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IEEE International Black Sea Conference on Communications and Networking

Generalized Schematic Successive Cancellation Decoding for Polar Codes

Karim El‐Abbasy, Ramy Taki Eldin, Salwa Elramly, Bassant Abdelhamid

Published May 24, 2021• 0 citations
Publisher

Abstract

Recently, generalized polar codes were introduced to provide more code lengths and rates without the need for shortening or puncturing. However, as the size of the kernel matrix increases, computing the Log-Likelihood Ratio (LLR) functions in successive cancellation decoder for these codes becomes mathematically more complicated. In this paper, a new simplified general method is proposed to calculate these LLR functions based on the schematic diagram of the selected kernel matrix for sizes higher than 2. Our proposed method fragments the main block into some basic components whose formulas have been previously proven by Arikan for $2 \times 2$ kernel matrix. The validity of the resulting LLR functions is tested on different kernel matrices and proven to be similar to those in literature.

DOI
10.1109/blackseacom52164.2021.9527800

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Recently, generalized polar codes were introduced to provide more code lengths and rates without the need for shortening or puncturing. However, as the size of the kernel matrix increases, computing the Log-Likelihood Ratio (LLR) functions in successive cancellation decoder for these codes becomes mathematically more complicated. In this paper, a new simplified general method is proposed to calculate these LLR functions based on the schematic diagram of the selected kernel matrix for sizes higher than 2. Our proposed method fragments the main block into some basic components whose formulas have been previously proven by Arikan for 2×22 \times 22×2 kernel matrix. The validity of the resulting LLR functions is tested on different kernel matrices and proven to be similar to those in literature.
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We present an LLR-based implementation of the successive cancellation list (SCL) decoder. To this end, we associate each decoding path with a metric which (i) is a monotone function of the path's likelihood and (ii) can be computed efficiently from the channel LLRs. The LLR-based formulation leads to a more efficient hardware implementation of the decoder compared to the known log-likelihood based implementation. Synthesis results for an SCL decoder with block-length of N = 1024 and list sizes of L = 2 and L = 4 confirm that the LLR-based decoder has considerable area and operating frequency advantages in the orders of 50% and 30%, respectively.
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A reduced complexity algorithm is presented for computing the log-likelihood ratios (LLRs) arising in the successive cancellation (SC) decoder for polar codes with large kernels of arbitrary dimension. The proposed algorithm exploits a recursive trellis representation of the codes generated by submatrices of the polarization kernel and enables polar codes based on large kernels to be decoded with lower complexity compared to the codes based on the Arıkan kernel with the same performance. Extensions to the case of non-binary kernels are presented.
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