Cepstral analysis discussion

If we take the DFT of a signal and then take the inverse DFT of that, we of course get back to where we started. The cepstrum calculation differs in two important respects. First, we are only using the magnitude of the spectrum - in effect we are throwing away the phase information. The inverse DFT of a magnitude spectnun is already very different from the inverse DFT of a normal (complex) spectrum. The log operation scales the harmonics, which emphasises the periodicity of the harmonics. It also ensures that the cepstrum is the sum of the source and filter components rather than their convolution. 

For many cases in which the spectral envelope is required, it is advantageous to keep the low part of the cepstrum as it is, instead of converting it back to a frequency-domain spectrum. The low cepstral coefficients form a very compact representation of the envelope, and have the highly desirable statistical modelling property of being independent so that only their means and variances, not their covariances, need be stored 

Linear prediction (LP) is another technique for source-filter separation, in which we use the techniques of LTI filters to perform an explicit separation of source and filter. In LP we adopt a simple system for speech production, where we have an input somce x[n], which passes through a hnear time-invariant filter A[n], to give the output speech signal y[n]. In tiie time domain this is 

The convolution makes it difficult to separate the somce and filter in the time domain, but this is easily achieved if we transform om system to the z-domain  

In Section 12.12, we showed that the lossless-tube model was a reasonable approximation for the vocal tract during the production of a vowel. If we assume for now that H z) can therefore be represented ly an aU-pole filter, we can write 

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