# 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
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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