Chapter 1 Week 2 EE 341 Clark Lecture Notes - 8 april

# Chapter 1 Week 2 EE 341 Clark Lecture Notes - 8 april - EE...

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EE 341 Discrete-Time Linear Systems – Week 1 Spring 2011 Page 20 Periodic Signals in Discrete Time How do we tell if a discrete-time signal [] x n is periodic? That is, given n and N are integers, is there some period 0 N such that [] [ ] x nx n N  Let’s examine a signal that did not necessarily come from sampling a continuous time signal: [] n x nC a Let 0 j ae  then 0 jn x e is a complex exponential. If [] x n is periodic, then [] [ ] x n N  and 00 0 0 () j nN jnjN Ce Ce Ce e   which implies 0 1 jN e . When does this happen? Only if 0 N is an integer multiple of 2 , because 2 1 j e and so, 2 1 jk e for k an integer. Therefore, 0 0 2 o r 2 k Nk N  0 2 is the normalized frequency – it must be a RATIONAL number for the complex sinusoid to be periodic. When it is rational, there are k cycles in N samples. If 0 2 is irrational, then 0 e is not periodic and we never get the samples repeated no matter how many samples we see. The same is true for real sinusoids and cosinusoids (since they are made up of complex exponentials). 2 2 Note that the frequency 0 is not always the same as the fundamental frequency. Since the period N must be an integer for a discrete-time signal, the fundamental frequency is 0 2 k N  , which is the same as 0 only for cases where 1 k . The fundamental period can be found as 0 2 where is the smallest integer such that is an integer k N or by normalizing frequency and reducing to the simplest ratio of integers 0 2 k N .

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EE 341 Discrete-Time Linear Systems – Week 2 Spring 2011 Page 21 Text reading assignment for this week: Chapter 1: Sec 3.2, 3.3, 5 – 7 (emphasis on discrete-time) Chapter 2: Sec 1, 3, 4 Ex. Determine which of the signals below are periodic. For the ones that are, find the fundamental period and fundamental frequency. 1. 6 1 [] jn x ne 2. 3 2 5 [] s i n ( 1 ) xn n  3. 3 [] c o s ( 2 ) n  4. 4 o s ( 12 ) x nn  5. 3 5 n j x
EE 341 Discrete-Time Linear Systems – Week 2 Spring 2011 Page 22 0 5 10 15 -1 1 Amplitude Time sample 0 = 2 1 / 8 0 5 10 15 -1 1 Time sample 0 = 2 2 / 8 0 5 10 15 -1 1 Time sample 0 = 2 3 / 8

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EE 341 Discrete-Time Linear Systems – Week 2 Spring 2011 Page 23 0 2 4 6 8 10 12 -1 1 Amplitude Time sample 0 = 2 1 / 7 0 2 4 6 8 10 12 -1 1 Time sample 0 = 2 2 / 7 0 2 4 6 8 10 12 -1 1 Time sample 0 = 2 3 / 7
EE 341 Discrete-Time Linear Systems – Week 2 Spring 2011 Page 24 There is a major difference between discrete and continuous time. For continuous time, distinct values of frequency produce distinct sinusoids. For discrete-time, complex exponentials (and cosinusoids and sinusoids), frequencies 0 and 0 2 k are indistinguishable, where k is any integer. Example: for integer n 97 44 4 4 (200 ) jn j n j n ee e e    but for real t 9 etc jt j t So for discrete-time, we only need to consider a frequency interval of length 2 such as [0 2 ) (0 <2 )   or [) ( )         .

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Chapter 1 Week 2 EE 341 Clark Lecture Notes - 8 april - EE...

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