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A Homomorphic Interpretation of the Complex FM Expansion

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A Complex Frequency Modulation (FM) signal is one whose instantaneous phase is time-varying according to a complicated dynamic function. This paper commences with the standard expansion for the spectrum of a Complex FM signal. It then explains how this can be interpreted in terms of a series of convolutions. The Homomorphic processing framework, in essence, provides a means by which a convolution operation can be related to a product operation which can then be transformed into an addition. This is very useful as it offers an approach for the fast computation of the theoretical spectra of complex FM signals, and further then leads to a cepstrum-like representation that will only display the modulation indices of the FM components. ‘Liftering’ of this representation can be carried out to alter the proportion of modulation components in the FM signal. Examples of the various stages of this processing will be given to illustrate its usefulness in the analysis and synthesis Complex FM signals.

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