Page 37 - Introduction to Information Optics

1. Entropy Information and Optics

Fig. 1.5. Symmetric pulse function.

It should be noted that if the symmetric pulse /(f) contains negative values,
as shown in Fig. 1.5, then the definitions of nominal duration and bandwidth
must be modified:

\f(t)\dt f(t)dt (1.79)

and

Av|F(0)| £ |F(v)|dv F(v)dv (1.80)

These definitions give rise to the uncertainty relation

1
Af At ^ 1,

which is essentially the condition of Eq. (1.69). In view of Fig. 1.5, we see that
the nominal duration is determined by equating the area under the rectangular
pulse function to the area under the curve of \f(t)\. It is evident that the
nominal duration Af is wider under this new definition, provided f(t) contains
negative values. Similarly, the nominal bandwidth Av can be interpreted in the
same manner.