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

Abstract

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.

Turbo Equalization: An Overview

“…The scaling of by has been neglected because of the normalization applied in the forward and backward recursion. Practical implementations of the BCJR algorithm often store the logarithms of the quantities in Table I [12], [81]. In this case, it is possible to use the terms directly in without the need for exponentiation.…”

“…However, when implementing the BCJR algorithm with the logarithms of , and , it is much easier to use in [12], [81]. Consequently, instead of the extrinsic soft information , it is also more practical to consider the extrinsic LLR (see second equation at the bottom of the page).…”

Joint complex diversity coding and channel coding over space, time and frequency

“…Several SISO algorithms can be applied to compute the ECC decoder output. For the purpose of this study, we consider the use of the Log-MAP algorithm [15].…”

Comparison of coded orthogonal frequency division multiplexing and multicarrier code division multiple access systems for power line communications

“…The decoding procedure is more complex; two RSC decoders are used with soft-input and softoutput decoding algorithms. There are three main algorithms on which the decoding procedure is based, namely, Log-MAP (Maximum A Posteriori) [36], Max-Log-MAP, and SOVA (Soft Output Viterbi Algorithm) [37]. The Turbo decoder operates in an iterative process, with each RSC decoder producing a soft output-estimation of the information data and an extrinsic output which provides information about the surrounding bits.…”