Digital Signal Processing Reference
In-Depth Information
q
(
n
)
M
M
s
(
n
)
s
(
n
)
H
(
z
)
d
Figure 4.10
. The all-discrete-time MIMO channel.
Defining the continuous-time noise vector
q
M−
1
(
t
)]
T
q
c
(
t
)=[
q
0
(
t
)
q
1
(
t
)
...
(4
.
25)
and the sampled noise vector
q
(
n
)=(
g
∗
q
c
)(
nT
)=
∞
−∞
g
(
τ
)
q
c
(
nT
−
τ
)
dτ,
(4
.
26)
we see that the discrete-time equivalent can be represented entirely using
H
d
(
z
)
and
q
(
n
) (Fig. 4.10). Note that the discrete-time model does not assume that
the continuous-time channel and filters are bandlimited.
4.3.2.A The digital MIMO transceiver
If
F
(
jω
)and
G
(
jω
) are designed appropriately, then
H
d
(
z
) is (approximately)
FIR:
L
h
d
(
n
)
z
−n
.
H
d
(
z
)=
(4
.
27)
n
=0
If this can be approximated by a constant:
H
d
(
z
)=
h
d
(0)
,
(4
.
28)
then the equivalent digital MIMO channel becomes
memoryless
. More generally
H
d
(
z
) has memory, and can be compensated for by using a digital precoder
F
d
(
z
) and a digital equalizer
G
d
(
z
)
,
as shown in Fig. 4.11.
The optimal choice of
F
d
(
z
)and
G
d
(
z
) is an important problem. The details
of this optimization problem depend on the objective function that we desire to
optimize. One example is the minimization of the total mean square error,
M−
1
|
2
]
,
E
[
|
s
k
(
n
)
−
s
k
(
n
)
(4
.
29)
k
=0
subject to the constraint
G
d
(
z
)
H
d
(
z
)
F
d
(
z
)=
I
.
(4
.
30)
Search WWH ::
Custom Search