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.

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.

In this paper we will show that the soft output of the soft-outputViterbi-decoder (SOVA) suffers from two distortions: Firstly, for bad channels the reliability information of the decoder output is too optimistic. The output can be considered as being multiplied by a factor, that depends on the current bit-error-rate (BER). To become closer to the true log-likelihood ratio the output has to be normalized. Secondly, the soft-output of the SOVA -when used in Turbo decoding -is effected by a correlation between the so called extrinsic and intrinsic information. Since the extrinsic information is fed forward to be the a-priori information in the next decoding stage and is treated as being uncorrelated with the systematic-information, a correcting term has to be introduced in order to compensate this correlation. We will show how these distortions can be corrected and we will apply the correcting measures on the SOVA that is used as component decoder of a Turbo-code scheme. Finally, we will provide simulation results that illustrate the gain due to normalizing the extrinsic information and exploiting the correlation in the soft output. Aa priori values for ' Decoder bib extrinsic information intrinsic information I I L(q, -De--inter leaver ConvolutionalDecoder -(MAPISOVA) .Figure 1: Horizontal component decoder (on the left) of a Turbo-decoding scheme.

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