Spectrum plot after DCT

[Yichao L]

In the video “The complete process”, it says we use DCT to decorrelate the spectrum after applying filterbank on it.

I wonder what the output would look like? Could you supply a figure?

[Simon]

The DCT provides us with a representation (a set of coefficients) that has less correlation within its dimensions (the coefficients) than the filterbank outputs have between themselves.

This representation is called the cepstrum, which is in a different domain to the spectrum (which is of course in the frequency domain). We can plot the cepstrum – for example Taylor (2009) figure 12.11(c). The horizontal axis is actually time, which is what happens when you take the DCT (or the very similar inverse Fourier transform) of something in the frequency domain. But this is not the key point.

Key points to note in Taylor’s figure are

• the lower order cepstral coefficients have larger magnitudes than the higher order ones: this is telling us that we can make a good approximation to the original spectrum using just a few coefficients – that is, by truncating the cepstrum; Taylor suggests about 30 coefficients are needed, but for ASR we usually go lower, to 12
• there is a small peak in the cepstrum at about 120 samples – this corresponds to the fundamental period, T0 (= 1/F0); this tells us that truncating the cepstrum will effectively remove evidence of F0, which is what we want to do for ASR

[Author]

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