Compressed Sensing MATLAB Implementation with Code Analysis

Compressed sensing is a revolutionary signal processing technique that breaks through the Nyquist sampling theorem by leveraging signal sparsity for efficient sampling and reconstruction. The MATLAB implementation typically involves three critical stages: First, sparse decomposition using FFT transforms signals from time domain to frequency domain. The Fast Fourier Transform effectively reveals sparsity characteristics in the frequency domain, forming the foundation for compressed sampling. In code implementation, we apply fft() function to the input signal and selectively retain the most significant frequency components through threshold-based component selection. Second, the design of random measurement matrices enables dimensional reduction in sampling. By constructing appropriate Gaussian random matrices or other measurement matrices satisfying the Restricted Isometry Property (RIP), we can substantially reduce sampling data while preserving essential information for signal recovery. The MATLAB code typically uses randn() function to generate Gaussian random matrices and validates RIP conditions through matrix property checks. Finally, signal reconstruction employs the Orthogonal Matching Pursuit (OMP) algorithm. This greedy algorithm iteratively selects the most correlated atoms from the dictionary, progressively approximating the original signal. OMP gains widespread application in compressed sensing due to its computational efficiency and reconstruction quality, particularly suited for sparse signal recovery problems. The MATLAB implementation involves iterative correlation calculation, residual update, and least-squares solution using matrix operations like pinv() for pseudoinverse computations.