Numerical Methods
Introduction to Fourier Series
Part: Introduction to Fourier
Series
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Lecture # 1
Chapter 11.01: Introduction to
Fourier Series
Major: All Engineering Majors
Authors: Duc Nguyen
http://numericalmethods.eng.usf.edu
Numerical Methods for STEM undergraduates
4/8/2015
http://numericalmethods.eng.usf.edu
5
Background
The following relationships can be readily
established
T
T
0
0
 sin( kw0t )dt   cos(kw0t ) dt  0
(1)
T
 sin ( kw0t )dt   cos ( kw0t ) dt 
0
0
2
(2)
T
2
T
2
1
6
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Background cont.
T
(3)
 cos(kw0t ) sin( gw0t ) dt  0
0
T
 sin( kw0t ) sin( gw0t ) dt  0
(4)
0
T
 cos(kw0t ) cos( gw0t ) dt  0
(5)
0
7
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Background cont.
w  2f
(6)
1
f 
T
(7)
0
Where f and
represents the frequency
in (cycles/time) and period (in seconds) respectively.
T
A periodic function with a period
the following equation:
f (t  T )  f (t )
8
T should satisfy
(8)
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Background cont.
Example 1
Let
T
A   sin( kw0t )dt
(9)
0
 1 
T
cos(kw0t )0
 
 kw0 
9
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Background cont.
 1 
cos(kw0T )  cos(0)
A  
 kw0 
 1 
cos(k 2 )  1
 
 kw0 
(10)
0
10
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Background cont.
Example 2
Let
T
B   sin (kw0t )dt
2
(11)
0
Recall
1  cos(2 )
sin ( ) 
2
(12)
1 1

B     cos(2kw0t ) dt
o 2
2

(13)
2
T
11
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Background cont.
T
 1   1  1 

 sin( 2kw0t )
  t   
 2   2  2kw0 
0
T

1
B 
sin( 2kw0T )  0
 2 4kw0

12
(14)
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Background cont.
T  1 
 sin( 2k * 2 )
  
2  4kw0 
T

2
Example 3
Let
T
C   sin( gw0t ) cos(kw0t )dt
(15)
0
13
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Background cont.
Recall that
sin(   )  sin( ) cos( )  sin(  ) cos( )
(16)
C   sin  g  k w0t   sin( kw0t ) cos(gw0t )dt
T
(17)
0
14
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Background cont.
  sin ( g  k ) w0t dt   sin( kw0t ) cos(gw0t )dt (18)
T
T
0
T
0
C  0   sin( kw0t ) cos( gw0t )dt
(19)
0
Adding Equations (15), (19),
T
T
0
0
2C   sin( gw0t ) cos(kw0t )dt   sin( kw0t ) cos(gw0t )dt
  sin  gw0t   (kw0t )dt   sin ( g  k ) w0t dt
T
0
15
T
0
(20)
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Background cont.
2C  0,
since the right side of the above equation is zero
Thus,
T
C   sin( gw0t ) cos(kw0t )dt  0
(21)
o
16
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THE END
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Acknowledgement
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undergraduate
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This material is based upon work supported by the National
Science Foundation under Grant # 0717624. Any opinions,
findings, and conclusions or recommendations expressed in
this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation.
The End - Really