Sparse channel estimation equalization by juanagui

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									Sparse channel estimation & equalization
   Motivation of Sparse channel estimation
     1. To obtain accurate channel estimate
         • Underwater acoustic channel is sparse and time-
            varying.
         • Channel should be estimated over the subspace
            with sparse constraint.
     2. Very good for channel tracking
         • Less number of channel taps  better tracking
            performance
     3. Low complexity equalization
         • Reduce receiver complexity
Approaches to sparse channel estimation
    Deterministic
     1. Basis selection approaches
         • Matching pursuit (MP), orthogonal MP, Basis
            pursuit, Order recursive LS MP
     2. Some Heuristics
         • Active tap detection based on independent
            assumption
         • Thresholding based on the coefficients of full-tap
            auxiliary RLS filter
    Stochastic
     1. Not known to my knowledge.
New approach
      What is the structure of the channel  non-active tap
       versus active tap
      STEP 1 - Structure detection
       1. Separate channel structure variable from channel
           model.
       2. Infer the channel structure via a message passing
           algorithm.
      STEP 2 - Projection into “structure embedded” channel
       space
       1. Apply channel estimation based on the inferred
           structure.
       2. Incorporate soft information on transmitted sequence
           (SISO channel estimation).
Channel model
    Time variant channel model
                L 1
                                                      h(n, 0) 
          yn   h  n, k  xn k  wn
                k 0                            hn  
                                                                 
                                                                   
              xn hn  w n
                H                                     h(n, L  1) 
                                                                  



    Channel state evolution model             Channel structure parameter

                                 First order    an   a0 ,, aL1 
                                                      
                                                         n     n
                                                                   
           g n  Ag n 1  v n
                                                ain  0,1
                                 AR model

                  a0 0
                    n
                        0 
                            
           h n   0  0  g n  diag  a n  g n
                  0 0 aL 1 
                        n
                            
Channel model
    Assume channel structure is slow-varying
                                                                   Channel
             a n  a n 1    a n  B 1  a

                     g k  Ag k 1  v n
                     yk  xk diag  ak  g k  wk
                           H




    Definitions
       X  x k ,, x k  F 1 , G  g k ,, g k  F 1   F is the length of the
       y   yk ,, yk  F 1 , A  a k ,, a k  F 1    processing block


    Objective: Find A, G from the above model !
What is factor graph
     Provide useful graph model which describes the factorization
    of a global function into local functions.
     Provide intuitive interpretation on estimation and detection
    of random variables
     Sum-produce algorithm (message-passing alg.) makes the
    marginalization of a function computationally efficient
What is factor graph
        Example
          p  x1, x2 , x3 , x4   pA  x1  pB ( x2 ) pC  x3 | x1, x2  PD  x4 | x3 


                 x1             x2
  pA                                               pB

                                                   x  f  x         
                      pC
                                                                                       h  x ( x)
                                                                     hn ( x )/{ f }

                           x3                      f  x  x    f ( X )                              y  f ( y )
                                                                     ~{ x}               yn ( f )/{ x}

                      pD

                                      x  p  x3    x  p
                                        3   c                4
                                                                x4 
                                                                   D

                           x4
                                      p  x
                                        c   3
                                                 x3    pc  x3 | x1 , x2   x  p  x1  x  p  x2 
                                                                                                1         c          2     c
                                                         ~{ x3 }
Channel model revisited !
     Assume channel structure is slow-varying
                                                                    Channel
              a n  a n 1    a n  B 1  a

                      g k  Ag k 1  v n
                      yk  xk diag  ak  g k  wk
                            H




     Definitions
        X  x k ,, x k  F 1 , G  g k ,, g k  F 1   F is the length of the
        y   yk ,, yk  F 1 , A  a k ,, a k  F 1    processing block


     Objective: Find A, G from the above model !
Factor graph representation
    For our application,
   p  G, X, A | y   p  y | G, A, X  p  X  p  A  p  G 
                                    n  F 1
                                    p y
                                     i n
                                                    n   | g n , a n , x n  I  X  C  p  A  p  g k | g k 1 
                                                                  p( A)
                                               a1                                   a2


     p  yn | gn , an , xn 

                                p  gk | gk 1 
                        g1              g2              g3          g4         g5          g6

                               x1              x2            x3           x4        x5          x6



                                                                  XC
Case 1. Known transmitted sequence
       Assume that transmitted sequence is known (training phase)
p  G, A | y   p  y | G, A  p  G 
                     n  B 1
                     p y
                      k n
                                     k   | g k , a k  p  g k | g k 1  p  g n  I  a n    a n  B 1  a  p  a 

                                         
                                                      a

                                                                               Tk  p  yk | gk , ak 
            ak                  a k 1                            a k  B 1
                                                                               Gk 1  p  gk 1 | gk 
       Tk              Tk 1                          Tk  B 1



            gk                  g k 1                            g k  B 1
                     Gk 1                Gk  2
Case 1. Known transmitted sequence
     Apply sum-product algorithm
       • Gaussian model – the messages are represented only by
         first and second statistics.

