Page 19 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

(a) (b)
Fig. 1-6 (a) Unit impulse function; (b) shifted unit impulse function.

Some additional properties of S(t) are

S(- t) = S(t)

x(t)S(t) = x(O)S(t)
if x(t) is continuous at t = 0.

x(t)S(t -to) =x(to)6(t - t,)
if x(t) is continuous at t = to.
Using Eqs. (1.22) and ( 1.241, any continuous-time signal x(t can be expressec

Generalized Derivatives:

If g( t ) is a generalized function, its nth generalized derivative g("Y t ) = dng( t )/dt " is
defined by the following relation:

where 4(t) is a testing function which can be differentiated an arbitrary number of times
and vanishes outside some fixed interval and @"'(t) is the nth derivative of 4(t). Thus, by
Eqs. ( 1.28) and (1.20) the derivative of S( t) can be defined as

where 4(t) is a testing function which is continuous at t = 0 and vanishes outside some
fixed interval and $(0) = d4(t)/dtl,=o. Using Eq. (1.28), the derivative of u(t) can be
shown to be S(t) (Prob. 1.28); that is,