Results 11  20
of
153
Restricted isometries for partial random circulant matrices
 APPL. COMPUT. HARMON. ANAL
, 2010
"... In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampl ..."
Abstract

Cited by 47 (8 self)
 Add to MetaCart
(Show Context)
In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a dataacquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the sth order restricted isometry constant is small when the number m of samples satisfies m � (s log n) 3/2, where n is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.
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 ..."
Abstract

Cited by 39 (5 self)
 Add to MetaCart
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.
Why Gabor frames? Two fundamental measures of coherence and their role in model selection
 J. Commun. Netw
, 2010
"... ar ..."
(Show Context)
Suprema of chaos processes and the restricted isometry property
 Comm. Pure Appl. Math
"... We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m � s log 2 s log 2 n. Key words. Compressive sensing, restricted isometry property, structured random matrices, chaos processes, γ2functionals, generic chaining, partial random circulant matrices, random Gabor synthesis matrices.
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 ..."
Abstract

Cited by 28 (2 self)
 Add to MetaCart
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.
ON THE IDENTIFICATION OF PARAMETRIC UNDERSPREAD LINEAR SYSTEMS
"... Identification of timevarying linear systems, which introduce both timeshifts (delays) and frequencyshifts (Dopplershifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as timevarying linear sy ..."
Abstract

Cited by 28 (9 self)
 Add to MetaCart
(Show Context)
Identification of timevarying linear systems, which introduce both timeshifts (delays) and frequencyshifts (Dopplershifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as timevarying linear systems whose responses lie within a unitarea region in the delay–Doppler space, by probing them with a known input signal. The main contribution of the paper is that it characterizes conditions on the bandwidth and temporal support of the input signal that ensure identification of ULSs described by a finite set of delays and Dopplershifts, and referred to as parametric ULSs, from single observations. In particular, the paper establishes that sufficientlyunderspread parametric linear systems are identifiable as long as the time–bandwidth product of the input signal is proportional to the square of the total number of delay–Doppler pairs in the system. In addition, the paper describes a procedure that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on subNyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. 1.
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 ..."
Abstract

Cited by 27 (8 self)
 Add to MetaCart
(Show Context)
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.
Asymptotic analysis of complex LASSO via complex approximate message passing
 IEEE Trans. Inf. Theory
, 2011
"... Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complexvalued. We study the popular reconstruction method of ℓ1regularized lea ..."
Abstract

Cited by 27 (9 self)
 Add to MetaCart
(Show Context)
Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complexvalued. We study the popular reconstruction method of ℓ1regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complexvalued signals and measurements to obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP. Our results are theoretically proved for the case of i.i.d. Gaussian sensing matrices. But we confirm through simulations that our results hold for larger class of random matrices. 1
Optimal phase transitions in compressed sensing
 IEEE TRANS. INF. THEORY
, 2012
"... Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonlinear, opti ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonlinear, optimal linear, and random linear encoders. Focusing on optimal decoders, we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise, respectively. The optimal phasetransition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms. In particular, we show that Gaussian sensing matrices incur no penalty on the phasetransition threshold with respect to optimal nonlinear encoding. Our results also provide a rigorous justification of previous results based on replica heuristics in the weaknoise regime.
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, ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
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.