A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain

Abstract

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.

Comparative study of turbo decoding techniques: an overview

“…However, much work has been done to reduce this complexity, and the Log-MAP algorithm [7], which will be described in Section V, gives the same performance as the MAP algorithm, but at a significantly reduced complexity and without the numerical problems described above. In the next section we will first describe the principles behind the iterative decoding of turbo codes, and how the MAP algorithm described in this section can be used in such a scheme, before detailing the Log-MAP algorithm.…”

“…In the forthcoming section, we first describe two related algorithms, the Max-Log-MAP [25], [26] and the Log-MAP [7], which are derived from the MAP algorithm, and then another, referred to as the SOVA [8], [9], [19], derived from the Viterbi algorithm.…”

“…Suffice to say at this stage that an iterative structure using a modified version of the classic minimum bit error rate (BER) Maximum Aposteriory Algorithm (MAP) due to Bahl et al [6] was invoked, in order to decode the codes. Since then, a large body of work has been carried out in the area, aiming, for example, to reduce the decoder complexity, as suggested by Robertson et al [7], Berrou et al [9], as well as by Battail [8]. Le Goff et al [10], Wachsmann and Huber [11], as well as Robertson and Worz [12] suggested to use the codes in conjunction with bandwidth efficient modulation schemes.…”

“…Because of this approximation its performance is sub-optimal compared to that of the MAP algorithm. However, Robertson et al [7] in 1995 proposed the Log-MAP algorithm, which corrected the approximation used in the Max-Log-MAP algorithm and hence gave a performance identical to that of the MAP algorithm, but at a fraction of its complexity. These two algorithms are described in this section.…”

Improving UMTS LTE Performance by UEP in High Order Modulation

“…The demodulated complex symbols are fed to a soft demapper (SDM) which delivers reliability information in form of log-likelihood ratios (LLR) L(x), L(p I ), and L(p II ) on the systematic (data) bits x and the parity bits of the two constituent encoders p I , p II , respectively. The LLRs are then passed on to a parallel turbo decoding structure consisting of two soft input soft output (SISO) channel decoders (CD) using the MaxLogMAP algorithm [4] for soft channel decoding. For the initial decoding step after soft demapping the LLRs are fed into CD I without using any a priori information.…”