Homomorphic deconvolution in reverberant and distortional channels: An analysis

Abstract

Homomorphic deconvolution is used to determine the difference in arrival times or time delays among the paths of a channel, to estimate both the impulse response of that channel, and the signal propagating through it. Here, an analysis of homomorphic deconvolution for processing the output of a distortional N-path channel is presented. Previous studies have been concentrated principally on establishing characteristics of homomorphic deconvolution when operating in a distortional two-path channel. With the N-path channel, further complications arise and these are addressed in the development. An analytical derivation is given to identify and interpret the constitutive elements at the output of each operation within the homomorphic filter. The origins of the new difficulties are traced to the interference generated by the various convolutions of the channel response with itself. Procedures are generated to deal with the interference problem in the complex cepstrum output or to circumvent it in the homomorphic filter output. The latter alternative is possible when the shortest time delay is estimated successfully. At the complex cepstrum output, conditions are set to enable a unique specification of the dominant peaks corresponding to the time delays. When measures on the channel are known a priori, these measures are incorporated to perform with great ease the time delays estimation, through matching of the cepstrum output to the known measures in the channel. The advantage of deriving the complex cepstrum, then deducing the power cepstrum output, is demonstrated through reduced computational complexity. Pitfalls in the separation of the channel response from the signal and their respective estimations are discussed. Finally, the various analytical results are illustrated through applications to passive source localization, restoration of cancelled peaks, identification, distortion in lumped parameter networks, and seismic echo sounding. It is shown that phase distortion in the channel precipitates the deterioration of the complex cepstrum over that of the power cepstrum, and processing through the latter technique is recommended. It is found that an eliminated peak in the complex cepstrum output may be restored and its corresponding time delay determined in the homomorphic filter output. It is noted that identification of the number of paths within a channel from the complex cepstrum output can be misleading whenever the composite signal z(t) contains similar zi(t) wavelets at multiples of a given delay. Examination of the homomorphic filter output would circumvent such ambiguity. Otherwise, it is found that attention needs to be paid to the amplitude and sign of the peak at the time delay in question, and at those of its harmonics, to differentiate between a two-path and multipath case.