Piecewise linear LLR approximation for non-binary modulations over Gaussian channels with unknown noise variance

Channel log-likelihood ratio (LLR) calculation on many communication channels is a challenging task especially when non-binary modulations are used. This is because LLRs are usually complicated functions of the channel output and their calculation also requires knowledge of the channel parameters. In this paper, we consider the problem of finding good approximate LLRs for the additive white Gaussian noise channel under non-binary modulations when the noise variance is unknown at the receiver. To this end, we propose piecewise linear LLR approximating functions and we use the LLR accuracy measure of to optimize the parameters. First, we assume that the noise variance is known at the receiver and later we generalize the method to the case of unknown noise variance. It is shown in the latter case that the optimum piecewise linear approximate LLRs depend on the code used on the channel. We observe that the optimized piecewise linear LLRs perform extremely close to exact LLRs.