Compressive Sensing Recovery of 1D Signals Using Orthogonal Matching Pursuit in MATLAB

This article demonstrates how to reconstruct one-dimensional signals using compressive sensing in MATLAB. Compressive sensing is a signal processing technique that enables perfect signal recovery from a small number of linear measurements. We implement this process using the Orthogonal Matching Pursuit (OMP) algorithm, a popular greedy approach for sparse signal reconstruction. The implementation involves several key MATLAB functions and algorithmic steps: First, we create a sensing matrix (typically random Gaussian or Bernoulli) that satisfies the Restricted Isometry Property. The OMP algorithm then iteratively selects the most correlated atoms from the sensing matrix using the inner product operation, solves a least-squares problem at each iteration to update the signal estimate, and removes the selected component from the residual. The algorithm terminates when either the desired sparsity level is reached or the residual falls below a predefined threshold. Key MATLAB functions used include matrix multiplication operations for projection calculations, QR factorization or pseudo-inverse (pinv) for efficient least-squares solutions, and iterative loops with conditional breakpoints. The code structure typically involves initializing a residual vector, maintaining an active set of indices, and progressively building the signal approximation through orthogonal projections. Through this tutorial, you will learn to implement OMP-based compressive sensing reconstruction for 1D signals in MATLAB, providing a valuable tool for your signal processing applications that combines mathematical foundations with practical coding implementation.