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Low Density Parity Check Codes

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					Low Density Parity Check Codes
LDPC ( Low Density Parity Check ) codes are a class of linear bock code. The term
“Low Density” refers to the characteristic of the parity check matrix which contains only
few ‘1’s in comparison to ‘0’s. LDPC codes are arguably the best error correction codes
in existence at present. LDPC codes were first introduced by R. Galager in his PhD
thesis in 1960 and soon forgotten due to introduction of Reed-Solomon codes and the
implementation issues with limited technological knowhow at that time. The LDPC
codes were rediscovered in mid 90s by R. Neal and D. Mackay at the Cambridge
University.


We can define N bit long LDPC code in terms of M number of parity check equations
and describing those parity check equations with a M x N parity check matrix H.

  Where,
  M – Number of parity check equations
  N – Number of bits in the codeword
    Low Density Parity Check Codes
  Consider the 6 bit long codeword in the form            c  c1 , c2 , c3 , c4 , c5 , c6 
   which satisfies 3 parity check equations as shown below.

     c1  c2  c5  0                                    We can now define 3x6 parity check matrix as,
     c1  c4  c6  0                                               c1  c2  c5
  c1  c2  c3  c6  0


The density of ‘1’s in LDPC code parity check matrix is very low              1 1 0 0 1 0
                                                                        H  1 0 0 1 0 1
                                                                                         
Row weight wc - number of ‘1’s in a row Number of
Number of symbols taking part in a parity check                               1 1 1 0 0 1
                                                                                         

                
Column weight wc - number of ‘1’s in a column
                                                                   wr and wc  changes, therefore this is an irregular
                                                                parity check matrix
Number of times a symbol taking part in parity checks


If the parity check matrix has uniform row weight and uniform column weight (same number of ‘1’ in a
column and same number of ‘1’ in a row) we call that a regular parity check matrix.
 Low Density Parity Check Codes
     1 1 0 0 1 0                     The H M  N parity check matrix defines a rate R  K N, N, K  code
 H  1 0 0 1 0 1
                
                                       where K  N  M

     1 1 1 0 0 1
                
                                       Codeword is said to be valid if it satisfies the syndrome calculation
                                                      z  c.H T  0
We can generate the codeword in by multiplying message       m     with generator matrix        G
                                     c  m.G

We can obtain the generator matrix    G from parity check matrix H        by,
                                                                                                  1 0 0 1 0 1
                                                                         
  1.) Arranging the parity check matrix in systematic form H sys  I M P M K          H sys    0 1 0 1 1 1
                                                                                                             
  using row and column operations                                                                 0 0 1 0 1 1
                                                                                                             
                                                                                      1 1 0 1 0 0
  2.) Rearranging the systematic parity check matrix G  P       T
                                                                  K M   IK      G  0 1 1 0 1 0 
                                                                                                  
                                                                                      1 1 1 0 0 1
                                                                                                  
  3.) We can verify our results as     G.H T  0
  Low Density Parity Check Codes
       1 1 0 0 1 0
   H  1 0 0 1 0 1
                                      Tanner graph is a graphical representation of parity check matrix
                                    specifying parity check equations.
       1 1 1 0 0 1
                                    Tanner graph consists of N number of variable nodes and M number
                                      of check nodes

In Tanner graph m th check node is connected to nth variable node if and only if nth element in
m th row hmn in parity check matrix H is a ‘1’.


                                 z1                 z2                 z3




           c1             c2               c3              c4               c5          c6

            The marked path z2 → c1 → z3 → c6 → z2 is an example for short cycle of 4

          The number of steps needed to return to the original position is known as the girth of
          the code
 Low Density Parity Check Codes
 1. Suppose we have codeword as follows: c  c1 , c2 , c3 , c4 , c5 , c6 
                                   c
   where each ci is either ‘0’ or ‘1’ and codeword now has three parity-check
    equations.
               c1  c2  c5  0
               c1  c4  c6  0
            c1  c2  c3  c6  0

a.) Determine the parity check matrix H by using the above equation
b.) Show the systematic form of H by applying Gauss Jordan elimination
c.) Determine Generator matrix G from H and prove G * HT = 0
d.) Find out the dimension of the H, G
e.) State whether the matrix is regular or irregular
   Low Density Parity Check Codes
   c1  c2  c5  0
                                            1 1 0 0 1 0
   c1  c4  c6  0                                                      6 columns and 3 rows
                                        H  1 0 0 1 0 1
                                                       
c1  c2  c3  c6  0
                                            1 1 1 0 0 1
                                                       
   Convert H into systematic form H sys using basic row and operations (try to avoid column operations)

                    1 1 0 0 1 0         1 1 0 0 1 0
                H  1 0 0 1 0 1 R2  R3 1 1 1 0 0 1
                                                   
                    1 1 1 0 0 1
                                        1 0 0 1 0 1
                                                     


                   1 1 0 0 1 0           1 1 0 0 1 0
                   1 1 1 0 0 1 R  R  R 0 1 1 1 0 0
                H             2    3   2           
                   1 0 0 1 0 1
                                         1 0 0 1 0 1
                                                      


                    1 1 0 0 1 0                   1 1 0 0 1 0
                H  0 1 1 1 0 0 R3  R3  R2  R1 0 1 1 1 0 0
                                                              
                    1 0 0 1 0 1
                                                  0 0 1 0 1 1 
                                                                
Low Density Parity Check Codes
       1 1 0 0 1 0                   1 0 0 1 0 1
   H  0 1 1 1 0 0 R1  R1  R2  R3 0 1 1 1 0 0
                                                 
