Page 41 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

common multiple of T, and T2, and it is given by Eq. (1.84) if the integers m and k are relative
prime. If the ratio T,/T, is an irrational number, then the signals x,(t) and x,(t) do not have a
common period and x(t) cannot be periodic.

1.15. Let x,[n] and x2[n] be periodic sequences with fundamental periods N, and N2,
respectively. Under what conditions is the sum x[n] =x,[n] +x2[n] periodic, and what
is the fundamental period of x[n] if it is periodic?

Since x,[n] and x2[n] are periodic with fundamental periods N, and N2, respectively, we
have
xI[n] =xI[n + N,] =x,[n +mN,] m = positive integer

x2[n] =x,[n + N,] =x,[n + kN,] k = positive integer
Thus,
~[n] =x,[n +mN,] +x2[n + kN,]

In order for x[n] to be periodic with period N, one needs
x[n + N] =x,[n + N] +x2[n + N] =x,[n + mN,] +x,[n + kN2]

Thus, we must have
mN, = kN2 = N
Since we can always find integers m and k to satisfy Eq. (1.861, it follows that the sum of two
periodic sequences is also periodic and its fundamental period is the least common multiple of
N, and N,.

1.16. Determine whether or not each of the following signals is periodic. If a signal is
periodic, determine its fundamental period.
2TT
(a) x(t) = cos (b) x(t)=sinpt
3
T TT
(c) x(t)=cos-I +sin -t (dl x(t)=cost+sinfit
3 4
(el x(t) = sin2 t (f) X(t) = eiI(r/2)f- 11
(g) x[n] = ej("/4)" (h) x[n]=cosfn
T T TT
(i) x[n] = cos -n + sin -n (j) x[n] = cos2 -n
3 4 8

x(t) is periodic with fundamental period T, = 27r/w0 = 27r.

x(r) is periodic with fundamental period TO = 27r/o,, = 3.
lr lr
(c) x(t) = cos --I + sin -t =x,(t) +x2(t)
3 4
where x,(t) = cos(7r/3)r = cos w,t is periodic with T, = 27r/w, = 6 and x2(t) =
sin(~/4)t = sin w2t is periodic with T2 = 21r/w2 = 8. Since T,/T, = = is a rational
number, x(t) is periodic with fundamental period To = 4T, = 3T2 = 24.