On the Rate of Channel Polarization (0807.3806v3)

Abstract: It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e \le 2^{-N^\beta}$ for any $\beta<\frac12$ when the block-length $N$ is large enough.

• The paper demonstrates that polar codes achieve exponential error reduction, with block decoding error bounded by 2^{-N^β} for any β < 0.5.
• It employs martingale and supermartingale processes (I_n and Z_n) in a channel transformation framework to rigorously analyze polarization behavior.
• The study’s findings significantly enhance understanding of polar code performance and pave the way for further research in coding theory.

On the Rate of Channel Polarization: An Overview

In the paper "On the Rate of Channel Polarization," Arıkan and Telatar explore the intricacies of channel polarization, a method leveraged to construct polar codes that achieve capacity over binary-input symmetric discrete memoryless channels (B-DMCs). Polar codes are of significant importance as they offer a structured, provably capacity-achieving coding mechanism without relying on trial-and-error schemes. This paper advances previous research by elucidating the behavior of the probability of block decoding error for polar codes under successive cancellation decoding.

Fundamental Contributions

The authors build upon a channel transform initially posited by Arıkan, examining the transformation $W \mapsto (W^-,W^+)$ which conserves symmetric capacity, leading to channel polarization. They define two processes: $I_n$, a bounded martingale converging to a 0-1 valued random variable, and $Z_n$, a bounded supermartingale with a similar convergence property. These constructs are central in proving that the transformation results in channels polarizing to either perfect or useless states as the block length grows.

Numerical Results and Claims

A pivotal result demonstrated is the upper bound of the block decoding error probability. For any binary-input DMC $W$ with symmetric capacity $I(W)$, and for any rate $R < I(W)$, the probability of block decoding error $P_e$ satisfies $P_e \le 2^{-N^\beta}$ for any $\beta < \frac{1}{2}$ when the block length $N$ is sufficiently large. This result not only strengthens previous findings, asserting that $P_e(N,R) = o(N^{-\frac{1}{4}})$, but also ensures exponential error probability reduction, illustrating more significant reliability gains as the block length increases.

Impact and Theoretical Implications

The theoretical implications laid out by this proof are profound in the field of information theory, offering a refined understanding of how channel polarization facilitates near-perfect data transmission reliability in practical scenarios. While the result is not finely sensitive to the specific rate $R$, it provides a robust framework for understanding the asymptotic behavior of polar codes as a function of the block length and coding rate.

Future Directions and Open Problems

The authors hint toward an array of open problems in the domain, such as deriving more explicit functions $E(n,R)$ for cumulative probabilities in channel polarization, and extending the theory to channels with non-binary inputs or more general channel transformations. Such advancements could pave the way for novel coding schemes and enhance existing communication systems' capacity and reliability.

In summary, the paper provides a rigorous analysis of polar code performance, presenting enhancements to the understanding of channel polarization rate. It sets a foundational stage for both theoretical explorations and practical applications in coding theory, inviting further research into optimizing and applying polar coding beyond binary-input channels.