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

13.5 Channel Capacity of a Conti nuous Memoryless Channel 771
Figure 13.8
Derivation of
channel
capacity.
near the surface of l s , Figure 13.8 shows the transmitted signal Sk, the received signal r, and
the noise n. We draw a sphere of radius ,/NT with r as the center. This sphere intersects the
sphere l s and forms a common lens-shaped region. The signal Sk lies on the surface of both
spheres. We shall use a maximum likelihood receiver. This means that when r is received,
we shall make the decision that "sk was transmitted," provided none of the remaining M - I
signal points are closer to r than s k • The probability of finding any one signal in the lens is
Viens/V s . Hence P e , the error probability in the detection of sk when r is received, is
M Vien s
< --
V s
From Fig. 13.8, we observe that Viens < V (h), where V(h) is the volume of the D­
dimensional sphere of radius h. Because r, Sk, and n form a right triangle,
hj(S +N)T = J(ST)(NT)
and
h = ✓ SNT
S+N
Hence,
( S NT ) BT
V(h) = --
S+N
V (l)
Also,
and
(
N
P e < M -­
S+N
) BT
If we choose