Digital Signal Processing Reference
In-Depth Information
Table I.11. All-digital SISO channel (Chap. 10)
The all-digital equivalent of the linear SISO transceiver is reproduced below. In what
follows
S
ss
(
e
jω
)and
S
qq
(
e
jω
) are the power spectra of the input and the noise, which
are assumed to be zero-mean and uncorrelated processes. The filters
F
(
e
jω
)and
G
(
e
jω
) optimized for MMSE are specified below; when magnitude squares are specified,
the filters are taken to be stable spectral factors.
q
(
n
)
y
(
n
)
s
(
n
)
F
(
z
)
H
(
z
)
+
s
(
n
)
G
(
z
)
prefilter
channel
postfilter
ZF-MMSE case (Sec. 10.3.2)
The optimal precoder is obtained as a spectral factor of
2
=
γ
S
qq
(
e
jω
)
F
(
e
jω
)
|
|
|
2
S
ss
(
e
jω
)
,
(I
.
41)
|
H
(
e
jω
)
and the zero-forcing equalizer is
G
(
e
jω
)=1
/
[
F
(
e
jω
)
H
(
e
jω
)]
.
Here
γ
is determined
from the power constraint. The minimized MSE is given by
π
2
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
1
p
0
dω
2
π
E
mmse
=
.
(I
.
42)
H
(
e
jω
)
|
2
−
π
Pure-MMSE case (Sec. 10.3.1)
The optimal prefilter is computed from the expression
γ
⎧
⎨
S
qq
(
e
jω
)
S
qq
(
e
jω
)
|
2
S
ss
(
e
jω
)
−
if this is
≥
0
|
H
(
e
jω
)
|
H
(
e
jω
)
|
2
S
ss
(
e
jω
)
|F
(
e
jω
)
|
2
=
(I
.
43)
⎩
0
otherwise.
The constant
γ
is computed from the power constraint. The equalizer is
H
∗
(
e
jω
)
F
∗
(
e
jω
)
S
ss
(
e
jω
)
S
ss
(
e
jω
)
G
(
e
jω
)=
(I
.
44)
|
H
(
e
jω
)
F
(
e
jω
)
|
2
+
S
qq
(
e
jω
)
The minimized MSE is given by
2
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|H
(
e
jω
)
|
2
dω
2
π
+
F
S
ss
(
e
jω
)
dω
2
π
F
p
0
+
F
E
mmse
=
(I
.
45)
S
qq
(
e
jω
)
|H
(
e
jω
)
|
2
dω
2
π
c
where
p
0
is the transmitted power,
F
is the set of frequencies in [
−π, π
]forwhich
|F
(
e
jω
)
|
2
is nonzero, and
F
c
is the complementary set.
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