A Comparative Study of Polar Code Constructions for the AWGN Channel (1501.02473v1)

Abstract: We present a comparative study of the performance of various polar code constructions in an additive white Gaussian noise (AWGN) channel. A polar code construction is any algorithm that selects $K$ best among $N$ possible polar bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit error rate (BER). Optimal polar code construction is hard and therefore many suboptimal polar code constructions have been proposed at different computational complexities. Polar codes are also non-universal meaning the code changes significantly with the design-SNR. However, it is not known which construction algorithm at what design-SNR constructs the best polar codes. We first present a comprehensive survey of all the well-known polar code constructions along with their full implementations. We then propose a heuristic algorithm to find the best design-SNR for constructing best possible polar codes from a given construction algorithm. The proposed algorithm involves a search among several possible design-SNRs. We finally use our algorithm to perform a comparison of different construction algorithms using extensive simulations. We find that all polar code construction algorithms generate equally good polar codes in an AWGN channel, if the design-SNR is optimized.

• The paper systematically compares major polar code construction algorithms for AWGN channels, including methods based on Bhattacharyya bounds, Monte-Carlo simulation, density estimation, and Gaussian approximation.
• It evaluates the performance of these constructions across varying design-SNRs, highlighting their computational complexities and theoretical underpinnings.
• The study introduces a heuristic algorithm to find the optimal design-SNR, demonstrating that different constructions can achieve comparable performance when optimally tuned for the channel.

Overview of Polar Code Constructions for AWGN Channel

The paper authored by Harish Vangala, Emanuele Viterbo, and Yi Hong systematically examines various polar code constructions tailored for an additive white Gaussian noise (AWGN) channel. As polar codes represent a significant advancement in error-correcting codes due to their capacity-achieving potential, the need to scrutinize their construction under practical constraints is crucial. This paper meticulously evaluates multiple polar code construction algorithms, highlighting their performance and computational complexities, and introduces an algorithm to determine the optimal design-SNR for superior code construction.

The primary focus of this paper is the comparative performance evaluation of the polar code construction algorithms in terms of their bit error rate (BER) across different design signal-to-noise-ratios (design-SNRs). Given that polar codes are non-universal, the code construction's efficacy is tightly linked with the chosen design-SNR, necessitating an in-depth analysis to determine the optimal design-SNR for achieving the best possible code performance.

Methodological Insights

The discourse begins with a comprehensive survey of existing polar code construction methods. It categorically investigates:

1. Arikan’s Bhattacharyya Bounds-Based Construction (PCC-0): This is one of the earliest techniques which employs simple bounds on Bhattacharyya parameters to identify good channels. The paper suggests a modification—utilizing Bhattacharyya parameters corresponding to the design-SNR instead of the default value, particularly targeted at AWGN channels.
2. Monte-Carlo Simulation-Based Construction (PCC-1): Proposed by Arikan, this robust simulation approach estimates the bit-channel metrics through empirical BER calculations. Despite its effectiveness, especially at high rates, the significant computational demand, specifically $O(MN\log N)$, where $M$ denotes Monte-Carlo iterations, remains a consideration.
3. Tal and Vardy’s Density Estimation (PCC-2): Utilizing transition probability matrices (TPMs), this approach presents a comprehensive technique combining accurate estimations with quantization strategies to manage complexity. The method is tailored for channels with substantial output alphabets and integrates well with AWGN channel modeling.
4. Trifonov’s Gaussian Approximation Method (PCC-3): This technique considers the intermediate log-likelihood ratios as Gaussian variables, simplifying the construction by evaluating the Gaussian approximation of BER. The algorithm efficiently calculates the polar code with $O(N)$ complexity.

Numerical and Theoretical Implications

A salient finding from extensive simulation studies is that while each of these algorithms has distinct theoretical postulates and computational implications, all can potentially achieve comparable performance in an AWGN channel when optimally tuned. The authors propose a heuristic algorithm designed to systematically discover the best design-SNR across various constructions, thereby harmonizing the algorithms’ performance at their optimal conditions.

Practical and Future Directions

The paper's implications resonate profoundly with both theoretical advancements and practical applications of polar codes. The convergence of construction methodologies when fine-tuned suggests potential for simplified practices in applications such as 5G implementations and future wireless systems. Additionally, the insights into polar codes' non-universality underscore the necessity for adaptive coding strategies that accommodate variability in channel conditions and operational constraints.

Future research avenues could explore:

• Enhanced algorithms that can dynamically adjust design-SNR settings based on real-time channel assessments.
• Investigations into universal polar codes without incurring excessive complexity, preserving the structure and low complexity of original polar codes.
• Further studies into concatenated polar codes for increasing robustness and performance in more generalized channel environments.

In conclusion, this paper enriches the polar code landscape by providing a rigorous examination of construction algorithms while inviting further inquiries into optimizing polar code designs and applications under various channel conditions.