# Closed-Form Expressions of Upper Bound for Polarization-MDCSK System
## 2 System Model of P-MDCSK  
A P-MDCSK constellation is characterized by a horizontal polarization state, a vertical polarization state, and a phase, which is shown in Figure 1. The information bit sequence consists of polarization and phase parts. By packing \(l_{b}\) bits on the sphere and \(n_{b}\) bits on the MDCSK phase, a total of \(m_{c}=l_{b}+n_{b}\) bits can be conveyed. Thus, the \(L_{b}=2^{l_{b}}\) symbols lie on the sphere and the \(N_{b}=2^{n_{b}}\) symbol is in constellation with MDCSK, where \(M=2^{m_{c}}\). We assume that \(L_{b}=2\) (This paper is based on the P-MDCSK in [8]). The \(l_{b}\) is set to 1. Symbols lie on the sphere and \(N_{b}\) symbols are with the MDCSK constellation. The transmitted signal with Stokes parameters [5] is written as  
\[ \begin{array}{l}\mathbf{S_{0}}=\mathbf{E_{0}s_{x}}\\ =\left[\begin{array}{c}E_{h}\\ E_{v}\end{array}\right]\mathbf{s_{x}}\quad=\left[\begin{array}{c}\cos\varphi e ^{i\varepsilon_{h}}\\ \sin\varphi e^{i\varepsilon_{v}}\end{array}\right]\mathbf{s_{x}},\end{array}  \tag{1}\]  
where \(\varphi\) is the angle of polarization, \(\epsilon_{h}\) and \(\epsilon_{v}\) are the phase of the signal in the horizontal and vertical state of polarization, respectively, considering \(\vartheta=\epsilon_{h}-\epsilon_{v}=0\), \(\varphi=0,\pi/2\), and \(E_{h}^{2}+E_{v}^{2}=1\). The \(\mathbf{s_{x}}\) is the symbol with MDCSK modulation, which is written as  
\[\mathbf{s_{x}}=[\mathbf{s_{ref}},\mathbf{s_{inf}}]=[\underbrace{\mathbf{c_{x }}}_{reference},\underbrace{\cos\theta\mathbf{c_{x}}+\sin\theta\mathbf{H(c_{x}) }}_{information-bearing}], \tag{2}\]  
Figure 1: The polarization constellation of (**a**) \(M=4\), (**b**) \(M=8\), (**c**) \(M=16\). where \(\mathbf{c}_{\mathbf{x}}=[c_{x,1},c_{x,2},\ldots,c_{x,i},\ldots,c_{x,\beta}]\) is \(\beta\)-length chaotic signal. The \(H(. )\) denotes Hilbert transform operator; thus, \(c_{y}=H(c_{x})\). And \(\theta\) is the phase of MDCSK modulation with \(\theta\in[0,2\pi)\), where \(\cos^{2}\theta+\sin^{2}\theta=1\). The received signals of P-MDCSK over the multipath Rayleigh fading channel are expressed as  
\[\mathbf{r}_{\mathbf{h}} =E_{h}\mathbf{s}_{\mathbf{x}}\otimes\mathbf{h}_{\mathbf{h}, \mathbf{h}}+E_{v}\mathbf{s}_{\mathbf{x}}\otimes\mathbf{h}_{\mathbf{h},\mathbf{ v}}+\mathbf{n}_{\mathbf{h}}, \tag{3}\] \[\mathbf{r}_{\mathbf{v}} =E_{v}\mathbf{s}_{\mathbf{x}}\otimes\mathbf{h}_{\mathbf{v}, \mathbf{v}}+E_{h}\mathbf{s}_{\mathbf{x}}\otimes\mathbf{h}_{\mathbf{v}, \mathbf{h}}+\mathbf{n}_{\mathbf{v}},\]  
where \(\mathbf{r}_{\mathbf{h}}=[\mathbf{r}_{\mathbf{h}_{\text{ref}}},\mathbf{r}_{ \mathbf{h}_{\text{ref}}}],\mathbf{r}_{\mathbf{v}}=[\mathbf{r}_{\mathbf{v}_{ \text{ref}}},\mathbf{r}_{\mathbf{v}_{\text{int}}}]\), are the received signals in the horizontal and vertical polarized states, and \(\mathbf{h}_{\mathbf{h},\mathbf{v}}\) and \(\mathbf{h}_{\mathbf{v},\mathbf{h}}\) are the composite gain of the input \(h/v\) and the output \(v/h\) polarization components. The \(\mathbf{n}_{\mathbf{h}}\) and \(\mathbf{n}_{\mathbf{v}}\) are the additive white Gaussian noise (AWGN) with zero mean and variance \(N_{0}/2\), and \(\otimes\) denotes the convolution operator. The \(\mathbf{h}_{\mathbf{h},\mathbf{v}}=\mathbf{h}_{\mathbf{v},\mathbf{h}}\) are set to \(0\)[8], and \(\mathbf{h}_{\mathbf{h},\mathbf{h}}\) and \(\mathbf{h}_{\mathbf{v},\mathbf{v}}\) have the same parameters as \(\mathbf{h}_{\mathbf{h},\mathbf{h}}=\mathbf{h}_{\mathbf{v},\mathbf{v}}=\sum_{ l=1}^{l}a_{l}\delta(t-\tau_{l})\), where \(L\) is the number of paths of the multipath channel, and \(\alpha_{l}\) and \(\tau_{l}\) are the channel coefficients and the path delay of the \(l\)th path. The demodulation of the receiver is implemented in two parts: MDCSK and polarization states. Each part is demodulated by an independent process.