Page 274 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

or

( jw + 2)Y(w) = (1 + jw) X(w )

Hence, by Eq. (5.67) the frequency response H(w) is

Taking the inverse Fourier transform of H(w), the impulse response h(t) is

Note that the procedure is identical to that of the Laplace transform method with s replaced
by jw (Prob. 3.29).

5.45. Consider a continuous-time LTI system described by

Using the Fourier transform, find the output y(t) to each of the following input
signals:

(a) x(t) = eP'u(t)
(b) x(r)=u(t)

(a) Taking the Fourier transforms of Eq. (5.1661, we have
jwY(w) + 2Y(w) = X(w)

Hence,

From Eq. (5.155)

and

Therefore,