Digital Signal Processing Reference
In-Depth Information
Now, assuming that all parameters of
b
(
z
)
are already optimized accord-
ing to the Wiener criterion, then
E
e
f
[
2
]
∂
n
π
1
2π
2
e
j
ω
e
j
ω
=
S
x
(
)
Bs
(
)
[
−
2cos
(
ω
−
ω
b
)
+
2ρ
b
]
d
ω
=
0
∂
ρ
b
−
π
(3.116)
e
j
ω
e
j
ω
2
will always be nonnegative, then ρ
b
−
Since both
S
x
(
)
and
|
b
(
)
|
cos
must, necessarily, assume positive and negative values in order
that (3.116) be valid. Therefore, since
(
ω
−
ω
b
)
|
cos
(
ω
−
ω
b
)
|≤
1 for any ω, ρ
b
should
be less than 1, which means that all zeros
z
i
(
i
,
L
) should necessarily
be located inside the unit circle in the complex
z
-plane. In conclusion, the
FEP is a minimum-phase filter.
=
1,
...
3.8.3 The Linear Prediction-Error Filter as a Constrained Filter
As previously mentioned, the PEF provides a direct mapping between the
input signal and the error signal. Such mapping can be implemented by an
FIR filter
K
e
(
n
)
=
x
(
n
)
−
w
f
,
k
x
(
n
−
k
)
k
=
1
w
f
x
=
(
n
)
(3.117)
where
w
K
]
T
w
f
=
[
1,
−
w
1
,
−
w
2
,
...
,
−
(3.118)
By doing so, the search for the optimal PEF can be carried out in an LCMV
context, in accordance with (3.87) and (3.88). Let us define
=
10
0
T
C
···
(3.119)
and
g
=
1
(3.120)
Then, from (3.92), the following solution is reached
R
−
1
C
C
T
R
−
1
C
w
f
=
(3.121)
