Sparse Frequency Analysis with Sparse-Derivative Instantaneous Amplitude and Phase Functions
This method addresses the challenge of narrow-band amplitude/phase jumps in signal processing, particularly for EEG data, but is incremental as it builds on existing convex optimization techniques.
The paper tackles the problem of representing signals with abrupt amplitude and phase changes as sums of sinusoids, proposing a convex variational method that regularizes total variation to avoid spectral spreading, and demonstrates its application on synthetic and EEG data for band-pass filtering and phase synchrony estimation.
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.