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

Abstract. Using analog, non-linear and highly parallel networks, we attempt to perform decoding of block and convolutional codes, equalization of certain frequency-selective channels, decoding of multi-level coded modulation and reconstruction of coded PCM signals. This is in contrast to common practice where these tasks are performed by sequentially operating processors. Our advantage is that we operate fully on soft values for input and output, similar to what is done in 'turbo' decoding. However, we do not have explicit iterations because the networks float freely in continuous time. The decoder has almost no latency in time because we are only restricted by the time constants from the parasitic RC values of integrated circuits. Simulation results for several simple examples are shown which, in some cases, achieve the performance of a conventional MAP detector. For more complicated codes we indicate promising solutions wirh more complex analog networks based on the simple ones. Furthermore, we discuss the principles of the analog VLSI implementation of these networks.

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