## 4 Probability of Chip Error in Non-Coherent O-QPSK
### Interferer and Useful Symbols are Synchronous  
When useful and interfering signals are synchronous, the received signal can be written as:  
\[R(\omega t)=\sqrt{C}e^{j(b_{C}\omega t+\phi_{C})}+\sqrt{I}e^{j(b_{I}\omega t+ \phi_{I})} \tag{20}\]  
where \(\omega t=[0,\pi/2]\), \(\phi_{C}=[0,2\pi]\) and \(\phi_{I}=[0,2\pi]\). \(b_{C}\) and \(b_{I}\) represent the turning directions of the phase of the complex envelope of the useful and the interferer chips on the complex domain in Figure 5b: \(b_{C},b_{I}=+1\) means a clockwise direction, whereas \(b_{C},b_{I}=-1\) means the counter-clockwise direction. We denote \(P_{eq}\) as the probability of erroneous detection when the phases of useful and interferer signals are turning to the same direction (\(b_{C}=b_{I}\)), while \(P_{dif}\) denotes the probability of erroneous detection when phase transitions are toward the different directions (\(b_{C}\
eq b_{I}\)). There are two cases; accordingly, \(P_{c}\) is:  
\[P_{c}=\frac{1}{2}P_{eq}+\frac{1}{2}P_{dif} \tag{21}\]  
Note that if the phase of the interferer signal turns toward the direction of the useful signal, the received signal in the Q-I plane will turn toward the direction of the useful signal, as well. As a consequence, \(P_{eq}\) will always be zero. To obtain \(P_{dif}\), we can consider the case where \(b_{C}=+1\) and \(b_{I}=-1\); in fact, this is mathematically identical to the case of \(b_{C}=-1\) and \(b_{I}=+1\). Motivated by this consideration, we can define the following success/error rule for the demodulator, based on the phase values at the beginning and at the end of the chip interval:  
\[\Theta_{R}(\pi/2)\gtrapprox e\Theta_{R}(0) \tag{22}\]  
where \(s\) means success and \(e\) means error. When \(b_{C}=+1\) and \(b_{I}=-1\), the received signal is:  
\[R(\omega t)=\sqrt{C}e^{j(\omega t+\phi_{C})}+\sqrt{I}e^{j(-\omega t+\phi_{I})} \tag{23}\]  
To obtain an analytical expression to \(\Theta_{R}(\omega t)\), we assume that during a chip interval, \(R(\omega t)\) always stays on the definition domain of \(\arctan\) (_i.e.,_\([-\pi/2\;\pi/2]\)). Starting from Equation (23), the instantaneous phase function can be written as:  
\[\Theta_{R}(\omega t)=\arctan\left(\frac{\sqrt{C}\sin(\omega t+\phi_{C})+\sqrt {I}\sin(-\omega t+\phi_{I})}{\sqrt{C}\cos(\omega t+\phi_{C})+\sqrt{I}\cos(- \omega t+\phi_{I})}\right) \tag{24}\]  
With no loss of generality, we can assume \(\phi_{c}=0\). Now, after some trigonometric simplifications, the phases of the received signal at the beginning and at the end of the chip interval can be written as:  
\[\Theta_{R}(0) =\arctan\left(\frac{\sqrt{I}\sin(\phi_{I})}{\sqrt{C}+\sqrt{I} \cos(\phi_{I})}\right)\] \[\Theta_{R}(\pi/2) =\arctan\left(\frac{\sqrt{C}-\sqrt{(}I)\cos(\phi_{I})}{\sqrt{I} \sin(\phi_{I})}\right)\]  
After replacing \(\Theta_{R}(0)\) and \(\Theta_{R}(\pi/2)\) in Equation (22), the decision rule becomes:\[\arctan\left(\frac{\sqrt{C}-\sqrt{I}\cos(\phi_{I})}{\sqrt{I}\sin(\phi_{I})}\right) \underset{e}{\overset{s}{\gtrless}}\arctan\left(\frac{\sqrt{I}\sin(\phi_{I})}{ \sqrt{C}+\sqrt{I}\cos(\phi_{I})}\right) \tag{25}\]  
by recalling that \(\arctan(\cdot)\) is monotonically increasing in \([-\pi/1\ \ \pi/2]\), Equation (25) becomes:  
\[\frac{\sqrt{C}-\sqrt{I}\cos(\phi_{I})}{\sqrt{I}\sin(\phi_{I})}\underset{e}{ \overset{s}{\gtrless}}\frac{\sqrt{I}\sin(\phi_{I})}{\sqrt{C}+\sqrt{I}\cos( \phi_{I})} \tag{26}\]  
to keep the trajectory of \(R(\omega t)\) in the definition domain of \(\arctan\), we can restrict the definition domain of \(\phi_{I}\) as \([0,\pi/2]\). With this assumption, \(\sqrt{C}+\sqrt{I}\cos(\phi_{I})\) and \(\sqrt{I}\sin(\phi_{I})\) will be always positive.