Analyzing the turbo decoder using the Gaussian approximation

Abstract

In this paper, we introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece. Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. Index Terms-Convergence-based design, graphical codes, iterative decoding, low-density parity-check (LDPC) codes, turbo codes. I. INTRODUCTION O NE of the main reasons behind the impressive performance achieved by graphical codes such as parallel concatenated convolutional code(s) (PCCC) is the elegant iterative decoding algorithm with the exchange of soft information between successive iterations. Recently, it has been shown that this iterative decoding algorithm is an instance of Pearl belief propagation in Bayesian networks [1]. Whereas belief propagation is known to converge to the maximum a posteriori (MAP) solution for graphs without loops, relatively little progress has been achieved to date in understanding the theoretical behavior of the algorithm on graphs with loops, especially as a suboptimal decoder for codes having such graphical representation [1]. The rediscovery of Gallager's low-density parity-check (LDPC) codes [2] and Wiberg's work on the graphical repre

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