  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.
