Signals and Systems - Electrical Engineering

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

The above representation ofx(t)corresponds to the Fourier representation of aperiodic signals, which

will be covered in Chapter 5. Again here, the eigenfunction property of LTI systems provides an effi-

cient way to compute the output. Furthermore, we also find that by lettingY()=X()H(j)the

above equation gives an expression to computey(t)fromY(). The productY()=X()H(j)cor-

responds to the Fourier transform of the convolution integraly(t)=x(t)∗h(t), and is connected with

the convolution property of the Laplace transform. It is important to start noticing these connections,

to understand the link between Laplace and Fourier analysis.

Remarks

n Notice the difference of notation for the frequency representation of signals and systems used above. If x(t)

is a periodic signal its frequency representation is given by{Xk}, and if aperiodic by X(), while for a

system with impulse response h(t)its frequency response is given by H(j).

n When considering the eigenfunction property, the stability of the LTI system is necessary to ensure that

H(j)exists for all frequencies.

n The eigenfunction property applied to a linear circuit gives the same result as the one obtained from phasors

in the sinusoidal steady state. That is, if

x(t)=Acos( 0 t+θ)=

Aejθ

2

ej^0 t+

Ae−jθ

2

e−j^0 t (4.5)

is the input of a circuit represented by the transfer function

H(s)=

Y(s)

X(s)

=

L[y(t)]

L[x(t)]

then the corresponding steady-state output is given by

yss(t)=

Aejθ

2

ej^0 tH(j 0 )+

Ae−jθ

2

e−j^0 tH(−j 0 )

=A|H(j 0 )|cos( 0 t+θ+∠H(j 0 )) (4.6)

where, very importantly, the frequency of the output coincides with that of the input, and the amplitude

and phase of the input are changed by the magnitude and phase of the frequency response of the system

for the frequency 0. The frequency response is H(j 0 )=H(s)|s=j 0 , and as we will see its magnitude

is an even function of frequency, or|H(j)|=|H(−j)|, and its phase is an odd function of frequency,

or∠H(j 0 )=−∠H(−j 0 ). Using these two conditions we obtain Equation (4.6).

The phasor corresponding to the input

x(t)=Acos( 0 t+θ)

is defined as a vector,

X=A∠θ

rotating in the polar plane at the frequency of 0. The phasor has a magnitude A and an angleθwith

respect to the positive real axis. The projection of the phasor onto the real axis, as it rotates at the given