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 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

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

4

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 Toc Back

Section 1: Theory

5

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 Toc

P E R IO D = L

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Section 1: Theory

6

q In this Tutorial, we consider working out Fourier series for functions f (x) with period L = 2π. Their fundamental frequency is then k = 2π = 1, and their Fourier series representations involve terms like L a1 cos x , a2 cos 2x , a3 cos 3x , b1 sin x b2 sin 2x b3 sin 3x

We also include a constant term a0 /2 in the Fourier series. This allows us to represent functions that are, for example, entirely above the x−axis. With a suﬃcient number of harmonics included, our approximate series can exactly represent a given function f (x)

f (x) = a0 /2

+ a1 cos x + a2 cos 2x + a3 cos 3x + ... + b1 sin x + b2 sin 2x + b3 sin 3x + ...

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Section 1: Theory

7

A more compact way of writing the Fourier series of a function f (x), with period 2π, uses the variable subscript n = 1, 2, 3, . . . f (x) = a0 + [an cos nx + bn sin nx] 2 n=1 ∞

q We need to work out the Fourier coeﬃcients (a0 , an and bn ) for given functions f (x). This process is broken down into three steps STEP ONE

a0

=

1 π

2π

f (x) dx

STEP TWO

an

=

1 π

2π

f (x) cos nx dx

STEP THREE

bn

=

1 π

2π

f (x) sin nx dx

where integrations are over a single interval in x of L = 2π Toc Back

Section 1: Theory

8

q Finally, specifying a particular value of x = x1 in a Fourier series, gives a series of constants that should equal f (x1 ). However, if f (x) is discontinuous at this value of x, then the series converges to a value that is half-way between the two possible function values " V e rtic a l ju m p " /d is c o n tin u ity in th e fu n c tio n re p re s e n te d

f(x )

x

F o u rie r s e rie s c o n v e rg e s to h a lf-w a y p o in t Toc

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Section 2: Exercises

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2. Exercises

Click on Exercise links for full worked solutions (7 exercises in total). Exercise 1. Let f (x) be a function of period 2π such that 1, −π < x < 0 f (x) = 0, 0 < x < π . a) Sketch a graph of f (x) in the interval...