# Temporal Encoding to Reject Background Signals
## 2 Results  
### Time Correlation Encoding Scheme  
The transmission scheme presented here, inspired by time-correlated single photon counting (TCSPC) techniques often used for fluorescence lifetime imaging [20], involves the use of a single SPAD to receive time correlated signals at a single photon level. Analysis of the SPAD response to incoming light over an interval \([-t_{1},t_{1}]\) shows that the correlation count density function \(g(\tau)\mathrm{d}t^{\prime}\) of recording two subsequent SPAD counts with temporal separation in the interval \([\tau,\tau+\mathrm{d}t^{\prime}]\) is given by:  
\[g(\tau)=\int_{-t_{1}}^{t_{1}}\mathrm{d}t\;f(t)f(t+\tau). \tag{1}\]  
Here, \(f(t)\) is the temporal probability distribution of received SPAD pulses, which is determined by the optical signal from the transmitter. Full analysis is given in the Supplementary Materials of Reference [21]. If a suitable optical source transmits pulses with a time separation of \(T\), \(g(\tau)\) will show a peak at \(\tau=T\), as the probability of observing SPAD pulses separated by \(T\) is increased. Figure 1: (**a**) Application scenario for inter-satellite communications: the proposed signal encodes data in trains of time-correlated optical pulses and is compatible with low power consumption LED emitters. It is resilient against constant background such as sunlight and also insensitive to most AC background such as conventional wireless optical signals. (**b**) Schematic of the transmission scheme used. LED output on transmission of “0” and “1” (**top**), SPAD response to the LED signal (**middle**) and the calculated correlation histograms for each data interval (**bottom**). Equation (1) is the autocorrelation of \(f(t)\), so it is expected that peaks in \(g(\tau)\) would have a width of \(2t_{pulse}\), where \(t_{pulse}\) is the width of the optical pulse. After detection of a photon, the output of a SPAD has a \"dead time\" (\(\tau_{d}\)) in which it is insensitive to further photons, typically 10 s of ns in length [6; 12]. It is important that \(T>\tau_{d}\), as otherwise the SPAD would not recover from the first pulse in time to see the second. This restriction can be lifted by using a SPAD array [22; 23], however here we consider the use of only a single SPAD. The presence and/or temporal position of peaks in \(g(\tau)\) directly depends on the sequence of optical pulses from the transmitter, and therefore can be used as a means of transmitting data. In reality, the SPAD output is not a continuous probability distribution, but a series of discrete photon detection events. These events can occur due to the optical pulses from the transmitter, background photons, or dark counts. SPADs typically have dark count rates (DCRs) ranging from 100 s of Hz to several kHz [2; 6; 7; 12], however, the SPAD used in the following experimental sections has an active cooling system, reducing the DCR to 25 Hz. The SPAD output signal will be sampled over a time period, into \(N_{s}\) time bins \(t_{i}\), \(i=1,\ldots,N_{s}\), chosen to be smaller than \(\tau_{d}\), so that bin contains a number of counts \(f_{i}\in[0,1]\). The correlation time will also be discretised into \(\tau_{j},j=1,\ldots,N_{\tau}\). For a single pair of pulses a correlation either is or is not detected, so the optical signal must be repeated many times to distinguish correlation counts from noise. Instead of a single pulse pair, the pulses are continuously repeated at a rate specified by the temporal separation \(R_{pulse}=1/T\). With correlation time bin size chosen as an integer multiple of sampling bin size, \(\tau_{bin}=kt_{bin}\), we can define start and stop indices for correlating across \(i\) as \(n_{start}=\tau_{1}/t_{bin}\) and \(n_{stop}=n_{start}+kN_{\tau}-1\).