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

672 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Fi g ure 12.4
Commonly used
receiver filter
matched instead
to the transmission
pulse.
y (t) Receiver filter z[n]
--- p (-t)
t = nT
Therefore, the receiver filter must not filter out any valuable signal component and should
have bandwidth equal to the bandwidth of P(f). On the other hand, if we let the receiver
filter have a bandwidth larger than P(f), then more noise will pass through the filter, with no
benefit to the signal. For these reasons, a good receiver filter should have bandwidth exactly
identical to the bandwidth of P(f). Of course many such filters exist. One is the filter matched
to the transmission pulse p(t) given by
p(-t) <===> P*(f)
Another consideration is that, if the channel introduces no additional distortions, then q(t) =
p(t). In this case, the optimum receiver would be the filter p(-t) matched to p(t). Consequently,
it makes sense to select p(-t) as a standard receiver filter (Fig. 12.4) for two reasons:
(a) The filter p( -t) retains all the signal spectral component in the received signal y(t).
(b) The filter p(-t) is optimum if the environment happens to exhibit no channel distortions.
Therefore, we often apply the receiver filter p( -t) matched to the transmission pulse shape
p(t). This means that the total channel impulse response consists of
h(t) = q(t) * p(-t)
Notice that because of the filtering z(t) = p(-t) * y(t). The signal z(t) now becomes
z(t) = L skh(t - kT) + w(t)
k
( 12.15)
in which the filtered noise term w(t) arises from
w(t) = p(-t) * n c (t)
( 12.16)
with power spectral density
Swlf) = IP(f) l 2 Sn,lf)
Finally, the relationship between the sampled output z[k] and the communication symbols sk is
z[n] = L h[n - k]sk + w[n]
/..:
= L h[k] Sn-k + w[n]
k
( 12.17)
where the discrete noise samples are denoted by wLn] = w (nT).