# Energy and Spectrally Efficient Modulation Scheme for IoT Applications
## 3 OSP-MSK Performance Analysis
### OSP-MSK Power Spectral Density  
In the following, a closed form expression for the PSD of the OSP-MSK is analytically derived to evaluate the effectiveness of the proposed OSP-MSK in boosting the spectral efficiency compared to the MIMO-CEM. As depicted in Figure 1, the only main difference between the OSP-MSK and the conventional MSK is the second branch (i.e., \(S_{2}(t)\)) in Equation (4). Thus, the PSD of the OSP-MSK mainly relies on Equation (4). As, \(cos\Big{(}\frac{\pi(t-T)}{2T}\Big{)}=sin\big{(}\frac{\pi t}{2T}\big{)}\) and \(sin\Big{(}\frac{\pi(t-T)}{2T}\Big{)}=cos\big{(}\frac{\pi t}{2T}\big{)}\), then Equation (4) can be rewritten as follows:  
\[S_{2}(t)=\pm\sqrt{\frac{2E_{b}}{T}}sin\bigg{(}\frac{\pi t}{2T}\bigg{)}cos(2\pi f _{c}t)\pm\sqrt{\frac{2E_{b}}{T}}cos\bigg{(}\frac{\pi t}{2T}\bigg{)}sin(2\pi f _{c}t). \tag{46}\]  
Therefore, the spectrum of \(S_{1}(t)\) or \(S_{2}(t)\) mainly relies on the bandwidth occupancy of the orthogonal shaping pulses \(cos\big{(}\frac{\pi t}{2T}\big{)}\) and \(sin\big{(}\frac{\pi t}{2T}\big{)}\). Accordingly, the PSD of the OSP-MSK can be obtained by deriving the PSD of the complex envelope representation of either \(S_{1}(t)\) provided by Equation (2) or \(S_{2}\) provided by Equation (4) [30]. As such, to obtain the PSD of \(S_{1}(t)\), we can rewrite Equation (2) as follows:  
\[S_{1}(t)=S_{even}(t)cos(2\pi f_{c}t)+S_{odd}(t)sin(2\pi f_{c}t), \tag{47}\]  
where \(S_{even}(t)\) and \(S_{odd}(t)\) are the in-phase and quadrature-phase components, respectively. Therefore, the complex representation of \(S_{1}(t)\) can be expressed as follows:  
\[\widetilde{S_{1}}(t)=S_{even}(t)+jS_{odd}(t). \tag{48}\]  
However, \(S_{even}(t)\) and \(S_{odd}(t)\) can be expressed as follows:  
\[S_{even}(t)=\sum\limits_{k=-\infty}^{k=\infty}S_{1I_{k}}(t)g(t-2 kT), \tag{49}\]  
\[S_{odd}(t)=\sum\limits_{k=-\infty}^{k=\infty}S_{1Q_{k}}(t)g(t-2 kT), \tag{50}\]where \(S_{1k}(t),\text{ and }S_{1Q_{k}}(t)\) are the in-phase and quadrature-phase bits, respectively, and \(g(t)\) is the half cycle sine shaping pulse, which can be expressed as follows:  
\[g(t)=\left\{\begin{array}{c}\sqrt{\frac{2E_{b}}{T}}sin\big{(}\frac{\pi t}{2T} \big{)}\ 0\leq t\leq 2T\\ \\ 0\qquad\text{otherwise}.\end{array}\right. \tag{51}\]  
Since the in-phase and quadrature-phase components of the OSP-MSK signal are independent and the PSDs of \(S_{even}(t)\) and \(S_{odd}(t)\) are equal, the baseband PSD of \(S_{1}(t)\) can be expressed as follows:  
\[S_{G}(f)=2\frac{\left|G(f)\right|^{2}}{2T}, \tag{52}\]  
where \(G(f)\) is the Fourier transform of the half-cycle shaping pulse \(g(t)\). However, the magnitude squared of \(G(f)\) can be expressed as follows:  
\[\left|G(f)\right|^{2}=\frac{4E_{b}[1+cos(4\pi fT)]}{T\big{(}\frac{\pi}{2}-8 \pi f^{2}T\big{)}^{2}}. \tag{53}\]  
Substituting Equation (53) into Equation (52), the baseband PSD of \(S_{1}(t)\) can be expressed as follows:  
\[S_{G}(f)=\frac{4E_{b}[1+cos(4\pi fT)]}{\big{(}\frac{\pi}{2}-8\pi f^{2}T^{2} \big{)}^{2}}. \tag{54}\]  
Using some mathematical manipulations, the baseband PSD of \(S_{1}(t)\) can be expressed as follows:  
\[S_{G}(f)=\frac{32E_{b}}{\pi^{2}}\bigg{(}\frac{cos(2\pi fT)}{1-16f^{2}T^{2}} \bigg{)}^{2}. \tag{55}\]