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

770 INTRODUCTION TO INFORMATION THEORY
Fi g ure 13.7
Volume of a shell
of a
D-dimensional
hypersphere.
As direct consequence of this result, when D is large, almost all of the volume of the sphere
is concentrated at the surface. This is because if R/ R < I, then (R / R) D -+ 0 as D -+ oo.
This ratio approaches zero even if R differs from R by a very small amount b. (Fig. 13.7). This
means that no matter how small b. is, the volume within radius R is a negligible fraction of
the total volume within radius R if D is large enough. Hence, for a large D, almost all of the
volume of a D-dimensional sphere is concentrated at the surface. Such a result sounds strange,
but a little reflection will show that it is reasonable. This is because the volume is proportional
to the Dth power of the radius. Thus, for large D, a small increase in R can increase the volume
tremendously, and all the increase comes from a tiny increase in R near the surface of the
sphere. This means that most of the volume must be concentrated at the surface.
The number of nonoverlapping spheres of radius that can be packed into a sphere
of radius ✓ (S + N) T is bounded by the ratio of the volume of the signal sphere to the volume
of the noise sphere. Hence,
< [✓(S +N)T] 2BT V (l)
BT
M _
(
2BT - 1+ )
N
() V(l)
(13.67)
Each of the M -ary signals carries the information of log 2 M binary digits. Hence, the transmission
of one of the M signals every T seconds is equivalent to the information rate C
given by
(13.68)
This equation gives the upper limit of C.
To show that we can actually receive error-free information at a rate of Blog (1 + S /N),
we use the argument proposed by Shannon. 8 Instead of choosing the M transmitted messages
at the centers of nonoverlapping spheres (Fig. 13.6b), Shannon proposed selecting the M
points randomly located in the signal sphere l s of radius ,/sf (Fig. 13.8). Consider one
particular transmitted signal sk - Because the signal energy is assumed to be :S S, point sk
will lie somewhere inside the signal sphere l s of radius Jsf. Because all the M signals are
picked randomly from this sphere, the probability of finding a signal within a volume b. V is
min(l, Mb. V /V,), where V s is the volume of 1, . But because for large D all of the volume
of the sphere is concentrated at the surface, all M signal points selected randomly would lie