Transcript Fourier Series

Fourier Series - QUIZ
Team A questions in white
Team B questions in red
1. What is 1  (  1) n when n = 3 ?
1  (  1)  0
2. What is 1  (  1) n when n = 52 ?
1  (  1)
3
3. What is 1  (cos 2 n ) when n = 1 ?
52
2
1  (1)  2
4. What is 1  (cos 2 n ) when n = 17 ?
1  (1)  2
5. What is 1  (cos 2 n ) when n = 52 ?
1  (1)  2
6. What is
1  (cos  n )
when n = 1 ?
1  (cos  )  1  (  1)  0
7. What is
1  (cos  n )
when n = 4 ?
1  (cos 4  )  1  (1)  2
Fourier Series - QUIZ
10
8. Team B: What is I 
10
 4 x dx ?
I 
 10
10

 4 x dx  2 x
 10
2

 200  200
 10
y axis
40
30
20
y=4x
10
0
-20
-15
-10
-5
-10
-20
-30
-40
0
5
10
15
x axis
20
Fourier Series - QUIZ
9. Team A: What is I 
10
10
10
 ( 2 x  5 ) dx ? I 
 ( 2 x  5 ) dx  x  5 x
0
0

2

 150
0
60
y axis
50
y=2x+5
40
30
20
10
0
-30
-20
-10
-10 0
-20
-30
-40
10
x axis
20
30
Fourier Series - QUIZ
10. Team B: Describe the following step function in terms of f(x) and x ?
x0
f ( x)  0
when
x0
f ( x )  50
y axis
when
step
100
80
60
40
20
x axis
0
-30
-20
-10
0
-20
-40
10
20
30
Fourier Series - QUIZ
10
11. Team A: What is I 

f ( x ) dx ?
 10
10
I 

0
f ( x ) dx 
0 dx 
 10
y axis
 10

10
step
 50 dx  0 x 10  50 x 0  500
0
10
0
100
80
60
40
20
x axis
0
-30
-20
-10
0
-20
-40
10
20
30
Fourier Series - QUIZ
12. Team B: Describe the following step function over one period in
terms of f(x) and x ?
when 5  x  0
f ( x)  0
when 10  x  5
f ( x )  50
y axis
100
step periods
80
60
40
20
0
-30
-20
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
13. Team A: What is the integral of f(x) over one period ?
10
I 

5
f ( x ) dx 
0
10
 0 dx 
 50 dx  0 x 0  50 x 5  250
0
5
5
10
y axis
100
step periods
80
60
40
20
0
-30
-20
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
14. Team B: Describe the following step function over one period in
terms of f(x) and x ?
when 5  x  0
y axis
when 10  x  5
f ( x )  20
f ( x )  70
100
80
60
40
20
0
-30
-20
step period raised
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
15. Team A: What is the integral of f(x) over one period ?
10
I 

5
f ( x ) dx 
 20 dx 
 70 dx  20 x 0  70 x 5  450
0
5
y axis
0
10
5
10
100
80
60
40
20
0
-30
-20
step period raised
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
1
f ( x) 
Fourier series
2

a0 

a n cos
n 1
2 nx
L
 b n sin
2 nx
L
16. Team B: If we were to represent the function below as a Fourier
series what could you say about the value of a0 ?
y axis
a0 is baseline shifter. Half way between 20 and 70 is 45. So ao = 90
100
80
60
40
20
0
-30
a0 
-20
2
period
-10
step
periodperiod raised 2

0
f ( x ) dx 
10
0
10
x axis
20
30
-20

5
0
20 dx 
2
-40
10

10
5
70 dx 
1
5
[ 20 x ] 0 
5
1
5
[ 70 x ] 5  20  70  90
10
Fourier Series - QUIZ
Fourier series
f ( x) 
1
2

a0 

a n cos
n 1
2 nx
L
 b n sin
2 nx
L
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 ?
y axis
odd function so all an terms are zero
100
80
60
40
20
0
-30
-20
step period raised
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
Fourier series
f ( x) 
1
2

a0 

a n cos
n 1
2 nx
L
 b n sin
2 nx
L
y axis
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 ?
100
80
60
40
20
0
-30
-20
step period raised
-10
0
-20
-40
10
x axis
20
30
Fourier Series - QUIZ
Fourier series
f ( x) 
1
2

a0 

a n cos
n 1
2 nx
L
 b n sin
2 nx
L
y axis
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
100
80
60
40
20
0
-30
step period
-20 raised
-10
0
-20
1st sine harmonic (fundamental)
-40
10
x axis 20
30
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?
x
f ( x)  
0
0 x
  x  2
Period is 2
2. Use equation to find a0?
a0 
2
L

L
f ( x ) dx 
0
2
2
3. Team A find coefficients an?
4. Team B find coefficients bn?

2
0
f ( x ) dx 
1



0
xdx 
1

2

0 dx 

1 x 


 
  2 0
2
2
Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
2

L
an 
2 nx
L
f ( x ) cos
dx
L
0
0 x
x
f ( x)  
0
  x  2
Period is 2
3. Team A find coefficients an?
an 
2

L
L
f ( x ) cos
2 nx
Integrate by parts
v 
 cos nxdx 
dx 
L
0
1
sin nx
n
 udv
2
2

2
f ( x ) cos nx dx 
0
 uv 
 vdu
and du = dx a n 

1



x cos nx dx 
0
so set u = x and
1



0
1

2

0 cos nx dx
cos (nx) dx = dv

1 x
1

x cos nx dx 
sin
nx


  n
0 


0
1
sin nxdx
n

1
1 
 x

 1

1
 
an  
sin nx    2 cos nx    sin n  
cos
n


0




2
2
n
n 
 n
 0  n
0  n
 
n=1
1 1
2

a1   0       
   


n=2
n=3
1   1 
1   1 
2


a2   0 

  0 a3   0 


4   4 
9   9 
9


n=4
a4  0
n=5
1   1 
2

a5   0 


25    25  
25 

Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
bn 
2
L

L
f ( x ) sin
2 nx
L
0
0 x
x
f ( x)  
0
dx
  x  2
Period is 2
4. Team B find coefficients bn?
bn 
2

L
L
f ( x ) sin
2 nx
Integrate by parts
v
 sin nxdx  
dx 
L
0
1
n
cos nx
 udv
2
2


f ( x ) sin nx dx 
0
 uv 
du = dx
2
 vdu
bn 
1

1



x sin nx dx 
0
so set u = x and


0
1

2

0 sin nx dx
sin (nx) dx = dv

1 x
1

x sin nx dx    cos nx  
 n
0 


0
1
cos nxdx
n

1
 x

 1

 1

bn   
cos nx    2 sin nx     cos n  
sin
n


2
n
 n
 0  n
0  n

n=1
b1  1
n=2
b2  
1
2
n=3
b3 
1
3
n=4
b4  
1
4
n=5
b5 
1
5