Digital Signal Processing Reference
In-Depth Information
Substituting this into Eq. (16.110) and simplifying, we get
E
mmse
=
σ
s
π
−π
1
dω
2
π
(16
.
117)
H
(
e
jω
)
|
|
Similarly, Eq. (16.111) can be rewritten by substituting from Eq. (16.116):
π
2
1
dω
2
π
σ
s
|
H
(
e
jω
)
|
−π
E
ZF
=
(16
.
118)
1
1
dω
2
π
−
1
dω
2
π
|
H
(
e
jω
)
|
|
H
(
e
jω
)
|
2
F
F
Substituting from Eqs. (16.117) and (16.118) into the left hand side of Eq.
(16.113) and simplifying we get
π
1
dω
2
π
σ
s
E
mmse
−
σ
s
E
ZF
|
H
(
e
jω
)
|
2
−π
=
(16
.
119)
π
2
1
dω
2
π
|
H
(
e
jω
)
|
−
π
Now, using Cauchy-Schwartz inequality (Appendix A) it readily follows that
the above right-hand side is
1
.
This proves Eq. (16.113) or equivalently
(16.112) under the assumption (16.115). Next consider the more general
case where
≥
H
(
e
jω
)
c
.
|
|≤
,
ω
∈F
(16
.
120)
In the expression (16.110) for
are involved, so
E
mmse
is not affected by this. From the expression (16.111) for
E
mmse
,
only integrals over
F
E
ZF
,
it is
clear that if we use Eq. (16.120) instead of Eq. (16.115), the result can only
get larger because 1
/
H
(
e
jω
)
|
|
gets larger in
F
c
.
Thus, Eq. (16.113) continues
to hold.
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