## 2 Materials and Methods
### Quaternion Model for RPW Communication (Q-RPW)  
Q-RPW model is motivated by the polarization of electromagnetic waves arriving at a receive antenna in multipath environment, given below as [20]:  
\[ \begin{array}{l}r(t)\quad=\sum\limits_{n}c_{n}e^{i\varphi_{n}}\cos(\omega_{ c}t+\theta_{n})\\ =\sum\limits_{n}c_{n}[\cos\varphi_{n}\cos\theta_{n}\cos(\omega_{c}t)+\cos \varphi_{n}\sin\theta_{n}\sin(\omega_{c}t)+j\sin\varphi_{n}\cos\theta_{n}( \omega_{c}t)+j\sin\varphi_{n}\sin\theta_{n}\sin(\omega_{c}t)]\end{array}  \tag{11}\]  
Since the change in polarization angle is independent of the phase of transmitted signal, the trigonometric coefficients of the carrier in the four terms of (11) are independent. That is exactly the case in a quaternion variable. Hence RPW can be expressed in quaternion form. In this section, new Q-RPW model for RPW communication is presented. The proposed model is based on quaternion representation of dual-polarized systems [17; 18]. Quaternion symbols for a dual-polarized system are represented as:  
\[q_{m}=s_{H}+s_{V}j \tag{12}\]  
Equation (12) states that two complex symbols \(s_{H}=a_{H}+ib_{H}\) and \(s_{V}=a_{V}+ib_{V}\)\((a_{H},b_{H},a_{V},b_{V}\in\mathbb{R})\) are transmitted through horizontal and vertical polarized elements of a dual-polarized antenna. To represent RP-MPSK symbols, we change (12) in accordance with (3). Since the baseband modulation frequencies have a phase difference of \(\pi/2\), we deduce that for RP-MPSK, \(s_{V}=s_{H}e^{-i\pi/2}\). We use (12) to define quaternion RP-MPSK symbol as:  
\[q_{m}\triangleq a+bi-bj+ak \tag{13}\]Here, \(s_{m}=s_{H}=a+ib\) is the conventional MPSK symbol. The imaginary numbers \(i,j,k\) are defined by:  
\[ij=-ji=k,jk=-kj=i,ki=-ik=j,i^{2}=j^{2}=k^{2}=ijk=-1\]  
A rotation of \(q_{m}\) in the polarization domain by an angle \(\varphi_{\ell}\) can be represented as \(q_{m,\ell}=q_{m}e^{-i\varphi_{\ell}}\)[17]. For RPW transmission, we rotate \(q_{m}\) by \(N_{\ell}\) polarization angles to generate the sequence \(\left\{q_{m,\ell}\right\}\), such that:  
\[q_{m,\ell}=q_{m}e^{-j2\ell\pi/N_{\ell}}\quad\ell=0,1,2,\ldots,N_{\ell}-1 \tag{14}\]  
From implementation perspective, this is equivalent to interpolating the transmitted symbol stream \(\left\{q_{m}\right\}\) by a factor \(N_{\ell}\). The digital quaternion stream \(\left\{q_{m,\ell}\right\}\) is up-converted to analog RPW signal for RF transmission through dual polarized antenna. The RF signal arrived at dual-polarized receive antenna is down-converted to analog baseband signal and sampled at a frequency \(f_{s}=N_{p}f_{r}\) specified in previous section. This is important to emphasize that \(N_{\ell}\) and \(N_{p}\) can be chosen independent of each other; however, for convenience, we take \(N_{\ell}=N_{p}\). The sampled discrete signal is converted to baseband quaternion stream \(r_{m,\ell}\) that can be represented in the form of (12):  
\[r_{m,\ell}=q_{m,\ell}H+w \tag{15}\]  
Here \(H\) represents the quaternion channel and \(w\) represents AWGN at receiver. The channel is described by the following equation:  
\[H\triangleq h_{0}+h_{i}i+h_{j}j+h_{k}k \tag{16}\]  
The coefficients \(h_{0},h_{i},h_{j},h_{k}\sim\mathcal{N}_{\mathbb{R}}\left(0,\sigma_{h}^{2}\right)\) are real i.i.d. normal random variables. We can also express the channel in a more compact form as \(H\sim\mathcal{N}_{\mathbb{H}}\left(0,\sigma_{H}^{2}\right)\) where \(\mathbb{H}\) is the quaternion or the Hamilton space. Similarly, the noise can be expressed as \(w\sim\mathcal{N}_{\mathbb{H}}\left(0,\sigma_{w}^{2}\right)\). The model of the received signal is related to conventional model of PD system, starting from the following expression consistent with (12):  
\[r_{m,\ell}=r_{H}+r_{V}j \tag{17}\]  
Here \(r_{H}\) is the signal received by the horizontal polarized antenna and \(r_{V}\) is the signal received by the vertical polarized antenna.