial estimate can further improve resolution.
  • 2) Targets falling off the grid points:: With 0.2° angle increments and 5m/s speed steps, the method captures three targets falling outside the grid in both dimensions.The targets move at 62.5m/s, 72.5m/s, and 82.5m/s toward −1.1°, 0.1°, and 1.1°.
  • 2) Targets falling off the grid points:: Increasing N_p or N_r does not necessarily improve off-grid performance because higher-dimensional basis vectors reduce column correlation.The closer-target case with d = 0.4° is evaluated in Fig. 14.
  • VI. CONCLUSIONS: The system uses network transmitters and receivers, uncorrelated transmit waveforms, receiver-side compressive sampling, and fusion-center ℓ1 optimization for sparse angle-Doppler targets.The optimization solution yields target angle and Doppler information from a small number of forwarded samples.
  • VI. CONCLUSIONS: For mild jammers, the method is at least as good as Capon, APES, GLRT, and MUSIC with significantly fewer samples; under strong noise and jamming, it is slightly worse than GLRT.For moving targets, it outperforms conventional matched filtering with single and multiple receive nodes.
  • VI. CONCLUSIONS: The CS implementation forwards M samples per receive node instead of L samples, where M is typically significantly smaller than L for a given performance.This reduces transmission energy and can significantly prolong wireless-network life.
  • VI. CONCLUSIONS: Future work includes range extraction, widely separated antennas, and wideband radar signals; the approach assumes synchronized nodes and perfect node-location information.Localization and synchronization errors, and mitigation methods, require further study.

APPENDIX I THE EFFECTS OF Nr, Np, Mt ON THE CORRELATION OF COLUMNS IN THE SENSING MATRIX … C. The effect of the number of transmit antennas on the column correlation in the sensing matrix

The appendix analyzes how pulse, receive-antenna, and transmit-node counts affect sensing-matrix column correlation. Increasing these quantities can reduce correlation or improve compressive-sensing estimation under specified conditions.

  • A. The effect of the number of pulses on the column correlation in the sensing matrix: For fixed T, h_kk′ measures the ratio of a column’s self-correlation to its cross-correlation and reveals the effect of Np.The analysis considers distinct column pairs k and k′.
  • A. The effect of the number of pulses on the column correlation in the sensing matrix: When (b_k − b_k′)NpT ≤ 1, h_kk′ increases with Np and reaches its maximum at (b_k − b_k′)NpT = 1.At the maximum, the cross-correlation of g_k and g_k′ becomes zero.
  • A. The effect of the number of pulses on the column correlation in the sensing matrix: If (b_k − b_k′)NpT ≤ 1 for every distinct column pair, increasing Np can always improve compressive-sensing estimation.The appendix also relates larger pulse counts to improved Doppler resolution in conventional radar.
  • B. The effect of the number of receive antennas on the column correlation in the sensing matrix: With a constant random measurement matrix across receive antennas, the sensing matrix is represented using the received data from one pulse.This simplifying assumption supports the receive-antenna correlation analysis.
  • B. The effect of the number of receive antennas on the column correlation in the sensing matrix: As Nr becomes large, the relevant column-correlation ratio approaches 0, so increasing the number of receive antennas reduces sensing-matrix column correlation.The correlation analysis is expressed through Bi,j = X^HD^H(b_i)Φ^HΦD(b_j)X.
  • C. The effect of the number of transmit antennas on the column correlation in the sensing matrix: For Nr = Np = 1, the transmit-node analysis rewrites v^H(a_i)B_i,jv(a_j) in terms of entries of v and X.The ratio of column correlations is then examined as Mt changes.
  • C. The effect of the number of transmit antennas on the column correlation in the sensing matrix: As Mt approaches infinity, the numerator approaches 0, so employing many transmit nodes reduces correlation between sensing-matrix columns.This conclusion follows from the transmit-node correlation ratio under the stated simplification.
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