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38
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 ..."
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Cited by 218 (48 self)
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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.
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 ..."
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Cited by 74 (41 self)
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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.
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 ..."
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Cited by 45 (14 self)
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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 ..."
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Cited by 42 (21 self)
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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 ..."
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Cited by 31 (19 self)
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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.
Nonideal Sampling and Regularization Theory
, 2008
"... Shannon’s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shiftinvariant” space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressi ..."
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Cited by 15 (3 self)
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Shannon’s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shiftinvariant” space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressing the issue of the specification of the best reconstruction space. We consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. We adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a Tikhonovlike (continuous domain) 2regularization to obtain the continuousspace solution. We present theoretical justification for the minimization of this cost functional and show that the globally minimal continuousspace solution belongs to a shiftinvariant space generated by a function (generalized Bspline) that is generally not bandlimited. When the sampling is ideal, we recover some of the classical smoothing spline estimators. The optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a Bsplinelike basis. To make the scheme practical, we specify the generating functions corresponding to the most popular families of regularization operators (derivatives, iterated Laplacian), as well as a new, generalized one that leads to a new brand of Matérn splines. We conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error (MMSE/Wiener) solutions of the generalized sampling problem.
Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals
"... Abstract—In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainabl ..."
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Cited by 15 (6 self)
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Abstract—In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific samplingapproach. Essential differences between the noisy and noisefree cases arise from this analysis. In particular, we identify settings in which noisefree recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related toaspecific typeofstructure,which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels for linear reconstruction, based on a generalization of the Karhunen–Loève transform. The results are illustrated for several types of timedelay estimation problems. Index Terms—Cramér–Rao bound (CRB), finite rate of innovation (FRI), sampling, timedelay estimation, union of subspaces. I.
Nonlinear and nonideal sampling: Theory and methods
 IEEE Trans. Signal Process
, 2008
"... Abstract—We study a sampling setup where a continuoustime signal is mapped by a memoryless, invertible and nonlinear transformation, and then sampled in a nonideal manner. Such scenarios appear, for example, in acquisition systems where a sensor introduces static nonlinearity, before the signal is ..."
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Cited by 14 (12 self)
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Abstract—We study a sampling setup where a continuoustime signal is mapped by a memoryless, invertible and nonlinear transformation, and then sampled in a nonideal manner. Such scenarios appear, for example, in acquisition systems where a sensor introduces static nonlinearity, before the signal is sampled by a practical analogtodigital converter. We develop the theory and a concrete algorithm to perfectly recover a signal within a subspace, from its nonlinear and nonideal samples. Three alternative formulations of the algorithm are described that provide different insights into the structure of the solution: A series of oblique projections, approximated projections onto convex sets, and quasiNewton iterations. Using classical analysis techniques of descentbased methods, and recent results on frame perturbation theory, we prove convergence of our algorithm to the true input signal. We demonstrate our method by simulations, and explain the applicability of our theory to Wiener–Hammerstein analogtodigital hybrid systems. Index Terms—Generalized sampling, interpolation, nonlinear sampling, Wiener–Hammerstein. I.
SubNyquist Sampling  Bridging theory and practice
, 2011
"... Signal processing methods have changed substantially over the last several decades. In modern applications, an increasing number of functions is being pushed forward to sophisticated software algorithms, leaving only delicate finely tuned tasks for the circuit level. Sampling theory, the gate to th ..."
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Cited by 14 (5 self)
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Signal processing methods have changed substantially over the last several decades. In modern applications, an increasing number of functions is being pushed forward to sophisticated software algorithms, leaving only delicate finely tuned tasks for the circuit level. Sampling theory, the gate to the digital world, is the key enabling this revolution, encompassing all aspects related to the conversion of continuoustime signals to discrete streams of numbers. The famous ShannonNyquist theorem has become a landmark: a mathematical statement that has had one of the most profound impacts on industrial development of digital signal processing (DSP) systems. Over the years, theory and practice in the field of sampling have developed in parallel routes. Contributions by many research groups suggest a multitude of methods, other than uniform sampling, to acquire analog signals [1]–[6]. The math has deepened, leading to abstract signal spaces and innovative sampling techniques. Within generalized sampling theory, bandlimited signals have no special preference, other than historic. At the same time, the market adhered to the Nyquist paradigm;