Inspired by the Auditory Periphery, it is proposed that signals be modeled as products of elementary signals, as opposed to sums of sinusoids. Exploiting such a product representation, new insights into Envelope and Instantaneous Frequency (IF) of multicomponent signals are provided. Specifically, it is shown that an analytic signal's IF's erratic behavior is dictated by the proximity of its complex signal-zeros to the unit-circle, and not by the band-occupancy of the signal's spectrum. Closed form expressions for IFs of signals consisting of multiple complex sinewaves are derived; it is shown that there exists a one-to-one correspondence between signals' Fourier coefficients and their IFs. Signals for which IFs tend to be impulsive are also addressed. This is followed by discussions on signals having positive IFs. While IF and log-envelope are known to be time-functions that typically have infinite spectral bandwidths, it is pointed out that their filtered versions sufficiently characterize a signal. Finally, issues related to computation of digital IF are addressed.
Next, a signal observed over T seconds is modeled using a pole-zero model, by considering periodic extensions of it. Several algorithms to decompose it into minimum and maximum phase components are proposed. A new algorithm for minimum phase--all phase signal-decomposition is also proposed. In this method, first the signal's envelope is approximated to desired accuracy using dual of the autocorrelation method of linear prediction, well known in spectral analysis. The criterion that is optimized is a waveform flatness measure as opposed to the spectral flatness measure used in spectral analysis. This algorithm provides a unique positive AM (amplitude modulation) and FM (frequency modulation) representation of a signal. These concepts are finally applied to real speech signals to extract time histories of "formant modulations".
Finally, as off-shoots of the above mentioned research, two new algorithms are developed. The first one is a parametric model-based procedure for computing the Hilbert transform of a real-valued signal. The second one is a simple technique for designing stable all-pass filters, given desired phase responses.
