Upper Bound on the Bit Error Probability of Systematic Binary Linear Codes via Their Weight Spectra

Jia Liu,Mingyu Zhang,Rongjun Chen,Chaoyong Wang,Yufei Wang,Xiaofeng An

doi:10.1155/2020/1469090

Abstract

In this paper, upper bound on the probability of maximum a posteriori (MAP) decoding error for systematic binary linear codes over additive white Gaussian noise (AWGN) channels is proposed. The proposed bound on the bit error probability is derived with the framework of Gallager’s first bounding technique (GFBT), where the Gallager region is defined to be an irregular high-dimensional geometry by using a list decoding algorithm. The proposed bound on the bit error probability requires only the knowledge of weight spectra, which is helpful when the input-output weight enumerating function (IOWEF) is not available. Numerical results show that the proposed bound on the bit error probability matches well with the maximum-likelihood (ML) decoding simulation approach especially in the high signal-to-noise ratio (SNR) region, which is better than the recently proposed Ma bound.

Highlights

Upper bounds on the maximum a posteriori (MAP) decoding error probability, as a key technique for evaluating the performance of the binary linear codes over additive white Gaussian noise (AWGN) channels, bring a profound impact on the reliable transmission of the next-generation mobile communication systems since they can be used to predict the performance without resorting to computer simulations and guide the design of coding [1]
Different from most of the existing bounds, we derive a tighter upper bound on the bit error probability of systematic binary linear codes via their weight spectra
In 1999, Divsalar derived a simple upper bound [2] on the bit error probability based on Gallager’s first bounding technique (GFBT), where the region R is chosen to be an n-dimensional sphere centered at a scaled transmitted signal vector

Summary

Introduction

Upper bounds on the maximum a posteriori (MAP) decoding error probability, as a key technique for evaluating the performance of the binary linear codes over additive white Gaussian noise (AWGN) channels, bring a profound impact on the reliable transmission of the next-generation mobile communication systems since they can be used to predict the performance without resorting to computer simulations and guide the design of coding [1]. Different from most of the existing bounds, we derive a tighter upper bound on the bit error probability of systematic binary linear codes via their weight spectra. (2) In Section 3, in a detailed way, we rederive the recently proposed bound on the bit error probability by Liu [5], in which the union bound is firstly truncated and amended for the systematic linear codes over AWGN channels. The proposed upper bound on the bit error probability is derived in a much more detailed way by considering more information of the Gallager region R and the truncated IOWEF of the code. (3) In Section 4, numerical examples are given to show that the proposed bound is helpful to evaluate the performance of the systematic binary linear codes which can predict the performance of the code in the high-SNR region, avoiding the time-consuming computer simulations.

Preliminaries

Upper Bound on the Bit Error Probability Based on GFBT

Numerical Examples

Conclusions