Fourier Transforms

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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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posted:5/30/2012
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