An instructive example of homomorphic signal processing

In this paper, a specific application of cepstral processing is considered to illustrate homomorphic signal processing. In particular, an analytic expression is derived for the frequency domain representation of a signal that has been recovered from a multi-path signal with cepstral processing. The multi-path signal is composed of time-delayed, scaled versions of the transmitted signal, and the transmitted signal is to be recovered. The recovered signal is expressed analytically in the frequency domain. By truncating an infinite series, the recovered signal can be expressed in the time domain. The analytic expressions can be used to predict the distortion that will result in the recovered signal when the cepstral processing is implemented with discrete Fourier transforms (DFTs). The analytic expression of a recovered signal is compared with the signal recovered with DFTs. The signal recovery from a multi-path signal that is composed of the transmitted signal and an amplified version of the transmitted signal is discussed and considered in two examples. The examples also demonstrate the scaling of nonminimum phase signals to minimum phase signals