## 3 Analytical Expressions
### Path and Antenna Losses  
The frequency band of 868 MHz is used as the radio channel for the wake-up signal. This frequency band is regulated in terms of transmission power and duty cycle and often used for many application, e.g., building automation. The use of high frequencies for the wake-up signal leads to high power consumption in the receiver due to switching losses. On the other hand, the use of low frequencies leads to larger antenna and lower gain [34; 36]. The amplitude shift keying (ASK) is known for its simplicity to modulate the  
Figure 4: Signal timings of wake-up intervals.  
carrier signal and create the WuPt sequence pattern. By switching the carrier frequency on and off--OOK--the pattern is generated. Therefore, the OOK modulation enables a simple and energy-efficient hardware design for demodulation, e.g., by using an envelope detector consisting only of diodes. Here, the frequency and phase remain unchanged and only the amplitude varies to represent the two binary values. When the WuPt is received, the envelope detector demodulates the high-frequency carrier signal to obtain a low-frequency wake-up signal [38]. A disadvantage of ASK modulation is that noise and interference affect the amplitude. Another modulation option for wake-up receivers is frequency shift keying (FSK), as it is less susceptible to noise compared to ASK. However, the FSK circuit consumes more power because a frequency synthesizer is needed to distinguish between the central frequency and the frequency deviation.  
In indoor applications, interferences, as well reflections, diffraction, and attenuation, are expected and should be taken into consideration. For that purpose, many research activities focused on the development of path loss models to predict the propagation of electromagnetic waves. The basic path loss model is the free space model (Equation (4)) [39], which predicts an inverse square dependence of the average received power on the distance between the transmitter and receiver.  
\[\frac{P_{\mathrm{r}}}{P_{\mathrm{t}}}=G_{\mathrm{t}}\cdot G_{\mathrm{r}}\cdot \left(\frac{\lambda}{4\pi d}\right)^{2} \tag{4}\]  
where \(P_{\mathrm{r}}\) and \(P_{\mathrm{t}}\) are the received and transmitted signal powers, \(G_{\mathrm{r}}\) and \(G_{\mathrm{t}}\) are the gains of the antennas, \(d\) is the distance between antennas, and \(\lambda\) is the wavelength. Modified models attempt to account for the complex nature of real wireless channels.  
The International Telecommunication Union (ITU) model for indoor attenuation considers loss through multiple floors, as well as transmission through walls. The coefficients from [40] implicitly take into account obstacles and other loss mechanisms that may occur inside a floor in buildings. The basic model is given by  
\[L_{\mathrm{total}}[\mathrm{dB}]=L(d_{0})+N\log_{10}\frac{d}{d_{0}}+L_{\mathrm{ f}}(n) \tag{5}\]  
where \(N\) is the distance power loss coefficient, \(L_{\mathrm{f}}\) the floor penetration loss factor, \(L(d_{0})=20\,\mathrm{dB}\cdot\log_{10}f-28\,\mathrm{dB}\) is basic transmission loss at reference distance \(d_{0}\) and carrier frequency \(f\) in MHz and assuming free-space propagation. Figure 5 shows the losses for a distance from up to \(40\,\mathrm{m}\) across one floor and through walls.  
Figure 5: Theoretical attenuation over a distance up to \(40\,\mathrm{m}\).