Page 255 - Schaum's Outline of Theory and Problems of Signals and Systems

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

5.12. If xJt) and x2(t) are periodic signals with fundamental period To and their complex
Fourier series expressions are

show that the signal x(t) =x,(t)x2(t) is periodic with the same fundamental period To
and can be expressed as

where ck is given by

Now x(t + To)=xl(t + To)x2(t + T,)=x1(t)x2(t)=x(t)
Thus, x(t) is periodic with fundamental period To. Let

Then

since

and the term in brackets is equal to e,-,.

5.13. Let xl( t) and x2( t) be the two periodic signals in Prob. 5.12. Show that

Equation ( 5.130) is known as Parseoal's relation for periodic signals.

From Prob. 5.12 and Eq. (5.129) we have

Setting k = 0 in the above expression, we obtain