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

544
PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
and based on Parseval's theorem, the output is
1_: q(r:)s;(r:) dr: = <q, s;>
Hence, <q, s;> is the output at t = TM of a filter matched to s; (t) when q(t) is applied to
its input.
Actually, we do not have q(t). The incoming signal r(t) is given by
r(t) = s;(t) + n w (t)
= s;(t) + n(t) + no(t)
'--.--'
q(t)
iJTclcvant
where no (t) is the (irrelevant) component of n w (t) orthogonal to the N-dimensional signal
space. Because no (t) is orthogonal to this space, it is orthogonal to every signal in this space.
Hence, it is orthogonal to the signal s;(t), and
and
1_: no(t)s;(t) dt = 0
<q, s;> = 1_: q(t)s;(t) dt + 1_: no(t)s;(t) dt
= 1_: [q(t) + no(t)]s;(t) dt
= 1_: r(t)s;(t) dt (10.91)
Hence, it is immaterial whether we use q(t) or r(t) at the input. We thus apply the incoming
signal r(t) to a parallel bank of matched filters, and the output of the filters is sampled at
t = \4 . Then a constant a; is added to the ith filter output sample, and the resulting outputs
are compared. The decision is made in favor of the signal for which this output is the largest.
The receiver implementation for this decision procedure is shown in Fig. 10.18a. Section l O. l
has already established that a matched filter is equivalent to a correlator. One may therefore
use correlators instead of matched filters. Such an arrangement is shown in Fig. 10.18b.
We have shown that in the presence of AWGN, the matched filter receiver is the optimum
receiver when the merit criterion is minimum error probability. Note that the optimum system
is found to be linear, although it was not constrained to be so. Therefore, for white Gaussian
noise, the optimum receiver happens to be linear. The matched filter obtained in Secs. 10.1
and 10.2, as well as the decision procedure are identical to those derived here.
The optimum receiver can be implemented in another way. From Eq. (10.91), we have
From Eq. (10.44), we can rewrite this as
<q, S; > = <r, S; >
N
<q, S;> = L r1s;1
j=!