New Bounds for Restricted Isometry Constant for the s-sparse Recovery via Compressed Sensing

The main purpose of this paper is to establish the sufficient condition for the restricted isometry constant δs in compressed sensing by using T. Cai and A. Zhang idea. Let h ≡ x* − x and h = (h1; h2; _ _ _ ; hn), where x is an unknown signal and x* is the CS-solution. For simplicity, we assume that the index of h is sorted by| h1 |≥ | h2| ≥ ….. ≥ |hn|. Let s be a fixed positive integer, T0 = {1; 2,…., s} and T1 ⊂ T0. In this paper, we focus the quality of hT0 and research good conditions for the recovery of sparse signals by investigating the difference between||hT1||1 and ||hTf||1. We shall show that if δs < 0:5 under an assumption for ||hT1||1, and similarly if δ 34 s < 0:414 or δ 24 25 s < 0:436, then we have stable recovery of approximately sparse signals.