# Closed-Form Expressions of Upper Bound for Polarization-MDCSK System
## 3 Closed Expression over Multipath Rayleigh Fading Channels  
A tight upper bound is used to calculate the BER of polarization modulation [5]. Thus, in our previous work [8], we derived a BER upper bound of P-MDCSK, which is described as  
\[P_{P-MDCSK}\leq P_{signal}+P_{polarization}+P_{joint}, \tag{7}\]  
where  
\[P_{signal}=\frac{n_{b}}{l_{b}+n_{b}}P_{MDCSK}, \tag{8}\]  
\[P_{polarization}=\frac{1}{L_{b}(l_{b}+n_{b})}\sum\limits_{l_{a}=1}^{L_{b}}\sum \limits_{l_{b}=1}^{L_{b}}D(l_{n}\to l_{s})Q\left(\sqrt{\frac{d_{l_{a},l_{b}}^ {2}}{2N_{0}}}\right), \tag{9}\]  
\[P_{joint}=\frac{1}{L_{b}N_{b}(l_{b}+n_{b})}\sum\limits_{l_{s}=1}^{L_{b}}\sum \limits_{l_{a}=1}^{L_{b}}\sum\limits_{l_{b}=1}^{N_{b}}(D(l_{n}\to l_{s})+D(n_{n} \to n_{s}))Q\left(\sqrt{\frac{d_{l_{a},l_{b},n_{b},n_{s}}^{2}}{2N_{0}}} \right), \tag{10}\]  
where \(D(l_{n}\to l_{s})\) is the Hamming distance, i.e., the number of different bits between symbols defined by \(l_{n}\) and \(l_{s}\), \(l_{n}\) and \(n_{n}\) denote the wrong symbols. The same definition is true of \(D(n_{n}\to n_{s})\). The generic distance of \(d_{l_{n},l_{a},n_{n},n_{s}}^{2}\) in the Euclidean space is expressed as  
\[d_{l_{n},l_{a},n_{n},n_{s}}^{2}=2E_{k}\left(1-\left(\cos\left(\Delta\varepsilon -\frac{\Delta\theta}{2}\right)\cos\frac{\varphi_{l_{a}}}{2}\cos\frac{\varphi_{ l_{b}}}{2}\right.\right.\left.+\cos\left(\Delta\varepsilon+\frac{\Delta\theta}{2} \right)\sin\frac{\varphi_{l_{a}}}{2}\sin\frac{\varphi_{l_{b}}}{2}\right)\right), \tag{11}\]  
where \(\Delta\varepsilon=\varepsilon_{n_{n}}-\varepsilon_{n_{s}},\varepsilon_{n_{s}}\) is the MDCSK component of symbol \(n\), \(\Delta\theta=\vartheta_{l_{n}}-\vartheta_{l_{s}}\), \((\vartheta_{l_{s}},\vartheta_{l_{s}})\) is the polarization of the \(l\) symbol. The \(d_{l_{n},l_{s}}^{2}\) is obtained from (11) when \(\Delta\varepsilon=0\)[8]. The closed expression is then calculated for each of these three components. The \(P_{MDCSK}\) (\(P_{DCSK}=\frac{1}{2}\text{erfc}\left(\left[\frac{4}{\gamma_{s}}+\frac{2\beta} {\gamma_{s}^{2}}\right]^{-\frac{1}{2}}\right)\), for \(n_{b}=1\)) for \(n_{b}\geq\)2 can be approximated to a simpler form in [12; 20] as  
\[P_{MDCSK}\approx\frac{2}{n_{b}}Q\left(\frac{\rho\pi}{2n_{b}}\right), \tag{12}\]  
where \(\rho=\sum\limits_{l=1}^{L}\alpha_{l}^{2}2E_{c}/\delta=\sum\limits_{l=1}^{L} \alpha_{l}^{2}E_{k}/\delta=2\gamma_{s}/\sqrt{2\gamma_{s}+\beta}\), and \(E_{c}=E_{\tilde{l}}^{2}(\sin^{2}\theta\sum_{i=1}^{\beta}c_{x,i}^{2}+\cos^{2} \theta\sum_{i=1}^{\beta}c_{y,i}^{2})=\sum_{i=1}^{\beta}c_{x,i}^{2}=\sum_{i=1}^ {\beta}c_{y,i}^{2}=\frac{E_{k}}{2}\), and \(E_{k}=2E_{\tilde{l}}^{2}\sum_{i=1}^{\beta}c_{x,i}^{2}=2(E_{\tilde{h}}^{2}+E_{ \tilde{v}}^{2})\sum_{i=1}^{\beta}c_{x,i}^{2}\) is the total energy of one symbol. The \(\gamma_{s}=\sum\limits_{l=1}^{L}\alpha_{l}^{2}E_{k}/N_{0}=\sum\limits_{l=1}^{L} \alpha_{l}^{2}n_{b}E_{b}/N_{0}\) and \(Q(x)=\frac{1}{\sqrt{2\pi}}\sum\limits_{x}^{\infty}\int e^{-\frac{\beta}{2}}dt\), for \(x\geq 0\). Then, the total system BER of the proposed scheme over Rayleigh fading channel is given by (Here we base our performance analysis more on the systemfrom a mathematical point of view in the original reference. In the real situation, it is necessary to consider whether there is a Rayleigh multipath fading in reality.) \[P_{multi}\approx\int\limits_{0}^{\infty}P_{P-MDCSK}f(\gamma_{s})\mathrm{d}\gamma _{s}. \tag{13}\]  
where \(\gamma_{s}=\sum\limits_{l=1}^{L}\alpha_{l}^{2}E_{k}/N_{0}\), and \(f(\gamma_{s})\) is the PDF of \(\gamma_{s}\) which can be found in [13]. In order to solve complex integrals of the BER expressions (13), the bounds of the Q-function are used. This part provides a generalized analytical expression of the closed form on approximate BER as the (7) is the upper bound of BER. However, this approach can still provide a trend of the BER. In the derivation of [19], a single-term exponential bound with adaptive parameters is considered.