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as the
reflection
coefficients, or
partial correlation
(PARCOR) coefficients. In the algorithm
description below, we denote the order of the filter using superscripts. Thus, the coefficients of
the fifth-order filter would be denoted by
a
(
5
)
i
{
}
. The algorithm also requires the computation
e
n
]
of the estimate of the average error
E
. We will denote the average error using an
m
th-order
filter by
E
m
. The algorithm proceeds as follows:
[
1.
Set
E
0
=
R
yy
(
0
)
,
i
=
0.
2.
Increment
i
by one.
3.
Calculate
k
i
i
−
1
j
1
a
(
i
−
1
)
=
R
yy
(
i
−
j
+
1
)
−
R
yy
(
i
)
/
E
i
−
1
.
=
j
4.
Set
a
(
i
)
i
k
i
.
5.
Calculate
a
(
i
)
j
=
a
(
i
−
1
)
j
k
i
a
(
i
−
1
)
=
+
for
j
=
1
,
2
,...,
i
−
1.
i
−
j
=
1
k
i
E
i
−
1
.
6.
Calculate
E
i
−
<
If
i
M
,gotostep2.
7.
In order to get an effective reconstruction of the voiced segment, the order of the vocal tract
filter needs to be sufficiently high. Generally, the order of the filter is 10 or more. Because
the filter is an IIR filter, error in the coefficients can lead to instability, especially for the high
orders necessary in linear predictive coding. As the filter coefficients are to be transmitted to
the receiver, they need to be quantized. This means that quantization error is introduced into
the value of the coefficients, and that can lead to instability.
This problem can be avoided by noticing that if we know the PARCOR coefficients, we can
obtain the filter coefficients from them. Furthermore, PARCOR coefficients have the property
that as long as the magnitudes of the coefficients are less than one, the filter obtained from
them is guaranteed to be stable. Therefore, instead of quantizing the coefficients
{
a
i
}
and
transmitting them, the transmitter quantizes and transmits the coefficients
. As long as we
make sure that all the reconstruction values for the quantizer have magnitudes less than one, it
is possible to use relatively high-order filters in the analysis/synthesis schemes.
The assumption of stationarity that was used to obtain (
7
) is not really valid for speech
signals. If we discard this assumption, the equations to obtain the filter coefficients change.
The term
E
{
k
i
}
[
y
n
−
i
y
n
−
j
]
is now a function of both
i
and
j
. Defining
c
ij
=
E
[
y
n
−
i
y
n
−
j
]
(14)
we get the equation
=
C
A
S
(15)
where
⎡
⎣
⎤
⎦
c
11
c
12
c
13
···
c
1
M
c
21
c
22
c
23
···
c
2
M
C
=
(16)
.
.
.
.
c
M
1
c
M
2
c
M
3
···
c
MM
