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Unformatted text preview: hysical e xplanation o f t he p eriodicity relationship. m (8.9a)
Because b oth m a nd No a re integers, Eq. (8.9a) implies t hat t he sinusoid cos n k is
periodic only i f
is a r ational number. In this case the period No is given by [Eq.
(8.9a)] * (8.9b)
To compute No, we m ust choose the smallest value of m t hat will make m (*)
a n integer. F or example, if n =
t hen t he smallest value of m t hat will make
= m a n integer is 2. Therefore * N, ¥ 271'
Using a similar argument, we c an show t hat t his discussion also applies t o a
discrete-time exponential e jOk . T hus, a discrete-time exponential e jOk is periodic
is a r ational number. t * Physical Explanation o f the Periodicity Relationship Q ualitatively, this result can be explained by recognizing t hat a discrete-time
sinusoid cos f2k c an be obtained by sampling a continuous-time sinusoid cos n t a t
u nit t ime interval T = 1; t hat is, cos n t s ampled a t t = 0, 1, 2, 3, . ... T his fact
t We c an also d emonstrate t his point by observing t hat if e jOk is No-periodic, t hen
e jOk T his r esult is possible only if n No = = e Jn(k+No) = d OkejONo 271'm (m, a n integer). T his conclusion leads t o Eq. (8.9b). means cos n t is t he envelope of cos n k. Since t he period of cos n t is 271' I n, t here
are 271' / n n umber of samples (elements) of cos n k in one cycle of its envelope. This
number m ayor m ay not be a n integer.
Figure 8.10 shows three sinusoids c os(ik), c os(Nk), a nd cos (0.8k). F igure
8.10a shows c os(ik), for which t here a re exactly 8 samples in each cycle of its
envelope ( (* = 8). Thus, cos ( ik) r epeats every cycle of its envelope. Clearly,
cos (4k/71') is periodic with period 8. O n t he o ther hand, Fig. 8.10b, which shows
cos ( Nk), has a n average of
= 8.5 samples (not a n integral number) in one cycle
of its envelope. Therefore, t he second cycle of t he envelope will n ot b e identical
t o t he first cycle. B ut t here are 17 samples (an integral number) in 2 cycles of
its envelope. Hence, t he p attern becomes repetitive every 2 cycles o f its envelope.
k) is also repetitive b ut its period is 17 samples (two cycles of its
envelope). This observation indicates t hat a signal cos n k is periodic only if we c an
fit a n integral number ( No) of samples in m integral number of cycles of its envelope
so t hat t he p attern becomes repetitive every m cycles of its envelope. Because t he
p eriod of t he envelope is
we conclude t hat * (N *, No = m (~) * which is precisely t he condition of periodicity in Eq. (8.9b). I f
is i rrational, i t is
impossible t o fit a n integral number (No) of samples in a n i ntegral number (m) of
cycles of its envelope, a nd t he p attern c an never become repetitive. For instance,
t he sinusoid cos (0.8k) in Figure 8.10c h as a n average of 2.571' samples (an irrational
number) per envelope cycle, a nd t he p attern c an never be made repetitive over any
integral number (m) of cycles of its envelope; so cos...
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