Digital Signal Processing Reference
In-Depth Information
⎧
⎨
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
π
2
1
p
0
dω
2
π
(with ZF)
H
(
e
jω
)
|
2
−π
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
2
E
mmse
=
dω
2
π
⎩
+
F
H
(
e
jω
)
|
2
S
ss
(
e
jω
)
dω
2
π
F
p
0
+
(no ZF).
S
qq
(
e
jω
)
|
dω
2
π
c
H
(
e
jω
)
|
2
F
(10
.
42)
Discussion
A number of points should be observed:
1.
Effective noise spectrum
. The channel transfer function
H
(
z
)andthe
noise spectrum
S
qq
(
e
jω
) appear together in these expressions, in the form
S
qq
(
e
jω
)
/
H
(
e
jω
)
|
2
.
They never appear separately.
|
The ratio given by
S
qq
(
e
jω
)
/
|
H
(
e
jω
)
|
2
can be regarded as the effective noise spectrum.
2.
ZF-MMSE error can be unbounded
. As seen from Eq. (10.39), for the ZF-
MMSE solution the prefilter magnitude
F
(
e
jω
)
|
|
is large where the channel
is bad (i.e.,
S
qq
(
e
jω
)
/
|
2
is large). So the ZF-MMSE solution pumps
more power in these bad regions. As a result, the minimized mean square
error (10.41) can be very large for bad channels. For example, if
H
(
e
jω
)
|
H
(
e
jω
)
|
|→
0 in some frequency regions, then the error is unbounded.
3.
Pure-MMSE error is bounded
. The pure-MMSE solution behaves differ-
ently. First, the filter magnitude
F
(
e
jω
)
|
|
does not keep increasing with
S
qq
(
e
jω
)
/
|
2
is larger than a
threshold, the filter becomes zero, as seen from Eq. (10.33). In such
regions, the reconstruction error is obtained from the second term in Eq.
(10.35) and is therefore bounded. In fact it is readily verified using Cauchy-
Schwartz inequality that the first term in Eq. (10.35) is bounded above by
H
(
e
jω
)
|
2
.
In regions where
S
qq
(
e
jω
)
/
H
(
e
jω
)
|
|
F
S
ss
(
e
jω
)
dω/
2
π
:
2
S
qq
(
e
jω
)
S
ss
(
e
jω
)
|
dω
2
π
S
qq
(
e
jω
)
|H
(
e
jω
)
|
2
dω
2
π
S
ss
(
e
jω
)
dω
2
π
H
(
e
jω
)
|
2
F
p
0
+
p
0
+
F
F
F
≤
S
qq
(
e
jω
)
|
dω
2
π
S
qq
(
e
jω
)
|
dω
2
π
H
(
e
jω
)
|
2
H
(
e
jω
)
|
2
F
S
ss
(
e
jω
)
dω
2
π
≤
F
Consequently,
+
F
=
π
−π
S
ss
(
e
jω
)
dω
2
π
S
ss
(
e
jω
)
dω
2
π
S
ss
(
e
jω
)
dω
2
π
=
σ
s
.
E
mmse
≤
F
c
Search WWH ::
Custom Search