# The Design of FIR Least Squares Inverse Filters by qok10781

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```									                       Digtal Signal Processing
And Modeling

The Design of FIR Least
Squares Inverse Filters
Chapter 4.4.5 Application

2006. 09. 21 / KIM JEONG JOONG / 20067168
www.themegallery.com
GOALS                                                                statistical

D.S.P
   Given      g ( n)
g 1 (n)
1
g (n) * g (n)   (n) or G ( z )G 1 ( z )  1

   The design of an inverse filter is important applications
   Signal is to be transmitted across a nonideal channel
   Channel is linear and has a system Funtion G(z)
   The chance of making errors at output of the receiver
channel equalization filter
   The goal is to find an equalizer H(z)
Inverse System                                                             statistical

D.S.P
   In Most applications, the inverse system

H ( z )  1 / G( z )         Not practical solution

   If G(z) is minimum phase,
   Inverse filter is causal and stable
   Or
   Inverse filter is not both causal and stable

   Constraining h(n) to be FIR requires
We find the best approximation to the inverse filter

hn(n)  g (n)  d (n)
Optimum inverse filter                                                    statistical

D.S.P
   Where        d ( n )   ( n)
N 1
e(n)  d (n)  hn(n)  g (n)  d (n)   hN (l ) g (n  l )
l 0

Shanks’ method                                                2
                           N 1
 N   e(n)   d (n)   hn (l ) g (n  l )
2

n 0          n 0            l 0

Optimum least squares inverse filter
N 1

h
l 0
N   (l ) rg (k  l )  rdg (k )   ;      k = 0,1,2,…., N-1
Least squares solution                                                        statistical

D.S.P
N 1

    h
l 0
N   (l ) rg (k  l )  rdg (k )

     d ( n)   ( n)
g (n)  0, n  0
rdg (k )  g (0) (k )
*
Rg H N  g (0)u1 *

 rg (0)       r * g (1)  r * g (2)         r * g ( N  1)   hN (0)   g * (0)
                                                                              
  rg (1)      rg (0)      r * g (1)        r g ( N  2)  hN (1)   0 
*

 rg (2)        rg (1)     rg (0)           r g ( N  3)   hN (2)    0 
*

                                                                             
                                                                 
r ( N  1) r ( N  2) r ( N  3)                rg (0)  hN ( N  1)  0 
g           g           g                                                    
Define the coefficients(1)                                             statistical

D.S.P
   Given                              N 1
EN m in  rd (0)   hn (k )rdg * (k )
k 0
rdg (k )  g * (0) (n)
EN m in  1  hN (0) g (0)
   e(n) = 0, ( n >= 0 )
 g (0)      0           0           0  hN (0)  1
 g (1)
          g (0)         0           0  hN (1)  0
              
 g (2)    g (1)       g (0)         0  hN (2)   0
                                                     
 g (3)    g (2)       g (1)         0             0 
 
                                     hN ( N  1)   
              
Define the coefficients(2)                                             statistical

D.S.P
   Given
e(n)   (n  n0 )  hN (n) * g (n)
d ( n )  ( n  n0 )

              
rdg (k )   d (n) g * (n  k )   (n  n0 ) g * (n  k )  g * (n0  k )
n 0            n 0

 g * ( n 0) 
 *           
 g ( n0  1)                     n0


      
   EN m in  1   hN (k ) g * (n0  k )
Rg hN   g * (0)                         k 0
             

0
   G0 hN  un 01
            
             
      0      
FIR least squares inverse filter                        statistical

D.S.P

hN (n) * g (n)   (n  n0 )
special case

hN (n) * g (n)  d (n)

Rg hN  rdg
   FIR least squares inverse filter
g 0 hN  d

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