This paper studies the joint support recovery of similar sparse vectors on the basis of a limited number of noisy linear measurements, i.e., in a multiple measurement vector (MMV) model. The additive noise signals on each measurement vector are assumed to be Gaussian and to exhibit different variances. The simultaneous orthogonal matching pursuit (SOMP) algorithm is generalized to weight the impact of each measurement vector on the choice of the atoms to be picked according to their noise levels. The new algorithm is referred to as SOMP-NS where NS stands for noise stabilization. To begin with, a theoretical framework to analyze the performance of the proposed algorithm is developed. This framework is then used to build conservative lower bounds on the probability of partial or full joint support recovery. Numerical simulations show that the proposed algorithm outperforms SOMP and that the theoretical lower bound provides a great insight into how SOMP-NS behaves when the weighting strategy is modified.
Gribonval have performed an analysis of SOMP for a problem similar to ours in @cite_29 . They were interested in providing a lower bound on the probability of correct support recovery when the signal to be estimated is sparse and its non-zero entries are statistically independent mean-zero Gaussian random variables exhibiting possibly different variances.
@cite_29
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Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.