Page 43 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

If x(t + T) =x(t), then letting t = 7 - T, we have
X(T-T+T)=x(r) =x(T-T)
and

Next, the right-hand side of Eq. (1.88) can be written as

By Eq. (1.87) we have

( t ) d = / x(t) dt
a+T
Thus.

1.18. Show that if x(t) is periodic with fundamental period To, then the normalized average
power P of x(t) defined by Eq. (1.15) is the same as the average power of x(0 over
any interval of length T,, that is,

By Eq. ( 1.15)
1
1'
P = lim - /T'2 1 x(t) dt
T-.r: T -7./2
Allowing the limit to be taken in a manner such that T is an integral multiple of the
fundamental period, T = kT,, the total normalized energy content of x(t) over an interval of
length T is k times the normalized energy content over one period. Then

1.19. The following equalities are used on many occasions in this text. Prove their validity.