# Maximum Likelihood Detection in a Four-Dimensional Stokes-Space Receiver

## TL;DR

This paper derives a maximum likelihood detection rule for a four-dimensional optical receiver, compares three detection algorithms, and evaluates their performance and achievable rates through simulations and approximations.

## Contribution

It introduces simplified detection algorithms with negligible performance loss and provides high-SNR approximations and achievable rate analyses for four-dimensional optical signals.

## Key findings

- Successive detection reduces complexity by a factor of 8 with minimal performance loss.
- High-SNR approximation is accurate even at low SNRs.
- Achievable rates of 10 bits per channel use at 25 dB SNR for specific constellations.

## Abstract

The maximum likelihood detection rule for a four-dimensional direct-detection optical front-end is derived. The four dimensions are two intensities and two differential phases. Three different signal processing algorithms, composed of symbol-by-symbol, sequence and successive detection, are discussed. To remedy dealing with special functions in the detection rules, an approximation for high signal-to-noise ratios (SNRs) is provided. Simulation results show that, despite the simpler structure of the successive algorithm, the resulting performance loss, in comparison with the other two algorithms, is negligible. For example, for an 8-ring/8-ary phase constellation, the complexity of detection reduces by a factor of 8, while the performance, in terms of the symbol error rate, degrades by 0.5 dB. It is shown that the high-SNR approximation is very accurate, even at low SNRs. The achievable rates for different constellations are computed and compared by the Monte Carlo method. For example, for a 4-ring/8-ary phase constellation, the achievable rate is 10 bits per channel use at an SNR of 25 dB, while by using an 8-ring/8-ary phase constellation and an error correcting code of rate 5/6, this rate is achieved at an SNR of 20 dB.