What is compressed sensing and how does it enable sub-Nyquist sampling?

Compressed sensing recovers sparse signals from fewer measurements than Nyquist requires. Theory: If signal x is K-sparse in some basis (K << N nonzero coefficients), it can be recovered from M measurements where M ~ K*log(N/K). Requirements: Sparsity (or compressibility), Incoherent sensing matrix (random measurements), and L1-minimization or matching pursuit for recovery. Applications: MRI (faster scans), radar (reduced data), single-pixel camera, and spectrum sensing. Limitations: Sparsity assumption must hold, computational complexity of recovery algorithms, and noise sensitivity. Active research area bridging information theory and optimization.