Page 42 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

(dl x(t) = cos r + sin fir =x,(r) +x2(r)
where x,(t) = cos r = cos o,t is periodic with TI = 27r/01 = 27r and x2(t) = sin fit =
sin w2t is periodic with T2 = 27r/02 = fir. Since T,/T2 = fi is an irrational number,
x(t) is nonperiodic.
(e) Using the trigonometric identity sin2 0 = t(l - cos 201, we can write

I
where x,(t) = $ is a dc signal with an arbitrary period and x2(t) = - $ cos2r = - cos 02t
is periodic with T2 = 2n/w2 = 7. Thus, x(t) is periodic with fundamental period To = T.
= -I 'e jwd ,
7T
(f) x(t) = ejt(r/2)r- 11 = e-jej(r/2)r Wo = Ir
L
x(t) is periodic with fundamental period To = 27r/w0 = 4.

Since R0/27r = $ is a rational number, x[nl is periodic, and by Eq. (1.55) the fundamen-
tal period is No = 8.
x[n] = cos fn = cos non --, Ro = $
is
Since n0/27r = 1/8~ not a rational number, x[n] is nonperiodic.
7r T
x[n] = cos -n + sin -n = x,[n] + x2[n 1
3 4
where

7r 7r
x2[n] = sin -n = cos f12n + 0, = -
4 4
Since R,/2~r = (= rational number), xl[n] is periodic with fundamental period N, = 6,
and since R2/27r = $ (= rational number), x2[n] is periodic with fundamental period
N2 = 8. Thus, from the result of Prob. 1.15, x[n] is periodic and its fundamental period is
given by the least common multiple of 6 and 8, that is, No = 24.
Using the trigonometric identity cos2 8 = i(l + cos28), we can write
T 1 1 7 r
x[n] = cost -n = - + - cos -n =x,[n] +x2[n]
8 2 2 4
where x,[n] = $ = $(l)" is periodic with fundamental period Nl = 1 and x2[n] =
1
cos(a/4)n = cos R2n --, Q2 = ~/4. Since R2/27r = ( = rational number), x2[n] is
periodic with fundamental period N2 = 8. Thus, x[n] is periodic with fundamental period
No = 8 (the least common multiple of N, and N,).

1.17. Show that if x(t + T) = x(t), then

for any real a, p, and a.