eurasip2007-groupdel.. - SRI Speech Technology and Research ...

6 EURASIP Journal on Audio, Speech, and Music Processing
production, the transfer function of such a system is given
by:
H(z) =
∑
∑ k b k z k
k a k z k . (20)
The transfer function of the same system for the production
of vowel assuming an all pole model is given by
1
H(z) = ∑ k=p
k=0 a kz , k (21)
1
H(z) =
1+ ∑ k=p
k=1 a kz . k (22)
Let the vowel be characterized by the frequencies F1, F2, F3.
Hence the poles of the system are located at
p = re ¦jωi . (23)
By substituting (23)in(21), the system function for production
of the ith formant now becomes
H i (z) =
But from resonance theory
1
1 2r cos ω i Tz 1 + r 2 z 2 . (24)
r = e πBiT . (25)
By substituting (25)in(24), the system function in (24)now
becomes
H i (z) =
1
1 2e πBiT cos ω i Tz 1 + e 2πBiT z 2 . (26)
In the above array of equations, ω i corresponds to the ith
formant frequency, B i to the bandwidth of the ith formant
frequency, and T to the sampling period. Using (26), we
generate a synthetic vowel with the following values: F1 =
500 Hz, F2 = 1500 Hz, F3 = 3500 Hz, B i = 10% of F i ,
and T = 0.0001 second corresponding to a sampling rate of
10 KHz. Note that F1, F2, and F3 are the formant frequencies
in Hz. We then extract the MODGDF, MFCC, and joint
features (MODGDF + MFCC) from the synthesized vowel.
To reconstruct the formants, we use the algorithm described
above. The reconstructed formant structures derived from
the MODGDF, MFCC, joint features (MODGDF + MFCC),
and also RASTA filtered MFCC are shown in Figures 3(a),
3(b), 3(c),and3(d), respectively. The illustrations are shown
as spectrogram like plots 2 where the data along the Y-axis
correspond to the DFT bins and the x-axis corresponds to
the frame number. It is interesting to note that while the
formants are reconstructed accurately by both the MOD-
GDF and the MFCC as in Figures 3(a) and 3(b),respectively,
joint features (MODGDF + MFCC) combine the formant
2 The differences between the subplots are better visualized in color than in
gray scale.
Frequency
(FFT bins)
Frequency
(FFT bins)
Frequency
(FFT bins)
Frequency
(FFT bins)
250
150
50
250
150
50
250
150
50
250
150
50
10 20 30 40 50 60 70 80 90
Frame number
(a)
10 20 30 40 50 60 70 80 90
Frame number
(b)
10 20 30 40 50 60 70 80 90
Frame number
(c)
10 20 30 40 50 60 70 80 90
Frame number
(d)
0.1
0.08
0.06
0.04
0.4
0.3
0.2
0.1
0.25
0.15
0.05
0.6
0.4
0.2
0
Figure 3: Spectrogram-like plots to illustrate formant reconstructions
for a synthetic vowel. (a) The short-time modified group delay
spectra reconstructed from MODGDF, (b) the short-time power
spectra reconstructed from MFCC, (c) the short-time composite
spectra reconstructed from joint features (MODGDF+MFCC), and
(d) the short-time power spectra reconstructed from RASTA filtered
MFCC.
information in the individual features as in Figure 3(c). Itis
expected that RASTA filtered MFCC shown in Figure 3(d)
does not capture the formant structure for the synthetic signal.
RASTA filtered MFCC reconstructions are illustrated
here only to show that while the MODGDF works well on
clean speech, RASTA MFCC fails, which is quite expected.
Although one may hypothesize that individual features capture
formant information well and the need for joint features
is really not there, it would be significant to note that joint
features combine information gathered from individual featuresasinFigure
3(c). Similar spectrogram-like plots to illustrate
formant reconstructions for a synthetic speech signal
with varying formant trajectories are shown in Figure 4.Itis
interesting to note that in Figure 4(a), all the 3 formants are
visible. In Figure 4(b), while the first 2 formants are visible,
the third formant is not clearly visible. In Figure 4(c), while
the first 2 formants are clear, the third formant is further emphasized.
Hence it is clear that joint features are able to combine
the information that is available in both the MODGDF
and MFCC.