High Resolution Spectral Estimation using BP via Compressive Sensing

In this paper we propose a method based on compressed sensing (CS) for estimating the spectrum of a signal written as a linear combination of a small number of sinusoids. In the case of finite-length signals, the Fourier coefficients are not exactly sparse due to the leakage effect if the frequency is not a multiple of the fundamental frequency; To overcome this problem our algorithm transform the DFT basis into a frame with a larger number of vectors, by inserting columns between some of the initial ones. The algorithm applies Basis Pursuit (BP) to estimate the sinusoids amplitude, phase and frequency.