Bit-wise combining for decode-and-forward relays

We consider the problem of combining the signals received from a decode-and-forward relay with those received from the source. As an alternative to the high-complexity maximum-likelihood combining strategy, we propose a bit-wise combining strategy in which the received signals are first converted to log-likelihood values before being combined. We show that, subject to the bit-wise constraint, the optimal combining strategy for binary alphabets - minimizing the probability of error - applies a sigmoidal nonlinearity to the relay log-likelihood values before adding to the source log-likelihood values. Unlike the maximal-ratio combining strategy, which has no diversity for the Rayleigh-fading decode-and-forward relay channel, the proposed strategy exhibits the maximum diversity order of two.