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.
Highlights
There are many applications in signal processing and communication systems where the discrete signals are sparse in some domain such as time, frequency, or space; i.e., most of the samples are zero, or alternatively, their transforms in another domain are sparse
The reconstruction methods used in each application domain have been introduced and the interconnections among them have been highlighted
This development has revealed, for example, that the iterative methods developed for random sampling can be applied to real-field block and convolutional channel coding for impulsive noise removal, Sparse Component Analysis (SCA), and channel estimation for orthogonal frequency division multiplexing systems
Summary
There are many applications in signal processing and communication systems where the discrete signals are sparse in some domain such as time, frequency, or space; i.e., most of the samples are zero, or alternatively, their transforms in another domain (normally called “frequency coefficients”) are sparse (see Figures 1 and 2). It is the intention of this tutorial to describe these methods in each field succinctly and show that these methods can be used in other areas and applications often with appreciable improvements Among these fields are 1—Sampling: random sampling of bandlimited signals [1], compressed sensing (CS) [2], and sampling with finite rate of innovation [3]; 2—Coding: Galois [4,5] and real-field error correction. As shown in column 2 of the table, depending on the topics, sparsity is defined in the time, space, or “frequency” domains. Spectral estimation (row 10) is the dual of error correction codes, i.e., the sparsity is in the frequency domain. For definition of ESPRIT on row 11 and column 7, see the footnote on page 41
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