Linear LLR approximation for iterative decoding on wireless channels

On a fading channel with no channel state information at the receiver, true log-likelihood ratios (LLR) are complicated functions of the channel output. It is assumed in the literature that the power of the additive noise is known and the expected value of the fading gain is used in a linear function of the channel output to find approximate LLRs. This approach, however, is not optimal in the sense of bit error rate performance. In this paper, we introduce a measure of accuracy for the approximate LLRs based on their probability density function and we show that this measure provides a very convenient tool for finding good approximate LLRs. Assuming that the power of the additive noise is known, and using the proposed measure, we find a linear LLR approximation whose performance is extremely close to that of the true LLR calculation on an uncorrelated Rayleigh fading channel. These results are then extended to the case that the noise power is also unknown and a performance almost identical to the previous case is obtained.