## 2 Electromagnetic Radiation Testing Method
### Determining the Test Antenna Type and Model
\[E_{\theta}(r,\theta,\pi)=\sum\limits_{n=0}^{N}\sum\limits_{m=-n}^{n}C_{mn}e^{jm \varphi}\Bigg{\{}a_{mn}\frac{jmP_{n}^{|m|}(\cos\theta)}{\sin\theta}H_{n}^{(2)}(kr )+b_{mn}\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}g_{n}(kr)\Bigg{\}} \tag{1}\]  
\[E_{\varphi}(r,\theta,\pi)=\sum\limits_{n=0}^{N}\sum\limits_{m=-n}^{n}C_{mn}e^{ jm\varphi}\Bigg{\{}-a_{mn}\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}H_{n}^{(2)}(kr )+b_{mn}\frac{jmP_{n}^{|m|}(\cos\theta)}{\sin\theta}g_{n}(kr)\Bigg{\}} \tag{2}\]  
When \(r\rightarrow+\infty\), \(H_{n}^{(2)}(kr)\approx\frac{\frac{\ell^{+1}e^{-jkr}}{kr}}{kr}\), and \(g_{n}(kr)\approx\frac{\ell^{n}e^{-jkr}}{kr}\), factors unrelated to \(\theta\), \(\varphi\) can be omitted to derive the electric field in the far-field region of the antenna:  
\[E_{\theta}(\theta,\pi)=\sum\limits_{n=0}^{N}\sum\limits_{m=-n}^{n}C_{mn}e^{ jm\varphi}j^{n}\Bigg{\{}-a_{mn}\frac{mP_{n}^{|m|}(\cos\theta)}{\sin\theta}+b_{mn} \frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}\Bigg{\}} \tag{3}\]  
\[E_{\varphi}(\theta,\pi)=\sum\limits_{n=0}^{N}\sum\limits_{m=-n}^{n}C_{mn}e^{ jm\varphi}j^{n+1}\Bigg{\{}-a_{mn}\frac{dP_{n}^{|m|}(\cos\theta)}{d\theta}+b_{mn} \frac{mP_{n}^{|m|}(\cos\theta)}{\sin\theta}\Bigg{\}} \tag{4}\]  
\[ \begin{array}{l}a_{mn}=\frac{-1}{H_{n}^{(2)}(kR_{0})}\Bigg{\{} \int_{0}^{\pi}\Big{[}\int_{0}^{2\pi}V_{\theta}(R_{0},\theta^{\prime},\varphi^ {\prime})e^{-jm\varphi^{\prime}}d\varphi^{\prime}\Big{]}\frac{jmC_{mn}P_{n}^{ |m|}(\cos\theta^{\prime})}{\sin\theta^{\prime}}\sin\theta^{\prime}d\theta^{ \prime}\\ \hskip 14.226378 pt+\int_{0}^{\pi}\Big{[}\int_{0}^{2\pi}V_{\varphi}(R_{0}, \theta^{\prime},\varphi^{\prime})e^{-jm\varphi^{\prime}}d\varphi^{\prime} \Big{]}\frac{C_{mn}dP_{n}^{|m|}(\cos\theta^{\prime})}{d\theta^{\prime}}\sin \theta^{\prime}d\theta^{\prime}\Bigg{\}}\\ \hskip 14.226378 ptb_{mn}=\frac{1}{g_{n}(kR_{0})}\Bigg{\{} \int_{0}^{\pi}\Big{[}\int_{0}^{2\pi}V_{\theta}(R_{0},\theta^{\prime},\varphi^ {\prime})e^{-jm\varphi^{\prime}}d\varphi^{\prime}\Big{]}\frac{C_{mn}P_{n}^{|m| }(\cos\theta^{\prime})}{\sin\theta^{\prime}}\sin\theta^{\prime}d\theta^{\prime }\\ \hskip 14.226378 pt-\int_{0}^{\pi}\Big{[}\int_{0}^{2\pi}V_{\varphi}(R_{0}, \theta^{\prime},\varphi^{\prime})e^{-jm\varphi^{\prime}}d\varphi^{\prime}\Big{]} \frac{jmC_{mn}dP_{n}^{|m|}(\cos\theta^{\prime})}{d\theta^{\prime}}\sin\theta^{ \prime}d\theta^{\prime}\Bigg{\}}\end{array}  \tag{6}\]  
where the radius of the smallest sphere surrounding the tested antenna is \(r_{\rm min}\), and \(r\) within the passive zone is \(r\geq r_{\rm min}\); the tangential electric field components at the near-field distance from the coordinate origin \(R_{0}\) are \(V_{\theta}(R_{0},\theta^{\prime},\varphi^{\prime})\), \(V_{\varphi}(R_{0},\theta^{\prime},\varphi^{\prime})\), \({\rm E}(r,\theta,\varphi)\), representing the vector sets of the spherical wave in three directions; \(H_{n}^{(2)}(kr)\) indicates the spherical Hankel function; \(P_{n}^{|m|}(\cos\theta)\) represents the associated Legendre function; \(a_{mn},\ b_{mn}\), and the respective coefficients \(M_{mn},\ N_{mn}\) are complex amplitude coefficients, containing both far- and near-field information; \(N\) is the order of the highest-order mode in the antenna expansion formula, \(N\approx kr_{\rm min}+10,N\), taking integers; and \(k\) is the wavenumber, \(k=2\pi/\lambda\). Based on the near-to-far-field algorithm formulas, we obtained the far-field radiation pattern, as shown in Figure 4. By analyzing the omnidirectional far-field radiation pattern, we can clearly see that the gain intensity of the vehicle antenna primarily concentrates in the direction forming a 45-degree angle with the horizontal plane. This finding provides an important theoretical basis for the subsequent selection of the base station and satellite placement in the anechoic chamber.