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On the Arikan Transformations of Binary-Input Discrete Memoryless Channels

Yadong Jiao, Xiaoyan Cheng, Yuansheng Tang, Ming Xu

TL;DR

A method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric is developed and a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is derived.

Abstract

The polar codes introduced by Arikan in 2009 achieve the capacity of binary-input discrete memoryless channels (BIDMCs) with low complexity encoding and decoding. Identifying the unreliable synthetic channels, generated by Arikan transformation during the construction of these polar codes, is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no efficient and practical approach to evaluate their reliability in general. To tackle this problem, by converting the generation of synthetic channels in polar code construction into algebraic operations, in this paper we develop a method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.

On the Arikan Transformations of Binary-Input Discrete Memoryless Channels

TL;DR

A method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric is developed and a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is derived.

Abstract

The polar codes introduced by Arikan in 2009 achieve the capacity of binary-input discrete memoryless channels (BIDMCs) with low complexity encoding and decoding. Identifying the unreliable synthetic channels, generated by Arikan transformation during the construction of these polar codes, is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no efficient and practical approach to evaluate their reliability in general. To tackle this problem, by converting the generation of synthetic channels in polar code construction into algebraic operations, in this paper we develop a method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.
Paper Structure (13 sections, 17 theorems, 88 equations, 5 figures)

This paper contains 13 sections, 17 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Binary-Input Discrete Memoryless Channels
  3. Likelihood Ratio Profile of BIDMC
  4. Random Switching of BIDMCs
  5. Arikan Transformations of BIDMCs
  6. Definition and LRPs of Arikan Transformations
  7. Arikan Transformations of Symmetric BIDMCs
  8. Synthetic Channels in Polar Codes
  9. Construction of Polar Codes
  10. Arikan Transformations $\Delta_{m}(W)$ and $\nabla_{m}(W)$
  11. Arikan Transformations $A_{\alpha}(\mathrm{E}(q))$ and $A_{\alpha}(\mathrm{B}(\varepsilon))$
  12. Number of BSCs in Arikan Transformations
  13. Conclusions

Key Result

Theorem 1

For any BIDMCs $W:x\in\mathcal{X}\mapsto y\in\mathcal{Y}$ and $W':x\in\mathcal{X}\mapsto y'\in\mathcal{Y}'$, we have $W\preccurlyeq W'\preccurlyeq W$ if and only if $W\cong W'$, i.e., $P_{W}(\varepsilon)$ equals $P_{W'}(\varepsilon)$ for any $\varepsilon\in[0,1]$.

Figures (5)

  • Figure 1: Transition probabilities of $\mathrm{B}_{a,b}$.
  • Figure 2: Degradation $W'\preccurlyeq W$ with intermediate channel $Q$.
  • Figure 3: An RSC of BIDMCs $\{W_j\}_{j\in[n]}$. For each transmission, the sub-channel $W_j$ is chosen with probability $q_j$ over which the data is transmitted.
  • Figure 4: A symmetric BIDMC which is equivalent to $\mathrm{B}(p)$.
  • Figure 5: Synthetic channels in polar code of order $k=4$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 15 more