  On a fading channel with no channel state information at the receiver,
calculating true log-likelihood ratios (LLR) is complicated. Existing work
assume that the power of the additive noise is known and use the expected value
of the fading gain in a linear function of the channel output to find
approximate LLRs. In this work, we first assume that the power of the additive
noise is known and we find the optimum linear approximation of LLRs in the
sense of maximum achievable transmission rate on the channel. The maximum
achievable rate under this linear LLR calculation is almost equal to the
maximum achievable rate under true LLR calculation. We also observe that this
method appears to be the optimum in the sense of bit error rate performance
too. These results are then extended to the case that the noise power is
unknown at the receiver and a performance almost identical to the case that the
noise power is perfectly known is obtained.
