An Analytical Study of the Min-Sum Approximation for Polar Codes

TL;DR

The paper analyzes the min-sum SC decoding of polar codes for binary-input symmetric channels, providing both finite-length and asymptotic insights under a labeling scheme $oldsymbol{ au}$ and its variants (fair/good).A posynomial framework is introduced to represent synthetic joint distributions, enabling a $O(N^{1.585})$-time procedure to compute exact per-bit error probabilities at finite length when using a good labeler.In the asymptotic regime, the authors prove the existence of rate thresholds $R_{ m L}(oldsymbol{ au})$ and $R_{ m U}(oldsymbol{ au})$ ensuring strong polarization for $R<R_{ m L}$ and failure for $R>R_{ m U}$, with $0<R_{ m L}igl(oldsymbol{ au}igr)\le R_{ m U}igl(oldsymbol{ au}igr)\le C$ and often $R_{ m U}<C$.The paper further strengthens thresholds by introducing a generalized tree-based construction $(oldsymbol{un G},oldsymbol{un E})$, defining $R_{ m L}(oldsymbol{un G},oldsymbol{un E})$ and $R_{ m U}(oldsymbol{un G})$, and proving monotonicity properties to ensure deeper analyses yield tighter polarization guarantees.

Abstract

The min-sum approximation is widely used in the decoding of polar codes. Although it is a numerical approximation, hardly any penalties are incurred in practice. We give a theoretical justification for this. We consider the common case of a binary-input, memoryless, and symmetric channel, decoded using successive cancellation and the min-sum approximation. Under mild assumptions, we show the following. For the finite length case, we show how to exactly calculate the error probabilities of all synthetic (bit) channels in time $O(N^{1.585})$, where $N$ is the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denoted $R_{\mathrm{L}} = R_{\mathrm{L}}(λ)$ and $R_{\mathrm{U}} =R_{\mathrm{U}}(λ)$, where $λ(\cdot)$ is the labeler of the channel outputs (essentially, a quantizer). For any $0 < β< \frac{1}{2}$ and any code rate $R < R_{\mathrm{L}}$, there exists a family of polar codes with growing lengths such that their rates are at least $R$ and their error probabilities are at most $2^{-N^β}$. That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceeding $R_{\mathrm{U}}$, the error probability approaches $1$ as the code-length increases, irrespective of which bits are frozen. We show that $0 < R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C$, where $C$ is the channel capacity. The last inequality is often strict, in which case the ramification of using the min-sum approximation is that we can no longer achieve capacity.

Figures (4)

• Figure 1: The capacity $C$ and the thresholds $R_{\mathrm{U}}$ and $R_{\mathrm{L}}$ of a BI-AWGN with 3-bit quantized output, using the labeling function $\lambda$ given in \ref{['eq: awgn labeler']}. For reference, the capacities of the corresponding non-quantized BI-AWGN and AWGN are also given.
• Figure 2: The capacity $C$ and the thresholds $R_{\mathrm{U}}$ and $R_{\mathrm{L}}$ for the $\mathrm{BSC}(p)$.
• Figure 3: A comparison between the non-approximated function $f(L_a,L_b)$ and the approximated function $\tilde{f}(L_a,L_b)$ for $L_a=1$.
• Figure 4: A full binary tree, where the nodes in $\mathcal{G}$ are red and the nodes in $\mathcal{E}$ are rectangular (leaves). The node $(d=2,j=2)$ is both in $\mathcal{G}$ and $\mathcal{E}$.

Theorems & Definitions (29)

• Theorem 1
• Definition 1: Good labeler
• Definition 2: Fair labeler
• Lemma 2: Transforms of synthetic joint distributions
• Lemma 3: Symmetry of synthetic joint distribution
• Lemma 4
• Corollary 5
• Lemma 6: Bhattacharyya-like bound
• Lemma 7: Bhattacharyya-like evolutions
• Lemma 8: Bound on posynomial minus transform