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

540 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
From Eqs. (10.73a) and (10.74), it is evident that we can filter out the component no(t) from
nw(t). This can be seen from the fact that the received signal, r(t), can be expressed as
r(t) = Sk (t) + nw (t)
= sk (t) + n(t) + no(t)
= q(t) + no(t)
(10.75)
where q(t) is the projection of r(t) on the N-dimensional space:
q(t) = Sk (t) + n(t)
(10.76)
We can obtain the projection q(t) from r(t) by observing that [see Eqs. (10.71b) and (10.73a)]
N
q(t) = )skj + nj)(f)j(t)
j=l
(10.77)
Figure 1 O. 17
Eliminating the
noise orthogonal
to signal space.
From Eqs. (10.71c), (10.74), and (10.77) it follows that if we feed the received signal r(t)
into the system shown in Fig. 10. 17, the resultant outcome will be q(t). Thus, the orthogonal
noise component can be filtered out without disturbing the message signal.
The question here is: Would such filtering help in our decision making? We can easily show
that it cannot hurt us. The noise n w (t) is independent of the signal waveform sk (t). Therefore,
its component no (t) is also independent of sk (t). Thus, no(t) contains no information about the
transmitted signal, and discarding such a component from the received signal r(t) will not cause
any loss of information regarding the signal waveform sk (t). This, however, is not enough.
We must also make sure that the noise being discarded [no(t)] is not in any way related to the
remaining noise component n(t). If no(t) and n(t) are related in any way, it will be possible to
obtain some information about n(t) from no (t), thereby enabling us to detect that signal with
less error probability. If the components no(t) and n(t) are independent random processes, the
component no(t) does not carry any information about n(t) and can be discarded. Under these
conditions, no (t) is irrelevant to the decision making at the receiver.
The process n(t) is represented by components n1 , n2, ... , nN along ({)I (t), q;2 (t), ... ,
(f)N (t), and n0(t) is represented by the remaining components (infinite number) along the
remaining basis signals in the complete set, { (f)k (t)}. Because the channel noise is white
Gaussian, from Eq. (10.68) we observe that all the components are independent. Hence,
l 'P1 (t)
r(t) q(t)
f - - - - - - - - - - - - - - - - - - - - - - - - - -
Jt:
- - - - - - -
► ,m► .-----., t
' sit) + n(t)