> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Abstract— Structurally random matrices (SRMs) have been proposed as a practical alternative to fully random matrices (FRMs) for generating compressive sensing measurements. If the compressive measurements are transmitted over a communication channel, they need to be efficiently quantized and coded and hence knowledge of the measurements’ statistics reequired. In this paper we study the statistical distribution of compressive measurements generated by various types of SRMs (and FRMs), give conditions for asymptotic normality and point out the implications for the measurements’ quantization and coding. Simulations on real-world video signals confirm the theoretical findings and show that the signal randomization of SRMs yields a dramatic improvement in quantization properties. Index Terms— Compressed Sensing, Quantization, Structurally Random Matrices. I. INTRODUCTION OMPRESSED sensing [1] is concerned with determining a signal n ∈ x ℝ from a vector =Φ y x (1) of compressive measurements, where m n × Φ∈ ℝ , m n ≪ , is a sensing matrix. This generally ill-posed problem is made solvable by constraining x to have a k-sparse representation in the column space of a sparsifying matrix Ψ , that is, Ψ is an orthonormal or a tight frame matrix, =Ψ x ξ , and ξ has only k non-zero entries. If Φ is incoherent with Ψ and meets some additional conditions, then x is the unique solution of the constrained minimization problem [2] 1 min s.t. , ξ =Φ =Ψ y x x ξ (2) Other results in the same vein give more relaxed conditions under which x is the solution of (2) only with a very high probability [2], or provide boundaries on the difference between solution of (2) and the true signal when the measurements are noisy [3]. The simplest way to generate good sensing matrices, i.e matrices which allow reconstruction of the signal from a relatively small number of measurements, is probabilistic. A fully random matrix (FRM) is defined as a matrix whose entries are independent, identically distributed (IID) Gaussian or Bernoulli random variables (RVs). If Φ is a FRM, then Manuscript received August 22, 2013. R. Haimi-Cohen is with Alcatel-Lucent Bell-Laboratories, Murray Hill, NJ 07974, USA (phone: 908-582-4159; e-mail: razi@ alcatel-lucent.com). Y. M. Lai is with University of Maryland, College Park, MD 20742 USA (e-mail: ylai@amsc.umd.edu). there is a constant α >0 (of moderate size) such that if log( ) m k nk α > and n is large, then with very high probability, x is the unique solution of (2) [4]. Furthermore, this result holds for any sparsifying matrix because any orthonormal or tight-frame matrix Ψ is incoherent with Φ with high probability. While FRMs are an excellent choice from a theoretical standpoint, they are unwieldy in large scale applications such as image and video processing. Some algorithms for solving (2) require fairly complex operations with Φ : For example, CoSaMP [5], requires computing pseudo-inverses of m m × minors of , which is computationally intractable for large scale applications. Even less demanding reconstruction algorithms, such as the alternate-direction augmented- lagrangian [6], require multiplying vectors by and T Φ . The resources needed for storing the sensing matrix and for repeated multiplication by it may be too high for large practical application. It is possible to avoid storing the matrix by generating its entries on the fly, but this significantly increases the computational complexity. Furthermore, since most random number generation algorithms are recursive, on- the-fly generation makes parallel processing implementation nearly impossible. A more practical alternative is to use a transform based sensing matrix (TBSM) n mSW Φ= (3) where nn W × ∈ ℝ is an orthonormal matrix having a fast transform, and mn S × ∈ ℝ randomly selects m out of the n rows of W, with equal probabilities for all possible selections. Thus, the rows of S are a random subset of the rows of the n n × identity matrix. Multiplying x by Φ can be done by computing the fast transform Wx and selecting a subset of the transform coefficients. With this sensing matrix, if Φ is incoherent with Ψ , x is the unique solution of (2) with very high probability [7]. However, the incoherence of Φ and Ψ depends on the incoherence of W and Ψ , which cannot be taken for granted. This problem was addressed by the introduction of structurally random matrices (SRM) [8], [9] : n mSWR Φ= (4) where S,W are as above and n n R × ∈ ℝ is a randomizing matrix - an orthonormal random matrix, designed to eliminate the coherence between W and Ψ . Two types of randomization were proposed: Local randomization (LR), where each entry of x is multiplied by ±1 with equal Φ Φ Distribution of Compressive Measurements Generated by Structurally Random Matrices Raziel Haimi-Cohen, Member, IEEE, and Yenming Mark Lai, Student Member, IEEE C