Spectral compressive sensing

Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals based on randomized dimensionality reduction. To recover a signal from its compressive measurements, standard CS algorithms seek the sparsest signal in some discrete basis or frame that agrees with the measurements. A great many applications feature smooth or modulated signals that are frequency sparse and can be modeled as a superposition of a small number of sinusoids. Unfortunately, such signals are only sparse in the discrete Fourier transform (DFT) domain when the sinusoid frequencies live precisely at the center of the DFT bins. When this is not the case, CS recovery performance degrades significantly. In this paper, we introduce a suite of spectral CS (SCS) recovery algorithms for arbitrary frequency sparse signals. The key ingredients are an over-sampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectral estimation. Using peridogram and eigenanalysis based spectral estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current state-of-the-art CS algorithms based on the DFT while providing provable bounds on the number of measurements required for stable recovery.