Digital Signal Processing Reference
In-Depth Information
shape of
H
c
(
jω
) does not matter in the formulation, as the solutions will be
expressed in terms of
H
c
(
jω
)
.
The shape of
H
c
(
jω
) will itself be optimized later
in Sec. 10.2.2.
Assumptions.
We will assume that the noise
q
(
t
) is WSS with zero mean and
power spectrum
S
qq
(
jω
). The signal
s
(
n
) is assumed to be a random process with
independent samples having zero mean and variance
σ
s
.
We also assume that
s
(
n
)
and
q
(
t
) are statistically independent for all integer
n
and real
t.
From Appendix E (Sec. E.3) we know that the output of
F
(
jω
) is a CWSS(
T
)
process with power spectrum
σ
s
|
|
2
/T
. So the transmitted power (average
variance at the output of
F
(
jω
)) is given by
F
(
jω
)
∞
−∞
|
p
0
=
σ
s
T
|
2
dω
2
π
F
(
jω
)
(10
.
3)
For fixed p ower
p
0
and fixed
H
c
(
jω
) satisfying the ZF constraint (10.2), we shall
minimize the mean square error
|
2
,
E
mse
=
E
|
e
(
n
)
where
s
(
n
)
.
In view of the zero-forcing constraint, the reconstruction error is entirely due to
the noise
q
(
t
) filtering through
G
(
jω
)
.
Thus
e
(
n
)=
∞
−∞
e
(
n
)=
s
(
n
)
−
g
(
τ
)
q
(
nT
−
τ
)
dτ.
Since
q
(
t
) is WSS, the mean square value of
e
(
n
) is independent of
n
and is given
by
E
mse
=
∞
−∞
dω
2
π
In view of the product constraint (10.1) we can rewrite this as
E
mse
=
∞
−∞
|
2
S
qq
(
jω
)
|
G
(
jω
)
|
H
c
(
jω
)
|
2
dω
2
π
S
qq
(
jω
)
(10
.
4)
|
F
(
jω
)
|
2
|
H
(
jω
)
|
2
Thus the product constraint (10.1) has been eliminated, and the goal is to opti-
mize
F
(
jω
) such that Eq. (10.4) is minimized subject to the constraint (10.3).
Theorem 10.1.
Optimal SISO transceiver.
Consider the digital communi-
cation system shown in Fig. 10.1. Under the power constraint (10.3) and the
product constraint (10.1), where
H
c
(
jω
) satisfies the ZF constraint (10.2), the
optimal combination of the filters
F
(
jω
)and
G
(
jω
)isgivenby
♠
1
/
2
H
c
(
jω
)
H
(
jω
)
F
(
jω
)=
βe
jθ
f
(
ω
)
S
1
/
4
qq
(
jω
)
(10
.
5)
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