Iterative turbo decoder analysis based on density evolution

We track the density of extrinsic information in iterative turbo decoders b y actual density evolution, and also approximate it by symmetric Gaussian d ensity functions, The approximate model is verified by experimental measure ments. We view the evolution of these density functions through an iterativ e decoder as a nonlinear dynamical system with feedback, Iterative decoding of turbo codes and of serially concatenated codes is analyzed by examing w hether a sign-to-noise ratio (SNR) for the extrinsic information keeps grow ing with iterations, We define a "noise figure" for the iterative decoder, such that tile turbo decoder will converge to the correct codeword if the n oise figure is bounded by a number below zero dB, By decomposing the code's noise figure into individual curves of output SNR versus input SNR corresp onding to the individual constituent codes, we gain many new insights into the performance of the iterative decoder for different constituents. Many m ysteries of turbo codes tire explained based on this analysis, For example, we show why certain codes converge better with iterative decoding than mor e powerful codes which are only suitable for maximum likelihood decoding, T he roles of systematic bits and of recursive convolutional codes as constit uents of turbo codes are crystallized. The analysis is generalized to seria l concatenations of mixtures of complementary outer and inner constituent c odes. Design examples are given to optimize mixture codes to achieve low it erative decoding thresholds on the signal-to-noise ratio of the channel.