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Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
Abstract

Cited by 309 (20 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
ADAPTIVE ITERATIVE REWEIGHTED LEAST SQUARES DESIGN OF ¢ ¡ FIR FILTERS
"... This paper presents an efficient adaptive algorithm for designing FIR digital filters that are efficient according to an £¥ ¤ error criteria. The algorithm is an extension of Burrus ’ iterative reweighted leastsquares (IRLS) method for approximating £¥ ¤ filters. Such algorithm will converge for mo ..."
Abstract
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This paper presents an efficient adaptive algorithm for designing FIR digital filters that are efficient according to an £¥ ¤ error criteria. The algorithm is an extension of Burrus ’ iterative reweighted leastsquares (IRLS) method for approximating £¥ ¤ filters. Such algorithm will converge for most significant cases in a few iterations. In some cases however, the transition bandwidth is such that the number of iterations increases significantly. The proposed algorithm controls such problem and drastically reduces the number of iterations required. 1.