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
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.
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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