12/2/2015 Fourier Series - Supplemental Notes A Fourier series is a sum of sine and cosine harmonic...

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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.

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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.

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Time-domain Waveform

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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:

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Fourier Series Approximation (n=1)

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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

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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.

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Fourier Series Approximation (n=3)

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Fourier Series Approximation (n=5)

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Fourier Series Approximation (n=7)

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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.

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Fourier Series Approximation (n=17)

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Higher Harmonic Content

• Pulse nearly square

• Oscillation where it should be flat

• Let’s see if adding more harmonics will improve this...

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Fourier Series Approximation (n=101)

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Fourier Series Approximation (n=1001)

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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

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