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

508 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
to be O or 1, then the optimum threshold detection is
dec{r(tm)} = { 0
1 if r(t m ) 2:: 0
if r(t m ) < 0
( IO.Sa)
whereas the probability of (bit) error is
P e = Q (p)
(10.5b)
in which
(IO.Sc)
To minimize P e , we need to maximize p because Q(p) decreases monotonically with p.
10. 1 .2 Optimum Receiver Filter-Matched Filter
Let the received pulse p(t) be time-limited to T 0 (Fig. 10.1 ). We shall keep the discussion as
general as possible at this point. To minimize the BER or P e , we should determine the best
receiver filter H (f) and the corresponding sampling instant t m such that Q(p) is minimized.
In other words, we seek a filter with a transfer function H (f) that maximizes
2 p(tm)
p = -­
a 2 n
(10.6)
which is coincidentally also the signal-to-noise ratio at time instant t = t m .
First, denote the Fourier transform of p(t) as P(f) and the PSD of the channel noise n(t)
as Sn (f). We will determine the optimum receiver filter in the frequency domain. Starting with
Po(t) = F - 1 [P(f)H(f)]
= 1_: P(f)H(f)d 2 nft df
we have the sample value at t = t m
On the other hand, the filtered noise has zero mean
Po (t m ) = 1_: P(f)H(f),l 2 7Tftm df (10.7)
while its variance is given by
n 0 (t) = fo 1 n('r)h(t - r) dr = fo' n(r)h(t - r)dr = 0
(10.8)