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Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
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Cited by 98 (15 self)
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
Y Eldar, Noise folding in compressed sensing
 IEEE Signal Process. Lett
, 2011
"... Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show th ..."
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Cited by 23 (4 self)
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Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show that, for the vast majority of measurement schemes employed in compressed sensing, the two models are equivalent with the important difference that the signaltonoise ratio (SNR) is divided by a factor proportional to,whereis the dimension of the signal and is the number of observations. Since is often large, this leads to noise folding which can have a severe impact on the SNR. Index Terms—Analog noise versus digital noise, compressed sensing, matching pursuit, noise folding, sparse signals. I.
Spatial Compressive Sensing in MIMO Radar with Random Arrays
 in Proc. CISS 2012
"... Abstract — We study compressive sensing in the spatial domain for target localization using MIMO radar. By leveraging a joint sparse representation, we extend the singlepulse framework proposed in [1] to a multipulse one. For this scenario, we devise a treebased matching pursuit algorithm to solv ..."
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Cited by 3 (2 self)
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Abstract — We study compressive sensing in the spatial domain for target localization using MIMO radar. By leveraging a joint sparse representation, we extend the singlepulse framework proposed in [1] to a multipulse one. For this scenario, we devise a treebased matching pursuit algorithm to solve the nonconvex localization problem. It is shown that this method can achieve high resolution target localization with a highly undersampled MIMO radar with transmit/receive elements placed at random. Moreover, a lower bound is developed on the number of transmit/receive elements required to ensure accurate target localization with high probability. I.
Robust Spike Train Recovery from Noisy Data by Structured Low Rank Approximation
"... Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approxima ..."
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Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approximation. We propose a new heuristic iterative optimization algorithm to solve it. Although it comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same implementation ease and speed. I. INTRODUCTION AND PROBLEM FORMULATION Reconstruction of signals lying in linear spaces, including bandlimited signals and splines, has long been the dominant
Subspace Recovery From Structured Union of Subspaces
"... Abstract — Lower dimensional signal representation schemes frequently assume that the signal of interest lies in a single vector space. In the context of the recently developed theory of compressive sensing, it is often assumed that the signal of interest is sparse in an orthonormal basis. However, ..."
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Abstract — Lower dimensional signal representation schemes frequently assume that the signal of interest lies in a single vector space. In the context of the recently developed theory of compressive sensing, it is often assumed that the signal of interest is sparse in an orthonormal basis. However, in many practical applications, this requirement may be too restrictive. A generalization of the standard sparsity assumption is that the signal lies in a union of subspaces. Recovery of such signals from a small number of samples has been studied recently in several works. Here, we consider the problem of only subspace recovery in which our goal is to identify the subspace (from the union) in which the signal lies using a small number of samples, in the presence of noise. More specifically, we derive performance bounds and conditions under which reliable subspace recovery is guaranteed using maximum likelihood (ML) estimation. We begin by treating general unions and then obtain the results for the special case in which the subspaces have structure leading to block sparsity. In our analysis, we treat both general sampling operators and random sampling matrices. With general unions, we show that under certain conditions, the number of measurements required for reliable subspace recovery in the presence of noise via ML is less than that implied using the restricted isometry property, which guarantees complete signal recovery. In the special case of block sparse signals, we quantify the gain achievable over standard sparsity in subspace recovery. Our results also strengthen existing results on sparse support recovery in the presence of noise under the standard sparsity model. Index Terms — Maximum likelihood estimation, union of linear subspaces, subspace recovery, compressive sensing, block sparsity. I.
A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
"... We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve ..."
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We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed. Key words and phrases: Dirac pulses, spike train, finite rate of innovation, superresolution, sparse recovery, structured low rank approximation, alternating projections, Cadzow denoising
Subspace Communication
, 2014
"... We are surrounded by electronic devices that take advantage of wireless technologies, from our computer mice, which require little amounts of information, to our cellphones, which demand increasingly higher data rates. Until today, the coexistence of such a variety of services has been guaranteed by ..."
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We are surrounded by electronic devices that take advantage of wireless technologies, from our computer mice, which require little amounts of information, to our cellphones, which demand increasingly higher data rates. Until today, the coexistence of such a variety of services has been guaranteed by a fixed assignment of spectrum resources by regulatory agencies. This has resulted into a blind alley, as current wireless spectrum has become an expensive and a scarce resource. However, recent measurements in dense areas paint a very different picture: there is an actual underutilization of the spectrum by legacy systems. Cognitive radio exhibits a tremendous promise for increasing the spectral efficiency for future wireless systems. Ideally, new secondary users would have a perfect panorama of the spectrum usage, and would opportunistically communicate over the available resources without degrading the primary systems. Yet in practice, monitoring the spectrum resources, detecting available resources for opportunistic communication, and transmitting over the resources are hard tasks. This thesis addresses the tasks of monitoring, de
FORMULATION
"... Superresolution consists in recovering the fine details of a signal from lowresolution measurements. Here we consider the estimation of Dirac pulses with positive amplitudes at arbitrary locations, from noisy lowpassfiltered samples. Maximumlikelihood estimation of the unknown parameters amount ..."
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Superresolution consists in recovering the fine details of a signal from lowresolution measurements. Here we consider the estimation of Dirac pulses with positive amplitudes at arbitrary locations, from noisy lowpassfiltered samples. Maximumlikelihood estimation of the unknown parameters amounts to a difficult nonconvex matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, yielding stateoftheart results. Index Terms — Dirac pulses, sparse spike deconvolution, superresolution, structured low rank approximation
SubNyquist Sampling for TDR Sensors: Finite Rate of Innovation with Dithering
"... is applied to time domain reflectometry and it is aimed at significantly reducing the data acquisition requirements. The sensitivity of FRI to quantisation noise is addressed given the stringent practical constraints on the resolution of the deployed analogue to digital converters in miniature refle ..."
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is applied to time domain reflectometry and it is aimed at significantly reducing the data acquisition requirements. The sensitivity of FRI to quantisation noise is addressed given the stringent practical constraints on the resolution of the deployed analogue to digital converters in miniature reflectometry sensors. Dithering with averaging is proposed to combat the effects of quantisation noise whilst maintaining remarkably low operational sampling rates. The substantial benefits of the adopted FRIbased reflectometry is demonstrated in the presented simulations. The tradeoff between the resolution of the quantiser, time averaging and sampling rate is also depicted in terms of the quality of the signal recovery attained from the subNyquist FRI samples. reference pulse impedance mismatch endofprobe reflection I.