MATLAB Implementation of Compressed Sensing Algorithms

Compressed Sensing (CS) is an emerging signal acquisition and processing technique that breaks the limitations of traditional Nyquist sampling theorem. It enables reconstruction of original signals from measurements taken at rates significantly below the Nyquist frequency, provided the signal has a sparse representation. MATLAB serves as an ideal platform for simulating compressed sensing algorithms, offering comprehensive matrix operations and optimization solvers for efficient implementation. The core concept of compressed sensing leverages signal sparsity. Assuming a signal is sparse in some transform domain (such as Fourier transform, wavelet transform, or DCT transform), dimensionality reduction sampling can be performed using random measurement matrices. Signal reconstruction is then achieved through optimization algorithms like basis pursuit or matching pursuit. MATLAB provides various optimization solvers including `l1eq_pd` for L1-minimization and OMP (Orthogonal Matching Pursuit) algorithms for effective signal recovery. In practical simulations, the implementation typically involves these key steps: First, generate sparse signals or obtain sparse representations of natural signals using sparsifying transforms. Second, design measurement matrices such as Gaussian random matrices or partial Fourier matrices. Third, acquire observation data through linear projections. Finally, employ optimization algorithms to solve for the sparse representation and reconstruct the original signal. MATLAB facilitates these steps with built-in functions for matrix operations and visualization tools for analyzing reconstruction results. Foreign research materials often include source code implementing various compressed sensing algorithm variants, such as smoothed L0-norm optimization, adaptive measurement matrix design, or deep learning-enhanced reconstruction methods. These advanced algorithms can further improve the accuracy and efficiency of compressed sensing. By comparing simulation results from different algorithms using MATLAB's comparative analysis capabilities, researchers can better understand the performance characteristics and application scope of compressed sensing techniques.