Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding

For estimating the states or outputs of a Markov process, the symbol-by-symbol maximum a posteriori (MAP) algorithm is optimal. However, this algorithm, even in its recursive form, poses technical diiculties because of numerical representation problems, the necessity of non-linear functions and a high number of additions and multiplications. MAP like algorithms operating in the logarithmic domain presented in the past solve the numerical problem and reduce the computational complexity, but are suboptimal especially at low SNR (a common example is the Max-Log-MAP because of its use of the max function). A further simpliication yields the soft-output Viterbi algorithm (SOVA). In this paper, we present a Log-MAP algorithm that avoids the approximations in the Max-Log-MAP algorithm and hence is equivalent to the true MAP, but without its major disadvantages. We compare the (Log-)MAP, Max-Log-MAP and SOVA from a theoretical point of view to illuminate their commonalities and diierences. As a practical example forming the basis for simulations, we consider Turbo decoding, where recursive systematic convolutional component codes are decoded with the three algorithms, and we also demonstrate the practical suitability of the Log-MAP by including quantization eeects. The SOVA is, at 10 ?4 , approximately 0.7 dB inferior to the (Log-)MAP, the Max-Log-MAP lying roughly in between. The channel capacities of the three algorithms-when employed in a Turbo decoder-are evaluated numerically. We also present some complexity comparisons and conclude that the three algorithms increase in complexity as they get closer to optimality.