## 2 Related Work
### Experiment Setup for the Investigation of the Impact of Attenuation on the Packet Error Rate
#### 3.1.2 Packet Error Rate Measurement of SPIRIT1 Transceiver  
To measure the reliability of the SPIRIT1 transceiver, a modification to the protocol was made. Because the WuRx was not tested, the SPIRIT1 transceiver directly sent the data packet sequence to the receiver without a WUC. The receiver was always in idle mode, listening to the channel for arriving packets. When the data were successfully transmitted, an acknowledgment (ACK) was sent back to the transmitter. At the end, the receiver counted the sum of successfully received packets. The sequence diagram in Figure 11 describes the communication protocol used to test the PER SPIRIT1 transceiver. The same setup for the previous experiment in Section 3.1.1 was used. The transmission output power was programmed to be \(-34\) dBm to reach the sensitivity limit level of the SPIRIT1 transceiver. The experiment was repeated several times, starting from 62 dB to the 90 dB level of attenuation, with a step size of 0.5 dB. The count of the lost packets is plotted in Figure 12 to demonstrate the impact of attenuation on the PER using the SPIRIT1 transceiver. When the attenuation rose from 70 dB to 75 dB, the PER changed from 6% to 100%. This showed the limit of the sensitivity of the SPIRIT1 transceiver; at this point, the transmission power was preserved, and the level of attenuation was 70 dB. The SPIRIT1 transceiver was attached to the spectrum analyzer to obtain the received output power, which was \(-107\) dBm. Figure 11: Communication sequence diagram using SPIRIT1 transceiver. Theoretical Model of the Packet Error Rate Variation for Wake-Up Receiver Based on AS3933 and SPIRIT1 Transceiver  
After collecting the measurement data, and inspecting, transforming, and cleansing the data, the second stage was to create a data model for the information system by applying certain formal techniques to extract the system behavior. Using the collected data explained in the previous section, we then theoretically modeled the PER, using a ML algorithm that uses nonlinear regression mathematical models based on supervised learning. A study [21] proposed a theoretical model based on PRR in IEEE802.15.4 2.4 GHz technology using the Q-function, which is the tail distribution function of the standard normal distribution. In this study, the model was built using a nonlinear regression function known as the error function, the error function--also called the Gauss error function--is a special (non-elementary) sigmoid function that often occurs in probability, statistics, and partial differential equations. The representation of the function is shown as follows:  
\[erf(x)=\frac{1}{\sqrt{2\pi}}\int_{0}^{x}e^{-\frac{t^{2}}{2}}dt \tag{4}\]  
Because different curves are applied for various hardware and the needs of the input parameters for the regression required us to define an error function applicable to the collected data, the input parameters were:  
* The transition interval, which is the interval of attenuation that passes the PER from 0% to 100% or the inverse. * The sensitivity level, which is the center of the transition interval to define the sensitivity level of the hardware. \[PER(RSSI|_{dBm})=\frac{1}{\sqrt{2\pi}}\int_{0}^{\frac{(RSSI|_{dBm})-d}{l}}e^{- \frac{t^{2}}{2}}dt+\frac{1}{2}\] (5)  
This equation can be applied to the extracted PER set of data for both the transceiver and WuRx, with \(d\) being the sensitivity level and \(l\) being the transition interval multiplied by 4. The built-in ML algorithm applied nonlinear regression and used the ordinary least square (OLS) method in order to minimize the deviation between the estimated and real curves. Figure 13 shows the estimated modeling function for the PER in the SPIRIT1 transceiver according to the RSSI. The calculation result showed a \(-111.27\) dBm sensitivity and an around 6.9 dB transition interval. Compared with the demonstrated measurement shown in  
Figure 12: Variation in the PER according to the attenuation factor using SPIRIT1 communication. Section 3.1.2 the rise in the PER started at \(-107\) dBm.