## 1 Introduction  
Pulse position modulation (PPM) is particularly appealing in optical wireless communications (OWC) over lossy and turbulent channels [1; 2], including space and underwater transmissions [3; 4; 5], since it provides a trade-off between bandwidth use and receiver sensitivity. Combined with optical amplification, PPM has the potential of providing a significant link gain but in OWCs it has been studied mainly for the binary modulation case [6; 7; 8; 9; 10]. The study of \(Q\)-ary PPM in pre-amplified OWC systems has received far less attention [11], even though binary PPM performs worse than on-off keying in terms of the receiver sensitivity that is required to target a specific probability of error (PER) [6].  
The goal of this work is to assess the performance of \(Q\)-ary PPM in a pre-amplified OWC receiver, where the signal and noise statistics are governed by \(\chi^{2}\) distributions. To this end, we first analyze the binary PPM case and derive analytical results for the PER using the non-central \(F\)-distribution. Similar to [6; 10], we provide a closed-form analytical relation for the binary PPM PER, which requires a simple and finite summation of the hypergeometric function and can be computed efficiently using standard mathematical software. We then use a tight upper bound [12], which has been previously reported for Gaussian noise statistics, to approximate the \(Q\)-ary PPM PER with a combination of the modulation order \(Q\) and the binary PPM PER. By comparing the approximation with the exact results, which are obtained via numerical integration, we demonstrate that the upper bound is highly accurate in \(\chi^{2}\) statistics. Moreover, the bound accuracy actually increases at lower PERs, which are of practical importance in any real-world OWC system.  
Given the accuracy of the presented bound, we use it further to evaluate the average PER of the OWC system performance in weak, strong, and saturated turbulence. The evaluation is performed via numerical integration with and without the proposed approximation; the corresponding approximated and exact results are in good agreement with each other for \(\gamma-\gamma\) and negative-exponential fading. In addition, the combination of \(Q\)-ary PPM, optical amplification, and spatial diversity is considered to be a means to further reduce the impact of turbulence. Spatial diversity is implemented by adding multiple identical pre-amplified receivers and adding the received signals in an equal-gain combiner (EGC) architecture that provides close to optimal PER performance. Despite the increased noise from the additional amplifiers and the split of the signal power over the diversity branches, the results show that very low average PERs can be achieved by combining the three proposed methods.  
The rest of the paper is organized as follows: in Section 2 we detail the PER performance of \(Q\)-ary PPM in \(\chi^{2}\) noise, and provide analytical relations and the approximation for calculating it. In Section 3 we present results for the average PER in weak, moderate, and strong fading and compare approximated and exact results. The approximation accuracy is discussed in all three fading regimes and it is shown that the accuracy improves with the diversity order and worsens with the modulation order. Finally, Section 4 summarizes the main results of this work and concludes the paper.