CS Compressed Sensing GPRS Algorithm Documentation

### Analysis of CS Compressed Sensing and GPRS Algorithm

Compressed Sensing (CS) is a mathematical framework that reconstructs sparse or compressible signals from observations obtained at rates significantly below the Nyquist sampling rate. The GPRS (Gradient Projection for Sparse Reconstruction) algorithm is an effective sparse signal reconstruction method in compressed sensing, particularly suitable for solving L1-norm minimization problems.

#### Core Concepts Sparse Representation: Signals exhibit sparsity in certain transform domains (such as Fourier or wavelet domains), meaning most coefficients approach zero. Incoherent Measurements: Linear projections of signals are obtained through random measurement matrices (like Gaussian matrices), breaking traditional sampling limitations. Optimized Reconstruction: Convex optimization techniques (such as L1-norm minimization) are employed to recover original signals from limited observations.

#### GPRS Algorithm Workflow GPRS iteratively solves for sparse coefficients using gradient projection, with key implementation steps including: Initialization: Set initial sparse coefficient values (e.g., zero vectors) and gradient step parameters. Gradient Descent: Update coefficients along the negative gradient direction to approach the optimal solution. Projection Operation: Project coefficients onto the feasible region constrained by L1-norm to ensure sparsity. Convergence Check: Terminate iterations based on residual errors or iteration counts, outputting the reconstructed signal.

#### Simulation and Validation Simulation Design Generate sparse signals (e.g., random K-sparse vectors). Construct Gaussian random measurement matrices to obtain compressed observations. Invoke GPRS algorithm for signal reconstruction and compare errors (e.g., Mean Squared Error, MSE) between original and reconstructed results.

Performance Metrics Reconstruction Success Rate: Signal recovery probability under different measurement numbers (M). Robustness Testing: Algorithm stability after adding Gaussian noise.

#### Results Analysis Ideal Conditions: When measurement number $M geq O(K log N)$ (where N is signal dimension), GPRS achieves high-probability accurate reconstruction. Noisy Environments: By adjusting regularization parameters, the algorithm maintains good anti-noise performance. Compared to other algorithms (like OMP, BP), GPRS achieves a balance between computational efficiency and accuracy.

#### Application Extensions GPRS applies to medical imaging (MRI acceleration), wireless communications (sparse channel estimation), and other fields. Future enhancements may integrate deep learning techniques (such as optimizing projection matrix design) to further improve performance.