Fourier Series

Fourier Series
2
0 0
0
2
0 0
0
1
( )cos(2 ) ( )
1
( )sin(2 ) ( )
n
n
a x t nf t d t
b x t nf t d t


 

 





0
0
0 0
0 0
0 0
1
2
1 2
( ) 2 2
f
T
t f t
d t f dt dt dt
T T
 

  


  
is the “fundamental frequency”
0 0
1
1
( ) cos(2 ) sin(2 )
2
N
n i n
n
x t a a nf t b nf t
 

  


Fourier Series
2
0 0
0
2
0 0
0
1
( )cos(2 ) ( )
1
( )sin(2 ) ( )
n
n
a x t nf t d t
b x t nf t d t


 

 





0
0
0 0
0 0
0 0
1
2
1 2
( ) 2 2
f
T
t f t
d t f dt dt dt
T T
 

  


  
is the “fundamental frequency”
0 0
1
1
( ) cos(2 ) sin(2 )
2
N
n i n
n
x t a a nf t b nf t
 

  


Fourier Series
Integration limits: when 0 2
t
 
 , then
0 0 0
2 2 1
2 /
t
T T
 
 
  
so we get:
0 0
1
1
( ) cos(2 ) sin(2 )
2
N
n i n
n
x t a a nf t b nf t
 

  

0
0
0
0 0
0
0 0
2
( )cos(2 )
2
( )sin(2 )
T
n
T
n
a x t nf t dt
T
b x t nf t dt
T







Fourier Series
Euler:
0 0
1
1
( ) cos(2 ) sin(2 )
2
N
n i n
n
x t a a nf t b nf t
 

  

2
cos(2 ) sin(2 )
i
j f t
i i
e f t j f t

 
 
0
2
( ) jn f t
n
n
x t c e 


 

Fourier Series
0
2
( ) jn f t
n
n
x t c e 


 
0
0
0
2
0
2
1
( )
T
jn t
n
T
c x t e dt
T



 
We can show
2 2
n n n
c a b
  1
tan ( / )
n n
b a
 

;
recall that
2 2 1
cos( ) sin( ) cos( tan ( ))
b
a b a b
a
   
   

Phasors:
Phasors
2 2
a b


Symmetry
Odd f(-t) =-f(t)
Fourier: sine terms
only
Even f(t) = f(-t)
Fourier: cosine terms
only
Neither

Half-wave symmetry:
0
( ) ( )
2
T
f t f t
  
has no even harmonics
| |
t t+T/2

Example of non-symmetric waveform:
0
( ) ( )
2
T
f t f t
  

Fundamental Signals
Unit Step:
1, 0
( )
0, 0
t
t
t


 
  

 

Fundamental Signals
Unit Step:
1, 0
( )
0, 0
t
t
t


 
  

 
Unit Impulse
0, 0
( )
, 0
t
t
undefined t


 
  

 
( ) ( )
t
t dt t
 


