# Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments
## 2 System Model
### Transmitter Architecture Design  
We exploited eight dipole antennas with ultra-wideband (UWB) to form an array in a circle for the transmitter, as seen in Figure 2.  
A UWB printed dipole antenna is the individual element for the array. The radiation pattern is omnidirectional in the azimuth plane. The pattern of the array antenna is closely related to the arrangement of the antenna, as well as to the amplitude and the phase of the element of the antenna. With N antennas arranged uniformly in a circular array, the antenna pattern is assumed to be isotropic, and the total field can be expressed as  
\[E_{tot}(r,\theta,\phi)=\sum\
olimits_{n=1}^{N}\frac{e^{-jkR_{n}}}{R_{n}},\ R_{n}= \left(r^{2}+d^{2}-2 dros\psi_{n}\right) \tag{1}\]  
where \(\theta\) and \(\phi\) represent the spherical coordinate system, \(a\) is the distance from original position to antenna position, and \(\theta_{n}\) and \(\phi_{n}\) are the spherical coordinates of the transmitting antenna (\(\theta_{n}=\theta+\psi_{n}\)). For the far field, the electrical field can be expressed as  
\[E_{tot}(r,\theta,\phi)\approx\sum\
olimits_{n=1}^{N}\frac{e^{-jk(r-d(r\ \tau^{\prime}))}}{r}=\sum \
olimits_{n=1}^{N}A_{n}\frac{e^{-jk\tau}}{r},\ \textit{where}\ A_{n}\ \textit{is the phase term}. \tag{2}\]  
Figure 1: Network model of the system.  
Figure 2: Circular ultra-wideband (UWB) dipole antennas.  
Note that  
\[\begin{array}{c}\dot{a}_{n}\dot{a}_{r}=(\dot{a}_{x}cos\phi_{n}sin\theta_{n}+ \dot{a}_{y}sin\phi_{n}sin\theta_{n}+\dot{a}_{z}cos\theta_{n})\cdot\\ (\dot{a}_{x}cos\phi\ sin\theta+\dot{a}_{y}sin\phi\ sin\theta+\dot{a}_{z}cos \theta)\\ =cos\phi_{n}sin\theta_{n}cos\phi\ sin\theta+sin\phi_{n}sin\theta_{n}sin\phi\ sin \theta+cos\theta_{n}cos\theta\\ =sin\theta\ sin\theta_{n}cos(\phi-\phi_{n})+cos\theta_{n}cos\theta.\end{array} \tag{3}\]  
The factor of this antenna array is written as  
\[AF_{t}(\theta,\varphi,f)=\sum\
olimits_{n=1}^{N_{t}}I_{n}e^{j\alpha_{n}(f)e^ {ikSin\theta cos(\varphi-\varphi_{n})}}. \tag{4}\]  
where \(I_{n}\) is the excitation current voltage, \(\mathrm{k}=2\pi\lambda\) is the wave-number (\(\lambda\) is the wavelength of the sinusoidal wave), and \(\mathrm{R}\) is the radius of the circle. \(\alpha_{n}(f)\) is the phase delay in different frequencies, which corresponds to the feed line length, \(l_{n}\). \(\alpha_{n}(f)=-\frac{2\pi f}{c}\sqrt{\varepsilon_{r}}l_{n}\), where \(\mathrm{C}\) is the light speed and \(\varepsilon_{r}\) is the permittivity of the feed line. We exploited eight antennas at the transmitter, positioned at \(\varphi_{\mathrm{n}}=0\), 45, 90, 135, 180, 225, 270, and 315 degrees in the circle. Equation (4) derives the array factor of the transmitter as the following equation:  
\[AF(\theta,\varphi,f)=\sum\
olimits_{n=1}^{N}I_{n}e^{jkRsin\theta cos(\varphi- \frac{2\pi}{N}(n-1))}. \tag{5}\]  
The total radiation field of the array antenna is the product of the element radiation pattern and the array factor. The vector of the total radiation at the transmitting site is described as follows:  
\[\mathrm{N}(\theta,\varphi,f)=\mathrm{AF_{t}(\theta,\varphi,f).N_{e}(\theta, \varphi,f)}. \tag{6}\]  
Here, the three-dimensional radiation field vector \(\mathrm{N_{e}(\theta,\varphi,f)}\) can be calculated by the structure simulation of the high-frequency.