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CHAP. 11
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in view of Eq. (1.76).
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Show that the complex exponential signal (t ) = ,j@d is periodic and that its fundamental period is 27r/00.
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By Eq. (1.7), x(t) will be periodic if
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e i @ d t + TI = e i w d
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Since
eiw~(r T )= eiqreiq,T +
we must have
eimoT =
(1.78)
If w,
= 0,
then x(t) = 1, which is periodic for any value of T. If o0# 0 Eq. (1.78) holds if , 27T m = positive integer ooT=m2r or T=ma0
Thus, the fundamental period To, the smallest positive T, of x(t) is given by 2 r / o o .
1.10. Show that the sinusoidal signal
x ( t ) = cos(w,t
is periodic and that its fundamental period is 27r/wo.
The sinusoidal signal x(l) will be periodic if cos[o,(t We note that cos[w,(t
+ T) + 81 = w s ( o o t + 8)
+ T) + 81 = cos[oot + 8 + woT] = cos(oot + 8 )
SIGNALS A N D SYSTEMS
[CHAP. 1
w0T=m2.rr
T=m-
= positive
integer
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
= , i n , , n , i ~ h p = ,inon
e i n ~ N =
Equation (1.79) holds only if floN= m 2 ~ or
a 0 -=
positive integer
2.rr
m - = rational number
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 That is, interval Ts.
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,
= , i w ~ n T , , i w u N , J ' , = ejwun-l;
Thus, we must have
T, - = - -m To
x(t)
rational number
Thus x [ n ] is periodic if the ratio T,/T,, of the sampling interval and the fundamental period of is a rational number. Note that the above condition is also true for sinusoidal signals x ( t ) = cos(o,,t + 8 ) .
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 ( n T , ) 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= 2 7 / 1 5 . By Eq. (1.81), x [ n ] = x ( n T s ) is periodic if
where m and No are positive integers. Thus, the required value of T, is given by
~ in Substituting T, = 0 . 1 = ~ / 1 0 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
x [ n l = x ( 0 . l ~ nis N , = 4. )
= 3.
Thus, the fundamental period of
.4. Let x , ( t ) and x , ( t ) be periodic signals with fundamental periods T, and T 2 , respectively. Under what conditions is the sum x ( t ) = x , ( t ) + x 2 ( 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 T I and T,, respectively, we have x l ( t ) = x , ( t + T I )= x , ( t + m T , ) m = positive integer
x 2 ( t ) = x 2 ( t + T 2 )= x 2 ( f+ k T 2 ) k
positive integer
Thus, In order for x ( t ) to be periodic with period T , one needs Thus, we must have
mT, = kT2 = T T- = -I - -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
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 x2[n] =x,[n Thus, ~ [ n =x,[n +mN,] + x 2 [ n + kN,] ] In order for x[n] to be periodic with period N, one needs x[n
+ N,] =x,[n +mN,] + N,] =x,[n + kN,]
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