Attenuation and dispersion compensation via cepstral processing

This paper describes the attenuation and dispersion compensation of the echo signal in dispersive media. The parameters needed for compensation are derived from the echo signature in the log-magnitude domain of the signal. A sharper power cepstrum display then results from the compensated echo signature, indicating improved echo location resolution. Consider a signal consisting of a transmitted signal and a delayed echo. In a nondispersive medium, successive invoking of 1) a Fourier transform, 2) a logarithmic operation, and then 3) a Fourier transform will result in a display which contains, to a first order, delta functions whose separation provides a measure of delay times between echos. This is the basis of cepstral processing [5], [9]. With dispersion, the expected delta functions become reduced in amplitude and increase in width [10], [13]. The media parameters which contribute to dispersion are readily observed in step 2), where the envelope of the Fourier transformed signal provides an easily extracted correction term. This correction term can be used to adjust the overall cepstrum display to compensate the echo amplitude, width, and position. An example taken from a laboratory cable study is presented, with compensation applied to the cepstral display. The resultant signal displays are more readily interpreted due to the narrower pulses marking echo locations and the amplitude compensation which restores the relative pulse heights to their expected amplitude. Since the parameters for compensation are extracted from the Fourier transform signal envelope, noise problems encountered with compensating in the original time domain are reduced. Time domain compensation, in contrast, uses increased contributions of higher derivatives which tend to emphasize local variation and local noise.