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88
Sensitivity to basis mismatch of compressed sensing,” preprint
, 2009
"... Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the c ..."
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Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in passive and active radar and sonar, this basis is usually taken to be a DFT basis. The compressed sensing measurements are then inverted using an ℓ1-minimization principle (basis pursuit) for the nonzero source amplitudes. This seems to make compressed sensing an ideal image inversion principle for high resolution modal analysis. However, in reality no physical field is sparse in the DFT basis or in an apriori known basis. In fact the main goal in image inversion is to identify the modal structure. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and there is always mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of basis pursuit to mismatch between the assumed and the actual sparsity bases and compare the performance of basis pursuit with that of classical image inversion. Our mathematical analysis and numerical examples show that the performance of basis pursuit degrades considerably in the presence of mismatch, and they suggest that the use of compressed sensing as a modal analysis principle requires more consideration and refinement, at least for the problem sizes common to radar/sonar. I.
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 time-frequency structured ..."
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Cited by 33 (6 self)
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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 time-frequency 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, γ2-functionals, 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 ..."
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Cited by 28 (2 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.
ON THE IDENTIFICATION OF PARAMETRIC UNDERSPREAD LINEAR SYSTEMS
"... Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as time-varying linear sy ..."
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Cited by 28 (9 self)
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Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as time-varying linear systems whose responses lie within a unit-area 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 Doppler-shifts, and referred to as parametric ULSs, from single observations. In particular, the paper establishes that sufficiently-underspread 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 sub-Nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. 1.
Online Sparse System Identification and Signal Reconstruction using Projections onto Weighted ℓ1 Balls
- IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2010
"... This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab is the geometric equivalent of a cost criterion, that quant ..."
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Cited by 24 (3 self)
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This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab is the geometric equivalent of a cost criterion, that quantifies “data mismatch”. Sparsity is imposed by the introduction of appropriately designed weighted ℓ1 balls and the related projection operator is also derived. The algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted ℓ1 balls. The resulting scheme exhibits linear dependence, with respect to the unknown system’s order, on the number of multiplications/additions and an O(L log2 L) dependence on sorting operations, where L is the length of the system/signal to be estimated. Numerical results are also given to validate the performance of the proposed method against the LASSO algorithm and two very recently developed adaptive sparse schemes that fuse arguments from the LMS / RLS adaptation mechanisms with those imposed by the lasso rational.
Application of Compressive Sensing to Sparse Channel Estimation
- IEEE communications magazine
"... Abstract-Compressive sensing is a topic that has recently gained much attention in the applied mathematics and signal processing communities. It has been applied in various areas, such as imaging, radar, speech recognition, and data acquisition. In communications, compressive sensing is largely acc ..."
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Cited by 18 (1 self)
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Abstract-Compressive sensing is a topic that has recently gained much attention in the applied mathematics and signal processing communities. It has been applied in various areas, such as imaging, radar, speech recognition, and data acquisition. In communications, compressive sensing is largely accepted for sparse channel estimation and its variants. In this paper we highlight the fundamental concepts of compressive sensing and give an overview of its application to pilot aided channel estimation. We point out that a popular assumption -that multipath channels are sparse in their equivalent baseband representation -has pitfalls. There are overcomplete dictionaries that lead to much sparser channel representations and better estimation performance. As a concrete example, we detail the application of compressive sensing to multicarrier underwater acoustic communications, where the channel features sparse arrivals, each characterized by its distinct delay and Doppler scale factor. To work with practical systems, several modifications need to be made to the compressive sensing framework as the channel estimation error varies with how detailed the channel is modeled, and how data and pilot symbols are mixed in the signal design.
A Message-Passing Receiver for BICM-OFDM over Unknown Clustered-Sparse Channels
"... We propose a factor-graph-based approach to joint channel-estimationand-decoding of bit-interleaved coded orthogonal frequency division multiplexing (BICM-OFDM). In contrast to existing designs, ours is capable of exploiting not only sparsity in sampled channel taps but also clustering among the lar ..."
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Cited by 16 (7 self)
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We propose a factor-graph-based approach to joint channel-estimationand-decoding of bit-interleaved coded orthogonal frequency division multiplexing (BICM-OFDM). In contrast to existing designs, ours is capable of exploiting not only sparsity in sampled channel taps but also clustering among the large taps, behaviors which are known to manifest at larger communication bandwidths. In order to exploit these channel-tap structures, we adopt a two-state Gaussian mixture prior in conjunction with a Markov model on the hidden state. For loopy belief propagation, we exploit a “generalized approximate message passing ” algorithm recently developed in the context of compressed sensing, and show that it can be successfully coupled with soft-input soft-output decoding, as well as hidden Markov inference. ForN subcarriers andM bits per subcarrier (and any channel length L < N), our scheme has a computational complexity of onlyO(N log 2N+N2 M). Numerical experiments using IEEE 802.15.4a channels show that our scheme yields BER performance within 1 dB of the known-channel bound and 4 dB better than decoupled channel-estimation-and-decoding via LASSO. 1.
Efficient high-dimensional inference in the multiple measurement vector problem.” arXiv:1111.5272 [cs.IT
, 2011
"... Abstract—In this work, a Bayesian approximate message passing algorithm is proposed for solving the multiple measurement vector (MMV) problem in compressive sensing, in which a collection of sparse signal vectors that share a common support are recovered from undersampled noisy measurements. The alg ..."
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Cited by 10 (3 self)
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Abstract—In this work, a Bayesian approximate message passing algorithm is proposed for solving the multiple measurement vector (MMV) problem in compressive sensing, in which a collection of sparse signal vectors that share a common support are recovered from undersampled noisy measurements. The algorithm, AMP-MMV, is capable of exploiting temporal correlations in the amplitudes of non-zero coefficients, and provides soft estimates of the signal vectors as well as the underlying support. Central to the proposed approach is an extension of recently developed approximate message passing techniques to the amplitude-correlated MMV setting. Aided by these techniques, AMP-MMV offers a computational complexity that is linear in all problem dimensions. In order to allow for automatic parameter tuning, an expectation-maximization algorithm that complements AMP-MMV is described. Finally, a detailed numerical study demonstrates the power of the proposed approach and its particular suitability for application to high-dimensional problems. Index Terms—Approximate message passing (AMP), belief propagation, compressed sensing, expectation-maximization algorithms, joint sparsity, Kalman filters, multiple measurement vector problem, statistical signal processing. I.
Sparsity Based Feedback Design: A New Paradigm in Opportunistic Sensing
"... Abstract — We introduce the concept of using compressive sensing techniques to provide feedback in order to control dynamical systems. Compressive sensing algorithms use l1regularization for reconstructing data from a few measurement samples. These algorithms provide highly efficient reconstruction ..."
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Cited by 8 (0 self)
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Abstract — We introduce the concept of using compressive sensing techniques to provide feedback in order to control dynamical systems. Compressive sensing algorithms use l1regularization for reconstructing data from a few measurement samples. These algorithms provide highly efficient reconstruction for sparse data. For data that is not sparse enough, the reconstruction technique produces a bounded error in the estimate. In a dynamical system, such erroneous stateestimation can lead to undesirable effects in the output of the plant. In this work, we present some techniques to overcome the aforementioned restriction. Our efforts fall into two main categories. First, we present some techniques to design feedback systems that sparsify the state in order to perfectly reconstruct it using compressive sensing algorithms. We study the effect of such sparsification schemes on the stability and regulation of the plant. Second, we study the characteristics of dynamical systems that produce sparse states so that compressive sensing techniques can be used for feedback in such scenarios without any additional modification in the feedback loop. I.
