Page 40 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

1.13. Consider the sinusoidal signal
x(t) = cos 15t

Find the value of sampling interval T, such that x[n] = x(nT,) is a periodic
sequence.
Find the fundamental period of x[n] = x(nT,) if TT = 0.1~ seconds.
The fundamental period of x(t) is To = 2*rr/wo = 27/15. By Eq. (1.81), x[n] =x(nTs) is
periodic if

where m and No are positive integers. Thus, the required value of T, is given by

Substituting T, = 0.1~ = ~/10 in Eq. (1.821, we have

Thus, x[n] =x(nT,) is periodic. By Eq. (1.82)

The smallest positive integer No is obtained with m = 3. Thus, the fundamental period of
is
x[nl = x(0.l~n) N, = 4.

.4. Let x,(t) and x,(t) be periodic signals with fundamental periods T, and T2, respec-
tively. Under what conditions is the sum x(t) =x,(t) + x2(t) periodic, and what is the
fundamental period of x( t) if it is periodic?
Since x,(t) and x,(t) are periodic with fundamental periods TI and T,, respectively, we
have
xl(t) =x,(t + TI) =x,(t + mT,) m = positive integer
x2(t) =x2(t + T2) =x2(f + kT2) k = positive integer
Thus,

In order for x(t) to be periodic with period T, one needs

Thus, we must have
mT, = kT2 = T

TI k
-- --= rational number
T2 m
In other words, the sum of two periodic signals is periodic only if the ratio of their respective
periods can be expressed as a rational number. Then the fundamental period is the least