Compressed Sensing of Complex Sinusoids off the Grid

Compressed sensing (CS) has received much attention in wide application recently. As complex sinusoids signal model is widely used in application, research of CS for complex sinusoids is very important. However parameter discretization brings off-gird problem in compressed sensing, which makes its performance degrade significantly. In 2011, Y. Chi studied the sensitivity of Basis Pursuit (BP) algorithm to off the grid, and pointed out it needs more consideration in application [1]. D. H. Chae proposed that building up CS basis element more finely can reduce the effect of off-grid [2]. However, the method needs more reconstruction time and more measurements. To decrease the computational complexity, the grid can be refined only around the regions where sources are present [3]. For widely separated objects, A. F. Jiang proposed a method based on band exclusion and local optimization [4]. Under some special conditions, the methods proposed in [5, 6] can be used. M. F. Duarte has proposed spectral compressed sensing (SCS) to solve off-grid problem [7]. L. Hu proposed a method based on basis refinement for complex sinusoids [8], which has better performance than SCS. As the computational cost is very high, L. Hu then developed a fast and accurate reconstruction algorithm [9], which applies a liner approximation to the true unknown dictionary. G. Tang has investigated compressed sensing off the gird and proposed an atomic norm minimization approach [10]. But it is provided that the frequencies are well separated. In this paper, a novel compressed sensing reconstruction algorithm for complex sinusoids is proposed, which selects dictionary adaptively according signal and can eliminate off-grid problem. Compared with the existing methods, the proposed algorithm has several advantages. The first advantage is that it is very simple and has low computational cost. The second advantage is that it does not require that the frequencies are well separated. The third advantage is that it can achieve high reconstruction accuracy.