Spectral compressed sensing of real spikes

This paper studies the reconstruction of a stream of real spikes, which has usually been studied in the context of finite rate of innovation (FRI), from compressive Fourier measurements. Two gridless sparse methods are proposed based on atomic norm and Toeplitz matrix completion, respectively, which are more precise results of and enjoy theoretical guarantees of their existing counterparts for line spectral estimation. Our analysis shows that atomic norm can better preserve signal structures and produce easy-to-interpret results. Numerical simulations suggest that Toeplitz matrix completion has higher resolution. The obtained results also shed light on the case of line spectral estimation.