Iterative decoding of binary block and convolutional codes

Abstract

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

Comparative study of turbo decoding techniques: an overview

“…1. Equations similar to (16) can be derived for the other component decoders which are used in iterative turbo decoding.…”

“…It can be shown [1] that, for a systematic code such as a RSC code, the output from the MAP decoder, given by (9), can be re-written as (16) where (17) Here, is the a-priori LLR given by (1), and is called the channel reliability measure and is given by (18) is the received version of the transmitted systematic bit and…”

“…Hence, it is called the extrinsic LLR for the bit . Equation (16) shows that the extrinsic information from a MAP decoder can be obtained by subtracting the a-priori information and the received systematic channel input from the soft output of the decoder. This is the reason for the subtraction paths shown in Fig.…”

“…Further advances in understanding the excellent preformance of the codes are due, for example, to Benedetto and Montorsi [13], [15], Perez et al [14]. Hagenauer et al [16], [17] extend the concept to use concatenated block codes. Jung and Naßhan [36], [34] characterized the coded performance under the constraints of short transmission frame length, which is characteristic of speech systems.…”

Error Rate Estimation Based on Soft Output Decoding and its Application to Turbo Coding

“…One of the first papers on stopping criteria is [11], which is based on cross entropy (CE) between the output distributions of the different decoder modules. Based on the same CE concept, Shao et al [12] introduced two simple methods called sign-change-ratio (SCR) criterion and hard-decision aided (HDA) criterion.…”