Iterative decoding of binary block and convolutional codes

TLDR

Turbo decoding refers to iterative decoding of two‑dimensional systematic convolutional codes. The paper presents optimal and suboptimal decoders with reduced complexity. The authors develop log‑likelihood‑domain iterative decoders that use soft inputs and extrinsic outputs, with a cross‑entropy stop criterion and interleaving, applicable to both convolutional and linear binary systematic block codes. Simulations demonstrate that simple component codes achieve near‑cutoff‑rate performance at BER 10⁻⁴ with 3–6 iterations, with block codes suited to high rates and convolutional codes to rates below 2/3, and any combination of the two is viable.

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 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10/sup -4/ the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations.