Page 36 - Introduction to Information Optics

1.4. Band-limited Analysis 2 !

Let us provide an example to show that the uncertainty relation is indeed
relevant for information transmission through a communication channel. We
assume a short symmetric pulse of duration of At, as applied to the channel;
that is.

f(tl
f(t) = (1.73)
0,

where
/(O) ^ f(t).

The corresponding signal spectrum can be obtained as given by

where J* denotes the Fourier transformation.
Let us define a nominal signal duration At and a nominal signal bandwidth
which are equivalent to the duration of a rectangular pulse of amplitude /(O)
and a rectangular pulse spectrum F(0), as given by

Ar/(0) f(t)dt, (1.74)
and

AvF(O) 4 F(v)dv. •1.75)

From the definition of the Fourier transformation, the nominal quantities can
be written as

(1.76)

(1.77)

for which we have the lower bound condition of the uncertainty relations; that
is,
(1.78)