Signals and Systems - Electrical Engineering

5.8 Additional Properties 345

It is important to realize that shifting in time does not change the frequency content of the signal—
that is, the signal does not change when delayed or advanced. This is clear when we see that the
magnitude of the two transforms, corresponding to the original and the shifted signals, is the same,

|X()|=|X()e±jt^0 |

and the effect of the time shift is only in the phase spectrum.

nExample 5.19

Consider the signal

x(t)=A[δ(t−τ)+δ(t+τ)]

Find its Fourier transformX(). Use this Fourier pair and the duality property to verify the Fourier

transform of a cos( 0 t)obtained before.

Solution

Applying the time-shift property, we have

X()=A[1e−jτ+ 1 ejτ]

= 2 Acos(τ)

giving the Fourier transform pair

x(t)=A[δ(t−τ)+δ(t+τ)] ⇔ X()= 2 Acos(τ)

Using the duality property, we then have

X(t)= 2 Acos(tτ) ⇔ 2 πx(−)= 2 πA[δ(−−τ)+δ(−+τ)]

Letτ= 0 , Use the evenness ofδ(t)to get

Acos( 0 t) ⇔ πA[δ(+ 0 )+δ(− 0 )] n

nExample 5.20

Consider computing the Fourier transform ofy(t)=sin( 0 t)using the Fourier transform of the

cosine signalx(t)=cos( 0 t)we just found.

Solution

Sincey(t)=cos( 0 t−π/ 2 )=x(t−π/( 2  0 )), applying the time-shifting property, we then get

F[sin( 0 t)]=F[x(t−π/ 2  0 )]

=π[δ(− 0 )+δ(+ 0 )]e−jπ/(^2 ^0 )

=πδ(− 0 )e−jπ/^2 +πδ(+ 0 )ejπ/^2

=−jπδ(− 0 )+jπδ(+ 0 )