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24
Sensing by Random Convolution
 IEEE Int. Work. on Comp. Adv. MultiSensor Adaptive Proc., CAMPSAP
, 2007
"... Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in a ..."
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Cited by 65 (4 self)
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Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in any fixed representation can be recovered from m � S log n measurements. We discuss two imaging scenarios — radar and Fourier optics — where convolution with a random pulse allows us to seemingly superresolve finescale features, allowing us to recover highresolution signals from lowresolution measurements. 1. Introduction. The new field of compressive sensing (CS) has given us a fresh look at data acquisition, one of the fundamental tasks in signal processing. The message of this theory can be summarized succinctly [7, 8, 10, 15, 32]: the number of measurements we need to reconstruct a signal depends on its sparsity rather than its bandwidth. These measurements, however, are different than the samples that
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 59 (13 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
Toeplitz compressed sensing matrices with applications to sparse channel estimation,” submitted
 Online]. Available: http://www.ece.wisc.edu/ ∼nowak/sub08 toep.pdf
"... Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entri ..."
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Cited by 43 (8 self)
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Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zeromean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, timeinvariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that timedomain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional leastsquares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies. Index Terms circulant matrices, compressed sensing, Hankel matrices, restricted isometry property, sparse channel estimation, Toeplitz matrices, wireless communications. I.
Compressed channel sensing
 in Proc. of Conf. on Information Sciences and Systems (CISS
, 2008
"... Abstract—Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many realworld ..."
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Cited by 40 (9 self)
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Abstract—Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many realworld channels of practical interest tend to exhibit impulse responses characterized by a relatively small number of nonzero channel coefficients. Conventional linear channel estimation strategies, such as the least squares, are illsuited to fully exploiting the inherent lowdimensionality of these sparse channels. In contrast, this paper proposes sparse channel estimation methods based on convex/linear programming. Quantitative error bounds for the proposed schemes are derived by adapting recent advances from the theory of compressed sensing. The bounds come within a logarithmic factor of the performance of an ideal channel
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 23 (8 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Learning sparse doublyselective channels
 in Proc. of Allerton Conf. on Communications, Control and Computing
, 2008
"... Abstract—Coherent data communication over doublyselective channels requires that the channel response be known at the receiver. Trainingbased schemes, which involve probing of the channel with known signaling waveforms and processing of the corresponding channel output to estimate the channel para ..."
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Cited by 14 (5 self)
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Abstract—Coherent data communication over doublyselective channels requires that the channel response be known at the receiver. Trainingbased schemes, which involve probing of the channel with known signaling waveforms and processing of the corresponding channel output to estimate the channel parameters, are commonly employed to learn the channel response in practice. Conventional trainingbased methods, often comprising of linear least squares channel estimators, are known to be optimal under the assumption of rich multipath channels. Numerous measurement campaigns have shown, however, that physical multipath channels tend to exhibit a sparse structure at high signal space dimension (timebandwidth product), and can be characterized with significantly fewer parameters compared to the maximum number dictated by the delayDoppler spread of the channel. In this paper, it is established that traditional trainingbased channel learning techniques are illsuited to fully exploiting the inherent lowdimensionality of sparse channels. In contrast, key ideas from the emerging theory of compressed sensing are leveraged to propose sparse channel learning methods for both singlecarrier and multicarrier probing waveforms that employ reconstruction algorithms based on convex/linear programming. In particular, it is shown that the performance of the proposed schemes come within a logarithmic factor of that of an ideal channel estimator, leading to significant reductions in the training energy and the loss in spectral efficiency associated with conventional trainingbased methods. I.
COMPRESSED REMOTE SENSING OF SPARSE OBJECTS
"... Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a logsq ..."
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Cited by 13 (5 self)
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Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a logsquare factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The prediction of theorems are confirmed by numerical simulations. 1.
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 13 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Sparsity in timefrequency representations
, 2007
"... We consider signals and operators in finite dimension which have sparse timefrequency representations. As main result we show that an Ssparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn / log(n) ..."
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Cited by 11 (6 self)
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We consider signals and operators in finite dimension which have sparse timefrequency representations. As main result we show that an Ssparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn / log(n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.