Digital Signal Processing Reference
In-Depth Information
3.
Memoryless case.
If the channel and equalizer are memoryless, as in Eq.
(4.76), then, by proceeding as in the proof of Theorem 4.3, we can show
(Problem 4.9) that the autocorrelation matrix of the minimized error is
R
ee
=
−
1
,
R
−
1
ss
+
C
†
R
−
1
qq
C
(4
.
83)
where
R
ss
=
E
[
s
(
n
)
s
†
(
n
)] and
R
qq
=
E
[
q
(
n
)
q
†
(
n
)]
.
Example 4.3: MMSE equalizer for scalar channel
Consider the special case of scalar channels, where all matrices become
scalars. In this case (4.79) yields
S
ss
(
e
jω
)
C
∗
(
e
jω
)
S
qq
(
e
jω
)+
S
ss
(
e
jω
)
G
(
e
jω
)=
(4
.
84)
|
C
(
e
jω
)
|
2
The noise spectrum
S
qq
(
e
jω
) is usually nonzero for all
ω
in the band of
interest. So, even if
C
(
e
jω
) = 0 for some frequencies, the equalizer is still
defined for all
ω
, unlike in the zero-forcing case. If
C
(
e
jω
)
=0forall
ω
,
then the equalizer can also be written as
1
/C
(
e
jω
)
G
(
e
jω
)=
(4
.
85)
S
qq
(
e
jω
)
S
ss
(
e
jω
)
1+
|
C
(
e
jω
)
|
2
The numerator 1
/C
(
e
jω
) is the zero-forcing part. The rest of the right-
hand side is the “Wiener part,” or correction factor, which makes the filter
statistically optimal. From (4.80) the power spectrum of the error is given
by
1
S
ee
(
e
jω
)=
(4
.
86)
|
2
S
qq
(
e
jω
)
C
(
e
jω
)
S
ss
(
e
jω
)
+
|
1
which can be rewritten as
S
ss
(
e
jω
)
1+
S
ss
(
e
jω
)
S
ee
(
e
jω
)=
(4
.
87)
C
(
e
jω
)
|
2
|
S
qq
(
e
jω
)
The Wiener filter (4.85) is in general
not realizable
as a causal stable filter even
if the channel is a rational FIR or IIR filter; it can only be approximated. See
Problems 4.13-4.15. A number of remarks are now in order.
1.
When is zero-forcing the best?
Observe first that
G
(
e
jω
0
)=1
/C
(
e
jω
0
)if
and only if the noise spectrum
S
qq
(
e
jω
0
) = 0 at the frequency
ω
0
.
Thus,
where the noise is zero, the best equalizer is certainly the “common sense
solution” 1
/C
(
e
jω
0
)
.
If the channel is noiseless (
q
(
n
)=0)then
S
qq
(
e
jω
)=0
for all
ω,
and the best equalizer is the zero-forcing equalizer.
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