# Temporal Encoding to Reject Background Signals
## 2 Results
### Signal-to-Noise Ratio  
To set an operation threshold for the system, a figure of merit similar to the classical signal-to-noise ratio (SNR) must be defined. In this method, it is the distinguishability of the correlation peak in the \(g(\tau)\) function that indicates the robustness of the classical information recovery to noise. Conventional SNR can be defined as \(SNR=10\log_{10}\frac{N_{signal}}{N_{noise}}\), where \(N_{signal}\) is average signal correlation counts and \(N_{noise}\) is average noise correlation counts. However, as the number of pulse repetitions increases, the correlation counting interval increases, causing both \(N_{signal}\) and \(N_{noise}\) to increase at linear rates. This results in a constant SNR, which does not reflect the observed increase in distinguishability of signal correlations with increasing pulse repetitions. Instead, it is more useful to consider the statistical distribution of correlation counts for signal and noise. Photon counting experiments were undertaken using the experimental setup described above, with a received power at the detector of 38 pW, corresponding to a detector count rate of \(1.07\times 10^{7}\) Hz, in a dark lab environment with an average background count rate of 619 Hz. Note that this count rate contains both dark counts and counts from the small amount of ambient light. The delay correlations of detected photons were binned with a resolution of 10 ns, with the transmitted pulse delay set at 40 ns. Figure 3a,b shows average histograms of received photon correlations for 5 and 100 pulse repetitions, respectively. Figure 3c shows the histogram for 100 pulse repetitions under high background conditions, displaying the correlation histogram due to background noise alone, and signal with noise. The background count rate for this measurement was \(10^{7}\) Hz. In Figure 3a-c, the signal is defined as the number of correlations in the 40 ns delay time bin, and the noise correlation count is taken from the 60 ns delay bin. Correlation counts follow a Poissonian distribution, as they are discrete independent events arising directly from shot noise limited photon counts. Figure 3d,e shows the measured Poissonian distributions for signal with noise and noise alone  
Figure 2: (**a**) Measured pulse pair from a micro-LED source. (**b**) Schematic of the experimental setup. correlation counts at 5 and 100 repetitions of the pulses, respectively, taken from 1500 independent measurements of each case. Figure 3f shows the distributions for background noise, signal alone (identical to Figure 3e) and combined signal and noise. At five repetitions, the probability distributions for signal and noise are strongly overlapped. Thus, a correlation count peak due to signal transmission is difficult to distinguish from a correlation count peak due to random background and dark counts. At 100 repetitions, the overlap of signal and noise distributions is significantly reduced, making distinction much easier. A histogram threshold equates to a point along the x-axis of the distribution plots. Evidently, a threshold of 2 would result in many erroneous detections at five repetitions, whereas, at 100 repetitions, the majority of signal correlation peaks would be correctly identified, and noise correlation peaks rejected. Under high background noise conditions, the number of correlations from noise is increased, so the threshold must increase to distinguish between the correlations due to noise from those due to the signal, as both will be present on transmission of \"1\". Distinguishability is therefore described by Equation (3), the overlap of the Poisson distributions for: (i) the total signal and noise contributions, \(P_{T}(k)\); and (ii) the noise alone, \(P_{n}(k)\). \(P_{T}(k)\) is related to the signal count distribution, \(P_{s}(k)\) and noise count distribution, \(P_{n}(k)\) via Equation (4).