Fourier-Series Tutorial

Series FOURIER SERIES
Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis

q Table of contents q Begin Tutorial

c 2004 [e-mail]

Table of contents
1. 2. 3. 4. 5. 6. 7. Theory Exercises Answers Integrals Useful trig results Alternative notation Tips on using solutions Full worked solutions

Section 1: Theory

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1. Theory q A graph of periodic function f (x) that has period L exhibits the same pattern every L units along the x-axis, so that f (x + L) = f (x) for every value of x. If we know what the function looks like over one complete period, we can thus sketch a graph of the function over a wider interval of x (that may contain many periods) f(x )

x

P E R IO D = L

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Section 1: Theory

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q This property of repetition defines a fundamental spatial frequency k = 2π that can be used to give a first approximation to L the periodic pattern f (x): f (x) c1 sin(kx + α1 ) = a1 cos(kx) + b1 sin(kx),

where symbols with subscript 1 are constants that determine the amplitude and phase of this first approximation q A much better approximation of the periodic pattern f (x) can be built up by adding an appropriate combination of harmonics to this fundamental (sine-wave) pattern. For example, adding c2 sin(2kx + α2 ) = a2 cos(2kx) + b2 sin(2kx) c3 sin(3kx + α3 ) = a3 cos(3kx) + b3 sin(3kx) (the 2nd harmonic) (the 3rd harmonic)

Here, symbols with subscripts are constants that determine the amplitude and phase of each harmonic contribution
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Section 1: Theory

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One can even approximate a square-wave pattern with a suitable sum that involves a fundamental sine-wave plus a combination of harmonics of this fundamental frequency. This sum is called a Fourier series
F u n d a m e n ta l F u n d a m e n ta l + 2 h a rm o n ic s

x

F u n d a m e n ta l + 5 h a rm o n ic s F u n d a m e n ta l + 2 0 h a rm o n ic s
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P E R IO D = L

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Section 1: Theory

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q In this Tutorial, we