# Energy and Spectrally Efficient Modulation Scheme for IoT Applications
## 3 OSP-MSK Performance Analysis
### OSP-MSK ABER Performance  
The ABER performance of the OSP-MSK modulation relies mainly on the correlation receiver (i.e., four-arms correlation receiver), as shown in Figure 1. For further illustration, one arm correlation receiver of the OSP-MSK demodulator is shown in Figure 2. It is assumed here that the received signals \(S_{I}(t)\) and \(S_{Q}(t)\) at the inputs of the four-arms correlator receiver are corrupted by AWGN distributed as \(\mathbf{n}(\mathbf{t})\sim\mathcal{CN}\left(0,\ \frac{N_{o}}{2}\right)\). Accordingly, the output of the sampler can be expressed as follows:  
\[r_{\text{I}}(2T)=\int_{0}^{2T}(S_{I}(t)+n(t))\phi(t)dt, \tag{10}\]  
\[r_{I}(2T)=\underbrace{\int_{0}^{2T}S_{I}(t)\phi(t)dt}_{S_{o}}\ +\underbrace{\int_{0}^{2T}n(t)\phi(t)dt}_{n_{o}}. \tag{11}\]  
However, in Equation (11), \(S_{o}\) is a deterministic signal, but \(n_{o}\) is an AWGN distributed as \(n_{o}\sim\mathcal{CN}\left(0,\frac{N_{o}}{2}\right)\). Therefore, the output of the sampler \(r_{\text{I}}(2T)\) provided by Equation (11) can be rewritten as follows:  
\[r_{\text{I}}(2T)=\left\{ \begin{array}{l}S_{o1}+n_{o}\ \text{when ``$1'' is transmitted}\\ \\ S_{o2}+n_{o}\ \text{when ``$0'' is transmitted}\end{array} \right., \tag{12}\]  
and the decision threshold \(\xi\) can be expressed as follows:  
\[\xi=\frac{S_{o1}+S_{o2}}{2}. \tag{13}\]  
Hence, to correctly recover \"0\" when \(S_{o2}\) is transmitted, the term \(\xi-r_{\text{I}}(2T)\) should be positive. That is:  
\[\frac{S_{o1}+S_{o2}}{2}-(S_{o2}+n_{o})>0. \tag{14}\]  
Therefore, the noise \(n_{o}\) should satisfy the following inequality:  
\[n_{o}<\frac{S_{o1}-S_{o2}}{2}, \tag{15}\]  
Figure 2: OSP-MSK correlator receiver. and the error will occur when the noise \(n_{o}\) satisfies the following inequality:  
\[n_{o}\geq\frac{S_{o1}-S_{o2}}{2}. \tag{16}\]  
Accordingly, the probability of error (i.e., the error in detecting \"0\" when \(S_{o2}\) is transmitted) is given by:  
\[p_{e}=p\bigg{(}n_{o}\geq\frac{S_{o1}-S_{o2}}{2}\bigg{)}=\int_{\frac{S_{o1}-S_{o 2}}{2}}^{\infty}\frac{1}{\sqrt{2\pi{\sigma_{o}}^{2}}}\;e^{-\frac{\sigma^{2}}{2 \sigma_{o}^{2}}}\;dq, \tag{17}\]  
where \({\sigma_{o}}^{2}\) is the noise variance, which is equal to \(\left(\frac{N_{o}}{2}\right)\). By letting \(r=\frac{q}{\sqrt{2}\;\sigma_{o}}\), the probability of error can be expressed as follows:  
\[p_{e}=\frac{1}{\sqrt{\pi}}\int_{\frac{S_{o1}-S_{o2}}{2\sqrt{2}\;\sigma_{o}^{2} }}^{\infty}e^{-r^{2}}dr, \tag{18}\]  
\[p_{e}=\frac{1}{2}erfc\bigg{(}\frac{S_{o1}-S_{o2}}{2\sqrt{2}\;{\sigma_{o}}^{2} }\bigg{)}, \tag{19}\]  
\[p_{e}=\frac{1}{2}erfc\left(\sqrt{\frac{\left(S_{o1}-S_{o2}\right)^{2}}{8{ \sigma_{o}}^{2}}}\right). \tag{20}\]  
However, in the proposed OSP-MSK, \(S_{o}\) in Equation (11) can be obtained by letting \(\phi(t)\) as follows:  
\[\phi(t)=\sqrt{\frac{2}{T}}cos\bigg{(}\frac{\pi t}{2T}\bigg{)}, \tag{21}\]  
substituting Equation (21) in Equation (11), \(S_{o}\) can be expressed as follows:  
\[\frac{S_{o1}+S_{o2}}{2}-(S_{o2}+n_{o})>0. \tag{14}\]0  
Hence, \(S_{o}\) will take one of two values as follows:  
\[S_{o}=\left\{ \begin{array}{c}\sqrt{E_{b}}\;\mbox{when ``1'' is transmitted}\\ \\ -\sqrt{E_{b}}\;\mbox{when ``0'' is transmitted}\end{array} \right.. \tag{23}\]  
Substituting Equation (23) in Equation (20), the probability of error can be expressed as follows:  
\[p_{e}=\frac{1}{2}erfc\Bigg{(}\sqrt{\frac{E_{b}}{2{\sigma_{o}}^{2}}}\bigg{)} =\frac{1}{2}erfc\Bigg{(}\sqrt{\frac{E_{b}}{N_{o}}}\Bigg{)}. \tag{24}\]  
The probability of error in Equation (24) is the probability of error of the first arm correlation receiver of the OSP-MSK demodulator. Likewise, the probability of error of the second arm correlation receiver can be acquired, and it is equal to the probability of error of the first arm. However, it should be mentioned here that the second arm correlation receiver constitutes a quadrature multiplexed version of the first arm.