Refined Analysis of Sparse MIMO Radar

• Published in: The Journal of fourier analysis and applications Vol. 23; no. 3; pp. 485 - 529
• Main Authors: Dorsch, Dominik, Rauhut, Holger
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2017; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Approximations and Expansions; Echoes; Fine structure; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Order parameters; Partial Differential Equations; Radar; Radar systems; Reconstruction; Recovery; Signal,Image and Speech Processing; Sparsity; Transmitters; Random matrix; MIMO radar; Restricted isometry property; minimization; 90C25; 65F22; LASSO; 60B20; 94A12; Compressed sensing; 94A20
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-016-9477-7

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—not only the size—affects the (nonuniform) recovery performance. We show that for certain “balanced” support sets reconstruction with essentially the optimal number of measurements is possible. Indeed, we introduce a parameter measuring the well-behavedness of the support set and resemble standard results from CS for near-optimal parameter choices. We prove recovery results for both perfect recovery of the support set in case of exactly sparse vectors and an
ℓ
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. Thereby—and apparently for the first time in CS theory—we prove theoretical results in which the difference between nonuniform and uniform recovery consists of more than just logarithmic factors.