Results 1  10
of
158
Robust Recovery of Signals From a Structured Union of Subspaces
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
Abstract

Cited by 221 (47 self)
 Add to MetaCart
(Show Context)
Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a blocksparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
, 2009
"... Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, alt ..."
Abstract

Cited by 158 (18 self)
 Add to MetaCart
Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.
Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
Abstract

Cited by 104 (16 self)
 Add to MetaCart
Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
Compressed Sensing of Analog Signals in ShiftInvariant Spaces
, 2009
"... A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that on ..."
Abstract

Cited by 74 (41 self)
 Add to MetaCart
(Show Context)
A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for lowrate sampling of continuoustime sparse signals in shiftinvariant (SI) spaces, generated by m kernels with period T. We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m/T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finitelength vectors, we consider sampling of analog signals for which no underlying finitedimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.
Multichannel sampling of pulse streams at the rate of innovation
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... We consider minimalrate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of ..."
Abstract

Cited by 51 (9 self)
 Add to MetaCart
We consider minimalrate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on subNyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimalrate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.
Robust recovery of signals from a union of subspaces
 IEEE TRANS. INFORM. THEORY
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees a unique signal consistent with the given measurements is that x lies in a known subspace. Recently, there has been growing interest in non ..."
Abstract

Cited by 44 (14 self)
 Add to MetaCart
Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees a unique signal consistent with the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x is assumed to lie in a union of subspaces. An example is the case in which x is a finite length vector that is sparse in a given basis. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a finite union of finite dimensional spaces and the samples are modelled as inner products with an arbitrary set of sampling functions. We first develop conditions under which unique and stable recovery of x is possible, albeit with algorithms that have combinatorial complexity. To derive an efficient and robust recovery algorithm, we then show that our problem can be formulated as that of recovering a block sparse vector, namely a vector whose nonzero elements appear in fixed blocks. To solve this problem, we suggest minimizing a mixed ℓ2/ℓ1 norm subject to the measurement equations. We then develop equivalence conditions under which the proposed convex algorithm is guaranteed to recover the original signal. These results rely on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. A special case of the proposed framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Specializing our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Xampling: Signal acquisition and processing in union of subspaces
, 2011
"... We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two: Analog compression that narrows down the input bandwidth prior to sampling with commercial devices followed by a nonlinear algorithm that ..."
Abstract

Cited by 43 (21 self)
 Add to MetaCart
(Show Context)
We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two: Analog compression that narrows down the input bandwidth prior to sampling with commercial devices followed by a nonlinear algorithm that detects the input subspace prior to conventional signal processing. A representative union model of spectrally sparse signals serves as a testcase to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy, and software complexities. We conduct a comprehensive comparison between two subNyquist acquisition strategies for spectrally sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address low rate signal processing and develop an algorithm for that purpose that enables convenient signal processing at subNyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.
Generalized sampling and infinitedimensional compressed sensing
"... We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demo ..."
Abstract

Cited by 33 (20 self)
 Add to MetaCart
(Show Context)
We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finitedimensional techniques are illsuited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.
Timedelay estimation from lowrate samples: a union of subspaces approach
 IEEE TRANS. SIGNAL PROCESS
, 2010
"... Timedelay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In ..."
Abstract

Cited by 33 (19 self)
 Add to MetaCart
(Show Context)
Timedelay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In this paper, we develop a unified approach to time delay estimation from lowrate samples. This problem can be formulated in the broader context of sampling over an infinite union of subspaces. Although sampling over unions of subspaces has been receiving growing interest, previous results either focus on unions of finitedimensional subspaces, or finite unions. The framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses, and allows for a variety of different sampling methods. The sampling rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. This result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces. By properly manipulating the lowrate samples, we show that the time delays can be recovered using the wellknown ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation, we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.