Digital Signal Processing Reference
In-Depth Information
⎡
⎣
⎤
⎦
W
P
−
1
(
z
)
W
P
−
2
(
z
)
K
−
1
z
−
i
w
(
z
)
=
w
(
i
)
=
(5.57)
.
.
.
i
=
0
W
0
(
z
)
where
w
is defined in (5.54).
Then, in the spirit of (5.10), the polynomial vector can be rewritten in
terms of its Smith form as
(
n
)
⎡
⎣
⎤
⎦
γ
(
z
)
0
.
.
.
h
(
z
)
=
Q
(
z
)
.
(5.58)
0
z
−
d
.
Hence, the condition for ZF equalization is reduced to γ
(
z
)
=
It is interesting to note that γ
(
z
)
represents the greatest common divisor
of all subchannels
h
i
(
are the common
zeros of all subchannels of the SIMO system. In other words, an FIR SIMO
channel is perfectly equalized by another FIR structure if and only if the sub-
channels have no zeros in common. This result, even though explained here
in terms of the Smith form of the channel, is also a consequence of
Bezout's
identity
[163].
z
)
, which means that the roots of γ
(
z
)
THEOREM 5.3 (Bezout's Identity)
Let the
h
, defined in (5.56) and (5.57), denote the polynomial
vectors associated with the channel and equalizer, respectively. If the poly-
nomials
H
0
(
(
z
)
and
w
(
z
)
z
)
...
,
H
P
−
1
(
z
)
,
do not share common zeros, then there exists a
set of polynomials
W
0
(
z
)
...
,
W
P
−
1
(
z
)
,
such that
w
H
z
−
d
(
z
)
(
z
)
=
h
(5.59)
where
d
is an arbitrary delay.
In the time domain, Bezout's identity implies
=
0
0
w
H
H
···
010
···
(5.60)
