## 4 3D-SWZCC System Performance  
In order to simplify the system analysis, four assumptions should be taken into account [29].  
First, the broadband light source is ideally unpolarized, and its spectrum belongs to the interval \([f_{0}-\Delta f,f_{0}+\Delta f]\) in the domain. '\(f_{0}\)' and '\(\Delta f\)', are, respectively, the central optical frequency and the optical source bandwidth, and they are estimated in Hertz. Second, each spectral component received by a user has the same power. Third, each spectral component produced by the spectral encoders has identical spectral width. Finally, the fourth assumption is represented by each bitstream from each user being synchronized. Depending on these assumptions and the Gaussian approximation, the bit error rate (BER) estimation can be realized by considering the thermal noise, PINN, and shot noise.  
The photo-current noise is given by [30]:  
\[\begin{array}{l}I_{noise}^{2}=\sigma_{thermal}^{2}+\sigma_{PIN}^{2}+\sigma_{ shot}^{2}\\ =4K_{b}T_{n}B/R_{L}+I_{dd}^{2}B\tau_{c}+2 eBI_{dd}\end{array} \tag{13}\]  
where \(K_{b}\) is Boltzmann's constant; \(T_{n}\) is the absolute temperature; \(B\) is the electrical bandwidth; \(R_{L}\) is the load resistance; \(I_{dd}\) is the average photo-current; \(\tau_{c}\) is the coherence time; and \(e\) is the electron charge.  
The coherence time can be expressed as [8]:  
\[\tau_{c}=\int_{0}^{\infty}r^{2}(f)dv\left/\left[\int_{0}^{\infty}r(f)dv\right] ^{2}\right. \tag{14}\]  
where \(r(v)\) represents the power spectral density (PSD) for each user. It can be expressed as [8]:  
\[r(f)=\frac{P_{r}}{w_{2}w_{3}\Delta f}\sum_{K=1}^{K}d_{K}R(i,j,k)\bigvee(f,i) \tag{15}\]  
where \(P_{sr}\) and \(d_{K}\) represent the effective source power at the receiver and the data bit of the \(K^{th}\) user, which may be (0) or (1), respectively.  
The \(\bigvee(v,i)\) function can be defined as [8]:  
\[\begin{array}{l}\bigvee(f,i)=u\Big{\{}f-f_{0}-\frac{fv}{2L_{1}}(-L_{1}+2i- 2)\Big{\}}-u\Big{\{}f-f_{0}-\frac{\Delta f}{2L_{1}}(-L_{1}+2i)\Big{\}}\\ \bigvee(f,i)=u\Big{\{}\frac{\Delta f}{L_{1}}\Big{\}}\end{array} \tag{16}\]  
where \(u(v)\) represents the unit step function, defined as  
\[u(f)=\begin{cases}1&f\geq 0\\ 0&f<0\end{cases} \tag{17}\]  
Figure 3: Proposed Spectral/Temporal/T patiial OCDMA rReceiver Architecture.  
The output photodiode current in the receiver can be written as [8]:  
\[\begin{array}{l}I_{output}=\Re\int_{0}^{\infty}r(f)dv\\ =\Re P_{sr}w_{1}/L_{1}\end{array} \tag{18}\]  
where \(\Re\) represents the photodiode responsivity and is equal to \(\frac{J\sigma}{h.v_{c}}\), where \(\eta\), \(h\), and \(v_{c}\) are the quantum efficiency, [PERSON]'s constant, and the central frequency of the original broadband optical pulse, respectively. Note that \(L_{1}=K_{1}\) and \(K_{1}=K/K_{2}K_{3}\), so Equation (18) can be written as:  
\[I_{output}=\frac{\Re P_{r}K_{2}K_{3}}{K} \tag{19}\]  
The PINN variance current can be defined as [8]:  
\[\begin{array}{l}\sigma_{PINN}^{2}=I_{dd}^{2}R\tau_{c}\\ =\Re^{2}\cdot B\int_{0}^{\infty}r^{2}(f)df\\ =B\cdot I_{output}\,^{2}\cdot L_{1}/\Delta f\end{array} \tag{20}\]  
Based on the results in Equations (19) and (20) and taking into account that the probabilities of transmitting (0) and (1) are the same, Equation (13) will become:  
\[I_{noise}^{2}=4K_{b}T_{n}B/R_{L}+B\cdot I_{dd}\,^{2}\cdot L_{1}/2\cdot\Delta v +2 eB\frac{\Re P_{r}K_{2}K_{3}}{K} \tag{21}\]  
The average signal-to-noise ratio (SNR) can be written as [31]:  
\[SNR=I_{output}^{2}/I_{noise}^{2} \tag{22}\]  
Using the Gaussian approximation, the BER is given by [31]:  
\[BER=\frac{1}{4}erfc\Big{(}\sqrt{SNR/2}\Big{)} \tag{23}\]  
The Q-factor can be derived from the BER, it can be expressed as [31]  
\[Q_{dB}=20\,log_{10}\Bigg{[}\frac{\sqrt{2}}{erfc(2\times BER)}\Bigg{]} \tag{24}\]