Page 33 - Introduction to Information Optics

18 1. Entropy Information and Optics

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0 ( 2 3 4

Fig. 1.3. The capacity of an additive white Gaussian channel as a function of bandwidth Av.

Since the average noise power within the channel bandwidth is N 0&v, as the
bandwidth increases to infinitely large the capacity channel approaches a
definite value, such as

C(oo) = Mm C(Av) = — log, e. (1.64)
"

This result provides an important physical significance: the measurement or
observation of any physical quantity is practically always limited by thermal
noise agitation. This thermal agitation is usually considered as an additive
white Gaussian noise within the channel bandwidth. The noise spectral density
N 0 is related to the thermal temperature T as given by

(1.65)

where k is Boltzmann's constant, and T is in degree Kelvins. It follows that the
signal energy transmitted through a physical communication channel must be
at least kT per nat of information. In other words, it takes at least kT energy
for a nat of information to be properly transmitted through the channel. A plot
of the channel capacity as a function of bandwidth is shown in Fig. 1.3, in
which the capacity started rapidly increases and then asymptotically ap-
proaches to C(oo) as Av becomes very large.