An analytic signal s(t) is modeled over a T second duration by a pole-zero model by considering its periodic extensions. This type of representation is analogous to that used in discrete-time systems theory, where the periodic frequency response of a system is characterized by a finite number of poles and zeros in the z-plane. Except, in this case, the poles and zeros are located in the complex-time plane. Using this signal model, expressions are derived for the envelope, phase, and the instantaneous frequency of the signal s(t). In the special case of an analytic signal having poles and zeros in reciprocal complex conjugate locations about the unit circle in the complex-time plane, it is shown that their instantaneous frequency ͑IF͒ is always positive. This result paves the way for representing signals by positive envelopes and positive IF ͑PIF͒. An algorithm is proposed for decomposing an analytic signal into two analytic signals, one completely characterized by its envelope and the other having a positive IF. This algorithm is new and does not have a counterpart in the cepstral literature. It consists of two steps. In the first step, the envelope of the signal is approximated to desired accuracy using a minimum-phase approximation by using the 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 method is called linear prediction in spectral domain ͑LPSD͒. The resulting residual error signal is an all-phase or phase-only analytic signal. In the second step, the derivative of the error signal, which is the PIF, is computed. The two steps together provide a unique AM-FM or minimum-phase/all-phase decomposition of a signal. This method is then applied to synthetic signals and filtered speech signals.PACS numbers: 43.72.Ar ͓JLH͔
INTRODUCTIONMany natural and man-made signals of interest are timevarying or nonstationary in nature, i.e., their frequency content or spectrum changes with time. Examples include speech signals, animal calls, biological/biomedical signals such as cardiac rhythms, etc. Techniques for characterizing such signals are of great importance in applications involving such signals. A collection of short-time Fourier spectra known as spectrogram is a common tool for analyzing such time-varying signals. Unfortunately, the spectrogram suffers from the need to compromise time and frequency resolution, i.e., a large time window is required to resolve closely spaced frequencies. To overcome this problem, a number of so-called time-frequency distributions or representations have been developed. 1,2 The time-frequency analysis tools are very useful in visualizing the time and frequency behavior of simple signals like a chirp. However, when the signals are complex, as in the case of speech, it is hard to interpret time-frequency representations because of the interactions between components in the signal. The time-frequency analysis ...
