# A Tutorial on Sparse Signal Reconstruction and Its Applications in Signal Processing

## Abstract

Sparse signals are characterized by a few nonzero coefficients in one of their transformation domains. This was the main premise in designing signal compression algorithms. Compressive sensing as a new approach employs the sparsity property as a precondition for signal recovery. Sparse signals can be fully reconstructed from a reduced set of available measurements. The description and basic definitions of sparse signals, along with the conditions for their reconstruction, are discussed in the first part of this paper. The numerous algorithms developed for the sparse signals reconstruction are divided into three classes. The first one is based on the principle of matching components. Analysis of noise and nonsparsity influence on reconstruction performance is provided. The second class of reconstruction algorithms is based on the constrained convex form of problem formulation where linear programming and regression methods can be used to find a solution. The third class of recovery algorithms is based on the Bayesian approach. Applications of the considered approaches are demonstrated through various illustrative and signal processing examples, using common transformation and observation matrices. With pseudocodes of the presented algorithms and compressive sensing principles illustrated on simple signal processing examples, this tutorial provides an inductive way through this complex field to researchers and practitioners starting from the basics of sparse signal processing up to the most recent and up-to-date methods and signal processing applications.

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## Notes

1. Consider a signal x(t) of a duration T and its samples $$x(n\Delta t)$$ satisfying the sampling theorem. The periodic extension of this signal can be written in a Fourier series (FS) form

\begin{aligned} x(t) = \frac{1}{N}\sum \limits _{k = 0}^{N-1} {{X(k)}{\hbox {e}^{j2\pi k \frac{t}{T}}}}, \end{aligned}

where the FS coefficients X(k) are equal to the DFT coefficients if we use the notation x(n) for $$x(n\Delta t)$$ and $$\Delta t=T/N$$ as the sampling interval. When the sampling theorem is satisfied then

\begin{aligned} X(k)=\frac{N}{T}\int \limits _0^Tx(t)\hbox {e}^{-j2 \pi kt/T}\hbox {d}t=\sum _{n=0}^{N-1}x(n)\hbox {e}^{-j2\pi kn/N}, \quad k=0,1,2,\dots ,N-1. \end{aligned}

This is the DFT of a signal x(n).

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Stanković, L., Sejdić, E., Stanković, S. et al. A Tutorial on Sparse Signal Reconstruction and Its Applications in Signal Processing. Circuits Syst Signal Process 38, 1206–1263 (2019). https://doi.org/10.1007/s00034-018-0909-2

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### Keywords

• Sparse signals
• Compressive sensing
• Signal sampling
• Signal representation
• Signal reconstruction
• Discrete Fourier transform