# Differential Signalling in Free-Space Optical Communication Systems
## 1 Introduction  
The received signal in a free-space optical (FSO) communication system is subject to the deterministic and random factors associated with the atmospheric channel, such as geometric loss, particle scattering due to fog, smoke, low clouds, snow, and the influence of the atmospheric turbulence [1; 2; 3; 4; 5]. Additionally, pointing errors (PE) due to building sway and thermal expansions can further deteriorate the FSO link performance [6; 7; 8], whereas fog, smoke, rain, etc. result in a constant power loss; both turbulence and PE lead to random fluctuations of the amplitude and the phase of the received signal. In non-return-to-zero on-off keying (NRZ-OOK) intensity-modulation/direct-detection (IM/DD) based FSO systems, an optimal detection threshold level (DTL) at the receiver (Rx) should be used to distinguish the received '0' and '1' bits [3; 4; 9]. However, turbulence and PE can result in signal fading [7], which necessitates adaptive setting of the DTL [9]. Recently, a differential signalling scheme (DSS) or differential detection was proposed for on-off keying- (OOK)-based FSO links, which allows using a pre-fixed DTL under various channel conditions such as fog, smoke [9], weak turbulence [3], and PE [4]. Compared to the already proposed detection techniques [10; 11; 12; 13] summarized hereafter, DSS offers the lower complexity of implementation and also does not require the channel state information (CSI). As in [11], a maximum-likelihood sequence detection (MLSD) scheme was proposed for an NRZ-OOK FSO link, and it was shown that provided the temporal turbulence correlation \(\tau_{0}\) is known, MLSD outperforms the maximum-likelihood symbol-by-symbol detection scheme. However, given that typically \(\tau_{0}=1-10\) ms the proposed MLSD suffers from high computational burden at the Rx, thus making its implementation too complex [14]. To reduce the computational complexity of MLSD, two suboptimal schemes based on the single-step Markov chain model were derived in [12]. However, aforementioned schemes require instantaneous CSI at the Rx. The classical approach for CSI estimation is to periodically insert some symbols in the data frames, which is usually referred to as pilot-symbol assisted modulation (PSAM) [13; 15]. The obvious drawback of PSAM is the reduced system throughput due to pilot overhead [13]. A decision-feedback detection scheme was proposed in [16] allowing data detection based on the knowledge of previous decisions within an observation window of \(\tau_{0}\). The drawback of this scheme lies in the dependence on \(\tau_{0}\) and on the data pattern (i.e., bits '0' and '1' in the underlying data stream). The \"fast multi-symbol detection\" was demonstrated in [10], which works based on block-wise decisions and a fast search algorithm. However, due to the dependency of the system on the search algorithm complexity, there is a trade-off between the throughput and performance of the system. Iterative channel estimation based on the expectation maximization was proposed in [17], but the proposed approach has a relatively high computational complexity. A blind detection scheme requiring no knowledge of CSI as well as the one using a sub-optimum maximum-likelihood detection were introduced in [18; 19], respectively. However, these schemes offer rather poor performance over short observation windows. Recently a maximum-likelihood sequence Rx with no knowledge of CSI and channel distribution was proposed in [20] for different channel conditions, which is however too complex to implement. The DSS investigated in this paper allows performing signal detection without the need to precise setting of DTL in a dynamic FSO channel. Provided that the received differential signals are highly correlated, the DTL of the received differential signal is not influenced by turbulence induced fluctuation [3; 4].