5.6 Spectral Representation 313
where we used the Laplace transform ofδ(t−ρ 0 )+δ(t+ρ 0 ), which ise−sρ^0 +esρ^0 defined over
the wholes-plane. Ats=j, we get 2 cos(ρ 0 ). According to the duality property, we thus have
the following Fourier pair:
2 cos(ρ 0 t) ⇔ 2 π[δ(−−ρ 0 )+δ(−+ρ 0 )]= 2 π[δ(+ρ 0 )+δ(−ρ 0 )]
Replacingρ 0 by 0 and canceling the 2 in both sides, and multiplying by A both sides, we have
x(t)=Acos( 0 t) ⇔ X()=πA[δ(+ 0 )+δ(− 0 )] (5.10)
indicating that it only exists at± 0. n
5.6 SPECTRAL REPRESENTATION
In this section, we consider first how to find the Fourier transform of periodic signals using the
modulation property, and then consider Parseval’s result for finite-energy signals. With these results,
we will unify the spectral representation of both periodic and aperiodic signals.
5.6.1 Signal Modulation
One of the most significant properties of the Fourier transform is modulation. Its application to
signal transmission is fundamental in communications.
n Frequency shift: IfX()is the Fourier transform ofx(t), then we have the pair
x(t)ej^0 t ⇔ X(− 0 ) (5.11)
n Modulation: The Fourier transform of the modulated signal
x(t)cos( 0 t) (5.12)
is given by
0.5[X(− 0 )+X(+ 0 )] (5.13)
That is,X()is shifted to frequencies 0 and− 0 , and multiplied by 0.5.
The frequency shifting property is easily shown:
F[x(t)ej^0 t]=
∫∞
−∞
[x(t)ej^0 t]e−jtdt
=
∫∞
−∞
x(t)e−j(−^0 )tdt
=X(− 0 )
