Pitch Detection via cepstral Method:
Cepstral analysis provides a way for the estimation of pitch. If we assume that a sequence of voiced speech is the result of convoluting the glottal excitation sequence e[n] with the vocal tractís discrete impulse response . †In frequency domain, the convolution relationship becomes a multiplication relationship. Then, using property of log function log AB = log A + log B, the multiplicationrelationship can be transformed into an additive relationship. Finally, the real cepstrum of a signal
†is defined as
††† ----------------- (8)
† ------------------ (9)
That is, the cepstrum is a Fourier analysis of the logarithmic amplitude spectrum of the signal. If the log amplitude spectrum contains many regularly spaced harmonics, then the Fourier analysis of the spectrum will show a peak corresponding to the spacing between the harmonics: i.e. the fundamental frequency. Effectively we are treating the signal spectrum as another signal, then looking for periodicity in the spectrum itself.
The cepstrum is so-called because it turns the spectrum inside-out. The x-axis of the cepstrum has units of quefrency, and peaks in the cepstrum (which relate to periodicities in the spectrum) are called rahmonics. To obtain an estimate of the fundamental frequency from the cepstrum we look for a peak in the quefrency region corresponding to typical speech fundamental frequencies (1/quefrency).
Waveform and unsmoothed pitch track with cepstral method.