# Exploiting Concurrent Wake-Up Transmissions Using Beat Frequencies
## 3 Concurrent Wake-Up Message
### More than Two Concurrent Senders  
In a wireless sensor network, it is probable that more than two senders are in wake-up range, and the superposition of more than two sinusoidal waves can occur. In Section 3.1 above, we discussed that it is possible to transmit a wake-up message from two concurrent senders having frequencies \(f_{1}\) and \(f_{2}=f_{1}+f_{w}/2\) and amplitudes \(A_{1}\) and \(A_{2}\). The results obtained in Section 3.1 demonstrate that the amplitude of the larger signal, for example \(A_{1}\), generates a constant offset on the wake-up signal. Consequently, if a third node has the same frequency as the node with the lower amplitude, the wake-up signal can be increased, as shown by Equation (4), which enhances Equation (3) by a third sinusoidal signal with amplitude \(A_{3}\) and with frequency \(f_{2}\):  
\[y(t)=e^{j\omega_{1}t}(A_{1}+(A_{2}+A_{3})e^{j(\omega_{2}-\omega_{1})t}). \tag{4}\]  
Figure 10: (a) Envelope for equal amplitudes \(A_{1}=1\) and \(A_{2}=1\); (b) envelope for \(A_{1}=0.7>A_{2}=0.4\). The signal \(y(t)\) expressed by Equation (4) is graphically reported in Figure 11. Note that the amplitudes in Plots (a) and (b) are \(A_{1}=0.8\), \(A_{2}=0.1\) and \(A_{3}=0.4\). The frequencies in (a) are \(f_{1}=868\,\mathrm{MHz}\), \(f_{2}=868.125\,\mathrm{MHz}\) and \(f_{3}=868\,\mathrm{MHz}\). In this case, where \(f_{3}=f_{1}\) and \(A_{1}+A_{3}>A_{2}\), the amplitude \(A_{3}\) of the third node adds to the DC-offset, but not to the beat. In Figure 10(b), the frequencies are \(f_{1}=868\,\mathrm{MHz}\), \(f_{2}=868.125\,\mathrm{MHz}\) and \(f_{3}=868.125\,\mathrm{MHz}\). In this case, \(A_{3}\) adds to the beat and increases its amplitude. However, due to inaccuracies of the crystal oscillators, frequency \(f_{3}\) of the third node will always be slightly different from \(f_{2}\) or \(f_{1}\). The utilized radio crystals often have an accuracy of 10 to 40 ppm in the kilohertz range, and Equation (4) has to be rewritten. Using \(\omega_{1}=\overline{\omega}-\Delta\omega\), \(\omega_{2}=\overline{\omega}+\Delta\omega\) and \(\omega_{3}=\overline{\omega}+\alpha\Delta\omega\), we get:  
\[y(t)=e^{i\overline{\omega}t}(A_{1}e^{-i\Delta\omega t}+A_{2}e^{i\Delta\omega t }+A_{3}e^{i\Delta\omega t}) \tag{5}\]  
where \(\alpha\) indicates the frequency offset of the third node. Equation (5) indicates that the frequency offset of the third node generates an additional beat (in the lower kilohertz range) that might disturb the wake-up signal. Additionally, the phase of the third node is probably shifted, compared to that of the other nodes, and destructive interference can occur. Due to these limitations, each additional node will likely generate additional frequencies and offsets that make an undisturbed wake-up signal more unlikely, although several wake-up receivers [20; 21] have a built-in automatic gain control unit that can help to mitigate this effect. By expanding Equation (3) by an additional pair of nodes sending at \(2\pi f_{3}=\omega_{3}\) and \(2\pi f_{4}=\omega_{4}\), we get Equation (6):  
\[y(t)=e^{i\omega_{1}t}(A_{1}+e^{i(\omega_{2}-\omega_{1})t}(A_{2}+e^{i(\omega_{3 }-\omega_{2})t}(A_{3}+A_{4}e^{i(\omega_{4}-\omega_{3})t}))) \tag{6}\]  
Considering Equation (6) reveals that four concurrently sending nodes generate the desired superposed beat frequencies in case \(f_{2}-f_{1}=f_{4}-f_{3}\). There is also an additional frequency generated by the offset of frequencies \(f_{3}-f_{2}\). This offset is present due to oscillator inaccuracies. By setting \(f_{3}-f_{2}\) purposefully in the MHz-range, these effects can be reduced, as the additional beats will also be in the MHz range and will be low-passed by the envelope detector. What is left are the inaccuracies of \(f_{4}-f_{3}\) not being exactly \(f_{2}-f_{1}\). Equation (6) indicates that \(n\) nodes can generate \(n/2\) beat wake-up frequencies. However, due to frequency inaccuracies, there will be superposed additional frequencies.