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A technique for decomposing a composite signal of unknown multiple wavelets overlapping in time is described. The computation algorithm incorporates the power cepstrum and complex cepstrum techniques. It has been found that the power cepstrum is most efficient in recognizing wavelet arrival times and amplitudes while the complex cepstrum is invaluable in estimating the form of the basic wavelet and its echoes, even if the latter are distorted. Digital data-processing problems such as the detection of multiple echoes, various methods of linear filtering the complex cepstrum, the picket-fence phenomenon, minimum-maximum phase situations, and amplitude- versus phase-smoothing for the additive-noise case are examined empirically and where possible theoretically, and are discussed. A similar investigation is performed for some of the preceding problems when the echo or echoes are distorted versions of the wavelet, thereby giving some insight into the complex problem of separating a composite signal composed of several additive stochastic processes. The threshold results are still empirical and the results should be extended to multi-dimensional data. Applications are the decomposition or resolution of signals (e.g., echoes) in radar and sonar, seismology, speech, brain waves, and neuroelectric spike data. Examples of results are presented for decomposition in the absence and presence of noise for specified signals. Results are tendered for the decomposition of pulse-type data appropriate to many systems and for the decomposition of brain waves evoked by visual stimulation.
Kemerait et al. (Wed,) studied this question.