Page 270 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Setting x,(t) = x(r) and x2(t) =x*(t) in Parseval's relation (5.631, we get

5.39. Show that Eq. (5.61a), that is,
X*(O) =X(-0)

is the necessary and sufficient condition for x(t) to be real.
By definition (5.31)

If x(t) is real, then x*(t) = x(t) and

Thus, X*(w) = X(-w) is the necessary condition for x(t) to be real. Next assume that
X * (o) = X( - o). From the inverse Fourier transform definition (5.32)

Then

which indicates that x(t) is real. Thus, we conclude that
X*(W) =X(-W)

is the necessary and sufficient condition for x(t) to be real.

5.40. Find the Fourier transforms of the following signals:
(a) x(t)=u(-t)
(b) x(t)=ea'u(-t), a >O