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Unformatted text preview: EE 3120 Linear Systems Analysis Lecture 4 Frequencies of Analog and Digital Sinusoids ( ) sin o f t t ϖ = 2 o P π ϖ = T [ ]: ( ) sin o f k f kT kT ϖ = = 0, 1, 2,... k = Analog sinusoid is periodic with fundamental (smallest) period P where: (repeats itself every P seconds) If we sample it with a sampling period: This is a discretetime sinusoid. for [ ] f k [ ] [ ] f k f k N = + ( , ) k [ ] [ ] [ 2 ] [ 3 ] [ ] f k f k N f k N f k N f k nN = + = + = + = + [ ] f k A DT signal is periodic with period N , where N is a positive integer if: for all k where 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 . and EE 3120 Linear Systems Analysis Lecture 4 Frequencies of Analog and Digital Sinusoids sin o t ϖ is periodic for every , its sampled sequence is NOT necessarily periodic for every and every sampling period T. o ϖ sin o kT ϖ is periodic, then there exists a positive integer N such that sin sin ( ) sin( ) o o o o kT k N T kT NT ϖ ϖ ϖ ϖ = + = + sin sin cos cos sin o o o o o kT kT NT kT NT ϖ ϖ ϖ ϖ ϖ = + for all k. This holds IFF cos 1 o NT ϖ = sin o NT ϖ = and i.e. 2 o NT n ϖ π = 2 o T n N ϖ π = 2 o n N for some integer n is rational o T ϖ π is irrational This equality will only hold if o ϖ or T are multiples of some rational and π o ϖ if For integer n and positive integer N : 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 ϖ π π π = = = 2 2 200 200 0.01 o n n N n T π π ϖ π = = = = 1 n = period choosing N = 20...
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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.
 Fall '08
 ARAVENA

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