                                           a
                                                                     Structure detection
        ak            a k 1                            a k  B 1

   Tk         Tk 1                         Tk  B 1
                                                                       Kalman filtering

        gk            g k 1                            g k  B 1
             Gk 1                 Gk  2
Case 1. Known transmitted sequence
   •   Approximation is needed to make a derivation easy.
           a   k   Tk    ak     ak  a 
                                               ˆ
           g   k   Tk    g k     g k  E  g k | Y 
                                                
                                                                a


                                ak            a k 1                        a k  B 1

                           Tk         Tk 1                     Tk  B 1



                                gk            g k 1                        g k  B 1
                                     Gk 1             Gk  2
Case 1. Known transmitted sequence
    Message passing scheduling
      Step 1. Forward passing
                                                          a


                      ak                a k 1                        a k  B 1

               Tk               Tk 1                     Tk  B 1
                           
                           1
                            k


                3k        
                           2
                            k


                      gk                g k 1                        g k  B 1
                               Gk 1             Gk  2
Case 1. Known transmitted sequence

                                                
     Define q  x, mx , Σ   exp   x  mx  Σ  x  mx 
                                                                       H
                                                                                       
             1k  g k    Tk  g k , a k  a    k   Tk   ak 
                           ak

                         Tk  g k , a k    a k  a   Tk  g k , a 
                                                      ˆ                ˆ
                           ak



             2  g k   3k  g k  1k  g k 
              k



                                        
                         q g k , m3 , Σ3 q yk , aT diag x k g k ,  2
                                   k    k
                                                 ˆ         H
                                                                                         
                          q g , m , Σ 
                                  k
                                        k
                                        2
                                                k
                                                2


                                      Σ(2)  Σ(3)   2diag(xk )aaT diag(xk )
                                                                ˆˆ        H



                                                                                                  
                                                                 1
                                      m   (2)
                                                 Σ       (2)          2
                                                                           yk diag(xk )a  Σ(3)m(3)
                                                                                       ˆ
Case 1. Known transmitted sequence

        3k  g k             Gk  g k , g k 1  2  g k 1 
                                                      k

                        g k 1

                                                         
                                 q  g k , Ag k 1 , V  q g k 1 , m k 1 ,  k 1
                                                                      2        2          
                        g k 1


                           
                    q g k , m 3 , 3
                               k    k
                                                
                                                                 
                                                                      1
                                               k 1
                            VA 
                                 k
                                 3             2       A VA
                                                           H
                                                                           k 1 A1
                                                                            2


                                                                             
                                               1                                   1
                                                           k 1
                         m     k
                                 3
                                           k
                                           3        VA    2       A VA   H
                                                                                         k 1mk 1
                                                                                          2    2
Case 1. Known transmitted sequence
     Step 2. Backward passing (Kalman smoothing)
                                                         a


                  ak                   a k 1                        a k  B 1
                        6k
             Tk                Tk 1                     Tk  B 1
                          k
                           5

            3k         4k
                  gk                   g k 1                        g k  B 1
                           Gk 1                Gk  2
Case 1. Known transmitted sequence

     5k  g k   3k  g k  4  g k 
                                k


     6k  a k    Tk  g k , a k 5k  g k 
                   gk



                   gk
                                    H
                                                      
                  q yk , aT diag x k g k ,  2 q g k , m 5 , 5
                            k
                                                           k    k
                                                                    
                                       
                  Tk  g k , a k  g k  m 5
                                              k
                                                   
                   gk


                        
                 Tk g k , m 5
                             k
                                  
Case 1. Known transmitted sequence
     Step 3. Structure detection
                                                             7
                            6k                               a

                                                    6k 1
                      ak                  a k 1                          a k  B 1

                 Tk               Tk 1                       Tk  B 1



                      gk                  g k 1                          g k  B 1
                            Gk 1                  Gk  2
Case 1. Known transmitted sequence
  7  a   6n  a n  a  6n  B 1  a n  B 1  a 
               n  B 1                           n  B 1
                                                 
                                                                                          2
                                                                                           
                                                                                           
               T                    
                              a, m5  exp   yk  m5                
                                                                           T
                         k
                                      k                                k
                                                                               diag(x k )a 
                k n                              k n
                                                                                          
                                                                                           
                              yn   z n   a0  
                                                  T
                                                                   2

                 
                                                         
            exp                                            
                              yn  B 1   zT  B 1   aL 1                         diag  x k 
                                                                                          T

                                          n                            z  m
                                                                                T     k
                                                                              k     5



  Approximate MAP structure detection
                                                                           1. Minimum distance
       a  max 7  a   min y  Za
                                                                 2
       ˆ                                                                      decoding.
                    a                               a
                                                                           2. Linear MMSE
                                    zT 
                                   
                                          n
                                                                             detection.
                                 Z  
                                    z T  B 1 
                                                                           3. Matching pursuit.
                                    n          
Case 1. Known transmitted sequence
     Step 4. Repeat the processing

                                                    a


                    ak            a k 1                        a k  B 1

               Tk         Tk 1                     Tk  B 1



                    gk            g k 1                        g k  B 1
                         Gk 1             Gk  2


     Kalman filter update which accounts for “channel structure”
Conclusions
    Detect the structure from the information drawn by factor
   graph.
    Reprocess the update step of Kalman filter based on the
   channel structure detected.
    It is necessary to incorporate a priori information on channel
   structure.
       1. The channel structure might exhibit high correlation in
            time and delay.

          ak



         a k 1
In the next talk
     Simulation results.
     How to use the soft information on data x.  Turbo
    principle
     How to use a priori information on structure a.  Use
    Markov model

								
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