# Asynchronous Physical-layer Network Coding: Symbol Misalignment Estimation and Its Effect on Decoding

## TL;DR

This paper proposes an optimal symbol-misalignment estimator and decoding schemes for asynchronous physical-layer network coding using double baud-rate sampling, significantly improving estimation accuracy and decoding performance in noisy and fading channels.

## Contribution

It introduces a novel double baud-rate estimator and optimal decoders for APNC, enhancing misalignment estimation accuracy and decoding robustness compared to baud-rate methods.

## Key findings

- Double baud-rate estimator outperforms baud-rate estimator with up to 8 dB MSE gain.
- Estimation-and-decoding scheme with double baud-rate sampling improves SER and PER performance.
- Simulation results demonstrate substantial performance gains in AWGN and Rayleigh fading channels.

## Abstract

In asynchronous physical-layer network coding (APNC) systems, the symbols from multiple transmitters to a common receiver may be misaligned. The knowledge of the amount of symbol misalignment, hence its estimation, is important to PNC decoding. This paper addresses the problem of symbol-misalignment estimation and the problem of optimal PNC decoding given the misalignment estimate, assuming the APNC system uses the root-raised-cosine pulse to carry signals (RRC-APNC). First, we put forth an optimal symbol-misalignment estimator that makes use of double baud-rate samples. Then, we devise optimal decoders for RRC-APNC in the presence of inaccurate symbol-misalignment estimates. In particular, we present a new whitening transformation to whiten the noise of the double baud-rate samples. Finally, we investigate the decoding performance of various estimation-and-decoding schemes for RRC-APNC. Extensive simulations show that: (i) Our double baud-rate estimator yields substantially more accurate symbol-misalignment estimates than the baud-rate estimator does. The mean-square-error (MSE) gains are up to 8 dB. (ii) An overall estimation-and-decoding scheme in which both estimation and decoding are based on double baud-rate samples yields much better performance than other schemes. Compared with a scheme in which both estimation and decoding are based on baud-rate samples), the double baud-rate sampling scheme yields 4.5 dB gains on symbol error rate (SER) performance in an AWGN channel, and 2 dB gains on packet error rate (PER) performance in a Rayleigh fading channel.