Signals and Systems - Electrical Engineering

238 C H A P T E R 4: Frequency Analysis: The Fourier Series

exponentials or sinusoids are used in the Fourier representation of periodic as well as aperiodic

signals by taking advantage of the eigenfunction property of LTI systems. The results of the Fourier

series in this chapter will be extended to the Fourier transform in Chapter 5.

n Steady-state analysis—Fourier analysis is in the steady state, while Laplace analysis considers both

transient and steady state. Thus, if one is interested in transients, as in control theory, Laplace is

a meaningful transformation. On the other hand, if one is interested in the frequency analysis,

or steady state, as in communications theory, the Fourier transform is the one to use. There will

be cases, however, where in control and communications both Laplace and Fourier analysis are

considered.

n Application of Fourier analysis—The frequency representation of signals and systems is extremely

important in signal processing and in communications. It explains filtering, modulation of mes-

sages in a communication system, the meaning of bandwidth, and how to design filters. Likewise,

the frequency representation turns out to be essential in the sampling of analog signals—the

bridge between analog and digital signal processing.

4.2 Eigenfunctions Revisited

As indicated in Chapter 3, the most important property of stable LTI systems is that when the input

is a complex exponential (or a combination of a cosine and a sine) of a certain frequency, the output

of the system is the input times a complex constant connected with how the system responds to the

frequency at the input. The complex exponential is called aneigenfunctionof stable LTI systems.

Ifx(t)=ej^0 t,−∞<t<∞, is the input to a causal and a stable system with impulse responseh(t),the

output in the steady state is given by

y(t)=ej^0 tH(j 0 ) (4.1)

where

H(j 0 )=

∫∞

0

h(τ)e−j^0 τdτ (4.2)

is the frequency response of the system at 0. The signalx(t)=ej^0 tis said to be aneigenfunctionof the LTI

system as it appears at both input and output.

This can be seen by finding the output corresponding tox(t)=ej^0 tby means of the convolution

integral,

y(t)=

∫∞

0

h(τ)x(t−τ)dτ=ej^0 t

∫∞

0

h(τ)e−j^0 τdτ

=ej^0 tH(j 0 )