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Capacity-Achieving BBT Polar Codes with Interleaver-Assisted BP Decoding

Xinyuanmeng Yao, Xiao Ma

Abstract

In this paper, we introduce a binary balanced tree (BBT) channel transformation that extends Arıkan's channel transformation to arbitrary block lengths. We prove that the proposed transformation induces channel polarization, thereby establishing that BBT polar codes achieve the capacity of binary-input memoryless symmetric (BMS) channels. To characterize the finite-length performance of BBT polar codes, we further develop an efficient method for estimating the weight spectrum by exploiting the hierarchical tree structure, and derive analytical upper and lower bounds on the frame error rate (FER) under maximum-likelihood (ML) decoding. For practical low-latency implementations, we propose interleaved BBT (IBBT) polar codes together with a belief-propagation (BP) decoding algorithm. Specifically, based on the normal-graph representation of BBT polar codes, interleavers are introduced between adjacent layers to modify the message-passing schedule. In addition, we propose to perform BP decoding on an IBBT sub-normal graph and replace partial BP processing modules with a posteriori probability (APP) calculation modules, thereby reducing the number of message-passing steps required per iteration. Numerical results demonstrate that the proposed interleaving strategy improves decoding convergence, while the sub-normal-graph-based BP decoding algorithm significantly reduces decoding latency while maintaining comparable error-rate performance.

Capacity-Achieving BBT Polar Codes with Interleaver-Assisted BP Decoding

Abstract

In this paper, we introduce a binary balanced tree (BBT) channel transformation that extends Arıkan's channel transformation to arbitrary block lengths. We prove that the proposed transformation induces channel polarization, thereby establishing that BBT polar codes achieve the capacity of binary-input memoryless symmetric (BMS) channels. To characterize the finite-length performance of BBT polar codes, we further develop an efficient method for estimating the weight spectrum by exploiting the hierarchical tree structure, and derive analytical upper and lower bounds on the frame error rate (FER) under maximum-likelihood (ML) decoding. For practical low-latency implementations, we propose interleaved BBT (IBBT) polar codes together with a belief-propagation (BP) decoding algorithm. Specifically, based on the normal-graph representation of BBT polar codes, interleavers are introduced between adjacent layers to modify the message-passing schedule. In addition, we propose to perform BP decoding on an IBBT sub-normal graph and replace partial BP processing modules with a posteriori probability (APP) calculation modules, thereby reducing the number of message-passing steps required per iteration. Numerical results demonstrate that the proposed interleaving strategy improves decoding convergence, while the sub-normal-graph-based BP decoding algorithm significantly reduces decoding latency while maintaining comparable error-rate performance.
Paper Structure (31 sections, 8 theorems, 56 equations, 13 figures, 2 algorithms)

This paper contains 31 sections, 8 theorems, 56 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. BBT Structure
  4. BBT Channel Transformation
  5. Tree-Graph Representation
  6. Normal-Graph Representation
  7. BBT Polar Codes with SC Decoding
  8. Encoding
  9. BBT Polar Codes
  10. SC Decoding
  11. BBT Channel Polarization
  12. Theoretical Results
  13. Numerical Results
  14. Weight Spectrum and Performance Bounds
  15. WEF Estimation for BBT Polar Codes
  16. ...and 16 more sections

Key Result

Lemma 1

Let $W_0$ and $W_1$ be two independent BMS channels, and let $W_0 \ast W_1$ and $W_0 \circ W_1$ denote the corresponding "upper" and "lower" synthesized channels under the polar transformation. Then, In other words, the polar transformation preserves the symmetric capacity.

Figures (13)

  • Figure 1: The parent-child relationship in a BBT.
  • Figure 2: The BBT structure for $N = 6$.
  • Figure 3: Tree-graph representation of the BBT channel transformation for $N=6$.
  • Figure 4: Transformation from the tree graph to the normal graph for the BBT channel transformation with $N=6$.
  • Figure 5: The BBT polar encoding for $N = 6$ and $K=2$, where the blue leaf nodes are frozen and red leaf nodes are active.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Remark 1
  • Lemma 1: Alsan2014PolarizationProof
  • Definition 1
  • Lemma 2: Alsan2014PolarizationProof
  • Definition 2
  • Lemma 3: Alsan2014PolarizationProof
  • Lemma 4
  • proof
  • Theorem 1: BBT Channel Polarization
  • proof
  • ...and 10 more