B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

510 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Figure 10.2
Optimum choice
for sampling
instant.
p(t)
p(-t)
tm < To
t--+p(tm
-t)
tm
t--+-
t--+p(tm-t)
tm = To
tm
tm > To
tm
t--+p(tm-t)
t--+-
The unit impulse response h(t) of the optimum filter is obtained from the inverse Fourier
transform
Note that p (-t) {::::::} P(-f) and e-j 2 :rrftm represents the time delay of t m seconds. Hence,
h(t) = k'p(t m - t)
(10. llc)
The response p(t m - t) is the signal pulse p (-t) delayed by t m . Three cases, t m < T 0 , t m = T 0 ,
and t m > T0 , are shown in Fig. 10.2. The first case, t m < T0 , yields a noncausal impulse
response, which is unrealizable.* Although the other two cases yield physically realizable
filters, the last case, t m > T0 , delays the decision-making instant t m unnecessarily. The case
* The filter unrealizability can be readily understood intuitively when the decision-making instant is tm < T0 • In this
case, we are forced to make a decision before the full pulse has been fed to the filter (tm < T0). This calls for a
prophetic filter, which can respond to inputs before they are applied. As we know, only unrealizable (noncausal)
filters can do this job.