Refined analysis of sparse MIMO radar

TL;DR

This paper provides a refined theoretical analysis of sparse MIMO radar, demonstrating stable recovery guarantees using compressed sensing techniques and revealing how support set structure influences reconstruction performance.

Contribution

It offers new RIP estimates for structured measurement matrices in MIMO radar and analyzes the impact of support set structure on recovery success.

Findings

RIP bounds are optimal up to logarithmic factors for the structured measurement matrix.

Recovery performance depends on the support set structure, with certain balanced sets allowing near-optimal measurements.

The analysis extends understanding of compressed sensing in MIMO radar beyond previous results.

Abstract

We analyze a multiple-input multiple-output (MIMO) radar model and provide recovery results for a compressed sensing (CS) approach. In MIMO radar different pulses are emitted by several transmitters and the echoes are recorded at several receiver nodes. Under reasonable assumptions the transformation from emitted pulses to the received echoes can approximately be regarded as linear. For the considered model, and many radar tasks in general, sparsity of targets within the considered angle-range-Doppler domain is a natural assumption. Therefore, it is possible to apply methods from CS in order to reconstruct the parameters of the targets. Assuming Gaussian random pulses the resulting measurement matrix becomes a highly structured random matrix. Our first main result provides an estimate for the well-known restricted isometry property (RIP) ensuring stable and robust recovery. We require more measurements than standard results from CS, like for example those for Gaussian random measurements. Nevertheless, we show that due to the special structure of the considered measurement matrix our RIP result is in fact optimal (up to possibly logarithmic factors). Our further two main results on nonuniform recovery (i.e., for a fixed sparse target scene) reveal how the fine structure of the support set affects the (nonuniform) recovery performance. We show that for certain "balanced" support sets reconstruction with essentially the optimal number of measurements is possible. We prove recovery results for both perfect recovery of the support set in case of exactly sparse vectors and an $\ell_2$-norm approximation result for reconstruction under sparsity defect.Our analysis complements earlier work by Strohmer & Friedlander and deepens the understanding of the considered MIMO radar model.