Sparse Signal Processing Concepts for Efficient 5G System Design

As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of future mobile communication systems, the question what could make up a 5G system, what are the crucial challenges and what are the key drivers is part of intensive, ongoing discussions. Partly due to the advent of compressive sensing, methods that can optimally exploit sparsity in signals have received tremendous attention in recent years. In this paper we will describe a variety of scenarios in which signal sparsity arises naturally in 5G wireless systems. Signal sparsity and the associated rich collection of tools and algorithms will thus be a viable source for innovation in 5G wireless system design. We will discribe applications of this sparse signal processing paradigm in MIMO random access, cloud radio access networks, compressive channel-source network coding, and embedded security. We will also emphasize important open problem that may arise in 5G system design, for which sparsity will potentially play a key role in their solution.Comment: 18 pages, 5 figures, accepted for publication in IEEE Acces

The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models

Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a small number of noisy linear measurements is an important problem in compressed sensing. In this paper, the high-dimensional setting is considered. It is shown that if the measurement rate and per-sample signal-to-noise ratio (SNR) are finite constants independent of the length of the vector, then the optimal sparsity pattern estimate will have a constant fraction of errors. Lower bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector. The tightness of the bounds in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing achievable bounds. Near optimality is shown for a wide variety of practically motivated signal models

Compressed Sensing over the Grassmann Manifold: A Unified Analytical Framework

It is well known that compressed sensing problems reduce to finding the sparse solutions for large under-determined systems of equations. Although finding the sparse solutions in general may be computationally difficult, starting with the seminal work of [2], it has been shown that linear programming techniques, obtained from an l_(1)-norm relaxation of the original non-convex problem, can provably find the unknown vector in certain instances. In particular, using a certain restricted isometry property, [2] shows that for measurement matrices chosen from a random Gaussian ensemble, l_1 optimization can find the correct solution with overwhelming probability even when the support size of the unknown vector is proportional to its dimension. The paper [1] uses results on neighborly polytopes from [6] to give a ldquosharprdquo bound on what this proportionality should be in the Gaussian measurement ensemble. In this paper we shall focus on finding sharp bounds on the recovery of ldquoapproximately sparserdquo signals (also possibly under noisy measurements). While the restricted isometry property can be used to study the recovery of approximately sparse signals (and also in the presence of noisy measurements), the obtained bounds can be quite loose. On the other hand, the neighborly polytopes technique which yields sharp bounds for ideally sparse signals cannot be generalized to approximately sparse signals. In this paper, starting from a necessary and sufficient condition for achieving a certain signal recovery accuracy, using high-dimensional geometry, we give a unified null-space Grassmannian angle-based analytical framework for compressive sensing. This new framework gives sharp quantitative tradeoffs between the signal sparsity and the recovery accuracy of the l_1 optimization for approximately sparse signals. As it will turn out, the neighborly polytopes result of [1] for ideally sparse signals can be viewed as a special case of ours. Our result concerns fundamental properties of linear subspaces and so may be of independent mathematical interest

Rank-Sparsity Incoherence for Matrix Decomposition

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the $\ell_1$ norm and the nuclear norm of the components. We develop a notion of \emph{rank-sparsity incoherence}, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature, with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems