Digital Signal Processing Reference
In-Depth Information
⎡
⎣
⎤
⎦
γ
0
(
z
)
0
0
···
0
···
0
0
γ
1
(
z
)
0
···
0
···
0
.
.
.
.
.
.
.
.
.
.
.
.
0
0
0
.
.
.
.
.
.
0
γ
ρ
−
1
(
z
)
0
···
0
(
z
)
=
,
(5.11)
.
.
.
0
0
0
0
···
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
···
0
0
···
0
where each element is a monic polynomial γ
i
(
z
)
. In addition to that, it is
verified that γ
i
(
z
)
divides γ
i
+
1
(
z
)
,0
≤
i
≤
ρ
−
2, ρ being the normal rank of
H
(
z
)
. For a given matrix
H
(
z
)
,
(
z
)
is unique, and represents the Smith form
of
H
(
z
)
.
Given the Smith form of
H
(
z
)
, the zeros of the channel are defined as the
roots of γ
i
(
[25]. Then, the following theorem provides the conditions for
the ZF equalization of MIMO-FIR channels [25].
z
)
THEOREM 5.2 (Equalizability of MIMO-FIR Channels)
Let
H
M
×
N
N
. Considering its asso-
ciated Smith form given by (5.10), the channel is perfectly equalizable, i.e.,
the ZF condition is attainable by means of MIMO-FIR equalizers if, and only
if, γ
i
(
(
z
)
denote a MIMO-FIR channel, with
M
≥
z
−
d
i
for some
d
i
≥
z
)
=
0.
Theorem 5.2 states that it is possible to obtain a ZF equalizer even if both
channel and equalizer are FIR structures, a situation very distant from that
found in the SISO case. Considering the definition of system zeros of a MIMO
system, the condition under which it is possible to perfectly equalize it is
equivalent to the nonexistence of finite zeros associated with the system [153].
It is important to mention that this condition evokes the idea of a trivial
SISO channel, i.e., one that only imposes a delay to the input signal. How-
ever, it is interesting to imagine that for a MIMO-FIR structure, even if the
sub-channels linking each transmitter-receiver are not trivial channels, the
global system can be perfectly inverted by an FIR equalizer.
5.2 SIMO Channel Equalization
Let us now consider a particular case of the general MIMO model, which is
illustrated in
Figure 5.3
. In this scenario, we have a SIMO channel model, in
which a single signal is transmitted by multiple subchannels.
