Analysis of the Asymptotic Performance of Turbo Codes

Abstract

Battail [1989] shows that an appropriate criterion for the design of long block codes is the closeness of the normalized weight distribution to Gaussian. A subsequent work by Biglieri and Volski [1994] shows that iterated product of single parity check codes satisfies this criterion. Motivated by these works, in the current article, we study the performance of turbo codes for large block lengths, N → ∞. We show that for a parallel concatenated code that consists of two component codes, for N → ∞, the normalized weight of the systematic sequence ŵ1 = w1 √ N , and the parity check sequences ŵ2 = w2 √ N and ŵ3 = w3 √ N become jointly Gaussian for the typical values of ŵi, i = 1, 2, 3, where the typical values of weight are defined as lim N→∞ wi N 6= 0, 1 for i = 1, 2, 3. To optimize the turbo code performance in the waterfall region, which is dominated by high-weight codewords, it is desirable to reduce the correlation coefficients between ŵi and ŵj , ρij , i 6= j = 1, 2, 3. We show that: (i) ρij > 0, i, j = 1, 2, 3, (ii) ρ12, ρ13 → 0 as N → ∞, and (iii) ρ23 → 0 as N → ∞ for “almost” any random interleaver. This indicates that for N → ∞, the optimization of the interleaver has a diminishing effect on the distribution of high-weight error events, and consequently, on the error performance in the waterfall region. We show that for the typical weights, the Gaussian weight distribution approaches the average spectrum defined by Poltyrev [1994]. We also apply the tangential sphere bound (TSB) on Gaussian distribution and show that the This work has been presented in part in the Canadian Workshop on Information Theory (CWIT’99) [1] and in part in the Conference on Information Sciences and Systems (CISS’02) [2] and is reported in part in [3]. Submitted to IEEE Transactions on Information Theory on March 2006. Coding & Signal Transmission Lab, Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, N2L 3G1

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Cite this paper

@inproceedings{Baligh2006AnalysisOT, title={Analysis of the Asymptotic Performance of Turbo Codes}, author={Mohammadhadi Baligh}, year={2006} }