Abstract

The recent note by Jin and Rogers (1983) presented examples of the failure of the homomorphic transform to invert properly. Since this transform is not only of interest in geophysics, but has also found applications in other fields (Oppenheim and Schafer, 1975), these results are of concern. We consequently attempted to reproduce Jin and Rogers' results. We failed to do so. In fact, in our experience, the transform has always inverted successfully. Our results using the first example of Jin and Rogers are shown in Figure 1. We used the algorithm of Tribolet (1977) with a modified Goertzel algorithm (Bonzanigo, 1978) for phase unwrapping. The figure is arranged as in Jin and Rogers' paper. Figure 1a shows the input: impulses separated by 20 samples, of magnitude 2000 and 1999. Figure 1b shows its complex cepstrum. We have set the zero-quefrency point to zero since this represents a scale factor and can dominate the plotting. Note the minimum delay cepstrum with a small amount of aliasing. The sequence returned by the inverse transform is shown in Figure 1c, demonstrating a successful inversion. The effect of noise is also shown. Noise with a standard deviation of 5 was added to the sequence of Figure 1a. This is shown in Figure 1d. Note that our noise realization is undoubtedly different from that of Jin and Rogers. The noise has changed the relative magnitude of the original spikes such that they are maximum delay. This is reflected in the cepstrum (Figure 1e). Figure 1f shows the returned sequence, again demonstrating the successful inversion.

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