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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
04/21/23