# Exploiting Concurrent Wake-Up Transmissions Using Beat Frequencies
## 3 Concurrent Wake-Up Message
### Two Concurrent Senders  
In case of two concurrently-sent wake-up packets, a possible and probable existing phase-shift \(\phi\) of one of the signals has to be considered. There will be a superposed signal \(y(t)\) of the form of Equation (1):  
\[y(t)=A_{1}e^{i\omega_{1}t}+A_{2}e^{i\omega_{1}t}=(A_{1}+A_{2})e^{i\omega_{1}t} \tag{1}\]  
with \(A_{1}\) and \(A_{2}\in\mathbb{C}\) such that \(A_{1}=r_{1}e^{i\phi_{1}}\) and \(A_{2}=r_{2}e^{i\phi_{2}}\) with \(r_{1},r_{2}\in\mathbb{R}\) are the signal amplitudes and \(\phi_{2}-\phi_{1}\) the phase shift between the two signals. Furthermore, \(\omega_{1}=2\pi f_{1}\) is the angular frequency of the two superposed sinusoidal waves. However, due to the periodicity of the sinusoidal signals, a destructive interference will take place in the range of \(2/3\pi<\phi_{2}-\phi_{1}<4/3\pi\), which will finally lead to an unreliable communication. The superposed signal \(y(t)\) as introduced in Equation (1) above is the sum of two sinusoidal signals having the same frequency \(f_{1}\). The following Equations (2) and (3) show the same sum of two sinusoidal functions having different frequencies \(f_{1}\) and \(f_{2}\), but highlight different aspects. Equation (2) expresses \(y(t)\) as the sum of two sinusoidal functions with angular frequencies \(\omega_{1}=2\pi f_{1}\) and \(\omega_{2}=2\pi f_{2}\):  
\[y(t)=A_{1}e^{i\omega_{1}t}+A_{2}e^{i\omega_{2}t}. \tag{2}\]  
By replacing \(\omega_{2}=\omega_{1}+(\omega_{2}-\omega_{1})\), Equation (2) can be rewritten to achieve Equation (3):  
\[y(t)=e^{i\omega_{1}t}(A_{1}+A_{2}e^{j(\omega_{2}-\omega_{1})t}). \tag{3}\]  
From Equation (3), it can be seen that the amplitudes \(A_{1}\) and \(A_{2}\) of the original sinusoidal signals are preserved by the addition. By expressing \(\omega_{1}\) as \(\omega_{1}=\overline{\omega}-\Delta\omega\) and \(\omega_{2}\) as \(\omega_{2}=\overline{\omega}+\Delta\omega\) and by replacing \(\Delta\omega\) with \((\omega_{2}-\omega_{1})/2\) and substituting \(\overline{\omega}\) with \((\omega_{1}+\omega_{2})/2\), it can be seen that the summation generates two new frequencies that we here call \(f_{c}\) and \(f_{b}\). The first frequency, \(f_{c}\), equals \((f_{1}+f_{2})/2\), and the second frequency, \(f_{b}\), equals \((f_{2}-f_{1})/2\). The lower frequency \(f_{b}\) is called the beat, as it appears on the left side in Figure 7, which illustrates the spectrum resulting from the sum of frequencies \(f_{1}\) and \(f_{2}\) of \(y(t)\). According to Equation (3), \(f_{c}\) is just between \(f_{1}\) and \(f_{2}\). Equation (3) also shows that a phase-shift \(\phi\) of one of the high-frequency signals is a phase-shift of the beat signal, but no destructive interference between \(A_{1}\) and \(A_{2}\) takes place. Figure 6: Node-1 and Node-2 send a concurrent wake-up to Node-3. Figure 7: Spectrum of the sum of two sinusoidal functions having frequencies \(f_{1}\) and \(f_{2}\) and the resulting carrier and beat frequencies \(f_{c}\) and \(f_{b}\), respectively. With the help of Equation (3), we just demonstrated that a low-frequency signal can be generated by adding two sinusoidal signals with \(f_{1}\) and \(f_{2}\). Now, assume that a wake-up receiver listens on a certain frequency \(f_{w}\), for example \(125\,\mathrm{kHz}\), for incoming messages. From the discussion above, it follows that \(f_{w}\) can be generated by adding two sinusoidal signals with frequencies \(f_{1}\) and \(f_{2}\) that are just separated about the right amount from each other. In the example above, with \(f_{w}=125\,\mathrm{kHz}\), this would be \(125\,\mathrm{kHz}\), since the envelope detector, as long as it consists of two diodes, doubles the frequency during rectification. By switching the \(f_{1}\) and \(f_{2}\) transmitting radios synchronously on and off, frequency \(f_{b}\) can be further on-off keying-modulated to achieve a certain required wake-up pattern. Changing only one of the frequencies \(f_{1}\) or \(f_{2}\) according to a certain timing, \(f_{b}\) can be frequency-shift keying modulated.