# Revisiting the Analysis of Radiative Mid-Range Wireless Link for Pacemakers
## 2 Materials and Methods
### Mid-Range Wireless Link Formulas
#### 2.2.3 Integral Coupling Formula  
In the formula, a fast Fourier transform (FFT) has been used to obtain the near-field coupling from both the magnitude and phase of the far-field pattern [32; 33; 34; 35; 36; 37]. The requirements of the formula are the complex far-field pattern and the antenna placements of both the on-axis and rotational scenarios. The formula requires the discretized sampling and implementation of the integral based on the FFT method. The use of the FFT method makes it possible to increase the computational efficiency, compared to a Fourier transform. The far-field patterns obtained at the phase center of an antenna are an important requirement for the FFT calculation. The first procedure is to transform the origin of the far-field pattern obtained from simulation or measurement into the global coordinate system. Eulerian angle transformation is used to allow origins of the far-field patterns to be located in the global coordinate system. The two-dimensional interpolation technique is used to obtain the complex far-field patterns of two antennas at sample points of the discretization. A distribution of far-field patterns confined in the rectangular plane provides the spatial bandwidth of the FFT. The sampling frequency is determined by the bandlimited characteristics using Nyquist theorem, and bandlimits the transverse displacement between two antennas.  
\[f_{s}=2\kappa\times(D_{t}+D_{r}) \tag{8}\]  
where \(\kappa\) is the oversampling ratio. The summation form of the coupling quotient presented in [33] can be defined as  
\[\begin{array}{c}\frac{b^{\prime}{}_{0}}{a_{0}}(\stackrel{{ \rightarrow}}{{R}})=\\ \times\sum\limits_{m}\sum\limits_{n}\frac{\stackrel{{\rightarrow} }{{f}_{TX}(k_{x}^{m},\,k_{y}^{m})\stackrel{{\rightarrow}}{{ \delta}}_{RX}(k_{x}^{m},\,k_{y}^{m})}}{k_{x}^{m}}\times e^{i\stackrel{{ \rightarrow}}{{k}}\stackrel{{\rightarrow}}{{\ldots}} \stackrel{{\rightarrow}}{{R}}}\end{array} \tag{9}\]  
where \(\stackrel{{\rightarrow}}{{f}}_{TX}\left(k_{x}^{mn},\,k_{y}^{m}\right)\) and \(\stackrel{{\rightarrow}}{{\delta}}_{RX}\left(k_{x}^{mn},\,k_{y}^{ mn}\right)\) are 3D vector far-field patterns of the transmitting and receiving antennas, respectively, and \(k=\hat{x}k_{x}^{mn}+\hat{y}k_{y}^{mn}+\hat{z}k_{z}^{mn}\) and \(\Delta k=\frac{2\pi}{f_{s}}=\frac{\pi}{\kappa(D_{t}+D_{r})}\cdot a_{0},b_{0}\) represent amplitudes of input wave to transmitting and output wave from receiving antenna, respectively, where there is a relationship such as \(S_{21}=\left|b^{\prime}{}_{0}/a_{0}\right|^{2}\). The impedance mismatch term \(C\) can be defined as  
\[C=\frac{Z_{FL,\,RX}}{Z_{0}}\frac{1}{(1-\Gamma_{0,\worth noting that the Equation (9) has \(e^{-i\mathrm{wt}}\) time dependence in free space, and multiple interactions are ignored.  
With an effort to extend the effective range, it is demonstrated that the far-field approximation to the integral coupling is well converged into the Friis formula. A valid range of the integral coupling, therefore, includes radiating both the near-field region and far-field region.