# Investigation of the Scattering Noise in Underwater Optical Wireless Communications
## 4 Scattering Noise Model  
Here, we introduce the scattering noise, which is due to the corpuscular nature of photons' transport, within the context of UOWC and outline the analytical model for the first time. Photons propagating in a real channel (i.e., water) will experience multiple scattering, which is considered as a random process, thus arriving at the Rx with delays compared with those propagating along the line-of-sight path. Therefore, photons with the random arrival time being collected at the Rx for a given fixed transmit optical power (i.e., photons per pulse) will have a Poisson distribution. The variation about the mean number of photons is the noise, which reflects itself as the channel-dependent scattering noise. We distinguish between the scattered and non-scattered photons received at the detector. The photocurrent at the detector due to the scattered photons-induced noise is given as:  
\[I=\sum_{j=1}^{N-N^{\prime}}q\delta(t-t_{j}), \tag{1}\]  
where \(t_{j}\) is the random arrival time of the \(j\)th scattered photon, \(q\) is the electron charge (i.e., \(1.602176634\times 10^{-19}\) C), \(N\) is the total number of received photons, and \(N^{\prime}\) is the number of photons not undergone scattering (\(N\) and \(N^{\prime}\) are determined from MC simulation). The Autocorrelation function is given as:  
\[R(t^{\prime})=\lim_{T\to\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}I(t )I(t+t^{\prime})\,dt=\lim_{T\to\infty}\frac{q^{2}}{T}\int_{-\frac{T}{2}}^{ \frac{T}{2}}\sum_{j=1}^{N-N^{\prime}}\delta(t-t_{j})\sum_{j^{\prime}=1}^{N-N^ {\prime}}\delta(t-t_{j^{\prime}}+t^{\prime})\,dt. \tag{2}\]  
The non-zero value for the first delta function is at \(t=t_{j}\), which is substituted in the second delta function in (2) to obtain:  
\[R(t^{\prime})=\lim_{T\to\infty}\frac{q^{2}}{T}\int_{-\frac{T}{2}}^{\frac{T}{2 }}\sum_{j=1}^{N-N^{\prime}}\sum_{j^{\prime}=1}^{N-N^{\prime}}\delta(t-t_{j}) \delta(t_{j}-t_{j^{\prime}}+t^{\prime})\,dt. \tag{3}\]  
Note, the integration of the first delta function is equal to 1, thus we have:  
\[R(t^{\prime})=\lim_{T\to\infty}\frac{1}{T}\sum_{j=1}^{N-N^{\prime}}\sum_{j^{ \prime}=1}^{N-N^{\prime}}q^{2}\delta(t_{j}-t_{j^{\prime}}+t^{\prime}). \tag{4}\]  
For \(j=j^{\prime}\) (i.e., \(t_{j}=t_{j^{\prime}}\)), we only have \(q^{2}\delta(t^{\prime})\), which will contribute \((N-N^{\prime})q^{2}\delta(t^{\prime})\) to the autocorrelation function. \[R(t^{\prime})=\lim_{T\to\infty}\frac{1}{T}((N-N^{\prime})q^{2}\delta(t^{ \prime})+\sum_{j=1}^{N-N^{\prime}}\sum_{j^{\prime}=1,j\
eq j^{\prime}}^{N-N^{ \prime}}q^{2}\delta(t_{j}-t_{j^{\prime}}+t^{\prime})). \tag{5}\]  
Wiener-Khintchine theorem states that, the noise spectrum is Fourier transform of the autocorrelation function in (5), thus the one-sided power spectral density (PSD) is given by:  
\[s(f)=2\int_{-\infty}^{+\infty}R(t^{\prime})\,e^{-i2\pi ft^{\prime}}\,dt^{ \prime}.f\geq 0. \tag{6}\]  
For \(t_{j}\
eq t_{j^{\prime}}\), (second term in (5)), the delta functions will occur at randomly distributed nonzero values of \(t^{\prime}\), which with suitable averaging contributions from the delta functions to Fourier transform in (5) will vanish and therefore not considered. Thus, the autocorrelation function can be written as:  
\[R(t^{\prime})=\lim_{T\to\infty}\frac{N-N^{\prime}}{T}\,q^{2}\delta(t^{\prime}). \tag{7}\]The mean photocurrent due to \(N\) photons received over a time interval \(T\) is defined as:  
\[\bar{I}=\frac{qN}{T}\Rightarrow T=\frac{qN}{\bar{I}}. \tag{8}\]  
Substituting for \(T\) in (7) we have:  
\[R(t^{\prime})=\frac{N-N^{\prime}}{N}q\bar{I}\delta(t^{\prime}). \tag{9}\]  
By substituting (9) in (6), we have the PSD, which is given as:  
\[PSD=2\frac{N-N^{\prime}}{N}q\bar{I}=2\kappa q\bar{I},0\leq\kappa\leq 1, \tag{10}\]  
where \(\frac{N-N^{\prime}}{N}\) is the scattering noise coefficient.