Page 18 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

Fig. 1-5

Fig. 1-5 and possesses the following properties:

But an ordinary function which is everywhere 0 except at a single point must have the
integral 0 (in the Riemann integral sense). Thus, S(t) cannot be an ordinary function and
mathematically it is defined by

where 4(t) is any regular function continuous at t = 0.
An alternative definition of S(t) is given by

Note that Eq. (1.20) or (1.21) is a symbolic expression and should not be considered an
ordinary Riemann integral. In this sense, S(t) is often called a generalized function and
4(t) is known as a testing function. A different class of testing functions will define a
different generalized function (Prob. 1.24). Similarly, the delayed delta function 6(t - I,) is
defined by
m
4(t)W - to) dt = 4Po) (1.22)

where 4(t) is any regular function continuous at t = to. For convenience, S(t) and 6(t- to)
are depicted graphically as shown in Fig. 1-6.