A unified approach to sparse signal processing

Abstract

A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing, and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The error locator polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method under noisy environments. The iterative methods developed for sampling and coding applications are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. Sparsity in unobservable source signals is also shown to facilitate source separation in sparse component analysis; the algorithms developed in this area such as linear programming and matching pursuit are also widely used in compressed sensing. Finally, the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.

Paper‐Based Electrical Respiration Sensor

“…By implementing additional compressed sensing / sparse sampling algorithms, the size of the files containing the collected data can be made significantly smaller, to enable use of networks with low speeds. This capability is required in the developing world 30,31 .…”

Graphical model driven methods in adaptive system identification

“…A closed form solution to (2.16) is not possible, and most techniques available to solve the problem are iterative, or approximate, or both [16], [17], [42], [82], [85], [124], [173], [186]. These techniques come from different fields and have various similarities and differences [110]. However, most of the available methods are still, in real terms, considerably slower than either solving (2.8) or (2.15), which is only to be expected.…”

Sparse representation of FHSS signals in the Hermite transform domain

“…A signal can be considered sparse in a certain transform domain if it can be represented with a small number of non-zero coefficients [1], [6]. Additionally, signals having a small number of significant coefficients while the influence of other coefficients is negligible although they are not zeros, can be considered as approximately sparse.…”

“…PARSE signal representation in a certain domain is commonly desirable in both signal processing and analysis [1]- [7]. A signal can be considered sparse in a certain transform domain if it can be represented with a small number of non-zero coefficients [1], [6].…”