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I'm relatively new to signal processing, so please excuse me if this is a trivial question. Why is the spectrum of a frame of speech samples periodic? What is the meaning of a periodic spectrum? And if a spectrum is periodic, doesn't that mean that it is not band-limited (i.e. it goes on to infinity)? How is the spectrum of a speech frame affected by the windowing function (does it change the amplitude or period of the periodic spectrum)? This is mostly out of curiosity, and to help me get a better qualitative understanding of cepstral analysis. Thanks in advance.

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The spectrum of a vowel sound typically contains many harmonics. These harmonics are of course evenly spaced in the frequency domain (i.e. you see peaks at f0, 2*f0, 3*f0, 4*f0, etc). The spectrum can therefore be considered to be "periodic", with a "period" equal to f0. (This is usually called quefrency in the cepstral analysis literature).

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  • $\begingroup$ With harmonics though, isn't there a decay in the amplitude of the harmonics as the frequency increases? Wouldn't the harmonics need to have a non-decaying amplitude in order to be considered harmonic (to satisfy X(w + W) = X(w) )? $\endgroup$ – Zetta Suro Aug 2 '12 at 21:32
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    $\begingroup$ I think you might be looking at two different things here - the DFT spectrum is periodic in a strict mathematical sense, since the whole spectrum repeats ad infinitum - I think this is what is being discussed in one of the other answers. However in the context of cepstral analysis I think we're talking about a loose periodicity within the spectrum due to the harmonics in a complex sound - the amplitudes are not particularly important - the important point is that we can identify f0 from this periodicity. $\endgroup$ – Paul R Aug 3 '12 at 6:36
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Discrete functions and periodic functions are Fourier transform pair, i.e. if one domain is periodic, the other is discrete, and vice versa. Applying the windowing function to the signal will always change the spectrum, i.e. the amplitude and period could both change.

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  • $\begingroup$ So the DFT and DTFT always result in a period spectrum? $\endgroup$ – Zetta Suro Aug 2 '12 at 21:34
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    $\begingroup$ @phoenixheart6 DTFT always produces periodic spectrum. Discretize the periodic spectrum, and choose one period, then you have the spectrum of DFT. $\endgroup$ – chaohuang Aug 2 '12 at 21:38

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