Performance Analysis of 3-D Turbo Codes

Abstract

In this work, we consider the minimum distance properties and convergence thresholds of 3-D turbo codes (3D-TCs), recently introduced by Berrou <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> Here, we consider binary 3D-TCs while the original work of Berrou <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> considered double-binary codes. In the first part of the paper, the minimum distance properties are analyzed from an ensemble perspective, both in the finite-length regime and in the asymptotic case of large block lengths. In particular, we analyze the asymptotic weight distribution of 3D-TCs and show numerically that their typical minimum distance <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d_{\min}$</tex></formula> may, depending on the specific parameters, asymptotically grow linearly with the block length, i.e., the 3D-TC ensemble is asymptotically good for some parameters. In the second part of the paper, we derive some useful upper bounds on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d_{\min}$</tex> </formula> when using quadratic permutation polynomial (QPP) interleavers with a quadratic inverse. Furthermore, we give examples of interleaver lengths where an upper bound appears to be tight. The best codes (in terms of estimated <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d_{\min}$</tex> </formula> ) obtained by randomly searching for good pairs of QPPs for use in the 3D-TC are compared to a probabilistic lower bound on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d_{\min}$</tex> </formula> when selecting codes from the 3D-TC ensemble uniformly at random. This comparison shows that the use of designed QPP interleavers can improve the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$d_{\min}$</tex></formula> significantly. For instance, we have found a (6144,2040) 3D-TC with an estimated <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d_{\min}$</tex></formula> of 147, while the probabilistic lower bound is 69. Higher rates are obtained by puncturing nonsystematic bits, and optimized periodic puncturing patterns for rates <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1/2,$</tex></formula> <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$2/3$</tex></formula> , and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$4/5$</tex> </formula> are found by computer search. Finally, we give iterative decoding thresholds, computed from an extrinsic information transfer chart analysis, and present simulation results on the additive white Gaussian noise channel to compare the error rate performance to that of conventional turbo codes.