12/2/2015 Fourier Series - Supplemental Notes A Fourier series is a sum of sine and cosine harmonic functions that approximates a repetitive (periodic)

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Presentation transcript:

12/2/2015 Fourier Series - Supplemental Notes A Fourier series is a sum of sine and cosine harmonic functions that approximates a repetitive (periodic) waveform. The amplitudes of the components terms of the series are the projections of the input onto the sine nd cosine harmonic functions.

12/2/2015 Mathematica ® Analysis Mathematica is a mathematics programming and graphics package available from Wolfram Research, Inc. A simple repeating square wave is analyzed to illustrate the properties of Fourier series approximation.

12/2/2015 Time-domain Waveform

12/2/2015 Mathematica ® Analysis The input waveform is periodic with a period of 1 second. A first approximation to the input would thus be a sinusoid in phase with it, as follows:

12/2/2015 Fourier Series Approximation (n=1)

12/2/2015 Fundamental Frequency Component Sinusoidal approximation Poor edge conformity at pulse transition Rounded peak - rather than flat Poor width control More harmonics of the 1 Hz input are needed for a better approximation

12/2/2015 Increasing Harmonic Content The following three slides shown the improvement in waveform approximation obtained by increasing the number of harmonics used in the Fourier series approximation.

12/2/2015 Fourier Series Approximation (n=3)

12/2/2015 Fourier Series Approximation (n=5)

12/2/2015 Fourier Series Approximation (n=7)

12/2/2015 Some Higher Harmonic Content Pulse takes on square shape, but top not flat Width becomes approximately correct The approximation will concinually show improvement as more harmonics are added.

12/2/2015 Fourier Series Approximation (n=17)

12/2/2015 Higher Harmonic Content Pulse nearly square Oscillation where it should be flat Let’s see if adding more harmonics will improve this...

12/2/2015 Fourier Series Approximation (n=101)

12/2/2015 Fourier Series Approximation (n=1001)

12/2/2015 Many Harmonics Even with a large number of harmonics, there are problems with the approximation Corner effects –Overshoot –Oscillations –This is the Gibbs phenomenon

Mathematica ® Notebook 12/2/2015