# Guard Band Protection Scheme to Facilitate Coexistence of 5G Base Stations and Radar Altimeters
## 2 Modelling for Guard Band Protection and Analysis
### Mathematical Model for Guard Band Protection  
The power of the interference signal received by the radar altimeter from the \(i\)th 5G base station is calculated as [20]  
\[\mathrm{I_{i}}=\mathrm{P_{t,i}}+\mathrm{G_{BS,i}}+\mathrm{G_{RA}}-\mathrm{L_{ total,i}}-\mathrm{ACIR}, \tag{1}\]  
where \(P_{t,i}\) is the transmitted power of a single 5G base station, \(G_{BS,i}\) is the antenna gain of the \(i\)th 5G base station, and \(G_{RA}\) is the antenna gain of the radar altimeter. \(ACIR\) is the adjacent channel interference ratio, and \(L_{total,i}\) is the total loss. The gain of a 5G base station is determined by its antenna gain and beamforming gain, expressed as  
\[\mathrm{G_{BS,i}}=\mathrm{A_{element}}+\mathrm{A_{array}}, \tag{2}\]  
where \(\mathrm{A_{element}}\) and \(\mathrm{A_{array}}\) denote the gains of the antenna element and beamforming, respectively [21]. The gain of the antenna element is given by:  
\[\mathrm{A_{element}}=\mathrm{G_{element}}-\mathrm{min}\{-\lfloor\mathrm{A_{ e,h}}(\varphi)+\mathrm{A_{e,v}}(\theta)\rfloor,\mathrm{A_{m}}\}, \tag{3}\]  
where \(\mathrm{G_{element}}\) is the maximum antenna element gain, and \(\mathrm{A_{m}}\) is the front-to-back ratio. \(\mathrm{A_{e,h}}(\varphi)\) represents the antenna radiation pattern in horizontal direction, which can be calculated as:  
\[\mathrm{A_{e,h}}(\varphi)=-\mathrm{min}\left[12\bigg{(}\frac{\varphi}{\varphi _{3\mathrm{dB}}}\bigg{)}^{2},\mathrm{A_{m}}\right], \tag{4}\]  
where \(\varphi\) denotes the azimuth angle, and \(\varphi_{3\mathrm{dB}}\) is the 3-dB beamwidth of the horizontal pattern. \(\mathrm{A_{e,v}}(\theta)\) denotes the antenna radiation pattern in vertical direction, which can be expressed as:  
\[\mathrm{A_{e,v}}(\theta)=-\mathrm{min}\left[12\bigg{(}\frac{\theta-90}{\theta _{3\mathrm{dB}}}\bigg{)}^{2},\mathrm{SLA_{v}}\right], \tag{5}\]  
where \(\theta\) is the elevation angle, \(\theta_{3\mathrm{dB}}\) is the 3-dB beamwidth of the vertical pattern, and \(\mathrm{SLA_{v}}\) denotes the front-to-back ratio. The antenna array gain, \(\mathrm{A_{array}}\), is expressed as:  
\[\mathrm{A_{array}}=10\log_{10}\bigg{(}\Big{|}\sum_{m=1}^{\mathrm{N_{H}}}\sum _{n=1}^{\mathrm{N_{V}}}w_{n,m}\cdot v_{n,m}\Big{|}^{2}\bigg{)}, \tag{6}\]  
where \(\mathrm{N_{H}}\) and \(\mathrm{N_{V}}\) are the number of antenna elements in the horizontal rows and vertical columns of the antenna array, respectively. \(\mathrm{v_{n,m}}\) is the super position vector, which is given by:  
\[\mathrm{v_{n,m}}=\exp\biggl{\{}\sqrt{-1}\cdot 2\pi\bigg{[}(n-1)\frac{d_{V}}{ \lambda}\cos\theta+(m-1)\frac{d_{H}}{\lambda}\sin\theta\sin\varphi\bigg{]} \biggr{\}}, \tag{7}\]  
where \(d_{V}\) and \(d_{H}\) are the radiating antenna element spacings, and \(\lambda\) is the wavelength. \(\mathrm{w_{n,m}}\) represents the beamforming weighting, which can be calculated as:  
\[\mathrm{w_{n,m}}=\frac{1}{\sqrt{N_{H}N_{V}}}\exp\biggl{\{}\sqrt{-1}\cdot 2\pi \bigg{[}(n-1)\frac{d_{V}}{\lambda}\sin(\theta_{\mathrm{etilt}})+(m-1)\frac{d_{ H}}{\lambda}\cos(\theta_{\mathrm{etilt}})\cos(\varphi_{\mathrm{escan}})\bigg{]} \biggr{\}}, \tag{8}\]  
where \(\theta_{\mathrm{etilt}}\) denotes down-tilt angle, and \(\varphi_{\mathrm{escan}}\) is the direction of the beamforming. The gain of the radar altimeter depends on the incident angle and 3-dB beamwidth of the altimeter antenna [22]:  
\[\mathrm{G_{RA}}=-\frac{12}{\varphi_{3\mathrm{dB}}}^{2}\cdot\varnothing^{2}+ \mathrm{G_{RA,max}}, \tag{9}\]  
where \(\varnothing\) is the incident angle, \(\varnothing_{3\mathrm{dB}}\) is the 3-dB beamwidth of the radar altimeter antenna, and \(\mathrm{G_{RA,max}}\) is the maximum antenna gain of the radar altimeter. Furthermore, the total loss between the 5G base station and radar altimeter is calculated considering the losses attributable to system components.