In this paper, a cross-layer framework to jointly optimize spectrum sensing and scheduling in resource constrained agile wireless networks is presented. A network of secondary users (SUs) accesses portions of the spectrum left unused by a network of licensed primary users (PUs). A central controller (CC) schedules the traffic of the SUs, based on distributed compressed measurements collected by the SUs. Sensing and scheduling are jointly controlled to maximize the SU throughput, with constraints on PU throughput degradation and SU cost. The sparsity in the spectrum dynamics is exploited: leveraging a prior spectrum occupancy estimate, the CC needs to estimate only a residual uncertainty vector via sparse recovery techniques. The high complexity entailed by the POMDP formulation is reduced by a low-dimensional belief representation via minimization of the Kullback-Leibler divergence. It is proved that the optimization of sensing and scheduling can be decoupled. A partially myopic scheduling strategy is proposed for which structural properties can be proved showing that the myopic scheme allocates SU traffic to likely idle spectral bands. Simulation results show that this framework balances optimally the resources between spectrum sensing and data transmission. This framework defines sensing-scheduling schemes most informative for network control, yielding energy efficient resource utilization.
@cite_4: Most of the existing sparse-recovery methods assume a static system: the signal is a finite-length vector for which a fixed set of measurements and sparse representation are available and an l1 problem is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the signal changes over time, and it is measured and reconstructed sequentially over small intervals. This is particularly desired when dividing signals into disjoint blocks and processing each block separately is infeasible or inefficient. In this paper, we discuss two streaming systems and a new homotopy algorithm for quickly solving the associated l1 problems: 1) recovery of smooth, time-varying signals for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation and 2) recovery of sparse, time-varying signals that follows a linear dynamic model. For both systems, we iteratively process measurements over a sliding interval and solve a weighted l1-norm minimization problem for estimating sparse coefficients. Since we estimate overlapping portions of the signal while adding and removing measurements, instead of solving a new l1 program as the system changes, we use available signal estimates as starting point in a homotopy formulation and update the solution in a few simple steps. We demonstrate with numerical experiments that our proposed streaming recovery framework provides better reconstruction compared to the methods that represent and reconstruct signals as independent, disjoint blocks, and that our proposed homotopy algorithm updates the solution faster than the current state-of-the-art solvers.
@cite_10: 
@cite_1: In this paper we introduce the Cognitive Compressive Sensing (CCS) problem, modeling a Cognitive Receiver (CR) that optimizes the @math projections of a @math dimensional vector dynamically, by optimizing the objective of correctly detecting the maximum number of idle entries, while updating each time its Bayesian beliefs on the future vector realizations. We formulate and study the CCS as a Restless Multi-Armed Bandit problem, generalizing the popular Cognitive Spectrum Sensing model, in which the CR can sense @math out of the @math sub-channels and propose a novel adaptive Finite Rate of Innovation (FRI) sampling method based on the CCS approach. While in general the optimum policy remains elusive, we provide sufficient conditions in which in the limit for large @math and @math the greedy policy is optimum. Numerical results corroborate our theoretical findings.
@cite_17: 
@cite_31: Recently, [3], [7] theoretically analyzed the success of a polynomial l 1 optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context [3], [7] proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that l 1 optimization succeeds in solving the system. In this paper, we consider the same problem while additionally assuming that all non-zero elements elements are equal to each other. We provide a performance analysis of a slightly modified l 1 optimization. As expected, the obtained recoverable sparsity proportionality constants improve on the equivalent ones that can be obtained if no information about the non-zero elements is available. In addition, we conducted a sequence of numerical experiments and observed that the obtained theoretical proportionality constants are in a solid agreement with the ones obtained experimentally.
@cite_16: 
@cite_13: In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, time-varying signals from sub-Nyquist, non-adaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the literature, the ability to perform inference on high-dimensional problems in a computationally efficient manner remains elusive. In response, we propose a probabilistic dynamic CS signal model that captures both amplitude and support correlation structure, and describe an approximate message passing algorithm that performs soft signal estimation and support detection with a computational complexity that is linear in all problem dimensions. The algorithm, DCS-AMP, can perform either causal filtering or non-causal smoothing, and is capable of learning model parameters adaptively from the data through an expectation-maximization learning procedure. We provide numerical evidence that DCS-AMP performs within 3 dB of oracle bounds on synthetic data under a variety of operating conditions. We further describe the result of applying DCS-AMP to two real dynamic CS datasets, as well as a frequency estimation task, to bolster our claim that DCS-AMP is capable of offering state-of-the-art performance and speed on real-world high-dimensional problems.
@cite_25: Spectrum sensing, which aims at detecting spectrum holes, is the precondition for the implementation of cognitive radio (CR). Collaborative spectrum sensing among the cognitive radio nodes is expected to improve the ability of checking complete spectrum usage. Due to hardware limitations, each cognitive radio node can only sense a relatively narrow band of radio spectrum. Consequently, the available channel sensing information is far from being sufficient for precisely recognizing the wide range of unoccupied channels. Aiming at breaking this bottleneck, we propose to apply matrix completion and joint sparsity recovery to reduce sensing and transmission requirements and improve sensing results. Specifically, equipped with a frequency selective filter, each cognitive radio node senses linear combinations of multiple channel information and reports them to the fusion center, where occupied channels are then decoded from the reports by using novel matrix completion and joint sparsity recovery algorithms. As a result, the number of reports sent from the CRs to the fusion center is significantly reduced. We propose two decoding approaches, one based on matrix completion and the other based on joint sparsity recovery, both of which allow exact recovery from incomplete reports. The numerical results validate the effectiveness and robustness of our approaches. In particular, in small-scale networks, the matrix completion approach achieves exact channel detection with a number of samples no more than 50 of the number of channels in the network, while joint sparsity recovery achieves similar performance in large-scale networks.
@cite_20: Model-based compressive sensing (CS) for signal-specific applications is of particular interest in the sparse signal approximation. In this paper, we deal with a special class of sparse signals with binary entries. Unlike conventional CS approaches based on l 1 minimization, we model the CS process with a bi-partite graph. We design a novel sampling matrix with unique sum property, which can be universally applied to any binary signal. Moreover, a novel binary CS decoding algorithm (BCS) based on graph and unique sum table, which does not need complex optimization process, is proposed. Proposed method is verified and compared with existing solutions through mathematical analysis and numerical simulations.
@cite_11: We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the least l1-norm solution of the underdetermined linear system Ax = b and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix A at any time. We consider two scenarios in which either the columns or the rows of A are distributed among the compute nodes. Our algorithm, named D-ADMM, is a decentralized implementation of the alternating direction method of multi- pliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the state-of-the-art algorithms.
There is significant prior work on cognitive radio and compressed sensing (CS); we have focused on the literature that is most relevant to our current problem framework. Centralized schemes for the tracking of sparse time-varying processes have been examined in @cite_13 @cite_17 @cite_4 and distributed CS has been studied in @cite_16 @cite_11 for signals. In contrast to these two veins, we study distributed CS for time-varying signals. Performance guarantees for recursive reconstruction of sparse signals under the assumption of slow support changes is studied in @cite_10 ; however, joint sensing and control is not examined. Recovery of static binary sparse signals via CS has been investigated in @cite_31 @cite_20 . Compressive spectrum sensing has been studied in @cite_25 , for a static setting, and in @cite_1 , for a dynamic setting with noiseless measurements, but without scheduling. We do not focus on recovery guarantees herein, but rather embed CS into a control framework wherein the number of measurements is adapted based on prior information in order to drive traffic scheduling decisions.