Analyzing the turbo decoder using the Gaussian approximation

In this paper, we introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined ov er graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic inf ormation from constituent maximum a posteriori (MAP) decoders is well appro ximated by Gaussian random variables when the inputs to the decoders are Ga ussian, The independent Gaussian model implies the existence of an iterativ e decoder threshold that statistically characterizes the convergence of the iterative decoder, Specifically, the iterative decoder converges to zero p robability of error as the number of iterations increases if and only if th e channel E-b/N-0 exceeds the threshold. Despite the idealization of the mo del and the simplicity of the analysis technique, the predicted threshold v alues are in excellent agreement with the waterfall regions observed experi mentally in the literature when the codeword lengths are large. Examples ar e given for parallel concatenated convolutional codes, serially concatenate d convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece. Convergence-based design of asymmetri c parallel concatenated convolutional codes (PCCC) is also discussed.