Digital Signal Processing Reference
In-Depth Information
q
(
n
)
F
(
z
)
d
H
(
z
)
d
s
(
n
)
+
G
(
z
)
d
s
(
n
)
precoder
channel
equalizer
Figure 4.7
. The all-discrete-time equivalent diagram of the transceiver, with a digital
precoder
F
d
(
z
) and an equalizer
G
d
(
z
)
.
4.3.2 Digital communication over a MIMO channel
Figure 4.8 shows the case where
M
symbol streams
s
k
(
n
)
,
0
1
,
have
to be transmitted over a communication medium with transfer matrix
H
(
jω
)
and additive noise sources
q
k
(
t
)
.
The reconstructed signal
≤
k
≤
M
−
s
m
(
n
) depends not
only on
s
m
(
n
)
,
but also on
s
k
(
)
,
where
k
=
.
The interference
between
s
m
(
n
)and
s
k
(
n
) for the same
n
is called
intrablock interference
,andthe
interference between
s
m
(
n
)and
s
k
(
)for
=
m
and/or
n
=
n
is called
interblock interference
.
These can be regarded as two forms of ISI. When we say that the transceiver is
free from ISI, or equivalently satisfies
zero forcing
, we imply that both interblock
and intrablock interferences have been eliminated.
The MIMO channel can often be approximated by a transfer matrix
H
(
jω
)
.
The (
m, k
)th element
H
mk
(
jω
) of this matrix represents the channel connecting
the output of
F
k
(
jω
) to the input of
G
m
(
jω
)
.
This transceiver can be made more
general by replacing the scalar prefilters and postfilters with matrix transfer
functions
F
(
jω
)and
G
(
jω
) as in Fig. 4.9. This implies that collaboration is
allowed between different “users”
s
k
(
n
)
.
Such a situation arises, for example,
in DMT systems where a single user is “divided” into different frequency bands
(Chap. 7). The situation also arises to a limited extent in the so-called broadcast
channels (Sec. 4.5). Denoting the cascaded system by
H
c
(
jω
)
,
we have
H
c
(
jω
)=
G
(
jω
)
H
(
jω
)
F
(
jω
)
.
(4
.
21)
Let
h
c
(
t
) denote the impulse response of this MIMO system, that is,
h
c
(
t
)=
∞
−∞
H
c
(
jω
)
e
jωt
dω
2
π
.
(4
.
22)
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