Lecture 7 Fourier Series

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Lecture 7Fourier Series
Remember homework 1 for submission 31/10/08
Best way to handle course

• Skim through notes before lecture
• Ask questions in lecture
• After lecture read notes and try some problems
• See me in E47 office if you have any questions

Where can I find stuff?

• ALL tutorial, problems class, homework in notes
  and web
• Completed solutions version of notes at Phils
  Problems
• All .ppt presentations from lectures at Phils
  Problems
• Loads of questions with worked answers at Phils
  Problems

http//www.hep.shef.ac.uk/Phil/PHY226.htm
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Fourier Series
Last lecture we learned how a Fourier series
formed from sine and cosine harmonics can
represent any periodically repeating function.
Whats a harmonic? Below are the first three
harmonics of sine and cosine
F(x) a1cos x
F(x) b1sin x
n 1
For these cases L is taken to be 2p to simplify
expressions
F(x) a2cos 2x
F(x) b2sin 2x
n 2
F(x) a3cos 3x
F(x) b3sin 3x
n 3
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Fourier Series
A key point to notice is that the summed output
will repeat with the period of the 1st harmonic
We must decide on the amplitude of each harmonic
term. This amplitude may be zero, positive,
negative, big or small..
http//www.falstad.com/fourier/
http//www.univie.ac.at/future.media/moe/galerie/f
ourier/fourier.html
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Fourier Series - QUIZ
Team A questions in white
Team B questions in red
1. What is when n 3 ?
2. What is when n 52 ?
3. What is when n 1 ?
4. What is when n 17
?
5. What is when n 52
?
6. What is when n 1 ?
7. What is when n 4 ?
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Fourier Series - QUIZ
8. Team B What is ?
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Fourier Series - QUIZ
9. Team A What is ?
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Fourier Series - QUIZ
10. Team B Describe the following step function
in terms of f(x) and x ?
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Fourier Series - QUIZ
11. Team A What is ?
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Fourier Series - QUIZ
12. Team B Describe the following step function
over one period in terms of f(x) and x ?
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Fourier Series - QUIZ
13. Team A What is the integral of f(x) over one
period ?
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Fourier Series - QUIZ
14. Team B Describe the following step function
over one period in terms of f(x) and x ?
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Fourier Series - QUIZ
15. Team A What is the integral of f(x) over one
period ?
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Fourier Series - QUIZ
Fourier series
16. Team B If we were to represent the function
below as a Fourier series what could you say
about the value of a0 ?
a0 is baseline shifter. Half way between 20 and
70 is 45. So ao 90
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Fourier Series - QUIZ
Fourier series
17. Team A If we were to represent the function
below as a Fourier series what could you say
about the values of the an terms ?
odd function so all an terms are zero
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Fourier Series - QUIZ
Fourier series
18. Team B If we were to represent the function
below as a Fourier series what could you say
about the sign of the b1 term ?
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Fourier Series - QUIZ
Fourier series
18. Team B If we were to represent the function
below as a Fourier series what could you say
about the value of the b1 term ?
It would have a negative amplitude
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Finding coefficients of the Fourier Series
Find Fourier series to represent this repeat
pattern.
Steps to calculate coefficients of Fourier series
1. Write down the function f(x) in terms of x.
What is period?
Period is 2p
2. Use equation to find a0?
3. Use equation to find an?
4. Use equation to find bn?
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Finding coefficients of the Fourier Series
4. Use equation to find bn?
5. Write out values of bn for n 1, 2, 3, 4,
5, .
6. Write out Fourier series with period L, an,
bn in the generic form replaced with values for
our example
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Finding coefficients of the Fourier Series
So what does this Fourier series look like if we
only use first few terms?
Use Fourier_checker on Phils problems website
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Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat
pattern.
Steps to calculate coefficients of Fourier series
1. Write down the function f(x) in terms of x.
What is period?
Period is 2p
2. Use equation to find a0?
Team A find coefficients an?
3.
4.
Team B find coefficients bn?
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Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat
pattern.
Period is 2p
Team A find coefficients an?
3.
Integrate by parts
so set u x and cos (nx) dx dv
and du dx
n1 n2 n3 n4 n5

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Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat
pattern.
Period is 2p
Team B find coefficients bn?
4.
Integrate by parts
so set u x and sin (nx) dx dv
du dx
n1 n2 n3 n4 n5

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Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat
pattern.
L is the period 2p
5. Write out the first few terms of Fourier
series
n1 n2 n3 n4 n5

n1 n2 n3 n4 n5

So
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Finding coefficients of the Fourier Series - QUIZ
Can we check our Fourier series using
Fourier_checker.xls at Phils Problems ???
Yes!!