Digital Signal Processing Reference
In-Depth Information
q
(
t
)
noise
y(t)
s
(
n
)
+
F
(
j
ω
)
H
(
j
ω
)
G
(
j
ω
)
s
(
n
)
C/D
D/C
T
T
postfilter
prefilter
channel
Figure 10.1
. A digital communication system to transmit messages through a continuous-
time channel.
However, some standard continuous-time filtering is required first: namely
a matched filter at the receiver, and a so-called “optimum compaction filter”
at the transmitter. Historically, Ericson's work unifies many earlier papers in
transceiver literature which solved the SISO transceiver optimization problems
under various settings.
In Sec. 10.5 we revisit SISO channels with oversampling at the receiver, and
some simple examples of optimization are considered. In Sec. 10.6 we briefly
mention a different kind of problem, namely the optimal pulse-shaping problem
for the case where a
single
pulse is transmitted over a continuous-time channel.
The criterion to be optimized is the signal-to-noise ratio at the receiver. This
“single shot” scenario creates a different solution compared to the earlier pulse-
shaping solutions because there is no intersymbol interference. Such pulse design
problems arise in pulsed radar systems.
10.2 Optimization of the SISO communication system
Consider the digital communication system shown in Fig. 10.1 which transmits
a sequence of symbols
s
(
n
) over a continuous-time channel. This system was de-
scribed in detail in earlier chapters. We now consider the problem of optimizing
the prefilter
F
(
jω
) and equalizer
G
(
jω
) to minimize the mean square error.
10.2.1 Minimizing MSE under the product constraint
Let us assume that the transfer function
G
(
jω
)
H
(
jω
)
F
(
jω
) is constrained to
be a fixed function of frequency:
G
(
jω
)
H
(
jω
)
F
(
jω
)=
H
c
(
jω
)
(product constraint).
(10
.
1)
Under this condition we will optimize
G
(
jω
)and
F
(
jω
) such that the mean
square error is minimized. If we wish to have the zero-forcing property, then
H
c
(
jω
) should be such that its impulse response is Nyquist(
T
), that is,
h
c
(
nT
)=
δ
(
n
)
(ZF constraint).
(10
.
2)
We will derive expressions for the optimal pair of filters
for fixed
H
c
(
jω
) satisfying the ZF constraint, and fixed transmitted power. The exact
{
F
(
jω
)
,G
(
jω
)
}
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