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

768 INTRODUCTION TO INFORMATION THEORY
Figure 13.5
Channel capacity
vs.
bandwidth for a
channel with
white Gaussian
noise and fixed
signal power.
C
1.44 S
N
0 2
B
SIN
3
Figure 13.6
(a) Signal space
representation of
transmitted and
received signals
and noise signal.
(b) Choice of
signals for
error-free
communication.
(b)
(a)
Verification of Error-Free Communication over
a Continuous Channel
Using the concepts of information theory, we have shown that it is possible to transmit errorfree
information at a rate of B log2 (1 + S /N) bit/s over a channel band-limited to B Hz. The
signal power is S, and the channel noise is white Gaussian with power N. This theorem can be
verified in a way similar to that used for the verification of the channel capacity of a discrete
case. This verification using signal space is so general that it is in reality an alternate proof of
the capacity theorem.
Let us consider M-ary communication with M equiprobable messages m1, m2, ... , mM
transmitted by signals s1 (t), s2 (t), ... , SM (t). All signals are time-limited with duration T and
have an essential bandwidth B Hz. Their powers are less than or equal to S. The channel is
band-limited to B, and the channel noise is white Gaussian with power N.
All the signals and noise waveforms have 2BT + 1 dimensions. In the limit we shall let
T ➔ oo. Hence 2BT » 1, and the number of dimensions will be taken as 2BT in our future
discussion. Because the noise power is N, the energy of the noise waveform of T -second
duration is NT. Given signal power S, the maximum signal energy is ST. Because signals and
noise are independent, the maximum received energy is (S + N)T. Hence, all the received
signals will lie in a WT-dimensional hypersphere of radius J(S + N)T (Fig. 13.6a). A typical
received signal s;(t) + n(t) has an energy (S; + N)T, and the point r representing this signal
lies at a distance of J(S; + N)T from the origin (Fig. 13.6a). The signal vector s;, the noise
vector n, and the received vector r are shown in Fig. 13.6a. Because
ls;l = M, lnl = m, lrl = J(S; + N)T (13.65)