lecture 4

lecture 4 - EE 3120 Linear Systems Analysis Lecture 4...

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EE 3120 Linear Systems Analysis Lecture 4 Frequencies of Analog and Digital Sinusoids () s in o f tt ω = Analog sinusoid is periodic with fundamental (smallest) period P where: 2 o P π = (repeats itself every P seconds) 0 T > If we sample it with a sampling period: 0, 1, 2,. .. k = ±± [] : ( ) s i n o fk fkT kT == for This is a discrete-time sinusoid. is periodic with period N , where N is a positive integer if: A D-T signal for all k where (, ) k →−∞∞ [ ] f k f kN =+ and [ ] [ 2 ] [ 3 ] [ ] f kf kNf k n N = += + = + = + for any k and every positive integer n . is periodic with period N. f is also periodic with period 2N, 3N The smallest N is the fundamental period .
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EE 3120 Linear Systems Analysis Lecture 4 Frequencies of Analog and Digital Sinusoids sin o t ω is periodic for every , its sampled sequence is NOT necessarily periodic for every o o and every sampling period T. sin if is periodic, then there exists a positive integer N such that o kT sin sin ( ) sin( ) oo o o kT k N T kT NT ωω = += + sin sin cos cos sin o o o kT kT NT kT NT = + for all k. i.e. 2 o NT n π = 2 o T n N = for some integer n and This holds IFF co sin 0 o NT = s 1 o NT = For integer n and positive integer N : o T 2 o n N is rational is irrational or T are multiples of some rational and This equality will only hold if o
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EE 3120 Linear Systems Analysis Lecture 4 Frequencies of Analog and Digital Sinusoids sin 0.01 k π 0.01 1 2 100 o T n N ω ππ == = 22 200 200 0.01 o nn T ωπ = = Nn period Æ 1 n = choosing
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sin 0.1 k π N = 20 sin 0.3 k N = 20 Both sequences have the same fundamental period and thus the same frequency. Obviously one sequence changes more rapidly than the other. The way we define frequency should not be used.
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We cannot use the fundamental period of a sequence to define its frequency.
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This note was uploaded on 02/06/2012 for the course EE 2120 taught by Professor Aravena during the Fall '08 term at LSU.

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lecture 4 - EE 3120 Linear Systems Analysis Lecture 4...

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