Accurate Spectral Estimation of Non-periodic Signals Based on Compressive Sensing

In this work we propose a method based on compressive sensing (CS) for estimating the spectrum of a signal written as a linear combination of a small number of sinusoids. In practice one deals with signals with finite-length and so the Fourier coefficients are not exactly sparse. Due to the leakage effect in the case where the frequency is not a multiple of the fundamental frequency of the DFT, the success of the traditional CS algorithms is limited. To overcome this problem our algorithm transform the DFT basis into a frame with a larger number of vectors, by inserting a small number of columns between some of the initial ones. The algorithm takes advantage of the compactness of the interpolation function that results from the ‘1 norm minimization of the Basis Pursuit (BP) and is based on the compressive sensing theory that allows us to acquire and represent sparse and compressible signals, using a much lower sampling rate than the Nyquist rate. Our method allow us to estimate the sinusoids amplitude, phase and frequency