## 2 Materials and Methods  
Previous works on RPW covered only RP-BPSK and RP-QPSK modulation [24]. RP-BPSK was the first modulation scheme used in RPW when it was invented. In this method, a binary data symbol \(s_{m}\) simultaneously modulates two orthogonal baseband sinusoids of low frequency \(\omega_{r}=2\pi f_{r}=2\pi/T_{r}\) as:  
\[\begin{array}{l}h_{t}(t)=\pm\sqrt{2E_{b}/T_{r}}\cos(\omega_{r}t)\\ v_{t}(t)=\pm\sqrt{2E_{b}/T_{r}}\sin(\omega_{r}t)\end{array}\bigg{\}}\quad m=0,1\]  
These signals further modulate a carrier of high frequency \(\omega_{c}=2\pi f_{c}\) for horizontal and vertical polarized transmissions through DP antenna, respectively. By transmitting the two baseband signals having a phase difference of \(\pi/2\) along two orthogonal polarizations, the polarization of the resultant electromagnetic signal is made to rotate at frequency \(\omega_{r}\), hence called RPW. In analogy with CP, we call this RPW as Right-Hand RPW (RHRPW) (Figure 1). On the receiver side, RPW signal is received by another DP antenna. The signals received on both antennas are sampled at frequency \(f_{s}\), that is an integral multiple of \(f_{r}\). Each sample obtained in this way on the two antennas is polarized at a different angle and is a replica of the transmitted symbol. For each sample, the signal with higher signal power out of the two polarizations is selected as the desired sample. Each selected sample is then demodulated to recover a replica of the transmitted symbol. If there is a greater number of 1's than 0's, the receiver decides in favour of binary 1. A binary 0 is decided otherwise.  
From classical theory of modulation, we know that QPSK is spectrally more efficient than BPSK, provides higher data rate and offers BER performance equivalent to BPSK. Motivated by the fact, RP-QPSK was proposed [24]. The main issue in moving to RP-QPSK was that RP-BPSK has already used two quadrature baseband carriers. Using the same carriers to send QPSK symbol as well as two orthogonal polarized signals with a phase difference of 90\({}^{\circ}\) at the same time cannot be realized. Also, the two QPSK transmitters should operate on the same baseband frequency to generate RPW. This problem was solved by manipulating the linear combination of the orthogonal basis functions (Figure 2). A symbol \(s_{m}\in\{s_{0},s_{1},s_{2},s_{3}\}\) with energy \(E_{s}\) is demultiplexed into even bit \(x_{e}\) and odd bit \(x_{o}\). The horizontal and the vertical polarized signals for RP-QPSK are described by the following equations:  
\[\begin{array}{l}h_{t}(t)=x_{e}\psi_{1}(t)+x_{o}\psi_{2}(t)\\ v_{t}(t)=x_{e}\psi_{2}(t)-x_{o}\psi_{1}(t)\end{array} \tag{1}\]  
Here, \(\psi_{1}(t)\) are \(\psi_{2}(t)\) are the orthogonal basis functions defined by  
\[\psi_{1}(t)=\sqrt{2/T_{r}}\cos(\omega_{r}t)\quad\psi_{2}(t)=\sqrt{2/T_{r}}\sin (\omega_{r}t)\]  
Note that even and odd bits have swapped positions in the signals to be transmitted through horizontal and vertical polarizations. Another difference is the addition of the two terms in the signal for horizontal polarization and subtraction in the signal for vertical polarization. RP-QPSK modulation can now be expressed in more general form as below:  
\[\left.\begin{array}{l}h_{t}(t)=\sqrt{\frac{2E_{s}}{T_{r}}}\cos(\omega_{r}t+ (2m+1)\frac{\pi}{4})\\ v_{t}(t)=\sqrt{\frac{2E_{s}}{T_{r}}}\sin(\omega_{r}t+(2m+1)\frac{\pi}{4})\end{array} \right\}\quad m\in 0,1,2,3 \tag{2}\]The demodulation process of RP-BPSK is adopted to recover transmitted symbol. Further details and the demodulation procedure will be covered in RP-MPSK modulation.