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⎡
⎤
a
1
a
2
a
3
a
N
⎣
⎦
a
=
(32)
⎡
⎣
⎤
⎦
R
xx
(
1
)
R
xx
(
2
)
R
xx
(
3
)
p
=
(33)
.
R
xx
(
N
)
where we have used the fact that
R
xx
(
−
for real-valued wide-sense stationary
processes. These equations are referred to as the discrete form of the Wiener-Hopf equations.
If we know the autocorrelation values
k
)
=
R
xx
(
k
)
{
R
xx
(
k
)
}
for
k
=
0
,
1
,...,
N
, then we can find the
predictor coefficients as
R
−
1
p
a
=
(34)
Example11.4.1:
For the speech sequence shown in Figure
11.7
, let us find predictors of orders one, two, and
three and examine their performance. We begin by estimating the autocorrelation values from
the data. Given
M
data points, we use the following average to find the value for
R
xx
(
k
)
:
M
−
k
1
R
xx
(
)
=
(35)
k
x
i
x
i
+
k
M
−
k
i
=
1
3
2
1
0
−1
−
2
−3
500
1000
1500
2000
2500
3000
3500
4000
F I GU R E 11 . 7
A segment of speech: a male speaker saying the word “test.”
