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113_1_chapter03 - Chapter 3 PERIODIC SEQUENCES Copyright c...

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Chapter 3 PERIODIC SEQUENCES Copyright c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected] Periodic sequences play an important role in the analysis of discrete-time signals and systems. A special role is also played by the particular complex exponential sequence x ( n ) = e 0 n . In this chapter, we introduce the reader to the notion of periodic sequences and highlight some properties of complex exponential sequences. Periodic Sequences . A sequence is said to be periodic of period N if N is the smallest positive integer such that x ( n ) = x ( n + N ) for all n In other words, if N is the smallest positive integer for which the sequence repeats itself. For example, consider the sinusoidal sequence x ( n ) = sin( ω 0 n + θ 0 ) for some given { ω o , θ o } . In order to verify whether this sequence is periodic or not, we need to find the smallest positive integer that satisfies sin( ω o n + θ o ) = sin( ω o ( n + N ) + θ o ) for all n, i.e., if and only if, sin( ω o n + θ o ) = sin( ω 0 n + θ o + ω o N ) for all n, We know from the properties of the sine function that this equality holds if, and only if, there exists an integer N such that ω o N = 2 for some integer k. That is, if and only if, ω o N is a multiple of 2 π . Actually, we need to find the smallest N that satisfies this equality. The difficulty is that there need not exist an integer value of k that results in an integer N ; in other words, not every sinusoidal sequence is periodic! This is just one of the subtle differences that exist between discrete-time and continuous-time signals. In continuous-time, all sinusoidal signals are periodic. But not in discrete-time. 18
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19 To continue with the above example, assume ω 0 = 5 π/ 3. Then N and k must be related via N = 6 5 k. In this case, the smallest integer k that results in an integer N is k = 5. It then follows that N = 6 and the sequence sin( 5 π 3 n + θ o ) repeats itself every 6 samples. We thus say that it is periodic with period N = 6. But what about a sinusoidal sequence with ω o = 2? In this case, N and k should be related via: N = 2 π k. It is clear that there does not exist any integer k that results in an integer N . For this reason, the sequence sin( 2 n + θ o ) is not periodic! [In contrast, the continuous-time signals sin( 2 t + θ o ) and sin ( 5 π 3 t + θ o ) are both periodic, since in continuous-time the period of a signal is allowed to be any positive real number.] Complex Exponential Sequences Consider now the complex exponential sequence x ( n ) = e o n This sequence will be periodic if we can find a smallest positive integer N such that e 0 n = e 0 ( n + N ) or, equivalently, if we can find a smallest positive integer N such that e 0 N = 1 We know from the properties of the exponential function that this equality holds if, and only if, ω o N = 2 for some integer k.
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