Digital Signal Processing Reference
In-Depth Information
to the unknown parameters. However, this criteria is not particularly well adapted
to our hearing system. A perception correction is very widely used to alleviate this
inconvenience [ATA 79]. A weighting function, in the form of a filter transfer function
W
(
z
), is added before the minimization criterion, as shown in Figure 6.6.
x
(
m
)
A
(
z
)
-
Min || . ||
A
(
z
/
γ
)
0
L-
1
0
r
(
m
)
1
A
(
z
)
x
(
m
)
N
-1
g
(
m
)
j
(
m
)
Figure 6.6.
Introducing a weighting function
W
(
z
)=
A
(
z
)
/A
(
z/γ
)
A presentation of the phenomena of masking one sound by another will be given
later. Let us simply say for the moment that the quantization noise is less perceptible
when the signal has plenty of energy. We can say that the signal masks the noise.
The total power of the quantization noise cannot be played about with. However, it is
possible to alter the spectral shape of the noise. We can therefore look for a weighting
function which attributes less importance to energetic frequency regions, that is, to
formant regions. We can show that the transfer function
W
(
z
)=
A
(
z
)
/A
(
z/γ
) with
0
<γ<
1 can fulfill this role. In effect, if we note:
P
A
(
z
)=1+
a
1
z
−
1
+
+
a
P
z
−P
=
p
i
z
−
1
)
···
−
(1
i
=1
where
p
i
specifies the
i
th root of the polynomial
A
(
z
), we observe that:
A
z
γ
=1+
a
1
γz
−
1
+
P
+
a
P
γ
P
z
−P
=
γp
i
z
−
1
)
···
(1
−
i
=1
The modulus of the frequency response of the filter 1
/A
(
z/γ
) shows peaks that
are less sharp than those of the filter 1
/A
(
z
) because the poles of the filter 1
/A
(
z/γ
)
are brought back toward the center of the unit circle compared with those of the
filter 1
/A
(
z
). The modulus of the frequency response filter
W
(
z
)=
A
(
z
)
/A
(
z/γ
)
therefore has the desired shape.
The diagram of the modeling principle is shown in Figure 6.7.
This diagram clearly shows the fact that we are attempting to model the
perceptual
signal
p
by
p
. The filter characterized by the transfer function 1
/A
(
z/γ
) is known as
