Signal Processing and Linear Systems-B.P.Lathi copy

86b also t he e xponential 05k decays faster t han 08k

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Unformatted text preview: hysical e xplanation o f t he p eriodicity relationship. m (8.9a) No 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 = m a n integer is 2. Therefore * N, ¥ 271' 17 N o=mn=2'2=17 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 only if 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. Therefore, cos 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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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