Digital Signal Processing Reference
In-Depth Information
and
σ
s
1+
σ
s
|
S
ee,mmse
(
e
jω
)=
(4
.
92)
C
(
e
jω
)
|
2
σ
q
where the extended subscript is to distinguish this from the zero-forcing case.
In the latter case the equalizer is
G
(
e
jω
)=1
/C
(
e
jω
) and the error spectrum is
σ
q
S
ee,ZF
(
e
jω
)=
(4
.
93)
|
C
(
e
jω
)
|
2
Note that
S
ee,mmse
(
e
jω
)
S
ee,ZF
(
e
jω
) for each frequency
ω.
Furthermore,
S
ee,mmse
(
e
jω
) is bounded above by
σ
s
,
whereas
S
ee,ZF
(
e
jω
) can get arbitrarily
large because the channel response
C
(
e
jω
) can be close to zero at some frequen-
cies. The mean square value of the reconstruction error can be calculated as the
integral
≤
2
π
1
2
π
|
2
=
S
ee
(
e
jω
)
dω,
E
=
E
|
s
(
n
)
−
s
(
n
)
0
where
S
ee
(
e
jω
) is as in Eq. (4.92) or Eq. (4.93).
Example 4.4: Zero-forcing versus MMSE equalizers
To consider a specific example with numbers, let
σ
s
=1
.
0
,
σ
q
=0
.
1
,
and
1
2
ρ
k
cos
θ
k
z
−
1
+
ρ
k
z
−
2
,
2
C
(
z
)=
−
k
=1
with
ρ
1
=0
.
94
,ρ
2
=0
.
96
,θ
1
=0
.
2
π,
and
θ
2
=0
.
32
π.
Figure 4.38 (top) shows the magnitude response
C
(
e
jω
)
.
The channel is
FIR, so the inverse 1
/C
(
z
) is IIR. It has poles inside the unit circle, but
they are rather close to the unit circle. This makes 1
/
|
|
rather large
as
ω
approaches
θ
k
.
The error spectrum
S
ee,ZF
(
e
jω
) is therefore large. Fig-
ure 4.38 shows the plots of the channel response
|
C
(
e
jω
)
|
C
(
e
jω
)
, the error spec-
trum for the
zero-forcing equalizer
, and the error spectrum for the
MMSE
equalizer
(Eqs. (4.92),(4.93)). Note that the peak error spectrum for the
zero-forcing equalizer is about 90 times larger than the error spectrum for
the MMSE equalizer. The integrals of these spectra yield the mean square
reconstruction errors:
|
|
|
2
=
7
.
7872
zero forcing
E
|
s
(
n
)
−
s
(
n
)
0
.
3351
MMSE.
Replacing a zero-forcing equalizer with an MMSE equalizer therefore results
in a reduction in MSE by a factor of about 23 (about 13
.
6dB).
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