JOURNAL OF L A T E X CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 Simultaneous Sparse Approximation Using an Iterative Method with Adaptive Thresholding Shahrzad Kiani, Sahar Sadrizadeh, Mahdi Boloursaz, Student Members, IEEE, and Farokh Marvasti, Senior Member, IEEE Abstract—This paper studies the problem of Simultaneous Sparse Approximation (SSA). This problem arises in many applications which work with multiple signals maintaining some degree of dependency such as radar and sensor networks. In this paper, we introduce a new method towards joint recovery of several independent sparse signals with the same support. We provide an analytical discussion on the convergence of our method called Simultaneous Iterative Method with Adaptive Thresholding (SIMAT). Additionally, we compare our method with other group-sparse reconstruction techniques, i.e., Simulta- neous Orthogonal Matching Pursuit (SOMP), and Block Iterative Method with Adaptive Thresholding (BIMAT) through numerical experiments. The simulation results demonstrate that SIMAT outperforms these algorithms in terms of the metrics Signal to Noise Ratio (SNR) and Success Rate (SR). Moreover, SIMAT is considerably less complicated than BIMAT, which makes it feasible for practical applications such as implementation in MIMO radar systems. Index Terms—Simultaneous Sparse Approximation; Iterative Method; Adaptive Thresholding; Joint Recovery. I. I NTRODUCTION S PARSE signal processing has recently been exploited in various fields of communication, due to fact that sparse signals can be approximated by only a few nonzero coefficients and hence sub-Nyquist sampling and Compressed Sensing (CS) [1], [2]. The general CS problem is formulated as follows: minimize ‖x‖ 0 subject to y = Ax + v (1) where x is the main sparse signal, y is the measurement vector, A is the sensing matrix, and v is the additive noise vector. Two main models are considered in CS for reconstruction of sparse signals. Models with one measurement vector are referred to as Single Measurement Vector (SMV) models, while the other models with at least two measurement vectors are called Multiple Measurement Vector (MMV) models. The problem investigated in MMV models, known as SSA, aims to jointly recover sparse representation of the mea- surement vectors. The SSA applications may be encountered in various fields such as sensor networks [3], [4], Elec- troencephalography and Magnetoencephalography (EEG and MEG) [5], source localization [6], and distributed MIMO radar systems [7]. S. Kiani, S. Sadrizadeh, M. Boloursaz and F. Marvasti are with the Advanced Communication Research Institute (ACRI), Electrical Engineer- ing Department, Sharif University of Technology, Tehran, Iran (email: shkianid@gmail.com; sadrizadeh s@ee.sharif.edu; marvasti@sharif.edu). [8] investigates the theory of MMV models. Some algo- rithms have been developed by extending the general SMV model into the MMV model to solve the SSA problems. Orthogonal Matching Pursuit (OMP) [9] as a greedy algorithm is one of the very first algorithms used for sparse recovery. At each iteration of this algorithm, the best local improvement to the current approximations is found in hope of obtaining a good overall solution. The extension of the OMP algorithm to The MMV paradigm, Simultaneous OMP or SOMP, has been developed in [10]–[12]. The Iterative Method with Adaptive Thresholding (IMAT) algorithm was originally proposed for sparse signal recon- struction from missing samples [13]–[15]. The Block Iterative Method with Adaptive Thresholding (BIMAT) [16] as an extension of IMAT is employed for block sparse recovery for distributed MIMO radar systems. In this paper, we propose SIMAT as a generalization of IMAT for simultaneous reconstruction of jointly sparse signals from their missing samples. A. Paper Overview The rest of this paper is structured as follows. In Section II, we first provide the description of SSA model. Then the pro- posed method is introduced and its convergence is analyzed. Numerical experiments of our method in comparison with the SOMP algorithm are presented in Section III. SIMAT is then demonstrated as a simple decoding algorithm for MIMO radar systems, and its performance is compared with BIMAT by means of simulation. Finally, the paper is concluded in Section IV. B. Notations Scalar variables, vectors, and matrices are denoted by italic lower-case, boldface lower-case, and boldface upper- case, respectively. The elements of a vector are denoted by subscript, i.e., x i is the i-th element of the vector x. |x| calculates the absolute value of each entries of the vector x. The pseudoinverse of matrix A is represented by A † . Finally, the output of the thresholding operator TH(x, thr) is defined as a diagonal matrix whose diagonal entries are determined as follows: TH(x, thr) ii =  1. |x i |≥ thr 0. |x i | < thr (2) arXiv:1707.08310v1 [cs.IT] 26 Jul 2017