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Polarization Decomposition and Its Applications

Tianfu Qi, Jun Wang

TL;DR

This work tackles the challenge of computing symmetric capacities for all polarized subchannels of arbitrary binary-input memoryless channels (BMCs). It introduces the polarization factor (PF), a conditional-entropy-based metric that captures input-relations among codeword bits, and shows how the usual subchannel mutual information $I(W_L^{(i)})$ can be decomposed into PF components. The authors derive explicit PF expressions across block lengths and subchannel indices, map each PF to an $n$-ary tree, and develop a pruning algorithm with complexity $\mathcal{O}(L^{\log 3} \log L)$, aided by closed-form channel-output entropy formulas and a polynomial-approximation of $h_Y(d=A)$ for Gaussian-like noise. The framework yields both theoretical insights (e.g., partial-order verification among subchannels) and practical benefits (efficient MI-based polar-code construction, direct rate-loss estimation, and polarization visualization) that extend to arbitrary BMCs. The PF-based decomposition thus provides a unifying, scalable approach to polar-code design and analysis with potential extensions to memory channels and more general kernels.

Abstract

The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input configurations, we demonstrate that the symmetric capacities of the polarized subchannels can be uniformly expressed as functions of the PF. The explicit formulation of the PF as a function of the block length and subchannel index is derived. Furthermore, an efficient algorithm is proposed for the computation of the PF. Notably, we establish a one-to-one correspondence between each PF and an $n$-ary tree. Leveraging this tree structure, we develop a pruning method to determine the conditional entropy associated with different input relationships. The proposed polarization framework offers both theoretical insights and practical advantages, including intuitive visualization of polarization behavior and efficient polar code construction. To the best of our knowledge, this is the first approach that enables the efficient computation of symmetric capacities for all subchannels in arbitrary BMCs.

Polarization Decomposition and Its Applications

TL;DR

This work tackles the challenge of computing symmetric capacities for all polarized subchannels of arbitrary binary-input memoryless channels (BMCs). It introduces the polarization factor (PF), a conditional-entropy-based metric that captures input-relations among codeword bits, and shows how the usual subchannel mutual information can be decomposed into PF components. The authors derive explicit PF expressions across block lengths and subchannel indices, map each PF to an -ary tree, and develop a pruning algorithm with complexity , aided by closed-form channel-output entropy formulas and a polynomial-approximation of for Gaussian-like noise. The framework yields both theoretical insights (e.g., partial-order verification among subchannels) and practical benefits (efficient MI-based polar-code construction, direct rate-loss estimation, and polarization visualization) that extend to arbitrary BMCs. The PF-based decomposition thus provides a unifying, scalable approach to polar-code design and analysis with potential extensions to memory channels and more general kernels.

Abstract

The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input configurations, we demonstrate that the symmetric capacities of the polarized subchannels can be uniformly expressed as functions of the PF. The explicit formulation of the PF as a function of the block length and subchannel index is derived. Furthermore, an efficient algorithm is proposed for the computation of the PF. Notably, we establish a one-to-one correspondence between each PF and an -ary tree. Leveraging this tree structure, we develop a pruning method to determine the conditional entropy associated with different input relationships. The proposed polarization framework offers both theoretical insights and practical advantages, including intuitive visualization of polarization behavior and efficient polar code construction. To the best of our knowledge, this is the first approach that enables the efficient computation of symmetric capacities for all subchannels in arbitrary BMCs.
Paper Structure (20 sections, 7 theorems, 91 equations, 10 figures, 2 tables, 2 algorithms)

This paper contains 20 sections, 7 theorems, 91 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Polar code
  4. Basic definitions
  5. Polarized subchannel decomposition
  6. Transformation of $I(W_L^{(i)})$
  7. Explicit expressions of PF
  8. $n$-ary tree and pruning algorithm
  9. Calculation of $h_S(p)$ and $h_P(p)$
  10. Constructing trees and pruning
  11. Complexity analysis
  12. Applications
  13. Theoretical insights
  14. Polarization visualization
  15. polar code construction
  16. ...and 5 more sections

Key Result

Theorem 1

Let the block length be $L$ and assume the subchannel index satisfies $i\bmod 4 = 0$. Define a constant sequence $\{Q_0,\cdots,Q_M\}$ and an index sequence $\{q_0,\cdots,q_M\}$, where $q_j\in\{1,\cdots,Q_j\},j=0,\cdots,M$. Let with $t_0=i$ and $j=1,\cdots,M-1$. The term $Q_j$ is defined as and $Q_0=C_L(i)$. The integer $M$ is the largest index such that $Q_{j},j=0,\cdots,M$ are all larger than 1

Figures (10)

  • Figure 1: The comparison of SC entropy and PC entropy for various values of $p$ is presented. The considered channel noise is white Gaussian noise (WGN). The channel output entropy $h(Y)$ and noise entropy $h(N)$ are also provided for comparison. It can be observed that both $h_S(p)$ and $h_P(p)$ are equal to $h(N)$ for both small and sufficiently large SNR. Moreover, the relationship $h(N) = h_P(+\infty) \leq \cdots \leq h_P(1) \leq h_S(1) \leq \cdots \leq h_S(+\infty) = h(Y)$ holds for all SNR values. The $h_S(p)$ is quite close to $h(Y)$ in the low SNR regime, but it decreases to $h(N)$ as the SNR continues to increase. Furthermore, the decay rate is much faster for larger values of $p$ compared to smaller ones.
  • Figure 2: We set $L = 32$ and $i = 12$, and the figure presents the submatrix of $\tilde{B}_{32}(12)$ due to space limitations. The missing parts of the matrix are all zeros. Additionally, the gray section represents bits that have been considered as prior information, and thus their influence can be omitted. An example of representations within a layer and across different layers is provided. Specifically, we aim to represent $\tilde{B}_{32}(12,[:,11])$ using columns that contain $\tilde{B}_{32}(12,[:,12])$, and to represent $\tilde{B}_{32}(12,[:,1])$ using columns that contain $\tilde{B}_{32}(12,[:,9])$. Elements within the blue and green boxes correspond to representations within a single layer, while elements across different layers are marked by yellow and purple boxes.
  • Figure 3: We set $L = 128$ and $i = 44$, and the figure shows the submatrix of $\tilde{B}_{128}(44)$. An example of representations both within a layer and across different layers is provided. For each case, we consider two representations for $\tilde{B}_{128}(44,[:,41])$. The purple dashed boxes highlight the common elements shared by the two representations. Note that we only present three 'padding-removing' steps due to space limitations.
  • Figure 4: The fitting results for the channel output entropy, SC entropy, and PC entropy under WGN are presented. We set that $\rho = 7$. In the legend, 'Sim' denotes the entropy values obtained via numerical integration, while 'Ana' refers to those derived from analytical expressions.
  • Figure 5: The merging process for a two-layer tree. The dotted line is utilized to denote the merging operation. Vertices in a dashed box represent the original elements during a merging operation.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Example 2
  • Example 3
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • ...and 5 more