1Sparse Frequency Analysis with Sparse-Derivative Instantaneous Amplitude and Phase Functions

Abstract—This paper addresses the problem of expressing a signal as a sum of frequency components (sinusoids) wherein each sinusoid may exhibit abrupt changes in its amplitude and/or phase. The Fourier transform of a narrow-band signal, with a discontinuous amplitude and/or phase function, exhibits spectral and temporal spreading. The proposed method aims to avoid such spreading by explicitly modeling the signal of interest as a sum of sinusoids with time-varying amplitudes. So as to accommodate abrupt changes, it is further assumed that the amplitude/phase functions are approximately piecewise constant (i.e., their time-derivatives are sparse). The proposed method is based on a convex variational (optimization) approach wherein the total variation (TV) of the amplitude functions are regularized subject to a perfect (or approximate) reconstruction constraint. A computationally efficient algorithm is derived based on convex optimization techniques. The proposed technique can be used to perform band-pass filtering that is relatively insensitive to narrow-band amplitude/phase jumps present in data, which normally pose a challenge (due to transients, leakage, etc.). The method is illustrated using both synthetic signals and human EEG data for the purpose of band-pass filtering and the estimation of phase synchrony indexes. Index Terms—sparse signal representation, total variation, discrete Fourier transform (DFT), instantaneous frequency, phase locking value, phase synchrony. I