# Fourier Transforms by 20Izr28

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```									                    Fourier Transforms

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier series
2
To go from f( ) to f(t) substitute      t  0t
T

f (t )   an cos(n0t )  bn sin(n0t )
n 0

To deal with the first basis vector being of
length 2 instead of , rewrite as
a0 
f (t )    an cos(n0t )  bn sin(n0t )
2 n 1

University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Fourier series
The coefficients become
t 0 T
2
ak 
T        f (t ) cos(k t ) dt
t0
0

t 0 T
2
bk 
T        f (t ) sin(k t ) dt
t0
0

University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Fourier series
Alternate forms
a0                  bn
f (t )    an (cos(n0t )  sin(n0t ))
2 n 1               an
a0 
   an (cos(n0t )  tan( n ) sin(n0t ))
2 n 1
a0 
   cn (cos(n0t )   n )
2 n 1
where
 bn 
cn  a  b   2
n
2
n     and n   tan  
a 
1

 n
University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Complex exponential notation
Euler’s formula e  cos(x)  i sin( x) ix

Phasor notation:
x  iy  z ei
where z  x 2  y 2

 zz
 ( x  iy)(x  iy)
 y
and   tan 1  
x
University of Texas at Austin CS395T - Advanced Image Synthesis    Spring 2006 Don Fussell
Euler’s formula
Taylor series expansions
x 2 x3 x 4
e x  1  x     ...
2! 3! 4!
Even function ( f(x) = f(-x) )
x 2 x 4 x 6 x8
cos(x)  1      ...
2! 4! 6! 8!
Odd function ( f(x) = -f(-x) ) 3
x  x5 x7 x9
sin( x)  x      ...
3! 5! 7! 9!
x 2 ix3 x 4 ix 5 x 6 ix 7
e ix  1  ix                       ...
2! 3! 4! 5! 6! 7!
 cos(x)  i sin( x)
University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Complex exponential form
Consider the expression
                   
f (t )     Fn ein0t 
n  
F
n  
n   cos(n0t )  iFn sin(n0t )

  ( Fn  F n ) cos(n0t )  i ( Fn  F n ) sin(n0t )
n 0

So an  Fn  Fn and bn  i( Fn  Fn )
Since an and bn are real, we can let Fn  Fn
and get an  2 Re( Fn ) and bn  2 Im( Fn )
an                              bn
Re( Fn )                 and Im( Fn )  
2                               2
University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Complex exponential form
10                                                  
t T                     t 0 T
Thus
Fn    f (t ) cos(n0t ) dt  i  f (t ) sin(n0t ) dt 
T  t0
                             t0


t 0 T
1

T       f (t )(cos(n t ) dt  i sin(n t ))dt
t0
0                          0

t 0 T
1

T      
t0
f (t ) e in0t dt

 Fn ei n


So you could also write                                f (t )     
n  
Fn e i ( n0t  n )

University of Texas at Austin CS395T - Advanced Image Synthesis      Spring 2006 Don Fussell
Fourier transform

We now have f (t )                          Fn e in0t
n  
t 0 T
1
Fn 
T        
t0
f (t ) e in0t dt

Let’s not use just discrete frequencies, n0 ,
we’ll allow them to vary continuously too

We’ll get there by setting t0=-T/2 and taking
limits as T and n approach 
University of Texas at Austin CS395T - Advanced Image Synthesis     Spring 2006 Don Fussell
Fourier transform
                                    T /2
in0t 1
f (t )      Fn ein0t
n  
 e
n         T T2
f (t ) e in0t dt
/
2                                  2

2 1
T /2
in
e
in      t                                   t

n  
T
T 2       f (t ) e
T /2
T
dt

 2 
lim          d         lim n d  
T 
 T                  n
              
1
f (t )   ei t d
2 
f (t ) e i t dt
              

1

 1           

e  2               f (t ) e
i t                             i t
                                                          dt  d
2                                                      

1

2           

ei t F ( ) d
University of Texas at Austin     CS395T - Advanced Image Synthesis               Spring 2006 Don Fussell
Fourier transform
So we have (unitary form, angular frequency)

1
F ( f (t ))  F ( ) 
2                
   f (t ) e i t dt


1
F ( F ( ))  f (t )                         F ( ) ei t d
- 1

2                    
Alternatives (Laplace form, angular frequency)

F ( f (t ))  F ( )         

f (t ) e i t dt


1
F ( F ( ))  f (t )                     F ( ) ei t d
- 1

2               

University of Texas at Austin CS395T - Advanced Image Synthesis        Spring 2006 Don Fussell
Fourier transform

Ordinary frequency                                                         
2

F ( f (t ))  F ( )       

f (t ) e i t dt


F - 1 ( F ( ))  f (t )       

F ( ) ei t d

University of Texas at Austin CS395T - Advanced Image Synthesis    Spring 2006 Don Fussell
Fourier transform
Some sufficient conditions for application
Dirichlet conditions


f (t ) dt  
f(t) has finite maxima and minima within any finite interval
f(t) has finite number of discontinuities within any finite
interval
Square integrable functions (L2 space)


[ f (t )] 2 dt  
Tempered distributions, like Dirac delta
1
F ( (t )) 
2
University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Fourier transform
Complex form – orthonormal basis functions for
space of tempered distributions
   e i1 t e  i2 t

2 2
dt   (1   2 )

University of Texas at Austin CS395T - Advanced Image Synthesis   Spring 2006 Don Fussell
Convolution theorem
Theorem                  F ( f * g )  F ( f )F ( g )
F ( fg)  F ( f ) * F ( g )
F - 1 ( F * G)  F - 1 ( F )F - 1 (G )
F - 1 ( FG)  F - 1 ( F ) * F - 1 (G )
                             
Proof (1)               F ( f * g )   f (t ' ) g (t  t ' )  e i t dt dt '
                            
                            
  f (t ' )e     i t '
dt '  g (t  t ' )e i (t t ') dt
                            
                            
  f (t ' )e     i t '
dt '  g (t ' ' )e it '' dt ' '
                            

 F ( f )F ( g )
University of Texas at Austin CS395T - Advanced Image Synthesis      Spring 2006 Don Fussell

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