# Low-Altitude Sensing Model: Analysis Leveraging ISAC in Real-World Environments
## 2 The Model of Single Base Station 3D Sensing
### Analysis of Single Base Station Sensing Capabilities
Therefore, we can obtain the following:  
\[N_{0} =P_{\text{noise}}+P_{\text{neighbor-sensing}}+P_{\text{uplink- communication}}+P_{\text{remote-interference}} \tag{3}\]  
The neighboring station's sensing interference power \(P_{\text{neighbor-sensing}}\) can be expressed as follows:  
\[P_{\text{neighbor-sensing}}=\sum_{k=1}^{N}\left(\frac{\sigma_{\text{n}}(k)P_{ \text{nt}}(k)G_{\text{nr}}(k)G_{\text{nt}}(k)\lambda^{2}n\tau}{(4\pi)^{3}R_{ \text{n}}^{4}(k)}\gamma(k)\right), \tag{4}\]  
where \(N\) is the total number of neighboring stations, \(\sigma_{\text{n}}(k)\) is the RCS of the target at the \(k^{\text{th}}\) neighboring station, \(P_{\text{nt}}(k)\) is the transmit power of the \(k^{\text{th}}\) neighboring station, \(G_{\text{nr}}(k)\) and \(G_{\text{nt}}(k)\) are the reception and transmission antenna gains of the \(k^{\text{th}}\) neighboring station, respectively, and \(R_{\text{n}}(k)\) is the distance between the sensing target and the \(k^{\text{th}}\) neighboring station; \(\gamma(k)\) is the signal attenuation coefficient of the sensing signal from the \(k^{\text{th}}\) neighboring station affected by multipath effects, typically ranging between 0 and 1. The uplink communication interference \(P_{\text{uplink-communication}}\) can be expressed as the sum of the power of \(N\) uplink users within the sensing bandwidth \(B\):  
\[P_{\text{uplink-communication}}=\int_{-B/2}^{B/2}\left(\sum_{k=1}^{N}(P_{\text{u }}(k,b))\right) \tag{5}\]  
Furthermore, in dense urban scenarios (Urban Macro), \(P_{\text{remote-interference}}\) can be further extended based on the path loss model for macro base station UAVs (RMa-AV). The RMa-AV model is derived from the RMa model [28], and \(P_{\text{remote-interference}}\) can be expressed as follows:  
\[P_{\text{remote-interference}}=\int_{-B/2}^{B/2}\left(\sum_{k=1}^{N}\left( \begin{matrix}10\text{max}\big{(}3.7-0.8\log_{10}(h_{\text{UT}}),2\big{)}\log _{10}(d_{\text{3D}}(k))\\ -35.5+18.5\log_{10}(h_{\text{UT}})+20\log_{10}\left(\frac{40\pi T_{\text{c}} (k)}{3}\right)\end{matrix} \right)\right), \tag{6}\]  
where \(h_{\text{UT}}\) is the target flight altitude, \(d_{\text{3D}}(k)\) is the radial distance between the target and the \(k^{\text{th}}\) remote base station, and \(f_{\text{c}}(\text{k})\) is the center frequency of the \(k^{\text{th}}\) remote base station. Given the minimum sensing threshold \(S_{\text{min}}\), the maximum sensing distance \(R_{\text{max}}\) for a single station can be expressed as follows:  
\[R_{\text{max}}=\left[\frac{\sigma P_{\text{t}}G_{\text{t}}G_{\text{r}}\lambda ^{2}n\tau}{(4\pi)^{3}LN_{0}S_{\text{min}}}\right]^{\frac{1}{2}}. \tag{7}\]  
It can be seen that station spacing is positively correlated with the maximum sensing distance of a single station. Meanwhile, the maximum sensing distance of a single station is positively correlated with parameters such as transmit power, antenna gain, target RCS, and gain accumulation, and negatively correlated with noise and SNR threshold.