Page 39 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

27
w0T=m2.rr or T=m- m = positive integer
*o
Thus. the fundamental period To of x(r) is given by 2.rr/wo.

1.11. Show that the complex exponential sequence
x[n] = e~"~"

is periodic only if fl0/2.rr is a rational number.
By Eq. (1.9), x[n] will be periodic if
,iflo(" +Nl = ,in,,n,i~hp = ,inon

or
ein~N = 1
Equation (1.79) holds only if
floN =m2~ m = positive integer
or

m
a0
-= - = rational number
2.rr N
Thus, x[n] is periodic only if R0/27r is a rational number

1.12. Let x(r) be the complex exponential signal

with radian frequency wo and fundamental period To = 2.rr/oo. Consider the
discrete-time sequence x[n] obtained by uniform sampling of x(t) with sampling
interval Ts. That is,

x[n] =x(nT,) =eJ"unT.
Find the condition on the value of T, so that x[n] is periodic.
If x[n] is periodic with fundamental period N,,, then

,iou(n+N,,)T, = ,iw~nT,,iwuN,J', = ejwun-l;
Thus, we must have

T, m
-=-- - rational number
To No
Thus x[n] is periodic if the ratio T,/T,, of the sampling interval and the fundamental period of
x(t) is a rational number.
Note that the above condition is also true for sinusoidal signals x(t) = cos(o,,t + 8).