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					Appendix 02
Linear Systems - Time-invariant systems




       f(t)        Linear         g(t)
                   System




                                          1
     Linear System                                                          g (t )  L f (t )




A linear system is a system that has the following two properties:


   Homogeneity:     L f1 (t )  f 2 (t )  L f1 (t )  L f 2 (t )  g1 (t )  g2 (t )


   Scaling:         Lf (t )  L f (t )  g (t )


The two properties together are referred to as superposition.


                    Lf1 (t )  f 2 (t )  L f1 (t )  L f 2 (t )
                                                                                        2
     Time-invariant System                             g (t )  L f (t )




A time-invariant system is a system that has the property
that the the shape of the response (output) of this system
does not depend on the time at which the input was applied.



                    L f (t  T )  g (t  T )




If the input f is delayed by some interval T,
the output g will be delayed by the same amount.                 3
                                                 f (t )  e jt
                                                 
     Harmonic Input Function
                                                 g (t )  L f (t )  H ( ) f (t )




Linear time-invariant systems have a very interesting (and useful)
response when the input is a harmonic.
If the input to a linear time-invariant system is a harmonic
of a certain frequency , then the output is also a harmonic
of the same frequency that has been scaled and delayed:



                  Le   e
                       jt       j ( t T )




                                                                          4
                                                                                           f (t )  e jt
                                                                                          
                  Transfer Function H()
                                                                                           g (t )  L f (t )  H ( ) f (t )


f1 (t )  e jt
f 2 (t )  x1 (t  T )  e j (t T )
                            
g1 (t )  L f1 (t )  L e jt  H ( , t )e jt
                                               
g 2 (t )  L f 2 (t )  L f1 (t  T )  L e j (t T )                               f (t )  e jt
                                       
         L e  jT e jt  e  jT L e jt  e  jT g1 (t )  e  jT H ( , t )e jt
                                                                                          
g 2 (t )  L f 2 (t )  L f1 (t  T )  g1 (t  T )
                                                                                          g (t )  L f (t )  H ( ) f (t )
         H ( , t  T )e j ( t T )  e  jT H ( , t  T )e jt

H ( , t )  H ( , t  T )  H  H ( )


    The response of a shift-invariant linear system to a harmonic input
    is simply that input multiplied by a frequency-dependent complex number
    (the transferfunction H()).
    A harmonic input always produces a harmonic output
    at the same frequency in a shift-invariant linear system.             5
                                                                        f (t )  e jt
                                                                        
                 Transfer Function                                                                           
                                                                        g (t )  L f (t )  H ( ) f (t )   h(t   ) f ( )d
                 Convolution                                                                                 

                                                                        h(t )  L (t )



    f (t )  e jt
                                         H ( )  H ( ) e  jT
    g (t )  L f (t )  H ( ) f (t )   H ( )  H ( ) e j ( )


f (t )  e jt

g (t )  L f (t )  H ( ) f (t )                                   f(t)                  H()              g(t)

G ( )  H ( ) F ( )                       Fourier tr ansform

                     
g (t )  h * f   h(t   ) f ( )d               Convolutio n      f(t)                  h(t)              g(t)
                                                                                                                   6
                                                                                                                     
                                                                                                g (t )  L f (t )   h(t   ) f ( )d
                                                                                                                    

                                                                                                h(t )  L (t )
                     Convolution

         
g (t )   h(t , ) f ( )d
        
                 
g (t  T )   h(t , ) f (  T )d          shift invariance
              

     t T t                               T  
g (t )       h(t  T , ) f (  T )d
             
                                                  h(t  T ,  T ) f ( )d
                                                  


h(t , )  h(t  T ,  T )
h(t , )  h(t   )
                                                                                                           
         
g (t )   h(t   ) f ( )d                                                          g (t )  h * f   h(t   ) f ( )d
                                                                                                        




                          f(t)                         h(t)                     g(t)
                                                                                                                             7
                                                                                                                                        
                                                                                                                   g (t )  L f (t )   h(t   ) f ( )d
                                                                                                                                       

                                                                                                                   h(t )  L (t )
                    Impulse Response [1/4]




                                                                      T
                              t  nt                    t  1 t 
f S (t )      f (nt )rect                       rect          2                                         t  nt 
                              t                         T  0 otherwise                               rect          
                                                                
                                                                                       (t  nt )  lim         t                  h(t )  L (t )
                                                                                                     t 0        t

                                                                                       T
                                                 t  nt               t  1 t 
f (t )  lim f S (t )  lim
        t 0          t 0
                                  f (nt )rect 
                                                 t 
                                                                  rect             2
                                                                         T  0 otherwise
                                                                              
