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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