Capacity-Approaching Polar Codes with Long Codewords and Successive Cancellation Decoding Based on Improved Gaussian Approximation
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This paper focuses on an improved Gaussian approximation (GA) based construction of polar codes with successive cancellation (SC) decoding over an additive white Gaussian noise (AWGN) channel. It has been theoretically proven by Arikan that polar codes with low-complexity SC decoder can approach the channel capacity of an arbitrary symmetric binary-input discrete memoryless channel, provided that the code length is chosen large enough. Nevertheless, how to construct such codes over an AWGN channel with low computational effort has been an open problem. The conventional GA is a powerful and low complexity technique, but due to numerical issues, it can fail to accurately trace the evolution of the mean log likelihood ratio (LLR) value as the code length increases. In this work, an improved GA approach is proposed where, by judiciously defining the functions associated with the GA, an accurate trace of mean LLR evolution becomes feasible. With this improved GA, through theoretical analysis and simulations with code lengths up to N=2^18, we explicitly demonstrate that various code-rate polar codes with long codeword and capacity approaching behavior can be easily constructed.
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