Digital Signal Processing Reference
In-Depth Information
E
e
f
,
L
(
]
T
E
e
f
,
L
(
)
=
a
L
−
i
[
x
n
)
e
f
,
L
−
i
(
n
−
i
n
)
(
n
−
i
), . . . ,
x
(
n
−
L
)
=
.
(2.34)
0
,
E
e
f
(
)
Therefore, we see that as
L
→∞
n
)
e
f
(
n
−
i
=
0, which means that
the sequence of forward errors
e
f
(
is asymptotically white. This means that a
sufficiently long forward prediction error filter is capable of
whitening
a stationary
discrete-time stochastic process applied to its input.
n
)
2.5.2 Backward Linear Prediction
In this case we start by trying to estimate
x
(
n
−
L
)
based on the next
L
samples, so
the backward linear prediction error can be put as
L
w
T
x
e
b
,
L
(
n
)
=
x
(
n
−
L
)
−
w
j
x
(
n
−
j
+
1
)
=
x
(
n
−
L
)
−
(
n
).
(2.35)
j
=
1
T
weminimize
the MSE. Following a similar procedure as before to solve the Wiener filter, the
augmented Wiener-Hopf equation has the form
R
x
To find the optimumbackward filter
w
b
,
L
=[
w
b
,
1
,
w
b
,
2
,...,
w
b
,
L
]
b
L
=
0
L
×
1
P
b
,
L
r
b
,
(2.36)
r
b
r
x
(
)
0
T
,
P
b
,
L
where
r
b
=
E
[
x
(
n
)
x
(
n
−
L
)
]
=[
r
x
(
L
),
r
x
(
L
−
1
), . . . ,
r
x
(
1
)
]
=
r
x
(
0
)
−
w
b
,
L
1
T
is the
backward prediction error filter
.
Consider now a stack of backward prediction error filters from order 0 to
L
.Ifwe
compute the errors
e
b
,
i
(
−
r
b
w
b
,
L
, and
b
L
=
n
)
for 0
≤
i
≤
L
, it leads to
⎡
⎤
⎡
⎤
1
0
1
×
(
L
−
1
)
e
b
,
0
(
n
)
⎣
⎦
⎣
b
1
⎦
0
1
×
(
L
−
2
)
e
b
,
1
(
n
)
b
2
e
b
,
2
(
n
)
0
1
×
(
L
−
3
)
e
b
(
n
)
=
=
x
(
n
)
=
T
b
x
(
n
).
(2.37)
.
e
b
,
L
−
1
(
.
.
w
b
,
L
−
1
n
)
−
1
The
L
L
matrix
T
b
, which is defined in terms of the backward prediction error filter
coefficients, is lower triangular with 1's along its main diagonal. The transformation
(
2.37
) is known as
Gram-Schmidt orthogonalization
[
3
], which defines a one-to-one
correspondence between
e
b
(
×
.
4
In this case, the principle of orthogonality states that
n
)
and
x
(
n
)
4
The Gram-Schmidt process is also used for the orthogonalization of a set of linearly independent
vectors in a linear space with a defined inner product.
