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

10.6 Optimum Receiver for White Gaussian Noise Channels 537
Fi g ure 1 0. 14
Mary communication
system.
m
s(t)
----J---1 Transmitter 1----+---1 r 1----+--1 Receiver
{sit) } r(t)
m
Fi g
ure 1 O. 1 5
Effect of
Gaussian
channel noise
on the received
signal.
·> :('\ :•· ..
••\f]:j}:
... ·' ·
corrupted by AWGN nw(t) (Fig. 10. 14) with PSD
At the receiver, the received signal r(t) consists of one of the M message waveforms Sk (t)
plus the channel noise,
r(t) = Sk (t) + nw(t)
(10.70a)
Because the noise n w (t) is white, we can use the same basis functions to decompose both
Sk (t) and n w (t). Thus, we can represent r(t) in a signal space by denoting r, Sk, and n w as the
vectors representing signals r(t), sk (t), and nw (t), respectively, Then it is evident that
r = Sk + llw
(10.70b)
The signal vector sk is a fixed vector, because the waveform sk (t) is nonrandom, whereas the
noise vector llw is random. Hence, the vector r is also random. Because n w (t) is a Gaussian
white noise, the probability distribution of n w has spherical symmetry in the signal space (as
shown in the last section). Hence, the distribution of r is a spherical distribution centered at a
fixed pointsk, as shown in Fig. 10.15. Whenever the message mk is transmitted, the probability
of observing the received signal r(t) in a given scatter region is indicated by the intensity of the
shading in Fig. 10.15. Actually, because the noise is white, the space has an infinite number of
dimensions. For simplicity, however, we have shown the space to be three-dimensional. This
will suffice to indicate our line of reasoning. We can draw similar scatter regions for various
points s1, s2, . .. , SM . Figure 10.16a shows the scatter regions for two messages m1 and mk