## 5 3D-VWZCC System Performance  
In order to analyze and evaluate the performance of our system, two essential parameters are used: bit error rate (BER), which is defined by the ratio between the number of erroneous bits and the total number of transmitted bits, and signal-to-noise ratio (SNR), which is also defined by the ratio between the total received power and the noise power that affected the system. In addition, the system is affected by three types of noise: thermal noise, shot noise, and PIN noise; further, our code has the property of zero cross-correlation ZCC. Therefore, we take into account the impact of these noises to calculate the signal-to-noise ratio, with the total photocurrent noise expressed as [10; 11; 13; 15]:  
\[\left\langle\sigma_{noise}^{2}\right\rangle=\left\langle\sigma_{Thermal}^{2} \right\rangle+\left\langle\sigma_{Shot}^{2}\right\rangle+\left\langle\sigma_{ PIN}^{2}\right\rangle \tag{19}\]  
\[\sigma_{noise}^{2}\frac{4K_{b}T_{n}B_{r}}{R_{l}}+2 eB_{r}I+B_{r}I^{2}\tau_{c} \tag{20}\]  
where \(K_{b}\) is Boltzmann's constant, \(T_{n}\) is the absolute temperature in Kelvin, \(B_{r}\) is the electrical bandwidth, \(R_{l}\), \(e\), \(I\) are load resistance, electron charge, and variance of the receiver photocurrent, respectively, and \(\tau_{c}\) denotes the light coherence time, which is defined by the following equation where \(G\) is the single sideband power spectral density (PSD) [10; 11; 12; 13]:  
\[\tau_{C}=\frac{\int_{0}^{+\infty}G^{2}(v)dv}{\left[\int_{0}^{+\infty}G(v)dv \right]^{2}} \tag{21}\]  
To simplify the analysis of 3D-VWZCC, we take into consideration four hypotheses presented as follows: first, the incoherent broadband light source has a non-variable spectrum during the light bandwidth [\(v_{0}-\Delta v/2\), \(v_{0}+\Delta v/2\)], where \(v_{0}\) is the central frequency and \(\Delta v\) is the bandwidth of the light source. Second, the spectral width is equi-spectro for all users. Third, equal power for all users at the receiver level. Finally, the fourth is each bit data stream of each user is synchronized. According to the above assumptions, the power spectral density (PSD) of the received signals \(r(v)\) can be written as [10; 11; 12; 13]:  
\[r(v)=\frac{P_{sr}}{W_{2}W_{3}\Delta v}\sum_{k=1}^{K}d_{k}\sum_{i=0}^{M-1}\sum _{j=0}^{N-1}\sum_{l=0}^{P-1}a_{i,j,l}\prod(v,i) \tag{22}\]  
\(P_{sr}\Delta v\),\(W_{2}\),\(W_{3}\) and \(\prod(v,i)\) are the effective received power, bandwidth of the light source, weight of temporal sequence, and weight of spatial sequence, respectively, and \(d_{k}\) is the data bit of the \(k^{th}\) user, which can be \"1\" or \"0\"and \(\prod(v,i)\) defined as:  
\[\prod(v,i)=\left\{u\left[v-v_{0}-\frac{\Delta v}{2M}(-2M+2i)\right]-\left[v- v_{0}-\frac{\Delta v}{2M}(-M+2i+2)\right]\right\}=u\left[\frac{\Delta v}{M}\right] \tag{23}\]  
\(u[v]\) is a unit step function given as:  
\[u(v)=\left\{ \begin{array}{ll}1&when\;\;v\in[0,+\infty[\\ 0&when\;\;v\in[-\infty,0]\end{array} \right. \tag{24}\]  
The output of the currents from the photodiode (PD) to the receiver is according to the cross-correlation between \(A^{(d)}0\),\(0\),\(0\), and \(A_{0,0,0}\) of the code 3D-VWZCC denoted by \(I_{r}\) and can be written as:  
\[I=\mathcal{R}\int_{0}^{+\infty}r(v)dv=\mathcal{R}\int_{0}^{+\infty}\frac{P_{sr }}{W_{2}W_{3}\Delta v}\sum_{k=1}^{K}d_{k}R_{i,j,l}^{(d)}\prod(v,i)dv \tag{25}\]By substituting Equations (17) and (23) into Equation (25) we find:  
\[I=\frac{\mathcal{R}P_{sr}}{W_{2}W_{3}\Delta v}\Big{[}W_{1}W_{2}W_{3}+\sum_{k=1}^ {K}d_{k}R_{0,0,0}^{(0)}\Big{]}\times\frac{\Delta v}{M}=\frac{\mathcal{R}P_{sr}W _{1}}{M} \tag{26}\]  
where \(\mathcal{R}\) is the photodiode responsivity given by \(\mathcal{R}=ne/hv_{0}\), \(\eta\) is the quantum efficiency of the photo-diode, and \(h\) is [PERSON]'s constant and \(e\) is the electron charge. \(M\), \(W_{1}\) are code length and code weight for spectral sequence \(X_{g}\), respectively.