IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 12, DECEMBER 2011 7877 Limits on Support Recovery of Sparse Signals via Multiple-Access Communication Techniques Yuzhe Jin, Student Member, IEEE, Young-Han Kim, Member, IEEE, and Bhaskar D. Rao, Fellow, IEEE Abstract—In this paper, we consider the problem of exact sup- port recovery of sparse signals via noisy linear measurements. The main focus is finding the sufficient and necessary condition on the number of measurements for support recovery to be reliable. By drawing an analogy between the problem of support recovery and the problem of channel coding over the Gaussian multiple-access channel (MAC), and exploiting mathematical tools developed for the latter problem, we obtain an information-theoretic framework for analyzing the performance limits of support recovery. Specif- ically, when the number of nonzero entries of the sparse signal is held fixed, the exact asymptotics on the number of measurements sufficient and necessary for support recovery is characterized. In addition, we show that the proposed methodology can deal with a variety of models of sparse signal recovery, hence demonstrating its potential as an effective analytical tool. Index Terms—Compressed sensing, Gaussian multiple-access channel (MAC), noisy linear measurement, performance tradeoff, sparse signal, support recovery. I. INTRODUCTION C ONSIDER the problem of estimating a sparse signal in high dimension via noisy linear measure- ments , where is the measurement matrix and is the measurement noise. A sparse signal in- formally refers to a signal whose representation in a certain basis contains a large proportion of zero elements. In this paper, we mainly consider signals that are sparse with respect to the canonical basis of the Euclidean space. The goal is to estimate the sparse signal by making as few measurements as possible. This problem has received much attention from many research principles, motivated by a wide spectrum of applications such as compressed sensing [1], [2], biomagnetic inverse problems [3], [4], image processing [5], [6], bandlim- ited extrapolation and spectral estimation [7], robust regression and outlier detection [8], speech processing [9], channel esti- mation [10], [11], echo cancellation [12], [13], and wireless communication [10], [14]. Manuscript received March 03, 2010; revised March 11, 2011; accepted June 09, 2011. Date of current version December 07, 2011. This work was supported in part by the National Science Foundation under Grants CCF-0830612 and CCF-0747111. The material in this paper was presented in part at the 2008 IEEE International Conference on Acoustics, Speech, and Signal Processing and the 2008 IEEE International Symposium on Information Theory. A short version of this paper was presented at the 2010 IEEE International Symposium on In- formation Theory. The authors are with the Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA 92093-0407 USA (e-mail: yuzhe.jin@gmail.com; yhk@ucsd.edu; brao@ece.ucsd.edu). Communicated by J. Romberg, Associate Editor for Signal Processing. Digital Object Identifier 10.1109/TIT.2011.2170116 Computationally efficient algorithms for sparse signal re- covery have been proposed to find or approximate the sparse signal in various settings. A partial list includes matching pursuit [15], orthogonal matching pursuit [16], LASSO [17], basis pursuit [18], FOCUSS [3], sparse Bayesian learning [19], finite rate of innovation [20], CoSaMP [21], and subspace pur- suit [22]. At the same time, many exciting mathematical tools have been developed to analyze the performance of these algo- rithms. In particular, Donoho [1], Donoho et al. [23], Candès and Tao [24], and Candès et al. [25] presented sufficient con- ditions for -norm minimization algorithms, including basis pursuit, to successfully recover the sparse signals with respect to certain performance metrics. Tropp [26], Tropp and Gilbert [27], and Donoho et al. [28] studied greedy sequential selection methods such as matching pursuit and its variants. In these papers, the structural properties of the measurement matrix , including coherence metrics [15], [23], [26], [29] and spectral properties [1], [24], are used as the major ingredient of the performance analysis. By using random measurement matrices, these results translate to relatively simple tradeoffs between the dimension of the signal , the number of nonzero entries in , and the number of measurements to ensure asymptoti- cally successful reconstruction of the sparse signal. When the measurement noise is present, i.e., , the performance of the sparse signal recovery algorithms has been measured by the Euclidean distance between the true signal and the estimate [23], [25]. In many applications, however, finding the exact support of the signal is important even in the noisy setting. For example, in applications of medical imaging, magnetoencephalography (MEG) and electroencephalography (EEG) are common ap- proaches for collecting noninvasive measurements of external electromagnetic signals [30]. A relatively fine spatial resolution is required to localize the neural electrical activities from a huge number of potential locations [31]. In the domain of cognitive radio, spectrum sensing plays an important role in identifying available spectrum for communication, where estimating the number of active subbands and their locations becomes a nontrivial task [32]. In multiple-user communication systems such as a code-division multiple-access (CDMA) system, the problem of neighbor discovery requires identification of active nodes from all potential nodes in a network based on a linear superposition of the signature waveforms of the active nodes [14]. In all these problems, finding the support of the sparse signal is more important than approximating the signal vector in the Euclidean distance. Hence, it is important to understand performance issues in the exact support recovery of sparse signals with noisy measurements. Information-theoretic tools 0018-9448/$26.00 © 2011 IEEE