                                                                                            t  nt                             t  nt  
                                                                                      rect                                rect  t  
                                                      t  nt                            t  t   lim
g (t )  lim L f S (t )  lim L      f (nt )rect            lim L  f (nt )
                                                       t  t 0                          t         t 0
                                                                                                                 f (nt ) L   t   t
                                                                                                                                               
         t 0              t 0
                                  
                                                                            
                                                                                                       
                                                                                                                            
                                                                                                                                               
                                                                                                                                                
       f ( )L (t   )d   f ( )h(t   )d   h(t   ) f ( )d

                                                                                                                                                8
                                                            
                                       g (t )  L f (t )   h(t   ) f ( )d
                                                            

                                       G ( )  H ( ) F ( )
Impulse Response [2/4]                 h(t )  L (t )
                                       H ( )  F [h(t )]




g (t )  L f (t )
                              f(t)   h(t)               g(t)
        h(t   ) f ( )d
         



G( )  H ( ) F ( )          F()   H()              G()




h(t )  L (t )
                                                                   9
                                                                        
                                                   g (t )  L f (t )   h(t   ) f ( )d
                                                                        

            Impulse Response [3/4]                 G ( )  H ( ) F ( )
                                                   h(t )  L (t )
            Convolution                            H ( )  F [h(t )]




                                        g(t)

                 
g (t )  h * f   h(t   ) f ( )d          t
                                                                             10
                
                                                                       
                                                  g (t )  L f (t )   h(t   ) f ( )d
                                                                       

Impulse Response [4/4]                            G ( )  H ( ) F ( )
                                                  h(t )  L (t )
Convolution                                       H ( )  F [h(t )]




          =                                 *




                           
          g (t )  h * f   h(t   ) f ( )d
                          


                                                                            11
Convolution
Rules




                                    
    f *g     f ( ) g (t   )d   f (t   ) g ( )d  g * f
                                   

    f * ( g  h)  f * g  f * h
    f * ( g * h)  ( f * g ) * h
    d
        f * g   f '*g  f * g '
    dt



                                                                     12
Some Useful Functions




                        A                 B




                            a/2                  b




                                                1 t
                                   (t )  lim13  
                                           a 0 a
                                                  a
                                                                                               1 t
               The Impulse Function [1/2]                                          (t )  lim    
                                                                                          a 0 a
                                                                                                  a



             1 t
 (t )  lim    
        a 0 a
                a




                         

 f ( ) (  t )d   f (  t ) ( )d  f (t )
                       

            1
 (at )       (t )
            a
                 
                                                   The impulse is the identity function
 (t ) * f (t )    ( ) f (t   )d  f (t )   under convolution                         14
                 
                                                                                                              1 t
               The Impulse Function [2/2]                                                         (t )  lim    
                                                                                                         a 0 a
                                                                                                                 a



            1
 (at )       (t )
            a



                                  t 1              1           t
                         (t ) f   dt    a0          (t ) f  dt a  0
                                   aa               a            a
  (at ) f (t )dt  
                                                     
                        (t ) f  t  1 dt            1            t

                      
                      
                                    
                                    aa
                                              a0
                                                       a 
                                                           
                                                              (t ) f  dt a  0
                                                                      a
                         
                                     t
                                                                             
                                                                                 1        
                                                           (t ) f t dt   a  (t ) f t dt
                     1                      1        1
                   
                     a     (t ) f  dt  f (0) 
                                     a    a        a                                    
                                                                                        

            1
 (at )       (t )
            a                                                                                               15
Step Function [1/3]   b




                          b




                              16
Step Function [2/3]                               b




                                              b




                             

    u(t   ) f (t )dt   f (t )dt
                        

                    t
                                  1   t 
   u (t   )    ( s   )ds  
                                0   t 

                du (t )
   u ' (t )              (t )
                 dt                                   17
                    Step Function [3/3]                                              b




              du (t )
u ' (t )               (t )
               dt




lim f (t )  0
     t  



                                                         
                                                 
 u' (t ) f (t )dt  u(t ) f (t )u (t ) f (t )

                                                 
                                                        u (t ) f ' (t )dt
                                                          
                                                                                        
                         u (t ) f ' (t )dt    f ' (t )dt  f ()  f (0)  f (0)    (t ) f (t )dt
                                                    0                                   


u ' (t )   (t )                                                                                            18
Smoothing a function by convolution
                                b




                                      19
Edge enhancement by convolution
                              b




                                                          / 2 2
                             h(t )  2 (t )  e t
                                                      2



                                                                                        / 2 2
                             g (t )  h * f  2 (t ) * f (t )  e t
                                                                                    2
                                                                                                 * f (t )
                                                
