lecture 7 fourier series skim through notes before lecture ask questions in lecture after lecture...
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Lecture 7Lecture 7Fourier SeriesFourier Series
•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
Best way to handle course
Remember homework 1 for submission 31/10/08
•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
Where can I find stuff?
http://www.hep.shef.ac.uk/Phil/PHY226.htm
Fourier SeriesFourier SeriesLast lecture we learned how a Fourier series formed from sine and cosine harmonics can represent any periodically repeating function.
What’s a harmonic? Below are the first three harmonics of sine and cosine F(x) = a1cos x
F(x) = b3sin 3x
F(x) = b2sin 2x
F(x) = b1sin x
F(x) = a3cos 3x
F(x) = a2cos 2x
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
For these cases L is taken to be
2to simplify
expressions
n = 1
n = 2
n = 3
SummarySummary
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
L
dxxfL
a00 )(
2L
n dxL
xnxf
La
0
2cos)(
2 L
n dxL
xnxf
Lb
0
2sin)(
2
The Fourier series can be written with period L as
The Fourier series coefficients can be found by:-
Fourier SeriesFourier Series
http://www.falstad.com/fourier/
Sine terms
Cosine terms
http://www.univie.ac.at/future.media/moe/galerie/fourier/fourier.html
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……..
Fourier Series - QUIZFourier Series - QUIZTeam A questions in white
1. What is when n = 3 ?
2. What is when n = 52 ?
3. What is when n = 1 ?
4. What is when n = 17 ?
n)1(1
n)1(1
)2(cos1 n
)2(cos1 n
5. What is when n = 52 ?)2(cos1 n
6. What is when n = 1 ?)(cos1 n
7. What is when n = 4 ?)(cos1 n
0)1(1 3
2)1(1 52
2)1(1
2)1(1
2)1(1
0)1(1)(cos1
2)1(1)4(cos1
Team B questions in red
Fourier Series - QUIZFourier Series - QUIZ
8. Team B: What is ?
-40
-30
-20
-10
0
10
20
30
40
-20 -15 -10 -5 0 5 10 15 20x axis
y ax
is
y=4x
2002002410
10
10
10
2
xdxxI
10
10
4 dxxI
Fourier Series - QUIZFourier Series - QUIZ
-40
-30
-20
-10
0
10
20
30
40
50
60
-30 -20 -10 0 10 20 30x axis
y ax
isy=2x+5
9. Team A: What is ? 1505)52(10
0
10
0
2 xxdxxI 10
0
)52( dxxI
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30
x axis
y ax
isstep
10. Team B: Describe the following step function in terms of f(x) and x ?
50)(0
0)(0
xfxwhen
xfxwhen
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30
x axis
y ax
isstep
11. Team A: What is ?
10
10
0
10
10
0
100
010 500500500)( xxdxdxdxxfI
10
10
)( dxxfI
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
isstep periods
12. Team B: Describe the following step function over one period in terms of f(x) and x ?
50)(510
0)(05
xfxwhen
xfxwhen
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
isstep periods
13. Team A: What is the integral of f(x) over one period ?
10
0
5
0
10
5
105
50 250500500)( xxdxdxdxxfI
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
14. Team B: Describe the following step function over one period in terms of f(x) and x ?
70)(510
20)(05
xfxwhen
xfxwhen
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
15. Team A: What is the integral of f(x) over one period ?
10
0
5
0
10
5
105
50 45070207020)( xxdxdxdxxfI
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
16. Team B: If we were to represent the function below as a Fourier series what could you say about the value of a0 ?
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
Fourier series
a0 is baseline shifter. Half way between 20 and 70 is 45. So ao = 90
907020]70[5
1]20[
5
170
10
220
10
2)(
2 105
50
10
5
5
000 xxdxdxdxxfperiod
aperiod
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
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 ?
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
Fourier series
odd function so all an terms are zero
Fourier Series - QUIZFourier Series - QUIZ
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
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 ?
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
Fourier series
Fourier Series - QUIZFourier Series - QUIZ
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 ?
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
Fourier series
It would have a negative amplitude
-40
-20
0
20
40
60
80
100
-30 -20 -10 0 10 20 30x axis
y ax
is
step period raised
1st sine harmonic (fundamental)
Finding coefficients of the Fourier SeriesFinding coefficients of the Fourier SeriesFind Fourier series to represent this repeat pattern.
