Gaussian approximation (GA) is widely used to construct polar codes. However when the code length is relatively long, the subchannel selection inaccuracy due to the calculation error of conventional approximate GA (AGA), which uses two-segment approximation function, results in a catastrophic performance loss. In this paper, new principles to design GA approximation functions are proposed. First, we introduce the concepts of polarization violation set (PVS) and polarization reversal set (PRS) to explain the essential reason that conventional AGA can not work well. In the AGA process, the subchannels belonging to these two sets lead to the reliability sorting error among the subsequent subchannels, which violates Arikan's fundamental polarization principle. Moreover, we propose a new metric, named cumulative-logarithmic error (CLE), to quantitatively evaluate the remainder approximation error of AGA in logarithm. We derive the upper bound of CLE to simplify its calculation. Guided by PVS, PRS and CLE bound, we further propose new rules to design multi-segment GA approximation functions, which obviously improve the calculation accuracy of AGA so as to guarantee the excellent performance of polar codes. Numerical results show that the new proposed GA approximation functions are critical to construct high-performance polar codes with long code lengths.
['Jincheng Dai', 'Kai Niu', 'Zhongwei Si', 'Jiaru Lin']
Evaluation and Optimization of Gaussian Approximation for Polar Codes