Digital Signal Processing Reference
In-Depth Information
The inverse of the pulse duration is called pitch frequency denoted in Figure 7.7 as
f
pitch
(
n
). These signal generators can be accessed either in a binary manner
g
(
n
)
[
{0, 1}
(7
:
1)
or continuously
0
g
(
n
)
1
:
(7
:
2)
For rebuilding the influence of the cavities a low-order all-pole filter with the
frequency response
s
(
n
)
A
(
e
jV
,
n
)
¼
s
(
n
)
1
P
H
(
e
jV
,
n
)
¼
(7
:
3)
N
pre
i¼
1
a
i
(
n
)
e
jVi
is employed.
1
The order
N
pre
of the all-pole model is chosen usually in the range of
10 to 20. Since the excitation signal is a spectrally flat signal, the transfer function
of this all-pole model represents the spectral envelope of the speech signal. The
parameters
˜
i
(
n
) of the all-pole model can be computed by solving the so-called
Yule-Walker equation system
2
3
2
3
2
3
r
ss
,0
(
n
)
r
ss
,1
(
n
)
r
ss
,
N
pre
1
(
n
)
r
ss
,1
(
n
)
r
ss
,2
(
n
)
.
r
ss
,
N
pre
(
n
)
a
1
(
n
)
a
2
(
n
)
.
a
N
pre
(
n
)
4
r
ss
,1
(
n
)
r
ss
,0
(
n
)
r
ss
,
N
pre
2
(
n
)
5
4
5
4
5
¼
:
(7
:
4)
.
.
.
.
.
.
r
ss
,
N
pre
1
(
n
)
r
ss
,
N
pre
2
(
n
)
r
ss
,0
(0)
|
{z
}
a
(
n
)
|
{z
}
R
ss
(
n
)
|
{z
}
r
ss
(
n
)
The coefficients
r
ss,i
(
n
) represent the short-term autocorrelation at lag
i
estimated
around the time index
n
. Finally, the gain parameter
s
(
n
) in (7.3) is computed as
the square root of the output power of a predictor error filter with coefficients
˜
i
(
n
)
t
r
ss
,0
(
n
)
X
N
pre
i¼
1
a
i
(
n
)
r
ss
,
i
(
n
)
s
(
n
)
¼
q
r
ss
,0
(
n
)
a
T
(
n
)
r
ss
(
n
)
:
¼
(7
:
5)
Due to the special character of the matrix
R
ss
(
n
) Eqs. (7.4) and (7.5) can be solved
in an order-recursive manner by using, for example, the Levinson-Durbin recursion
1
We have used the tilde notation for the coefficients
˜
i
(
n
) in order to avoid a conflict with the definitions of
standard transformations such as Fourier or
z
-transform. The Fourier transform is defined as
A
(
e
jV
,
n
)
¼
P
i¼1
a
i
(
n
)
e
jVi
. By comparing the coefficients we get
a
0
(
n
)
¼
1,
a
i
(
n
)
¼
2
˜
i
(
n
) for 1
i N
pre
, and
a
i
(
n
)
¼
0 else.
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