Table 1 BP detection algorithm

Step 1: Initialize the posterior probability values

\(v_{ji}(s)=1/\sqrt {M}\)

Step 2: Iterative message passing along edges

a) The messages passed from the i ^th FN to the j ^th VN

\(a_{ij}(s)= \frac {1}{\sigma _{ij}\sqrt {2\pi }}\exp {(\frac {-(y_{i}-u_{ij}-h_{ij}s)^{2}}{2\sigma _{ij}^{2}})}\)

b)The message passed from the j ^th VN to the i ^th FN

\(v_{ji}(s)=\prod _{l=1,l\neq i}^{N} a_{lj}(s)\)

where u _ij and σ _ij denote the mean and variance of the interference,

respectively.

Step 3 LLR output of the j ^th VN after the iteration

\(LLR_{j}=\log (P_{x_{j}}(s)/P_{x_{j}}(s_{0}))\)

where \(P_{x_{j}}(s)\) denotes the probability of the symbol s, s ₀ means the

symbol corresponding to 0.