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Sparse signal processing is a framework for representing, acquiring, and reconstructing signals that have a sparse or compressible representation in some dictionary or transform basis, meaning that only a small fraction of the coefficients are significant while most are zero or near-zero. The central result of compressed sensing is that such signals can be reconstructed exactly from far fewer linear measurements than the Nyquist-Shannon sampling theorem requires, provided the measurement matrix satisfies the restricted isometry property and the signal is sufficiently sparse ^[1]. The reconstruction problem is formulated as an underdetermined linear system solved through L1-norm minimization, which promotes sparsity by penalizing the sum of absolute coefficients, or through greedy algorithms such as orthogonal matching pursuit that iteratively select the dictionary atoms most correlated with the residual. The framework applies to image processing, MRI, radar, and communication systems where the cost or bandwidth of acquiring full Nyquist-rate samples is prohibitive. The mathematical conditions for exact recovery are governed by incoherence between the measurement basis and the sparsity basis, and the number of measurements required scales logarithmically with signal dimension and linearly with sparsity level ^[2]. The theory unifies signal representation, sampling, and inverse problem solving under the principle that exploiting sparsity is equivalent to exploiting low-dimensional structure, and it provides deterministic recovery guarantees under random measurement ensembles ^[3].