0 x
1
20
01)(
x
xxf
Steps to calculate coefficients of Fourier series
1. Write down the function f(x) in terms of x. What is period?
2. Use equation to find a0?1][
10
11
1)(
10
2
0
2
00
xdxdxdxxfa
Period is 2
3. Use equation to find an?
0sin1
cos1
cos)0(1
cos)1(1
cos)(1
00
2
0
2
0
n
nxdxnxdxnxdxnxdxnxxfan
4. Use equation to find bn?
00
2
0
2
0
cos1sin
1sin)0(
1sin)1(
1sin)(
1
n
nxdxnxdxnxdxnxdxnxxfbn
nn
n
nn
n
n
nxbn
1cos10coscos1cos1
0
Finding coefficients of the Fourier SeriesFinding coefficients of the Fourier Series
5. Write out values of bn for n = 1, 2, 3, 4, 5, ….
4. Use equation to find bn?
n
n
nnn
n
n
nxbn
cos110coscos1cos1
0
2
1
)1(1
1
1
1cos
1
111
b 0
2
)1(
2
11
2
2cos
2
112
b
3
2
3
)1(
3
11
3
3cos
3
113
b 0
4
)1(
4
11
4
4cos
4
114
b
5
2
5
)1(
5
11
5
5cos
5
115
b
6. Write out Fourier series
with period L, an, bn in the generic form replaced with values for our example
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
...5sin5
23sin
3
2sin
2
2
1
2
2sin
2
2cos
2
1)(
10
xxxxn
bxn
aaxfn
nn
Finding coefficients of the Fourier SeriesFinding 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
...5sin5
23sin
3
2sin
2
2
1)( xxxxf
Fourier series (5 terms only for sine and cosine)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 60 120 180 240 300 360
degrees
tota
l am
plit
ud
e
0 x
1
Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
20
0)(
x
xxxf
Steps to calculate coefficients of Fourier series
1. Write down the function f(x) in terms of x. What is period?
2. Use equation to find a0?
22
10
11)(
2
2)(
2
0
22
0
2
000
xdxxdxdxxfdxxf
La
L
3.
4.
Team A find coefficients an?
Team B find coefficients bn?
Period is 2
Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
20
0)(
x
xxxf
3. Team A find coefficients an?Period is 2
vduuvudv
nxn
nxdxv sin1
cos
220
20
10cos
1sin
1cos
1sin
nn
nn
nnx
nnx
n
xan
Integrate by parts so set u = x and cos (nx) dx = dv
and du = dx
L
n dxL
xnxf
La
0
2cos)(
2
2
0
2
00cos0
1cos
1cos)(
2
22cos)(
2dxnxdxnxxdxnxxfdx
L
xnxf
La
L
n
000
sin11
sin1
cos1
nxdxn
nxn
xdxnxxan
n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=5
211
01
a 0
4
1
4
102
a
9
2
9
1
9
103
a 04 a 25
2
25
1
25
105
a
Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
20
0)(
x
xxxf
4.
Period is 2
vduuvudv
nxn
nxdxv cos1
sin
nn
nn
nxn
nxn
xbn sin
1cos
1sin
1cos
20
20
Integrate by parts so set u = x and sin (nx) dx = dv
du = dx
L
n dxL
xnxf
Lb
0
2sin)(
2
2
0
2
00sin0
1sin
1sin)(
2
22sin)(
2dxnxdxnxxdxnxxfdx
L
xnxf
Lb
L
n
000
cos11
cos1
sin1
nxdxn
nxn
xdxnxxbn
n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=511 b
2
12 b
Team B find coefficients bn?
3
13 b
4
14 b
5
15 b
Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
5. Write out the first few terms of Fourier series
20
a
1
0
2sin
2cos
2
1)(
nnn L
xnb
L
xnaaxf
n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=511 b
2
12 b
3
13 b
4
14 b
5
15 b
L is the period = 2
n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=5
211
01
a 0
4
1
4
102
a
9
2
9
1
9
103
a 04 a 25
2
25
1
25
105
a
So .....5sin5
14sin
4
13sin
3
12sin
2
11sin15cos
25
23cos
9
21cos
2
4)( xxxxxxxxxf
Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ
Can we check our Fourier series using Fourier_checker.xls at Phils Problems ???
Yes!!
.....5sin5
14sin
4
13sin
3
12sin
2
11sin15cos
25
23cos
9
21cos
2
4)( xxxxxxxxxf
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