[Paper Review] Performance of Polar Codes on wireless communications Channel

Peng Shi, Shengmei Zhao+1 more|arXiv (Cornell University)|Apr 27, 2012

Error Correcting Code Techniques11 references4 citations

TL;DR

This paper proposes a generalized Bhattacharyya parameter for continuous wireless channels—specifically additive white Gaussian noise (AWGN) and Rayleigh fading channels—enabling polar code design and performance evaluation on practical wireless systems. Simulation results show polar codes outperform LDPC codes in image and speech transmission over Rayleigh fading channels, with a 3.8 dB PSNR gain at 8192 code length and 6 dB SNR, and a 2.4 dB SD improvement at 4096 code length, indicating superior error correction and scalability with code length.

ABSTRACT

We discuss the performance of polar codes, the capacity-achieving channel codes, on wireless communication channel in this paper. By generalizing the definition of Bhattacharyya Parameter in discrete memoryless channel, we present the special expression of the parameter for Gaussian and Rayleigh fading the two continuous channels, including the recursive formulas and the initial values. We analyze the applications of polar codes with the defined parameter over Rayleigh fading channel by transmitting image and speech. By comparing with low density parity-check codes(LDPC) at the same cases, our simulation results show that polar codes have better performance than that of LDPC codes. Polar codes will be good candidate for wireless communication channel.

Motivation & Objective

• To extend the Bhattacharyya parameter from discrete to continuous channels like AWGN and Rayleigh fading for polar code design.
• To evaluate the performance of polar codes in real-world wireless communication scenarios, particularly Rayleigh fading channels.
• To compare polar codes with LDPC codes in image and speech transmission over Rayleigh fading channels under identical conditions.
• To demonstrate that polar codes achieve better error correction performance and faster convergence with increasing code length compared to LDPC codes.

Proposed method

• Generalize the Bhattacharyya parameter for continuous channels using probability density integration instead of summation: $ Z(W) = \int \sqrt{W(y|0)W(y|1)} \, dy $.
• Derive recursive formulas for the Bhattacharyya parameter in polar code construction: $ Z(W_{2N}^{(2i-1)}) \leq 2Z(W_N^{(i)}) - Z(W_N^{(i)})^2 $ and $ Z(W_{2N}^{(2i)}) = Z(W_N^{(i)})^2 $.
• Define initial values and recursive relations for AWGN and Rayleigh channels using numerical integration and iterative approximation.
• Construct polar codes using the derived Bhattacharyya parameters and apply them to image and speech transmission over Rayleigh fading channels.
• Use PSNR and spectral distortion (SD) as objective metrics to evaluate image and speech reconstruction quality.
• Perform Monte Carlo simulations with 1000 repetitions to ensure statistical reliability of performance comparisons.

Experimental results

Research questions

• RQ1 How can the Bhattacharyya parameter be generalized for continuous wireless channels such as AWGN and Rayleigh fading?
• RQ2 What recursive formulas and initial values are required to accurately model the Bhattacharyya parameter for polar code construction on continuous channels?
• RQ3 How does polar code performance compare to LDPC codes in image transmission over Rayleigh fading channels?
• RQ4 How does polar code performance compare to LDPC codes in speech transmission over Rayleigh fading channels?
• RQ5 Does the performance gap between polar codes and LDPC codes increase with code length in wireless fading environments?

Key findings

• At 6 dB SNR and code length 2048, polar codes achieve a 1 dB higher PSNR than LDPC codes in image reconstruction over Rayleigh fading channels.
• At code length 8192 and 6 dB SNR, the PSNR gain of polar codes over LDPC codes increases to 3.8 dB, indicating superior performance with longer codes.
• For speech transmission at 5 dB SNR and code length 2048, polar codes achieve a spectral distortion (SD) of 5.4 dB, compared to 7.2 dB for LDPC codes, showing a 1.8 dB improvement.
• When code length increases to 4096 at 5 dB SNR, the SD performance gap between polar codes and LDPC codes widens from 1.8 dB to 2.4 dB, demonstrating faster convergence of polar codes.
• Polar codes exhibit a more significant performance improvement with increasing code length than LDPC codes in both image and speech transmission over Rayleigh fading channels.
• The results confirm that polar codes are a superior candidate for wireless communication channels, especially in fading environments where high reliability and scalability are required.