1.4 Representation Using Basic Signals 105
FIGURE 1.14
AM signal. − 5 05
− 5
0
5
t
z(
t)
n
1.4.5 Generic Representation of Signals..............................................
Consider the following integral:
∫∞
−∞
f(t)δ(t)dt
The product off(t)andδ(t)gives zero everywhere except at the origin where we get an impulse of
areaf( 0 )—that is,f(t)δ(t)=f( 0 )δ(t)(lett 0 =0 in Figure 1.15). Therefore,
∫∞
−∞
f(t)δ(t)dt=
∫∞
−∞
f( 0 )δ(t)dt=f( 0 )
∫∞
−∞
δ(t)dt=f( 0 ) (1.40)
since the area under the curve of the impulse is unity. This property of the impulse function is appro-
priately called thesifting property. The result of this integration is to sift outf(t)for alltexcept fort=0,
whereδ(t)occurs. If we delay or advance theδ(t)function in the integrand, the result is that all values
off(t)are sifted out except for the value corresponding to the location of the delta function—that is,
∫∞
−∞
f(t)δ(t−τ)dt=
∫∞
−∞
f(τ)δ(t−τ)dt=f(τ)
∫∞
−∞
δ(t−τ)dt
=f(τ) for anyτ
since the last integral is still unity. Figure 1.15 illustrates the multiplication of a signalf(t)by an
impulse signalδ(t−t 0 ), located att=t 0.
