Homomorphic modulation spectra

Physical evidence points to the importance of a concept called "modulation frequency". This dimension exists jointly with standard Fourier or acoustic frequency. Thus, akin to other time-varying analysis, we seek a two-dimensional representation, the "modulation spectrum", where the first dimension is the well-known acoustic frequency and the second dimension is modulation frequency. We describe some deficiencies in previous discussions of this concept, and then address those deficiencies via a homomorphic approach. We also reduce previous difficulties in homomorphic demultiplication by integrating this processing into modulation spectra and, in particular, show how assumption of analytic and relatively narrowband sub-bands allows more accurate and practical use of homomorphic demultiplication. Lastly, we show how an unambiguous demultiplication concept is only consistent with complex modulator envelopes. The assumption of complex envelopes is necessary for accurate modulation spectral analysis and filtering.