Digital Signal Processing Reference
In-Depth Information
and
⎡
⎤
w
P
−
1
(
n
)
⎣
⎦
w
P
−
2
(
n
)
w
(
n
)
=
,
n
=
0,
...
,
K
−
1
(5.54)
.
.
.
w
0
(
n
)
Using this formulation, obtaining the Wiener solution is a process similar
to that developed in Section 3.2. We can express the Wiener-Hopf equations
(3.13) for the multichannel model as
R
x
)
−
1
p
R
x
w
opt
=
p
⇒
w
opt
=
(
(5.55)
where the autocorrelation matrix is given by
E
x
x
H
R
x
=
(
n
)
(
n
)
and the cross-correlation vector between the desired and the received
signals is
E
x
s
∗
n
d
=
n
−
p
(
)
5.2.5 Bezout's Identity and the Zero-Forcing Criterion
The condition stated in Theorem 5.2 is also valid for this particular case of
a SIMO channel, thus providing a suitable condition for ZF equalization.
Let the polynomial vector associated with the channel be defined as
⎡
⎣
⎤
⎦
H
0
(
z
)
H
1
(
z
)
L
−
1
z
−
i
h
(
z
)
=
h
(
i
)
=
(5.56)
.
.
.
i
=
0
H
P
−
1
(
z
)
where
h
(
i
)
is given in (5.34), and the polynomial vector related to the
equalizer is
