04/21/23

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.

04/21/23

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.

04/21/23

Time-domain Waveform

04/21/23

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:

04/21/23

Fourier Series Approximation (n=1)

04/21/23

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

04/21/23

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.

04/21/23

Fourier Series Approximation (n=3)

04/21/23

Fourier Series Approximation (n=5)

04/21/23

Fourier Series Approximation (n=7)

04/21/23

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.

04/21/23

Fourier Series Approximation (n=17)

04/21/23

Higher Harmonic Content

• Pulse nearly square

• Oscillation where it should be flat

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

04/21/23

Fourier Series Approximation (n=101)

04/21/23

Fourier Series Approximation (n=1001)

04/21/23

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