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; for such signals, the discrete Fourier transform (DFT) basis is a natural choice for CS recovery. Unfortunately, such signals are only sparse in the DFT domain when the sinusoid frequencies live precisely at the centers of the DFT bins; when this is not the case, CS recovery performance degrades significantly. In this paper, we introduce the spectral CS (SCS) recovery framework for arbitrary frequency-sparse signals. The key ingredients are an over-sampled DFT frame and a restricted union-of-subspaces signal model that inhibits closely spaced sinusoids. We demonstrate that SCS significantly outperforms current state-of-the-art CS algorithms based on the DFT while providing provable bounds on the number of measurements required for stable recovery. We also leverage line spectral estimation methods (specifically Thomsonʼs multitaper method and MUSIC) to further improve the performance of SCS recovery.