Digital Signal Processing Reference
In-Depth Information
6.3.3.7.
Introducing an adaptive codebook
We know that to determine coefficients for the filter
A
(
z
) we minimize the
prediction error energy:
D
1
=
n
P
i
)]
2
[
x
(
n
)
−
a
i
x
(
n
−
i
=1
with respect to
P
unknown parameters
a
i
. This is known as short-term prediction
because, to predict the value of the signal at time
n
,weusethe
P
previous samples.
Once the calculation has been done directly by solving the linear system obtained by
deriving
D
1
with respect to the
P
unknown parameters, the signal
x
(
n
) is filtered
by the filter
A
(
z
) with order
P
. The short-term residual signal
y
(
n
) is obtained. The
display of the signal graphically, especially in the case of voiced sounds, shows that
not all the redundancy in the speech signal has been removed. A certain periodicity
remains as shown in Figure 6.2. This periodicity corresponds, physiologically, to
the vibration period of the vocal cords. We can characterize this information by
introducing two new parameters
b
and
Q
then minimizing the energy of a new
prediction error:
D
2
=
n
Q
)]
2
[
y
(
n
)
−
by
(
n
−
with respect to these two unknown parameters. This is known as long-term prediction.
We can remark that this minimization cannot by realized in the same way as the
previous one since for
D
1
,
P
is fixed, whereas for
D
2
,
Q
is a parameter to be
determined.
Open-loop minimization
To calculate the optimum values of
B
and
Q
, we need to only derive
D
2
with
respect to
b
to obtain the optimum
b
as a function of
Q
, carry this value forward into
D
2
then choose a value for
Q
which minimizes the criterion. This is not strictly the
best adapted solution. In general, a closed-loop solution is preferred.
Closed-loop minimization
Let us introduce the transfer function:
bz
−Q
B
(
z
)=1
−
In the synthesis stage, the inverse filter 1
/B
(
z
) is used which must be placed
upstream from the filter 1
/A
(
z
).
The diagram that corresponds to this function is
given in Figure 6.9.
Let us revisit the previous exposition, this time assuming that
b
and
Q
are
predetermined. We need to find the vectors
c
j
(
k
)
in the excitation codebook and the
