For estimating the states or outputs of a Markov process, the symbol-by-symbol MAP algorithm is optimal. However, this algorithm, even in its recursive form, poses technical difficulties because of numerical representation problems, the necessity of nonlinear 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 simplification 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 differences. 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 effects. The SOVA is, at l o p 4 , approximately 0.7 dB inferior to the (Log-)MAP, the Max-Log-MAP lying roughly in between. We also present some complexity comparisons and conclude that the three algorithms increase in complexity in the order of their optimality.

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Abstract. 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 difficulties 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 simplification 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 differences. As a practical example, we consider Turbo decoding, and we also demonstrate the practical suitability of the Log-MAP by including quantization effects. The SOVA is, at IOJ, 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.

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joined the University of Kiel, Germany, where he is currently a Professor in Electrical Engineering. His research interests are in the general area of communications theory, including digital modulation techniques, soft-output decoding, equalization and channel estimation, with applications in mobile radio. Dr. Hoeher received the Hugo-Denkmeier-Award '90. John Lodge (S'80-M'81-SM'90) received the B.Sc. and Ph.D. degrees in electrical engineering from Queen'