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Fourier series tutorial

Fourier series tutorial

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Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis ● Table of contents ● Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 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 3 1. Theory ● 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 Toc JJ II J I Back Section 1: Theory 4 ● This property of repetition defines a fundamental spatial frequency k = 2π L that can be used to give a first approximation to 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 ● 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 t ...
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A frequency distribution is a tabular arrangement for classifying data intodifferent groups and the number of observations ...

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