## 4 Probability of Chip Error in Non-Coherent O-QPSK  
Normally, O-QPSK requires coherent detection [26]; however with the half-sine pulse shaping in 2.4 GHz PHY of IEEE 802.15.4, the information-bearing part of the signal is not only the carrier phase, but also the complex-envelope of the signal. Thus, without recovering the carrier phase, just observing the phase changes of the complex envelope, transmitted information can be extracted. In 2.4 GHz PHY of IEEE 802.15.4, the phase of the complex envelope increases or decreases by an amount of \(\pi/2\) every \(T_{c}\) seconds. In this sense, the behavior of the modulation is similar to that of minimum shift keying (MSK). The main difference between the two modulations is that they use different symbol mappings. In particular, symbol mapping in IEEE 802.15.4 is shown in Figure 5.  
Figure 4: Chip error rate: the case of coherent O-QPSK demodulation. CER, chip error rate; SIR, signal-to-interference ratio.  
Figure 5: Phase transitions of the complex envelope. (**a**) Example phase transition sequence; (**b**) All possible phase transitions.  
Chip intervals in Figure 5a are marked as \(T_{C1},T_{C2},...\). In each even chip interval, \(T_{C2},T_{C4},...\), a symbol (\(m_{0},m_{1},m_{2},m_{3}\)) is transmitted, and in the odd chip intervals, \(T_{C3},T_{C5},...\), there are phase changes (\(t_{4},t_{5},t_{6},t_{7}\)), which can be considered as the transitions between two sequential O-QPSK symbols. For instance, in Figure 5a, a sequence of phase changes are shown. At the beginning of the \(T_{C2}\) chip interval (_i.e., \(\omega t=0\)_), the phase of the signal is equal to the carrier phase, \(\phi_{C}\); then at the end of the same chip interval, the phase reaches \(\phi_{C}+\pi/2\), which is indicated in Figure 5b by the arc, \(m_{3}\). In the next chip interval, \(T_{C3}\), the phase reaches the \(\phi_{C}+\pi\) value, which is represented by the arc, \(t_{5}\). The sequential chip intervals send the phase of the complex envelope, \(\pi/2\), further or back with respect to the transmitted symbols. All possible phase transitions are shown in Figure 5b. This relationship between phase transition and transmitted symbol suggests the idea of constructing a non-coherent detector, which identifies the transmitted chip based on the phase transitions of the complex envelope. Using such a detector, \(P_{c}\) will refer to the probability of the erroneous detection of phase transitions during a chip interval. In the absence of interference, the complex envelope of the received chip through a chip duration in the Q-I plane can be expressed as:  
\[R(\omega t)=\sqrt{C}e^{j(\pm\omega t+\phi_{C})},\ \ \ \omega t\epsilon[0,\pi/2],\ \ \ \phi_{c} \epsilon[0,2\pi] \tag{17}\]  
where \(C\) is the energy of the signal and \(\phi_{C}\) is the uniformly distributed carrier phase. The definition domain of \(\omega t\) is from zero to \(\pi/2\), because the signal can only move \(\pi/2\) further on the Q-I plane during a chip interval. The instantaneous phase of \(R(\omega t)\) in Equation (17) is:  
\[\Theta_{R}(\omega t)=\pm\omega t+\phi_{C} \tag{18}\]  
and the phase of the received signal at the beginning and at the end of the chip interval is:  
\[\Theta_{R}(0) =\phi_{C}\] \[\Theta_{R}(\pi/2) =\pm\pi/2+\phi_{C}.\]  
Since the carrier phase can be eliminated by subtracting \(\Theta_{R}(\pi/2)\) and \(\Theta_{R}(0)\), the following discriminator demodulator can be applied [28]:  
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