1. 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.

2. 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.

3. 12/2/2015 Time-domain Waveform

4. 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:

5. 12/2/2015 Fourier Series Approximation (n=1)

6. 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

7. 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.

8. 12/2/2015 Fourier Series Approximation (n=3)

9. 12/2/2015 Fourier Series Approximation (n=5)

10. 12/2/2015 Fourier Series Approximation (n=7)

11. 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. 12/2/2015 Fourier Series Approximation (n=17)

13. 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...

14. 12/2/2015 Fourier Series Approximation (n=101)

15. 12/2/2015 Fourier Series Approximation (n=1001)

16. 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

17. Mathematica ® Notebook 12/2/2015