## 2 Theoretical Background
The accumulated error over these bits is called periodic or phase jitter and quantifies phase measurement accuracy. In general, total jitter (TJ) present in the signal is the convolution of random jitter (RJ) and deterministic jitter (DJ) [40]. The RJ component is inherently present due to shot and thermal noise and cannot be ignored. The DJ can be removed using software-based algorithms. The probability density function of RJ follows the Gaussian distribution given by Equation (7), where \(\sigma\) denotes the standard deviation of normal Gaussian distribution, \(\tau\) is the jitter. Random jitter is characterized by its root-mean-square value (\(\sigma\)) or standard deviation. The DJ is introduced by ambient interfering sinusoidal signal [40]. Time deviations depending on the used data pattern (e.g., PRBS) are regarded as one of the types of DJ results in duty cycle distortion arisingfrom the rising and falling edge timing differences or single-ended system ground shifts, and inter-symbol interference (ISI), arising from limitations of component and system bandwidth. DJ is measured in terms of peak-to-peak values since it has been defined to have upper and lower bounds [41; 42]. \[\text{RJ}(\tau)=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\bigg{(}-\frac{\tau^{2}}{2 \sigma^{2}}\bigg{)} \tag{7}\]  
The TJ is mathematically represented in Equation (8):  
\[\text{TJ}(\tau)=\text{RJ}(\tau)*\frac{\text{A}_{\text{L}}\delta(\tau-\mu_{ \text{L}})+\text{A}_{\text{R}}\delta(\tau-\mu_{\text{R}})}{\text{A}_{\text{L}} +\text{A}_{\text{R}}} \tag{8}\]  
\(\text{A}_{\text{L}}\) and \(\text{A}_{\text{R}}\) designate the amplitudes of the left and right DJ components, and \(\mu_{\text{L}}\) and \(\mu_{\text{R}}\) denote the mean of left and right jitter, respectively. Dirac-delta function is indicated by \(\delta\). Equation (9) represents the TJ for a bit error rate of \(10^{-12}\). DJ is often measured in peak-to-peak values [40; 42; 43; 44]. \[\text{TJ}=\text{RJ}+\text{DJ}=14\sigma+\text{DJ} \tag{9}\]  
The tolerance of two signals to additive white Gaussian noise can be compared when signals have the same energy per bit, \(E\), but the noise level can differ for the exact value of \(E\) or the same average power level. Error probability for binary unipolar signal elements is given by Equation (10) [45]. \[P_{e}=Q\sqrt{\frac{E}{N_{o}}} \tag{10}\]  
\(Q\) is the quality factor, and _No/2_ is the magnitude of the power spectral density of Gaussian white noise. The idea from cellular systems defined in [37] can be used for \(n\) number of sources here, seven LEDs, and Equation (10) can be modified to Equation (11). The bigger the number of interferers, the lower the error probability, as a Gaussian interferer is better than a single interferer. \[R_{a}=\frac{1}{8}\sum_{i=1}^{8}R_{i} \tag{3}\]0  
The theoretical bit rate of LED is estimated by [46]. \[R_{a}=\frac{1}{8}\sum_{i=1}^{8}R_{i} \tag{3}\]1  
Signal rise time is the time it takes the leading edge to reach 10% to 90% amplitude. Similarly, fall time is the trailing edge's 90% to 10% change. The human detection thresholds for flickering of light sources at different sinusoidal frequencies are defined in CIE recommendations [47].