Stable Recovery of Sparse Signal in Compressed Sensing via the RIP of Order less than s

Abstract

Our goal is to reconstruct an unknown sparse signal. In this paper, we consider the feature of the sparse signal and research good conditions for the recovery of sparse signals. In detail, we assume that h ≡ x*−x and h = (h1; h2; · · · ; hn), where x is an unknown signal and x* is the CS-solution. Furthermore for simplicity, we assume that the index of h is sorted by |h1| ≥ |h2| ≥ · · · ≥ |hn| and T0 = {1; 2; · · · ; s}. In this paper, we focus the quality of hT0 . In more details, we shall reseach good conditions for the recovery of sparse signals by investigating the difference between the mean |h1|+|h2|+···+|hs| / s and the mean |h1|+|h2|+···+|hr| / r , r = 1; 2; · · · ; s. We shall show that if δs < 0:366 by the quality of x, and similarly if δ2 / 3*s < 0:436, then we have stable recovery of approximately sparse signals.