Signals and Systems - Electrical Engineering

328 CHAPTER 5: Frequency Analysis: The Fourier Transform

This can be shown by considering the eigenfunction property of LTI systems. The Fourier

representation ofx(t), if aperiodic, is an infinite summation of complex exponentialsejtmultiplied

by complex constantsX(), or

x(t)=

1

2 π

∫∞

−∞

X()ejtd

According to the eigenfunction property, the response of an LTI system to each termX()ejtis

X()ejtH(j)whereH(j)is the frequency response of the system, and thus by superposition the

responsey(t)is

y(t)=

1

2 π

∫∞

−∞

[X()H(j)]ejtd

=

1

2 π

∫∞

−∞

Y()ejtd

so thatY()=X()H(j).

Ifx(t)is periodic of periodT 0 (or fundamental frequency 0 = 2 π/T 0 ), then

X()=

∑∞

k=−∞

2 πXkδ(−k 0 )

so that the outputy(t)has as its Fourier transform

Y()=X()H(j)

=

∑∞

k=−∞

2 πXkH(j)δ(−k 0 )

=

∑∞

k=−∞

2 πXkH(jk 0 )δ(−k 0 )

Therefore, the output is periodic—that is,

y(t)=

∑∞

k=−∞

Ykejk^0 t

whereYk=XkH(jk 0 ).

An important consequence of the convolution property, just like in the Laplace transform, is that the

ratio of the Fourier transforms of the input and the output gives the frequency response of the system,