# Hiding Stealth Optical CDMA Signals in Public BPSK Channels for Optical Wireless Communication
## 4 System Performance Analysis with Stealth Signal  
We assume that the broadband light source of the stealth channel is ideally unpolarized and its spectrum is flat in the bandwidth range [\(v_{0}-\Delta v/2\), \(v_{0}+\Delta v/2\)], where \(v_{0}\) is the central optical frequency and \(\Delta v\) is the optical source bandwidth. From the aforementioned assumptions, we can easily calculate the proposed system performance using Gaussian approximation and \(u(v)\), the unit step function, which is expressed as  
\[u(v)=\left\{ \begin{array}{ll}1,\ v\geq 0\\ 0,\ v<0\end{array} \right.. \tag{27}\]  
Let \(C_{k}(i)\) be the \(i\)th element of the \(k\)th low of the Walsh-Hadamard matrix. The power spectral density (PSD) of the received optical signal can be written as  
\[s(v)=\frac{P_{sr}}{\Delta v}\underset{i=1}{\overset{N}{\sum}}[b_{k}c_{k}(i)+b _{k}\overline{c_{k}}(i)]rect(i), \tag{28}\]  
where \(P_{sr}\) is the effective power from a single source at the receiver, \(b_{k}\) is the data bit of the stealth signal, and \(N\) is the length of the codeword. The _rect_(\(i\)) function in Equation (28) is given by  
\[rect(i)=u[v-v_{0}-\frac{\Delta v}{2N}(-N+2i-2)]-u[v-v_{0}-\frac{\Delta v}{2N}( -N+2i)]. \tag{29}\]  
The PSD at PD\({}_{1}\), PD\({}_{2}\), PD\({}_{3}\), and PD\({}_{4}\) of the stealth receiver during the 1-bit period can be written as follows:  
\[G_{1}(v)=\frac{P_{sr}}{\sqrt{2}\Delta v}\underset{i=1}{\overset{N}{\sum}}b_{k} [c_{k}(i)\cdot c_{k}(i)]\{rect(i)\}, \tag{30}\]  
\[G_{2}(v)=\frac{P_{sr}}{\sqrt{2}\Delta v}\underset{i=1}{\overset{N}{\sum}}b_{k}  
\[G_{4}(v)=\frac{P_{sr}}{\sqrt{2}\Delta v}\sum_{i=1}^{N}b_{k}[\overline{c_{k}}(i) \cdot\overline{c_{k}}(i)]\{rect(i)\}, \tag{33}\]  
The coefficient \(\sqrt{2}\) results from using a PBS. Using Equations (5) and (6), the detected photocurrent from the stealth signal at PD\({}_{1}\)-PD\({}_{4}\) can be written as \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\), respectively:  
\[I_{1}=R\int_{0}^{\infty}G_{1}(v)dv=\frac{RP_{sr}}{\sqrt{2}N}[b_{k}\frac{N}{2}], \tag{34}\]  
\[I_{2}=R\int_{0}^{\infty}G_{2}(v)dv=0, \tag{35}\]  
\[I_{3}=R\int_{0}^{\infty}G_{3}(v)dv=0, \tag{36}\]  
\[I_{4}=R\int_{0}^{\infty}G_{4}(v)dv=\frac{RP_{sr}}{\sqrt{2}N}[b_{k}\frac{N}{2}], \tag{37}\]  
The signal from the stealth channel is given by the difference between the photodiode current outputs:  
\[I_{1}-I_{2}=R\int_{0}^{\infty}G_{1}(v)dv-R\int_{0}^{\infty}G_{2}(v)dv=\frac{ RP_{sr}}{\sqrt{2}N}[b_{k}\frac{N}{2}], \tag{38}\]  
\[I_{3}-I_{4}=R\int_{0}^{\infty}G_{3}(v)dv-R\int_{0}^{\infty}G_{4}(v)dv=-\frac{ RP_{sr}}{\sqrt{2}N}[b_{k}\frac{N}{2}]. \tag{39}\]  
After the second differential detection, the signal can be expressed as  
\[I=(I_{1}-I_{2})-(I_{3}-I_{4})=\left\{ \begin{array}{c}\frac{RP_{sr}}{\sqrt{2} },b_{k}=1\\ 0\quad,b_{k}=0\end{array} \right.. \tag{40}\]  
Noise existing in the proposed SPC-OCDMA system comprises phase-induced intensity noise (PIN), shot noise, thermal noise, and system noise in the public channel. The frequency band of the public noise is similar to that of the stealth noise. Public noise is difficult to eliminate using normal optical steganography. Using the duplicate process of public signal cancellation at the OCDMA decoder, public noise can be reduced because white noise is unpolarized. Therefore, we use a factor \(\alpha\) to denote the decrease in public noise. Using Equation (40), we can obtain the variance of the photocurrent of noise as  
\[ \begin{array}{ll}\left\langle i^{2}\right\rangle&=\left\langle I_{shot}^{2 }\right\rangle+\left\langle I_{PINI}^{2}\right\rangle+\left\langle I_{thermal }^{2}\right\rangle+\left(\alpha I_{pn}\right)^{2}\\ &=2 eIB+2I^{2}B\tau_{c}+4K_{b}T_{n}B/R_{L}+\left(\alpha I_{pn}\right)^{2}\\ &=2e\left(\frac{RP_{sr}}{\sqrt{2}}\right)B+\frac{BR^{2}P_{sr}^{2}}{\Delta v}+ \frac{4K_{b}T_{n}B}{R_{L}}+\left(\alpha I_{pn}\right)^{2}\end{array}  \tag{41}\]  
where the notation \(<\) \(>\) represents the time averaging operator.