       0 0 1 0 1 1 
                                     0 0 1 0 1 1 
                                                   

       1 0 0 1 0 1              1 0 0 1 0 1
   H  0 1 1 1 0 0 R2  R2  R3 0 1 0 1 1 1
                                           
       0 0 1 0 1 1 
                                0 0 1 0 1 1
                                             

                  1 0 0 1 0 1
          H sys  0 1 0 1 1 1                  
                                          H sys  I M P M K          
                                                                   G  PK M I K
                                                                        T
                                                                                   
                             
                  0 0 1 0 1 1
                             
                                                              1       1 1
                                                              1       0 1
       1 1 0 1 0 0                            1 1 0 1 0 0             0 0 0 
   G  0 1 1 0 1 0 
                                                              0       0 1 
                                        G.H T  0 1 1 0 1 0.                      
                                                                            0 0 0 
                                                          0        1 0
                                                1 1 1 0 0 1                0 0 0 
       1 1 1 0 0 1
                   
                                                            1       0 0         
                                                                         
                                                              0
                                                                      1 1
                                                                          
 Dimensions of H = 3x6
                            Since the column weight and row weight changes, this
 Dimensions of G = 3x6      is an irregular parity check matrix
  Low Density Parity Check Codes
2. The parity check matrix H of LDPC code is given below:

               1 0       1 0 0 1 1 0 1 0 0 1
               0 1       1 1 1 0 1 0 0 0 0 1
                                            
               0 1       01 0 1 0 1 1 1 0 0
            H                              
               1 1       10 0 0 0 1 1 0 1 0
               0 0       0 1 1 0 1 0 0 1 1 1
                                            
               1 0
                         0 0 1 1 0 1 0 1 1 0
                                             

  a.) Determine the degree of rows and column
  b.) State whether the LDPC code is regular or irregular
  c.) Determine the rate of the LDPC code
  d.) Draw the tanner graph representation of this LDPC code.
  e.) What would be the code rate if we make rows equals to column
  f.) Write down the parity check equation of the LDPC code
Low Density Parity Check Codes
    1 0       1 0 0 1 1 0 1 0 0 1
    0 1       1 1 1 0 1 0 0 0 0 1                  Dimensions of the H matrix = 6x12
                                 
    0 1       01 0 1 0 1 1 1 0 0                   M  6, N  12, K  N  M  6
 H                              
    1 1       10 0 0 0 1 1 0 1 0
    0 0       0 1 1 0 1 0 0 1 1 1                  Row weight    wr  6
                                 
    1 0
              0 0 1 1 0 1 0 1 1 0
                                                    Column weight    wc  3

Since the parity check matrix has a uniform row weight and uniform column weight, this
is a regular LDPC parity check matrix

Since the parity check matrix has a uniform row weight and uniform column weight, this
is a regular LDPC parity check matrix

 Code rate R  K N  612  1 2 ,   R 1
                                          2

  If we make the number of rows equal to columns code rate will be equal to ‘0’
Low Density Parity Check Codes
    1 0     1 0 0 1       1 0 1      0 10
    0 1     1 1 1 0       1 0    0 0 0 1
                                         
    0 1     0 1 0 1       0 1    1 1 0 0 
H                                       
    1 1     1 0 0 0       0 1    1 0 1 0 
    0 0     0 1    1 0    1 0    0 1 1 1
                                         
    1 0
            0 0 1 1       0 1    0 1 1 0 
                                          

  The parity check equations of the matrix are

 z1  c1  c3  c6  c7  c9  c12
 z2  c2  c3  c4  c5  c7  c12
 z3  c2  c4  c6  c8  c9  c10
 z4  c1  c2  c3  c8  c9  c11
 z5  c4  c5  c7  c10  c11  c12
 z6  c1  c5  c6  c8  c10  c11
     Low Density Parity Check Codes
   3. Consider parity check matrix H generated in question 1,


   a.) Determine message bits length K, parity bits length M, codeword length N
   b.) Use the generator matrix G obtained in question 1 to generate all possible codewords c.

         1 1 0 0 1 0         1 1 0 1 0 0
     H  1 0 0 1 0 1
                         G  0 1 1 0 1 0 
                                                     Dimensions of the H matrix = 3x6
                                                   M  3,        N  6,   K  N M 3
         1 1 1 0 0 1
                             1 1 1 0 0 1
                                           
All possible message set               All possible codeword set
                  msg_1 000                                    
                                                         codeword_1  msg_1.G  000000
                  msg_2  001                                   
                                                        codeword_2 msg_2.G  111001
                                                                               
                  msg_3 010                                    
                                                        codeword_3 msg_3.G  011010
                                                                               
                   msg_4  011                          codeword_4 msg_4.G  100011
               m                                   c                     
                  msg_5 100                                    
                                                        codeword_5 msg_5.G  110100
                                                                               
                  msg_6 101                                   
                                                          codeword_6 msg_6.G   001101
                 msg_7 110                                     
                                                        codeword_7 msg_7.G  101110
                                                                               
                  msg_8 111
                                                    codeword_8 msg_8.G   010111
                                                                   
                                                                                     
    Low Density Parity Check Codes
 Question 4.
 a.) What is the difference between regular and irregular LDPC codes?
 b.) What is the importance of cycles in parity check matrix?
 c.) Identify the cycles of 4 in the following tanner graph.

If the parity check matrix has uniform row weight and uniform column weight (same number of ‘1’ in a
column and same number of ‘1’ in a row) we call that a regular parity check matrix.

The decoding algorithm assumes that the LDPC code is cycle free ( large girth sizes ). The short cycles
in the code ( cycles with a girth of 4 and cycles of girth of 6 ) weakens the code. Therefore the codes
must be carefully constructed to be free of short cycles.
                                              C heck Nodes




                                              Variable Nodes

				
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