Cepstra

Figure 12.12 Series of spectra calculated from cepstra. The 512 point cepstrum has all its values after a cut off point K set to 0 and then a 512 point DFT is apphed to generate a spectrum. The figure shows this spectrum for 4 values of K, 10, 20, 30, 50. (In the fignre each has been graphically separated for clarity, with = 10 as the bottom spectmm and AT = 50 as the top. In reality all lie in the same amplitude range.) As the number of coefficients increases, so does the level of spectral detail, until effects of harmonics start appearing. The choice of optimal cut-off point depends on whether spectral precision or elimination of harmonics is the top priority.

Figure 12.13b shows the spectra predicted from linear prediction filters for a range of orders. It is quite clear that as the order increases, so does the detail in the spectrum. It is interesting to compare this to Figure 12.12 which shows the same investigation but with cepstra. It is clear that the cepstral spectra are much more heavily influenced by the harmonics as the order of analysis increases. 

In the early days of speech research, formants held the centre stage as the representation for speech. Today however, it is extremely rare to find formants used in speech recognition, synthesis or other systems. This is partly because robust formant trackers have always proved hard to develop, and so researchers moved towards cepstra and LP representations that could be robustly found from speech, even if their form didn t explicitly show the t5q)e of information that... 

Transform die representation into a space that has more desirable properties log magnitude spectra follow the ear s dynamic range mel-scaled cepstra scale according to the frequency sensitivity to the ear log area ratios are amenable to simple interpolation and line-spectral frequencies show the formant patterns robustly. 

This spectral section is Fourier transformed to give a mathematical function called a cepstrum [12] as shown in the top of Figure 3.23. A cepstrum is similar to but not identical to an interferogram. To perform a deconvolution we take advantage of one of the properties of cepstra illustrated in Figure 3.24. 

FIGURE 3.24 The cepstra of a narrow and a broad infrared band. Note that the featnres in the narrow band cepstrnm are larger than the featnres in the broad band cepstrum. 

Cepstrum The result obtained when a Fourier transform is performed on a spectrum. Cepstra pi) are used in the deconvolution process. 

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