Databases Reference
In-Depth Information
0.40
0.35
0.30
0.25
AMDF (
P
)
0.20
0.15
0.10
0.05
0
20
40
60
80
Pitch period (
P
)
100
120
140
160
F I GU R E 18 . 7
AMDF function for the sound /s/ in
test
.
Obtaining the Vocal Tract Filter
In linear predictive coding, the vocal tract is modeled by a linear filter with the input-output
relationship shown in Equation (
1
). At the transmitter, during the analysis phase we obtain the
filter coefficients that best match the segment being analyzed in a mean squared error sense.
That is, if
{
y
n
}
are the speech samples in that particular segment, then we want to choose
{
a
i
}
to minimize the average value of
e
n
where
y
n
−
2
M
e
n
=
a
i
y
n
−
i
−
G
n
(3)
i
=
1
If we take the derivative of the expected value of
e
n
with respect to the coefficients
{
a
j
}
,we
get a set of
M
equations:
⎡
2
⎤
⎦
=
y
n
−
M
∂
⎣
E
a
i
y
n
−
i
−
G
n
0
(4)
∂
a
j
i
=
1
2
E
y
n
−
y
n
−
j
M
⇒−
a
i
y
n
−
i
−
G
n
=
0
(5)
i
=
1
M
a
i
E
y
n
−
i
y
n
−
j
=
E
y
n
y
n
−
j
⇒
(6)
i
=
1
where in the last step we have made use of the fact that
E
[
n
y
n
−
j
]
is zero for
j
=
0. In
order to solve (
6
) for the filter coefficients, we need to be able to estimate
E
. There
are two different approaches for estimating these values, called the
autocorrelation
approach
and the
autocovariance
approach, each leading to a different algorithm. In the autocorrelation
approach, we assume that the
[
y
n
−
i
y
n
−
j
]
{
y
n
}
sequence is stationary and therefore
E
y
n
−
i
y
n
−
j
=
R
yy
(
|
i
−
j
|
)
(7)
