Fourier Series - QUIZ
Team A questions in white
Team B questions in red
1. What is
1  (1) n when n = 3 ?
1  (1)3  0
2. What is
1  (1) n when n = 52 ?
1  (1)52  2
3. What is
1  (cos 2n) when n = 1 ?
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
1  (1)  2
Fourier Series - QUIZ
10
I
8. Team B: What is
10
 
10
 4 x dx ?
I
2
4
x
dx

2
x

10
10
 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
10
10
10
0
0
0
I   (2 x  5) dx ? I   (2 x  5) dx  x 2  5 x   150
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 ?
when x  0 f ( x)  0
y axis
when x  0 f ( x)  50
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 
10
y axis
10
step

10
0 dx   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
5
10
I   f ( x) dx   0 dx   50 dx  0 x0  50 x5  250
5
0
0
10
5
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 f ( x)  20
y axis
when 10  x  5 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
5
10
I   f ( x) dx   20 dx   70 dx  20 x0  70 x5  450
5
0
5
y axis
0
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
2nx
2nx
f ( x)  a0   an cos
 bn sin
2
L
L
n 1
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 ?
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 
2
period
-20
-10
step
period raised 2
period

0
f ( x)dx 
0
-20
5
10 0
10
x axis
20
30
2 10
1
1
5
10
70
dx

[
20
x
]

[
70
x
]
0
5  20  70  90

5
-40
10
5
5
20dx 
Fourier Series - QUIZ
Fourier series

1
2nx
2nx
f ( x)  a0   an cos
 bn sin
2
L
L
n 1
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

1
2nx
2nx
f ( x)  a0   an cos
 bn sin
2
L
L
n 1
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

1
2nx
2nx
f ( x)  a0   an cos
 bn sin
2
L
L
n 1
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? f ( x)   x

0
0 x 
  x  2
Period is 2
2. Use equation to find a0?
a0 
2
2
f
(
x
)
dx

L 0
2
L
3. Team A find coefficients an?
4. Team B find coefficients bn?

2
0
f ( x)dx 
1



0
xdx 
1

2


1  x2 

0dx    
  2 0 2
Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
2 L
2nx
an   f ( x) cos
dx
0
L
L
x
f ( x)  
0
0 x 
  x  2
Period is 2
3. Team A find coefficients an?
2 L
2nx
2
an   f ( x) cos
dx 
L 0
L
2
Integrate by parts
1
v   cos nxdx  sin nx
n

2
0
f ( x) cos nx dx 
 udv  uv   vdu
and du = dx an 

1


0
x cos nx dx 
so set u = x and
1



0
2
1
 
0 cos nx dx
cos (nx) dx = dv

1 x
1 1

x cos nx dx   sin nx   sin nxdx
 n
0  0 n

1
1 
x
  1

1
 
an   sin nx   2 cos nx   sin n  2 cos n    0  2 
n
 n
 0  n
0  n
  n 
n=1
n=2
1 1
2
1


a1   0       
a2   0 

4
    

  1

  4
n=3
1


  0 a3   0 
9


  1

  9
n=4
2


9

a4  0
n=5
1

a5   0 
25

  1

  25
2


25

Finding coefficients of the Fourier Series - QUIZ
Find Fourier series to represent this repeat pattern.
bn 
2 L
2nx
f
(
x
)
sin
dx

0
L
L
x
f ( x)  
0
0 x 
  x  2
Period is 2
4. Team B find coefficients bn?
2 L
2nx
2
bn   f ( x) sin
dx 
L 0
L
2
Integrate by parts
1
v   sin nxdx   cos nx
n

2
0
f ( x) sin nx dx 
 udv  uv   vdu
du = dx

bn 
1

1


0
x sin nx dx 
so set u = x and


0
1
2
 
0 sin nx dx
sin (nx) dx = dv

1 x
1 1

x sin nx dx    cos nx   cos nxdx
 n
0  0 n

1
 x
  1

 1

bn   cos nx   2 sin nx    cos n  2 sin n 
n
 n
 0 n
0  n

n=1
n=2
b1  1
b2  
1
2
n=3
b3 
1
3
n=4
b4  
1
4
n=5
b5 
1
5