## 3 Probability of Chip Error in Coherent O-QPSK
### Half-Sine Pulse Shaping  
Using the pulse shaping given by Equation (4), the received signal in Equation (8) can be written, on the Q\({}_{d}\)-I plane, as:  
\[\tilde{r}(t)=\left(\pm\sqrt{\frac{C}{2}}\pm j\sqrt{\frac{C}{2}}\right)+\left( \pm\sqrt{\frac{I}{2}}\pm j\sqrt{\frac{I}{2}}\right)\cos\left(\frac{\pi t}{2T_ {c}}+\phi_{I}\right)\ \ -T_{c}\leq t<T_{c},\ \ \ 0\leq\phi_{I}<2\pi \tag{12}\]  
where the first addend represents the useful symbol and the second addend represents the interferer symbol. In the variable, \(t\), in Equation (12), is the sampling instance for the interferer symbol, and \(\phi_{I}\) is the carrier phase of the interferer, which shifts the pulse shaping function on the Q\({}_{d}\)-I domain. When there is perfect knowledge of the useful carrier phase and symbol timing, there is no difference, from the demodulator point of view, between half-sine-shaped or not shaped useful symbols, since the sampling is done at the maximum amplitude instance. As a consequence, the first addends in Equation (10) and Equation (12) are equal; this can also be observed by comparing Figure 1b with Figure 3b. When the pulse shaping function, \(\cos(\frac{\pi t}{2T_{e}}+\phi_{I})\), of the interferer is included to the scenario, the interferer signal will be positioned on one of the diagonal lines shown in Figure 3d. As a consequence, the received signal plus interferer will be positioned over the dashed lines of Figure 3e. To obtain the error probability, we need to derive the probability distribution of the amplitude of the interferer at the sampling instance. The random symbol sampling instance, \(t\), and carrier phase, \(\phi_{I}\), of the interferer can be modeled as uniformly distributed random variables. To simplify the notation in a pulse shaping function, we substitute the random variables as \(X=\frac{\pi t}{2T_{e}}\), \(Y=\phi_{I}\) and \(Z=X+Y\) and denote the Probability Density Functions (PDFs) of \(X\) and \(Y\) as \(f_{X}(x)\) and \(f_{Y}(y)\); then the PDF of Z, which is our interest, will be the convolution integral [27] given as:  
\[f_{Z}(z)=\int_{-\infty}^{\infty}f_{X}(x)f_{Y}(z-x)dx = \begin{cases}\frac{\pi+2z}{4\pi^{2}}&\frac{-\pi}{2}\leq z<\frac{ \pi}{2}\\ \frac{1}{2\pi}&\frac{\pi}{2}\leq z<\frac{3\pi}{2}\\ \frac{5\pi-2z}{4\pi^{2}},&\frac{3\pi}{2}\leq z<\frac{5\pi}{2}\\ 0&\text{otherwise}\\ \end{cases} \] \[=\frac{1}{2\pi} 0\leq z<2\pi \tag{13}\]  
where:  
Figure 3: The impact of of the interferer: the asynchronous case. (**a**) Useful signal; (**b**) Q\({}_{d}\)-I domain representation of useful signal; (**c**) Interferer signal; (**d**) Q\({}_{d}\)-I domain representation of interferer signal; (**e**) Q\({}_{d}\)-I domain representation of received signal. \[f_{X}(x)= \begin{cases}\frac{1}{\pi},&\frac{-\pi}{2}\leq x<\frac{\pi}{2}\\ 0&\text{otherwise}\end{cases} \quad f_{Y}(y)= \begin{cases}\frac{1}{2\pi},&0\leq y<2 \pi\\ 0&\text{otherwise}\end{cases} \]  
The solution of the convolution integral in Equation (13) indicates that \(Z\) is a uniformly-distributed random variable in the interval \(0\leq z<2\pi\). Thus, the sampled point of the interferer signal is \((\pm\sqrt{\frac{I}{2}}\pm j\sqrt{\frac{I}{2}})\cos(z)\), and it always stays over the dashed lines in Figure 3d. If the interferer is transmitting \(m_{3}\) or \(m_{0}\), the slope of the diagonal line is \(+1\); on the other hand, if the interferer is transmitting \(m_{1}\) or \(m_{2}\) then the slope is \(-1\). The probability distribution of the positions of the interferer sampling point on the dashed lines are symmetric on the opposite sides of the origin, because of the cosine function.