The implementation of the belief propagation algorithm is based on the decoding algorithm presented by Gallager [2].

For transmitted LDPC-encoded codewordc=c₀,c₁, …,c_n-1, the input to the LDPC decoder is the log-likelihood ratio (LLR) value $$L(c_i)=\log \left(\frac{\mathrm{Pr}(c_i=0|\text{channel output for}{c}_i)}{\mathrm{Pr}(c_i=1|\text{channel output for}{c}_i)}\right)$$.

In each iteration, the key components of the algorithm are updated based on these equations:

$$L(r_{ji})=2\text{atanh}\left({\displaystyle \underset{i^{\prime }\in V_j\backslash i}{\prod }\tanh \left(\frac{1}{2}L(q_{i^{\prime }j})\right)}\right)$$,

$$L(q_{ij})=L(c_i)+{\displaystyle \underset{j^{\prime }\in C_i\backslash j}{\sum }L(r_{j^{\prime }i})}$$, initialized as $$L(q_{ij})=L(c_i)$$before the first iteration, and

$$L(Q_i)=L(c_i)+{\displaystyle \underset{j^{\prime }\in C_i}{\sum }L(r_{j^{\prime }i})}$$.

At the end of each iteration,L(Q_i) contains the updated estimate of the LLR value for transmitted bitc_i. The valueL(Q_i) is the soft-decision output forc_i. IfL(Q_i) < 0, the hard-decision output forc_iis 1. Otherwise, the hard-decision output forc_iis 0.

If decoding is configured to stop when all of the parity checks are satisfied, the algorithm verifies the parity-check equation (H c'= 0) at the end of each iteration. When all of the parity checks are satisfied, or if the maximum number of iterations is reached, decoding stops.

Index sets $$C_i\backslash j$$and $$V_j\backslash i$$are based on the parity-check matrix (PCM). Index setsC_iandV_jcorrespond to all nonzero elements in columniand rowjof the PCM, respectively.

This figure shows the computation of these index sets in a given PCM fori= 5 andj= 3.

To avoid infinite numbers in the algorithm equations,atanh(1)andatanh(–1)are set to 19.07 and –19.07, respectively. Due to finite precision, MATLAB returns 1 fortanh(19.07)and –1 fortanh(-19.07).