                                   2 f (t )   e (t  )            / 2 2
                                                                                f ( )d
                                                                   2



                                                




                                                                                  20
                                                                                             
                                                                        g (t )  L f (t )   h(t   ) f ( )d
                                                                                            

           Discrete 1-Dim Convolution [1/5]                             h(t )  L (t )
           Matrix



                                                   
g (t )  h(t ) * f (t )   h( ) f (t   )d   h(t   ) f ( )d
                                                  



g (i )  h(i ) * f (i )   h( j ) f (i  j )   h(i  j ) f ( j )
                           j                    j

g i  hi * f i   h j f i  j   hi  j f j
                   j               j

gHf
 g1   h1        hN      ... h2   f1 
 g  h           h1      ... h3   f 2 
 2 2                            
 ...   ...      ...     ... ...   ... 
                                 
 g N  hN       hN 1    ... h1   f N                                                          21
                                                        
                                   g (t )  L f (t )   h(t   ) f ( )d
                                                       

Discrete 1-Dim Convolution [2/5]   h(t )  L (t )
Example




                                                               22
                                                        
                                   g (t )  L f (t )   h(t   ) f ( )d
                                                       

Discrete 1-Dim Convolution [3/5]   h(t )  L (t )
Discrete operation




                                                               23
                                                        
                                   g (t )  L f (t )   h(t   ) f ( )d
                                                       

Discrete 1-Dim Convolution [4/5]   h(t )  L (t )
Graph - Continuous / Discrete




                                                               24
                                                        
                                   g (t )  L f (t )   h(t   ) f ( )d
                                                       

Discrete 1-Dim Convolution [5/5]   h(t )  L (t )
Wrapping h index array




                                                               25
          Two-Dimensional Convolution




                                                          
g ( x, y )  h * f      h(u, v) f ( x  u, y  v)dudv    h( x  u, y  v) f (u, v)dudv
                                                        




                                                                                      26
          Discrete Two-Dimensional Convolution [1/3]


                                                                             
g ( x, y )  h * f      h(u, v) f ( x  u, y  v)dudv    h( x  u, y  v) f (u, v)dudv
                                                                            




                   g (i, j )  h(i, j ) * f (i, j )   h(i  m, j  n) f (m, n)
                                                        m    n

                   gi, j         hi , j * f i , j     hi  m , j  n f m ,n
                                                         m   n



                   G  H F
                   [ g1 ]  [h1 ] [hN ]            ... [h2 ]  [ f1 ] 
                    [ g   [h ] [h ]               ... [h3 ]  [ f 2 ] 
                    2 2            1                                
                    ...   ...     ...             ... ...   ... 
                                                                     
                   [ g N  [hN ] [hN 1 ]          ... [h1 ]  [ f N ]
                                                                                       27
        Discrete Two-Dimensional Convolution [2/3]


           1 2 0                   1 1 0                       1  1 2
   1 2                 1 1 
F      3 4 0
                    H           2 2 0
                                                     G  H * F    5  3 8
                                                                            
   3 4 0 0 0          2 2  0 0 0                          6  2 8 
                                                                        



             1 0 1  0 0  0 2 0   2  1    1 
             1 1 0  0 0  0  2  2 0   2   1 
                                          
             0 1 1 0 0  0 0   2  2 0   2 
                                          
             2 0 2 1 0  1 0   0  0   3   5 
g  H  f   2 2 0  1 1 0 0   0  0    4    3
                                          
             0  2 2 0 1 1 0   0  0  0   8 
            0   0 0 2 0  2 1 0   1  0    6 
                                          
             0  0 0  2 2 0 1  1 0  0    2
            0          2 2 0   1  1  0   8 
                0 0  0                                          28
          Discrete Two-Dimensional Convolution [3/3]


                  Kernel matrix




Input image                            Output image




 Array
                             
                                                      Output pixel
 of products                           xC

                          Summer   Scaling factor         29
                                                                                   
                                                              g (t )  L f (t )   h(t   ) f ( )d
                                                                                   

                                                              G ( )  H ( ) F ( )
     Linear System - Fourier Transform                        h(t )  L (t )
                                                              H ( )  F [h(t )]



                             Impulse respons


Input function        f(t)                      g(t)       Output function
                                    h(t)

Spectrum of                        H()                    Spectrum of
                     F()                       G()
input function                                             output function

                             Transfer function


G( )  H ( ) F ( ) H ( )  G ( )  F g (t )
                                   F ( )    F  f (t )
                                    F g (t ) 
                                   1
                         h(t )  F             
                                    F  f (t )                                       30
End




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