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14
Spectral Compressive Sensing
, 2010
"... Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency do ..."
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Cited by 39 (5 self)
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Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency domain. In practical applications, the standard frequency domain signal representation is the discrete Fourier transform (DFT). Unfortunately, the DFT coefficients of a frequencysparse signal are themselves sparse only in the contrived case where the sinusoid frequencies are integer multiples of the DFT’s fundamental frequency. As a result, practical DFTbased CS acquisition and recovery of smooth signals does not perform nearly as well as one might expect. In this paper, we develop a new spectral compressive sensing (SCS) theory for general frequencysparse signals. The key ingredients are an oversampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectrum estimation. Using peridogram and eigenanalysis based spectrum estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current stateoftheart CS algorithms while providing provable bounds on the number of measurements required for stable recovery.
Wideband spectrum sensing from compressed measurements using spectral prior information
 IEEE Trans. Signal Process
, 2013
"... Abstract—Wideband spectrum sensing (WSS) encompasses a collection of techniques intended to estimate or to decide over the occupancy parameters of a wide frequency band. However, broad bands require expensive acquisition systems, thus motivating the use of compressive schemes. In this context, prev ..."
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Cited by 10 (6 self)
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Abstract—Wideband spectrum sensing (WSS) encompasses a collection of techniques intended to estimate or to decide over the occupancy parameters of a wide frequency band. However, broad bands require expensive acquisition systems, thus motivating the use of compressive schemes. In this context, previous works in compressive WSS have already realized that great compression rates can be achieved if only secondorder statistics are of interest in spectrum sensing. In this paper, we go a step further by exploiting spectral prior information that is typically available in practice in order to reduce the sampling rate even more. The signal model assumes that the acquisition is done by means of an analogtoinformation converter (A2I). The input signal is the linear combination of a number of signals whose secondorder statistics are known and the goal is to estimate/decide over the coefficients of this combination. The problem is thus a particular instance of the wellknown structured covariance estimation problem. Unfortunately, the algorithms used in this area are extremely complex for inexpensive spectrum sensors so that alternative techniques need to be devised. Exploiting the fact that the basis matrices are Toeplitz, we use the asymptotic theory of circulant matrices to propose a dimensionality reduction technique that simplifies existing structured covariance estimation algorithms, achieving a similar performance at a much lower computational cost. Index Terms—Analogtoinformation converters, compressed sensing, covariance matching, wideband spectrum sensing.
SubNyquist radar via Doppler focusing
 IEEE Transactions on Signal Processing
"... Abstract—We investigate the problem of a monostatic pulseDoppler radar transceiver trying to detect targets sparsely populated in the radar’s unambiguous timefrequency region. Several past works employ compressed sensing (CS) algorithms to this type of problem but either do not address sample rate ..."
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Cited by 10 (5 self)
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Abstract—We investigate the problem of a monostatic pulseDoppler radar transceiver trying to detect targets sparsely populated in the radar’s unambiguous timefrequency region. Several past works employ compressed sensing (CS) algorithms to this type of problem but either do not address sample rate reduction, impose constraints on the radar transmitter, propose CS recovery methods with prohibitive dictionary size, or perform poorly in noisy conditions. Here, we describe a subNyquist sampling and recovery approach called Doppler focusing, which addresses all of these problems: it performs low rate sampling and digital processing, imposes no restrictions on the transmitter, and uses a CS dictionary with size, which does not increase with increasing number of pulses. Furthermore, in the presence of noise, Doppler focusing enjoys a signaltonoise ratio (SNR) improvement, which scales linearly with, obtaining good detection performance even at SNR as low as 25 dB. The recovery is based on the Xampling framework, which allows reduction of the number of samples needed to accurately represent the signal, directly in the analogtodigital conversion process. After sampling, the entire digital recovery process is performed on the low rate samples without having to return to the Nyquist rate. Finally, our approach can be implemented in hardware using a previously suggested Xampling radar prototype. Index Terms—Compressed sensing, rate of innovation, radar, sparse recovery, subNyquist sampling, delayDoppler estimation. I.
1 Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
, 2013
"... 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 a difficult, even believed NPhard, matrix problem of structured low rank approximation. T ..."
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Cited by 7 (4 self)
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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 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. Index Terms—Recovery of Dirac pulses, spike train, finite rate of innovation, superresolution, spectral estimation, maximum
Xampling in Ultrasound Imaging
, 2011
"... Recent developments of new medical treatment techniques put challenging demands on ultrasound imaging systems in terms of both image quality and raw data size. Traditional sampling methods result in very large amounts of data, thus, increasing demands on processing hardware and limiting the flexibil ..."
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Cited by 3 (1 self)
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Recent developments of new medical treatment techniques put challenging demands on ultrasound imaging systems in terms of both image quality and raw data size. Traditional sampling methods result in very large amounts of data, thus, increasing demands on processing hardware and limiting the flexibility in the postprocessing stages. In this paper, we apply Compressed Sensing (CS) techniques to analog ultrasound signals, following the recently developed Xampling framework. The result is a system with significantly reduced sampling rates which, in turn, means significantly reduced data size while maintaining the quality of the resulting images.
SubNyquist sampling for power spectrum sensing in cognitive radios: A unified approach
 CoRR
"... Abstract—In light of the everincreasing demand for new spectral bands and the underutilization of those already allocated, the concept of Cognitive Radio (CR) has emerged. Opportunistic users could exploit temporarily vacant bands after detecting the absence of activity of their owners. One of the ..."
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Cited by 2 (1 self)
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Abstract—In light of the everincreasing demand for new spectral bands and the underutilization of those already allocated, the concept of Cognitive Radio (CR) has emerged. Opportunistic users could exploit temporarily vacant bands after detecting the absence of activity of their owners. One of the crucial tasks in the CR cycle is therefore spectrum sensing and detection which has to be precise and efficient. Yet, CRs typically deal with wideband signals whose Nyquist rates are very high. In this paper, we propose to reconstruct the power spectrum of such signals from subNyquist samples, rather than the signal itself as done in previous work, in order to perform detection. We consider both sparse and non sparse signals as well as blind and non blind detection in the sparse case. For each one of those scenarios, we derive the minimal sampling rate allowing perfect reconstruction of the signal’s power spectrum in a noisefree environment and provide power spectrum recovery techniques that achieve those rates. The analysis is performed for two different signal models considered in the literature, which we refer to as the analog and digital models, and shows that both lead to similar results. Simulations demonstrate power spectrum recovery at the minimal rate in noisefree settings and the impact of several parameters on the detector performance, including signaltonoise ratio, sensing time and sampling rate. Index Terms—Cognitive radio, spectrum sensing, subNyquist sampling, compressed sensing (CS), power spectrum reconstruction. 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
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
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
1Channel Capacity under SubNyquist Nonuniform Sampling
"... Abstract—This paper investigates the effect of subNyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear timeinvariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of rig ..."
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Abstract—This paper investigates the effect of subNyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear timeinvariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of rightinvertible timepreserving sampling methods which include irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signaltonoise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while typically complicated to realize, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain, which is in contrast to the benefits obtained from random mixing in spectrumblind compressive sampling schemes. Index Terms—nonuniform sampling, irregular sampling, sampled analog channels, subNyquist sampling, channel capacity, Beurling density, timepreserving sampling systems I.