# Revisiting the Analysis of Radiative Mid-Range Wireless Link for Pacemakers
## 2 Materials and Methods
### Mid-Range Wireless Link Formulas
#### 2.2.1 Asymptotic Friis Formula  
The Friis formula, derived from plane-wave characteristics, has been widely used for the long-range wave propagations [27]. A configuration of the power transmission between  
\begin{table}
\begin{tabular}{c c c c} \hline
**Tissue** & **Permittivity (\(\epsilon_{r}\))** & **Conductivity (\(\sigma\)[S/m])** & **Mass Density (\(\rho\)[kg/m\({}^{3}\)])** \\ \hline Skin & 46.74 & 0.69 & 1100 \\ \hline \end{tabular}
\end{table}
Table 1: Electrical properties of human tissue (skin) at 400 MHz.  
Figure 1: Varied radiation patterns of the parabolic reflector antenna with a radius of 10\(\lambda\) at the different positions of the non-radiative near-field region and the radiative near-field region.  
two antennas is depicted in Figure 2. A ratio between transmitted power \(P_{t}\) and received power \(P_{r}\) between two antennas can be obtained using the Friis formula, as follows:  
\[\frac{P_{r}}{P_{t}}=\left(\frac{\lambda}{4\pi R}\right)^{2}G_{t}(\theta_{t}, \varphi_{t})G_{r}(\theta_{t},\varphi_{t})|\hat{\rho}_{t}\cdot\hat{\rho}_{r}|^{2} \tag{4}\]  
where \(G(\theta,\varphi)\) is the far-field gain in terms of angular variables of \((\theta,\varphi)\), and \(R\) is the separation distance between two antennas. \(\hat{\rho}\) is the polarization vector. The subscripts \(t\) and \(r\) represent the transmitting and receiving antenna, respectively. Although it predicts well in the far-field region, a comparison between the Friis formula and measurements imply that the Friis formula is inaccurate in a short distance [28; 29]. The discrepancy in the radiative near-field region is owing to the plane-wave approximation that is suitable for the far-field region. The Friis formula can be incorporated with an asymptotic correction term for accurately calculating the mid-range wave propagation. The asymptotic term, as a function of distance \(R\) and far-field distance \(2D^{2}/\lambda\), has been used to include the gain reduction effect in the radiative near-field region. The problem of the formula is the intense computation of the gain reduction factor required for each antenna. It has been found in [29] that the gain reduction factors tend to converge through taking the slightly modified form of the asymptotic term presented in Equation (3). The modification is beneficial in terms of its applicability to a variety of antennas with rapid computations. The use of the asymptotic expression, however, requires slightly larger minimum ranges where all of the radiative near-field region is not covered. The Friis formula with the gain reduction factors \(\gamma\) can be defined as  
\[\frac{P_{r}}{P_{t}}=\frac{\lambda^{2}}{16\pi^{2}R^{2}}G_{t}\gamma_{t}(R)G_{r} \gamma_{r}(R) \tag{5}\]  
In order to include both cases of low and high gain antennas, the switching function \(F(G)\) is incorporated with the gain reduction factor \(\gamma\). The gain reduction factors of the transmitted antenna \(\gamma_{t}\) and received antennas \(\gamma_{r}\) can be defined as  
\[\gamma_{t}=1-\alpha_{E}F(G_{t})\left(\frac{R}{2\lambda G_{t}/\pi^{2}}\right)^{ -2},\ \ \gamma_{r}=1-\alpha_{E}F(G_{r})\left(\frac{R}{2\lambda G_{r}/\pi^{2}}\right)^{ -2} \tag{6}\]  
where \(F(G)=2.5-\frac{\pi}{\pi}\cdot\arctan[c\cdot(G-b)]\) and \(\alpha_{E}=0.066\).  
In this work, the parameters, \(a=3\), \(b=5\), and \(c=1\), are selected for smooth switching. The other technique based on the Gaussian beam has been applied to both the radiative near-field region and far-field region. It is worth noting that \(G_{t}\) and \(G_{r}\) included in the correction term are substituted with the directivity of the transmitting and receiving antenna, respectively.  
Figure 2: Configuration of the power transmission between a Tx antenna and a Rx antenna.