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Exact joint sparse frequency recovery via optimization methods
, 2014
"... Frequency recovery/estimation from samples of superimposed sinusoidal signals is a classical problem in statistical signal processing. Its research has been recently advanced by atomic norm techniques which deal with continuousvalued frequencies and completely eliminate basis mismatches of existing ..."
Abstract

Cited by 4 (3 self)
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Frequency recovery/estimation from samples of superimposed sinusoidal signals is a classical problem in statistical signal processing. Its research has been recently advanced by atomic norm techniques which deal with continuousvalued frequencies and completely eliminate basis mismatches of existing compressed sensing methods. This work investigates the frequency recovery problem in the presence of multiple measurement vectors (MMVs) which share the same frequency components, termed as joint sparse frequency recovery and arising naturally from array processing applications. `0 and `1normlike formulations, referred to as atomic `0 norm and the atomic norm, are proposed to recover the frequencies and cast as (nonconvex) rank minimization and (convex) semidefinite programming, respectively. Their guarantees for exact recovery are theoretically analyzed which extend existing results with a single measurement vector (SMV) to the MMV case and meanwhile generalize the existing joint sparse compressed sensing framework to the continuous dictionary setting. In particular, given a set of N regularly spaced samples per measurement vector it is shown that the frequencies can be exactly recovered via solving a convex optimization problem once they are separate by at least (approximately) 4N. Under the same frequency separation condition, a random subset of N regularly spaced samples of size O (K logK logN) per measurement vector is sufficient to guarantee exact recovery of the K frequencies and missing samples with high probability via similar convex optimization. Extensive numerical simulations are provided to validate our analysis and demonstrate the effectiveness of the proposed method.
COMPRESSIVE SENSING BASED TARGET COUNTING AND LOCALIZATION EXPLOITING JOINT SPARSITY
"... ABSTRACT One of the fundamental issues in Wireless Sensor Networks (WSN) is to count and localize multiple targets accurately. In this context, there has been an increasing interest in the literature in using Compressive Sensing (CS) based techniques by exploiting the sparse nature of spatially dis ..."
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ABSTRACT One of the fundamental issues in Wireless Sensor Networks (WSN) is to count and localize multiple targets accurately. In this context, there has been an increasing interest in the literature in using Compressive Sensing (CS) based techniques by exploiting the sparse nature of spatially distributed targets within the monitored area. However, most existing works aim to count and localize the sparse targets utilizing a Single Measurement Vector (SMV) model. In this paper, we consider the problem of counting and localizing multiple targets exploiting the joint sparsity feature of a Multiple Measurement Vector (MMV) model. Furthermore, the conventional MMV formulation in which the same measurement matrix is used for all sensors is not valid any more in practical timevarying wireless environments. To overcome this issue, we reformulate the MMV problem into a conventional SMV in which MMVs are vectorized. Subsequently, we propose a novel reconstruction algorithm which does not need the prior knowledge of the sparsity level unlike the most existing CSbased approaches. Finally, we evaluate the performance of the proposed algorithm and demonstrate the superiority of the proposed MMV approach over its SMV counterpart in terms of target counting and localization accuracies.
LU et al.: DISTRIBUTED COMPRESSED SENSING OFF THE GRID 1 Distributed Compressed Sensing off the Grid
"... Abstract—This letter investigates the joint recovery of a frequencysparse signal ensemble sharing a common frequencysparse component from the collection of their compressed measurements. Unlike conventional arts in compressed sensing, the frequencies follow an offthegrid formulation and are con ..."
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Abstract—This letter investigates the joint recovery of a frequencysparse signal ensemble sharing a common frequencysparse component from the collection of their compressed measurements. Unlike conventional arts in compressed sensing, the frequencies follow an offthegrid formulation and are continuously valued in [0, 1]. As an extension of atomic norm, the concatenated atomic norm minimization approach is proposed to handle the exact recovery of signals, which is reformulated as a computationally tractable semidefinite program. The optimality of the proposed approach is characterized using a dual certificate. Numerical experiments are performed to illustrate the effectiveness of the proposed approach and its advantage over separate recovery. Index Terms—compressed sensing, basis mismatch, joint sparsity, atomic norm, semidefinite program I.
SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization
"... Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory w ..."
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Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory was then extended to the cases with partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as offgrid/continuous compressed sensing. However, a major problem of existing atomic norm methods is that the frequencies can be recovered only if they are sufficiently separated, prohibiting commonly known high resolution. In this paper, a novel nonconvex optimization method is proposed which guarantees exact recovery under no resolution limit and hence achieves high resolution. A locally convergent iterative algorithm is implemented to solve the nonconvex problem. The algorithm iteratively carries out ANM with a sound reweighting strategy which enhances sparsity and resolution, and is termed as reweighted atomicnorm minimization (RAM). Extensive numerical simulations are carried out to demonstrate the performance of the proposed method with application to direction of arrival (DOA) estimation. Index Terms—Continuous compressed sensing (CCS), DOA estimation, frequency estimation, gridless sparse method, high resolution, reweighted atomic norm minimization (RAM). I.