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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

31

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