Page 266 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Let
sgn(t) -X(w)
Then applying the differentiation property (5.551, we have
2
jwX(w) = F[26(t)] = 2 +X(w) = -
iw

Hence,

Note that sgn(t) is an odd function, and therefore its Fourier transform is a pure imaginary
function of w (Prob. 5.41).

530. Verify Eq. (5.481, that is,
1
~(t) H ~s(0.l) + y-
I
As shown in Fig. 5-25, u(t) can be expressed as

Note that $ is the even component of u(t) and sgn(t) is the odd component of dt). Thus, by
Eqs. (5.141) and (5.153) and the linearity property (5.49) we obtain

Fig. 5-25 Unit step function and its even and odd components.

531. Prove the time convolution theorem (5.581, that is,
*
-4) -41) -X,(4X*(4
By definitions (2.6) and (5.311, we have

Changing the order of integration gives