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DECODING RECEIVED BLOCKS


Notation:
H  :  The IG parity check matrix of dimension M by N
 r   :   Received vector of length N
 t   :   # of 1's per column in H ( # of 1's per row in HT )
tr  :  #  of 1's per row in H (# of 1's per column in HT )

r should satisfy the conditions given by  r.HT = 0

     i.e., 
 
 

Hence, each column of  HT (each row of H )defines a Check. There are a total of M checks.
Bits of r and the checks form a Bayesian Network. Each bit of r participates in t checks and hence, every bit is the parent of t checks in the Bayesian Network.
Since each check verifies tr bits, each check is the child of tr bits in the Bayesian Network.
The decoding is done by the method of Probability Propogation or the Sum-Product Algorithm.[3] gives an excellent tutorial
on this topic.

More Notation: (This notation is the same as the notation used by the authors in [2],[3] )

N (m)º{l : Hmn=1}    --- This is the set of bits participating in check m i.e., the columns which have 1's in row m of H
                                                belong to this set. For each m, there are tr elements in this set.

 M(n)º{m: Hmn=1}   --- This is the set of checks bit n of  r participates in i.e., the rows which have 1's in column l of H
                                                belong to this set. For each n, there are t elements in this set.

N (m)\n   --- The set of (tr-1) elements participating in check m apart from bit n

qxmn       ---The probability that bit l of  r has a value x given the information obtained via the M-1 checks apart from check m

rxmn         ---Probability of check m being satisfied by bit n of r, if rn is considered fixed at the value xÎ {0,1} and the
                    other  N-1 bits have a seperable distribution given by {qxmn' : n'ÎL(m)\n}

DECODING ALGORITHM

Step 1. Initialization

Initialize q0mn and q1mn to the likelyhoods of xn,  fn0 and fn1 .
For the AWGN channel,
                         fn1 = 1/(1+exp(-2ayn/s2)
and                    fn0 =1- fn1    where,

the input to the Gaussian channel is ±a, s2 =N0/2 is the variance of the additive noise and yn is the soft output of the
Gaussian Channel.

Step 2. Horizontal Step

For each row/check m and every nÎN(m), the 2 probabilities r0mn and r1mn are computed.
For example , say we are looking at a check m where the 2nd , 3rd and 5th columns are 1(here tr=3).
i.e.,  the check is r2År3År5=0
Suppose we wanted to calculate r0m2,

                      r0m2 = Prob{ r2År3År5=0| r2=0,r3=0,r5=0} x  q0m3   x q0m5
                                          + Prob{ r2År3År5=0| r2=0,r3=0,r5=1} x  q0m3   x q1m5
                                          + Prob{ r2År3År5=0| r2=0,r3=1,r5=0} x  q1m3   x q0m5
                                 + Prob{ r2År3År5=0| r2=0,r3=1,r5=1} x  q1m3   x q1m5

                   |||ly    r0m2 can be calculated.

Therefore, generalizing the above equations we have,

The conditional probabilities in the above equations are 0 or 1 depending on whether check m is satisfied by the
corresponding values of xn and xn'.
Using the above equations to calculate the r0mn and r1mn  will require the enumeration of all possible combinations
of  xn' which may be computationally intensive. The authors in [2] suggest a couple of different methods to compute
the above probabilities and one of the effecient schemes they suggest is as follows:

compute   dqmn= q0mn-q1mn

then compute              drmn= r0mn-r1mn= Pdqmn'n'ÎL(m)\n

and then we get      r0mn=(1+drmn)/2    and r1mn=(1-drmn)/2

Step 3. Vertical Step
In this step we update the values of q0mnand q1mnusing the r's obtained in step 3.

q0mn= amn fnPr0m'n :m'ÎM(n)\m
q1mn= amn fnPr1m'n :m'ÎM(n)\m

where amn is chosen such that  q0mn +q1mn=1
The pseudoposterior probabilities are then obtained.

q0n= an fnPr0mn :mÎM(n)
q1n= an fnPr1mn :mÎM(n)

Step 4. Tentative Decoding

set rn =1 if  q1n >0.5

and rn=0 otherwise.

If  r.HT = 0  stop
Else go to step 2.

In the software implementation the algorithm iterates 1000  times.
The function decode_ldpc(rx_waveform,No,amp,h) does the decoding.
The function takes 4 parameters
                                          (i) rx_waveform : The soft output of the gaussian channel
                                          (ii) No: Twice the magnitude of PSD of the AWGN (No/2)
                                          (iii)amp: the amplitude of the bpsk waveform (amp representing the voltage for 1and -amp
                                                                                                                representing the voltage for 0)
                                          (iv)h: the parity check matrix to be used for decoding.
The function returns the decoded codeword and not the message bits.
The function can be used along with the modulation and channel transmission functions as shown below:

Function Usage Example:
  % assume the codeword c, the parity check matrix H and No are already loaded into the MATLAB memory
  tx_waveform=bpsk(c,1);  %amp= 1
  rx_waveform=awgn(tx_waveform,No);
  vhat=decode_ldpc(rx_waveform,No,1,h) %since amp=1;
 

Once the decoding is done ( the termination condition is satisfied or the maximum number of iterations are performed), the message bits need to be extracted.  At the last step of encoding the message blocks, the reordering done to get a non-
singular A was undone. If this reordering was done again, the message bits would be located at the end of the decoded codeword and can be extracted easily. The rearranged_cols array (output by the rearrange_cols( ) function) that holds the
reordering information is used to do the reordering. The extract_mesg(vhat,rearranged_cols) works in this manner.

Function Usage Example:
 % assume we have the rearranged_cols array output by the rearrange_cols( ) function already loaded into MATLAB memory
 uhat = extract_mesg(vhat,rearranged_cols);
 



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