of Microprocessors

SOURCE-SIGNAL ANALYSIS
577
SYSTEM
FUNCTION
~
:
IIi,l
,
EXCITATION
FUNCTION
50 jLsec
"TIME"
lCI
51 msec
20 Hz
FREQUENCY
(0)
5 kHz
Fig. 16-16. Homomorphic spectral analysis (cont.).
cepstrum. (0) Smoothed log spectrum.
(C) Magnitude of the
recovered excitation function is usually discarded, except for its fundamental
frequency, and the recovered system function is utilized.
Of course, the Fourier transform can also be used to implement the
separation filter with the advantage of zero phase shift. Phase shift (delay) in
the separation filter is undesirable because it shifts all of the recovered
frequencies upward (or downward depending on the direction of filtering).
The forward Fourier transform of a decibel spectrum, however, is a
mathematical absurdity and so the word cepstrum was coined to refer to it.
The word is formed by reverse spelling of the first half of the word "spectrum"
and tacking on the last half. It is only fitting, then, to call the
independent variable of a cepstral plot, which has the dimension of time,
quefrency! A cepstral plot is shown in Fig. 16-16C.
Low-quefrency values in the cepstrum are due to the system function
shape, while high-quefrency values are due to the excitation function. To
recover the system function shape, all quefrency values above a certain cutoff
point are set to zero and the inverse Fourier transform is taken. Figure
16-16D shows a cepstrally smoothed spectrum. To recover the excitation
function, low-quefrency values are omitted and the inverse transform is
taken.