# Environmental Impacts on Hardware-Based Link Quality Estimators in Wireless Sensor Networks
## 4 Typical Hardware-Based LQEs: A Survey
### Mapping Models between Communication Distance and PRR  
According to Figure 3, the theoretical model and fitting model are usually used as mapping models between communication distance and \(PRR\). In [12], a theoretical model between communication distance and \(PRR\) is proposed as follows:  
\[PRR=\left(1-Q\left(\sqrt{2\cdot 10^{\left(P_{t}+L_{C}-PL(d_{0})-10n\log_{10} \left(\frac{d}{d_{0}}\right)+X_{o}-P_{n}\right)/10}\cdot B_{N}/R}\right) \right)^{SI} \tag{2}\]  
where \(Q(\cdot)\) represents the Q function, \(P_{t}\) is the transmit power (dBm), \(L_{c}\) is the signal strength gain (or loss, if its value is negative) in the hardware circuit (dB), \(n\) is the path loss exponent characterizing the attenuation of wireless signals in the environment, \(d\) is the distance between the transmitter and receiver (m), \(d_{0}\) is the reference distance and its value is usually 1 m, \(PL(d_{0})\) is the free-space path loss (dBm) at the reference distance, \(X_{o}\) is a normally distributed random variable with a mean of zero and a standard deviation of \(\sigma\) (dB), \(P_{n}\) is the background noise power (dBm), \(B_{N}\) is the noise bandwidth of the transceiver (kHz), \(R\) is the communication data rate (kb/s), and \(l\) is the length of the data packet (bytes). Among which, the value of \(PL(d_{0})\) can be calculated as follows [12]:  
\[PL(d_{0})=32.44+20\log_{10}(d_{0}\cdot f) \tag{3}\]  
where \(f\) is the carrier frequency (MHz).  
As CC2420 was used in [12], the values of \(B_{N}\) and \(R\) are 384 kHz and 250 kb/s, respectively. Further, the transmit power \(P_{t}\), carrier frequency \(f\), and packet length \(l\) were set to 0 dBm, 2480 MHz (corresponding to channel 26), and 17 bytes, respectively. Therefore, substituting Equation (3) into Equation (2), we have  
\[PRR=\left(1-Q\left(\sqrt{3\cdot 10^{\left(L_{C}-32.44-20\log_{10}\left(2400 \right)-10n\log_{10}\left(d\right)+X_{o}-P_{n}\right)/10}}\right)\right)^{136} \tag{4}\]  
With the measured data in the runway, relevant parameters of Equation (4) were determined, as shown in Table 2. Standard deviation \(\sigma\) of the random variable \(X_{o}\) was calculated from the variances of \(RSSI\) at different distances, the measured background noise power was used as \(P_{n}\), and \(Lc\) and \(n\) are obtained by the least squares fitting method.  
Therefore, the mapping model between the communication distance \(d\) and \(PRR\) in the runway could be obtained by substituting the above parameters into Equation (4), as expressed as  
\[PRR=\left(1-Q\left(\sqrt{3\cdot 10^{\left(97.6199-20\times\log_{10}\left(2400 \right)-15.871\times\log_{10}\left(d\right)+X_{o}\right)/10}}\right)\right)^{136} \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(L_{c}\) & 31.6899 dB \\ \(n\) & 1.5871 \\ \(\sigma\) & 1.0081 dB \\ \(P_{n}\) & \(-98.37\) dBm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Relevant parameters of Equation (4).  
In [11], the mapping model between communication distance and \(\mathit{PLR}\) was constructed by data fitting, expressed as follows  
\[\mathit{PLR}=\begin{cases}0,&0\leq d<d_{1}\\ a_{1}\text{atan}(a_{2}d+a_{3})+a_{4},&d_{1}\leq d<d_{2}\\ 1,&d>d_{2}\end{cases} \tag{6}\]  
where \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\) are fitting parameters. With the measured data in the runway, the fitting model between communication distance \(d\) and \(\mathit{PRR}\) was obtained, as shown in Equation (7).  
\[\mathit{PRR}=\begin{cases}1,&0\leq d<18\\ 0.5094-0.3401\times\text{atan}(0.3316\times d-13.72),&18\leq d<82\\ 0,&d>82\end{cases} \tag{7}\]