# Why does the excitation signal appear, separated, at high quefrencies in the cepstrum?

So, I've just begun a speech and language processing course and have found the explanation of the process of getting the cepstrum of a signal and its properties a little confusing. The following is a description of my current understanding and an explanation of confusion it's causing me:

1. start with the speech signal. we can think of it as the formant signal convolved with a the excitation signal which is a dirac comb (approximately).
2. take the FFT, giving the spectrum of the excitations multiplied with the spectrum of the formants. the FFT of the excitation signal is another dirac comb with period 1/T
3. take logs. so the 2 signals above are now added
4. inverse fourier transform - the two signals from 1 should now be combined in addition (the FT transform is linear)

So if those 4 step are right, then why does the excitation appear at a particular region in the quefrency domain? it should emerge as a Dirac comb, added to the formant impulse response shouldn't it?

When using an FFT, an evenly spaced sequence of events in one domain usually produces a strong component in the other domain at a location related to the spacing of the events in the first domain.

A voiced speech signal usually includes a lot of harmonics which are evenly spaced in the frequency domain. These evenly spaced events in the frequency domain will thus produce a component in the quefrency domain whose location is related to the spacing of the harmonics. And the spacing of the harmonics in the frequency domain is related to the spacing of the exciter events in the time domain (the period of the fundamental).

• hi, thank-you for your answer, sorry for my slow response. I'm a little confused by your first statement. Do you think you could elaborate or give some examples?
– Sam
Feb 17, 2013 at 18:42

From your description, it looks like the log operation will have the effect of shrinking the amplitude range at different frequencies. Generating the cepstrum by taking the IFFT of the log(FFT) should generate a sharper signal. You can experiment with using different log bases and nth roots to understand the effect.

In the source-filter model of speech production, the filter (formants) is very smooth in the frequency domain - it is just a bunch of relatively large-band bumps. Hence, in the quefrency domain, its energy is mostly spread over a handful of the lower cepstrum coefficients.

• why is it that the excitation frequency can be identified in the quefrency domain? the excitation signal should be spread as many 'delta' functions across the whole quefrency domain (or a large amount of it)
– Sam
Jan 27, 2013 at 16:42