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103
Compressed Sensing Off the Grid
, 2012
"... This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized ..."
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Cited by 25 (1 self)
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This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(s log s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Numerical experiments are performed to illustrate the effectiveness of the proposed method.
Compressed Synthetic Aperture Radar
, 2010
"... In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a highresolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, ..."
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Cited by 24 (3 self)
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In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a highresolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, requires no new hardware components and allows the aperture to be compressed. It also presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced onboard storage requirements.
A compressive sensing data acquisition and imaging method for stepped frequency GPRs
 IEEE Trans. Geosci. Remote Sens
, 2009
"... Abstract—A novel data acquisition and imaging method is presented for steppedfrequency continuouswave ground penetrating radars (SFCW GPRs). It is shown that if the target space is sparse, i.e., a small number of point like targets, it is enough to make measurements at only a small number of ran ..."
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Cited by 18 (1 self)
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Abstract—A novel data acquisition and imaging method is presented for steppedfrequency continuouswave ground penetrating radars (SFCW GPRs). It is shown that if the target space is sparse, i.e., a small number of point like targets, it is enough to make measurements at only a small number of random frequencies to construct an image of the target space by solving a convex optimization problem which enforces sparsity through minimization. This measurement strategy greatly reduces the data acquisition time at the expense of higher computational costs. Imaging results for both simulated and experimental GPR data exhibit less clutter than the standard migration methods and are robust to noise and random spatial sampling. The images also have increased resolution where closely spaced targets that cannot be resolved by the standard migration methods can be resolved by the proposed method. Index Terms—Compressive sensing, minimization, ground penetrating radar (GPR), sparsity, stepped frequency systems, subsurface imaging. I.
Sparse representation of a polytope and recovery of a sparse signals and lowrank matrices
 IEEE Transactions on Information Theory
, 2014
"... This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and lowrank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex comb ..."
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Cited by 17 (1 self)
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This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and lowrank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant t ≥ 4/3, in compressed sensing δAtk < (t − 1)/t guarantees the exact recovery of all k sparse signals in the noiseless case through the constrained `1 minimization, and similarly in affine rank minimization δ M tr < (t − 1)/t ensures the exact reconstruction of all matrices with rank at most r in the noiseless case via the constrained nuclear norm minimization. Moreover, for any > 0, δAtk < t−1 t + is not sufficient to guarantee the exact recovery of all ksparse signals for large k. Similar result also holds for matrix recovery. In addition, the conditions δAtk < (t − 1)/t and δMtr < (t − 1)/t are also shown to be sufficient respectively for stable recovery of approximately sparse signals and lowrank matrices in the noisy case.
Performance Analysis for Sparse Support Recovery
, 2009
"... In this paper, the performance of estimating the common support for jointly sparse signals based on their projections onto lowerdimensional space is analyzed. Support recovery is formulated as a multiplehypothesis testing problem and both upper and lower bounds on the probability of error are deri ..."
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Cited by 17 (1 self)
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In this paper, the performance of estimating the common support for jointly sparse signals based on their projections onto lowerdimensional space is analyzed. Support recovery is formulated as a multiplehypothesis testing problem and both upper and lower bounds on the probability of error are derived for general measurement matrices, by using the Chernoff bound and Fano’s inequality, respectively. The form of the upper bound shows that the performance is determined by a single quantity that is a measure of the incoherence of the measurement matrix, while the lower bound reveals the importance of the total measurement gain. To demonstrate its immediate applicability, the lower bound is applied to derive the minimal number of samples needed for accurate direction of arrival (DOA) estimation for an algorithm based on sparse representation. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions to guarantee a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into the sparse signal reconstruction based on their projections. For example, as far as support recovery is concerned, the wellknown bound in compressive sensing is generally not sufficient if the Gaussian ensemble is used. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in compressive sensing as well as other application areas taking advantage of signal sparsity.
Sampling and Recovery of Pulse Streams
"... Compressive Sensing (CS) is a new technique for the efficient acquisition of signals, images, and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the Ndimensional basis representation has just K ≪ N significant coefficients; in this case, the ..."
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Cited by 13 (1 self)
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Compressive Sensing (CS) is a new technique for the efficient acquisition of signals, images, and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the Ndimensional basis representation has just K ≪ N significant coefficients; in this case, the CS theory maintains that just M = O (K log N) random linear signal measurements will both preserve all of the signal information and enable robust signal reconstruction in polynomial time. In this paper, we extend the CS theory to pulse stream data, which correspond to Ssparse signals/images that are convolved with an unknown Fsparse pulse shape. Ignoring their convolutional structure, a pulse stream signal is K = SF sparse. Such signals figure prominently in a number of applications, from neuroscience to astronomy. Our specific contributions are threefold. First, we propose a pulse stream signal model and show that it is equivalent to an infinite union of subspaces. Second, we derive a lower bound on the number of measurements M required to preserve the essential information present in pulse streams. The bound is linear in the total number of degrees of freedom S + F, which is significantly smaller than the naive bound based on the total signal sparsity K = SF. Third, we develop an efficient signal recovery algorithm that infers both the shape of the impulse response as well as the locations and
A compressive beamforming method
"... Compressive Sensing (CS) is an emerging area which uses a relatively small number of nontraditional samples in the form of randomized projections to reconstruct sparse or compressible signals. This paper considers the directionofarrival (DOA) estimation problem with an array of sensors using CS. ..."
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Cited by 13 (1 self)
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Compressive Sensing (CS) is an emerging area which uses a relatively small number of nontraditional samples in the form of randomized projections to reconstruct sparse or compressible signals. This paper considers the directionofarrival (DOA) estimation problem with an array of sensors using CS. We show that by using random projections of the sensor data, along with a full waveform recording on one reference sensor, a sparse angle space scenario can be reconstructed, giving the number of sources and their DOA’s. The number of projections can be very small, proportional to the number sources. We provide simulations to demonstrate the performance and the advantages of our compressive beamformer algorithm.
Compressive mechanism: utilizing sparse representation in differential privacy
 In Proc. of WPES ’11, ACM
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Compressive sensing in nonstationary array processing using bilinear transforms
 in Proc. IEEE
, 2012
"... Compressive sensing (CS) has successfully been applied to reconstruct sparse signals and images from few observations. For multicomponent nonstationary signals characterized by instantaneous frequency laws, the sparsity exhibits itself in the timefrequency domain as well as the ambiguity domain ..."
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Cited by 11 (10 self)
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Compressive sensing (CS) has successfully been applied to reconstruct sparse signals and images from few observations. For multicomponent nonstationary signals characterized by instantaneous frequency laws, the sparsity exhibits itself in the timefrequency domain as well as the ambiguity domain. In this paper, we examine CS in the context of nonstationary array processing. We show that the spatial averaging of the ambiguity function across the array improves the CS performance by reducing both noise and crossterms. The corresponding timefrequency distribution which is reconstructed through L1 minimizations yields significant improvement in timefrequency signature localizations and characterizations. 1.
Petropulu A: Stepfrequency radar with compressive sampling (SFRCS
 In Proc. IEEE Int Acoustics Speech and Signal Processing (ICASSP) Conf 2010:1686–1689
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