228_w06_Fourier_Series_demo

228_w06_Fourier_Series_demo - Fourier Series Any periodic...

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1 Fourier Series Any periodic function X P (t) can be represented by an infinite sum of properly weighted sine and cosine functions of the proper frequencies X P (t) = a 0 + a k cos (2 π k f 0 t ) + b k sin (2 π k f 0 t) k =1 Where f 0 is the fundamental frequency = 1 / T and T is the period of x(t) i.e. x (t) = x (t + T)
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2 Fourier Series Illustration Let us show how we can approximate a periodic Square wave of amplitude 1 and frequency of f 0 =1 Hz by adding sinusoids of varying frequencies and amplitudes b k sin (2 π k f 0 t ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 ( 1/1 ) sin( 2 π * 1 *1* t ) Approximated square wave base on 1 sinusoid b k k f 0
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3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 (1/1) sin(2 π * 1 *1* t ) + (1/3) sin(2 π * 3 *1* t ) Approximated square wave base on 2 sinusoid
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4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 (1/1) sin(2 π * 1 *1* t ) + (1/3) sin(2 π * 3 *1* t ) + (1/5) sin(2 π * 5 *1* t ) Approximated square wave base on 3 sinusoid
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This note was uploaded on 10/06/2010 for the course EE EE 228 taught by Professor Saghri during the Spring '10 term at Cal Poly.

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228_w06_Fourier_Series_demo - Fourier Series Any periodic...

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