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
We introduce an extension of the relay channel that we call two-way relay channel. The two-way relay channel consists of two users which want to communicate to each other with the help of one relay. We consider the time-division twoway relay channel without power control, where the broadcast channels are orthogonalized in time and where the two users and the relay use the same transmission power. We describe a joint network-channel coding method for this channel model, where channel codes are used at both users and a network code is used at the relay. The channel code of one user and the network code form a distributed turbo code which we call turbo network code and which can be iteratively decoded at the other user. Moreover, we conjecture closed expressions for lower bounds for the channel capacities of the time-division relay and twoway relay channel without power control and deliver simulation results of the proposed turbo network code.
Abstract-An overview of the many practical applications of channel coding theory in the past 50 years is presented. The following application areas are included: deep space communication, satellite communication, data transmission, data storage, mobile communication, file transfer, and digital audio/video transmission. Examples, both historical and current, are given that typify the different approaches used in each application area. Although no attempt is made to be comprehensive in our coverage, the examples chosen clearly illustrate the richness, variety, and importance of error-control coding methods in modern digital applications.
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