LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
486CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING
a short-time Fourier analysis is done first, resulting in a DFT, Xm[k], for the
m th frame. Then the DFT values are grouped together in critical bands and
weighted by triangular weighting functions such as those depicted in Fig. 8.41.
Note that the bandwidths in Fig. 8.41 are constant for center frequencies below
0.01
0.005
0
0 500 1000 1500 2000 2500 3000 3500 4000
frequency in Hz
Figure 8.41: DFT weighting functions for mel-frequency-cepstrum computations.
1 kHz and then increase exponentially up to half the sampling rate of 4 kHz
resulting in 24 “filters”. The mel-spectrum of the mth r = 1, 2, . . . , R as
frame is defined for
MFm[r] = 1
Ur
|Vr[k]Xm[k]| 2
(8.117a)
Ar
k=Lr
where Vr[k] is the weighting function for the the rth filter ranging from DFT
index Lr to Ur, and
Ur
Ar = |Vr[k]| 2
(8.117b)
k=Lr
is a normalizing factor for the r th mel-filter. This normalization is built into the
plot of Fig. 8.41. It is needed so that a perfectly flat input Fourier spectrum
will produce a flat mel-spectrum. For each frame, a discrete cosine transform
of the logarithm of the magnitude of the filter outputs is computed to form the
function mfcc[n] as
mfccm[n] = 1
R
R
r=1
2π
log (MFm[r]) cos r +
R
1
n . (8.118)
2
Typically, mfccm[n] is evaluated for n = 1, 2, . . . , Nmfcc, where Nmfcc is less than
the number of mel-filters, e.g., Nmfcc = 13 and R = 24. Figure 8.42 shows the
result of mfcc analysis of a frame of voiced speech in comparison with the shorttime
spectrum, LPC spectrum, and a homomorphically smoothed spectrum. 21
The large dots are the values of log (MFm[r]) and the line interpolated between
21 The speech signal was pre-emphasized by convolution with δ[n] − 0.97δ[n − 1] prior to
analysis so as to equalize the levels of the formant resonances.