Digital Signal Processing Reference
In-Depth Information
and
1
/
2
G
(
jω
)=
e
−j
[
θ
(
ω
)+
θ
f
(
ω
)]
β
H
c
(
jω
)
H
(
jω
)
S
−
1
/
4
qq
(
jω
)
.
(10
.
6)
The minimized mean square error per symbol is given by
∞
2
S
1
/
2
σ
s
p
0
T
qq
(
jω
)
dω
2
π
E
mmse
=
|
H
c
(
jω
)
|
.
(10
.
7)
|
H
(
jω
)
|
−∞
In these expressions
θ
f
(
ω
) is arbitrary (e.g., it can be taken to be zero for
simplicity),
θ
(
ω
) is the phase of
H
(
jω
)
/H
c
(
jω
)
,
and the constant
β
is computed
using the power constraint (10.3).
♦
Proof.
From Cauchy-Schwartz inequality (Appendix A) we have
∞
∞
−∞
|
|
H
c
(
jω
)
|
2
dω
2
π
|
2
dω
2
π
S
qq
(
jω
)
F
(
jω
)
|
F
(
jω
)
|
2
|
H
(
jω
)
|
2
−∞
∞
2
qq
(
jω
)
H
c
(
jω
)
H
(
jω
)
dω
2
π
S
1
/
2
≥
.
(10
.
8)
−∞
Substituting from Eq. (10.3) and rearranging this we get
∞
|
2
|F
(
jω
)
|
2
|H
(
jω
)
|
2
|
H
c
(
jω
)
dω
2
π
S
qq
(
jω
)
−∞
σ
s
∞
−∞
dω/
2
π
2
S
1
/
2
qq
(
jω
)
|
H
c
(
jω
)
/H
(
jω
)
|
≥
(10
.
9)
p
0
T
Equality is achieved when the two integrands on the left-hand side of Eq.
(10.8) are equal up to a scale factor. So we can take the optimal prefilter
solution to be as in Eq. (10.5), where
θ
f
(
ω
) is an arbitrary phase factor
and the constant
β
is such that the power constraint (10.3) is satisfied. The
optimum equalizer
G
(
jω
) can then be found from the product constraint
(10.1), and yields Eq. (10.6). The minimized MSE is given by the right
hand side of Eq. (10.9), which proves Eq. (10.7).
Remarks
1.
Bandlimiting.
The optimal solution assumes that the channel
H
(
jω
)and
the noise spectrum
S
qq
(
jω
) are nonzero for all
ω
(because these appear
in certain denominators). The error expression (10.7) also requires the
assumption
H
(
jω
)
=0
.
In practice, the channel is bandlimited and so are
the filters
G
(
jω
)and
F
(
jω
). So these assumptions are required to hold
only in this band. All the infinite integrals in the preceding results are
then replaced with finite integrals limited to the baseband.
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