Iterative decoding of two-dimensional systematic convolutional codes has been termed "turbo" (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 213. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of lop4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple blockkonvolutional component codes, interleaver sizes less than 1000 and for three to six iterations. Index Terms-Concatenated codes, product codes, iterative decoding, "soft-idsoft-out" decoder, "turbo" (de)coding. I. INTRODUCTION INCE the early days of information and coding theory the S goal has always been to come close to the Shannon limit performance with a tolerable complexity. The results achieved so far show that it is relatively easy to operate at signalto-noise ratios of &,/NO above the value determined by the channel cutoff rate. For a rate 1/2 code and soft decisions on a binary input additive white Gaussian noise (AWGN) channel the cutoff rate bound is at 2.5 dB, as opposed to the capacity limit which for rate 1/2 is at 0.2 dB. It is generally held that between those two values of ,?&/NO the task becomes very complex. Previously known methods of breaking this barrier were a) sequential decoding with the drawback of time andor storage overflow and b) concatenated coding using Viterbi and Reed-Solomon decoders which achieve 1.6 dB at the cost of a large interleaver and feedback between two decoders [l]. Recently, interest has focused on iterative decoding of product or concatenated codes using "soft-inlsoft-out" decoders Manuscript

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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Orthogonal frequency division multiplexing (0FDM)-CDMA is an interesting multiple access technique for future mobile radio systems. Especially in the down-link OFDM-CDMA enables low complex mobile receivers since OF DM can prevent intersymbol interference (ISI) and with that, the complexity of a RAKE receiver in a multipath channel. In this paper an OFDM-CDMA system combined with convolutional and Turbo channel coding is investigated for the down-link. We derive the optimal soft value -the so called log-likelihood ratio -for the Viterbi decoding algorithm, applied in the OFDM-CDMA systcm. Based on this derivation we present the performance of an OFDM-CDMA system with channel coding. The investigations are carried out for conventional detection with minimum mean square error (MMSE) equalization and for joint detection with maximum likelihood sequence estimation (MLSE). The various combinations between detection techniques and decoding schemes enable a comparison between achievable system performance an'd necessary system complexity.

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