Signals and Systems - Electrical Engineering

5.6 Spectral Representation 321

This energy conservation property is shown using the inverse Fourier transform. The finite-energy
signal ofx(t)can be computed in the frequency domain by

∫∞

−∞

x(t)x∗(t)dt=

∫∞

−∞

x(t)



^1

2 π

∫∞

−∞

X∗()e−jtd



dt

=

1

2 π

∫∞

−∞

X∗()





∫∞

−∞

x(t)e−jtdt



d

=

1

2 π

∫∞

−∞

|X()|^2 d

nExample 5.9

Parseval’s result helps us to understand better the nature of an impulseδ(t). It is clear from its

definition that the area under an impulse is unity, which meansδ(t)is absolutely integrable, but

does it have finite energy? Show how Parseval’s result can help resolve this issue.

Solution

Let’s consider this from the frequency point of view, using Parseval’s result. The Fourier transform

ofδ(t)is unity for all values of frequency and as such its energy is infinite. Such a result seems

puzzling, becauseδ(t)was defined as the limit of a pulse of finite duration and unity area. This is

what happens if

p 1 (t)=

1

1

[u(t+1/ 2 )−u(t−1/ 2 )]

is a pulse of the unity area from which we obtain the impulse by letting 1 →0. The signal

p^21 (t)=

1

12

[u(t+1/ 2 )−u(t−1/ 2 )]

is a pulse of area 1/1. If we then let 1 →0, the squared pulsep^21 (t)will tend to infinity with an

infinite area under it. Thus,δ(t)is not finite energy. n

nExample 5.10

Consider a pulsep(t)=u(t+ 1 )−u(t− 1 ). Use its Fourier transformP()and Parseval’s result to

show that

∫∞

−∞

(

sin()



) 2

d=π