Page 235 - Schaum's Outline of Theory and Problems of Signals and Systems

224 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

where Y(w), X(o), and H(w) are the Fourier transforms of y(f), dt), and h(t),
respectively. From Eq. (5.66) we have

The function H(o) is called the frequency response of the system. Relationships repre-
sented by Eqs. (5.65) and (5.66) are depicted in Fig. 5-3. Let
H(w) = I H(w)I ei@~(O) (5.68)

Then IH(o)l is called the magnitude response of the system, and 0,(0) the phase response
of the system.

X(w) Y(w)=X(w)H(w)
Fig. 5-3 Relationships between inputs and outputs in an LTI system.

Consider the complex exponential signal

with Fourier transform (Prob. 5.23)

X(w) = 2dqw - 0,)
Then from Eqs. (5.66) and ( I .26) we have

Y(o) = 27rH(wo) 6(w - too)
Taking the inverse Fourier transform of Y(w), we obtain

y(f ) = H(wo) eioll'
which indicates that the complex exponential signal ei"l)' is an eigenfunction of the LTI
system with corresponding eigenvalue H(w,), as previously observed in Chap. 2 (Sec. 2.4
and Prob. 2.171. Furthermore, by the linearity property (5.491, if the input x(t) is periodic
with the Fourier series
m
~(1) C ckejkw,+ (5.73)
=
&= - m
then the corresponding output y(l) is also periodic with the Fourier series