# Towards mmWave Altimetry for UAS: Exploring the Potential of 77 GHz Automotive Radars
## 4 Theoretical Basis of Performance Metrics
### Link Budget for Radar Altimeter  
The maximum range of an FMCW radar is a function of the IF BW, as well as the SNR available at the receiver. Despite a beat frequency being in the range of the IF filter BW, if the SNR is below the requisite threshold, the requisite range is compromised. Careful estimation is, therefore, prudent in this context. It can be conveniently estimated using the radar range equation, which is a manipulation of the Friss transmission formula [5]. The power received at a monostatic radar reflected from a point target is  
\[P_{r}=\frac{P_{t}G_{t}G_{r}\sigma\lambda^{2}}{(4\pi)^{3}R^{4}}, \tag{21}\]  
where \(P_{t}\) is the transmitted power with \(G_{t}\) and \(G_{r}\) being the gains of the transmit and receive antennae, respectively, \(\sigma\) is the RCS of the point target, and \(\lambda\) is the wavelength. However, the exception for a radar altimeter use case is that the target is the surface of the ground. Therefore, (21) must be extended for a distributed target to accurately approximate the RCS. Backscattering or clutter from the ground surface is an unwanted phenomenon in surveillance radars; however, it is the intended target for altimeters. For a distributed horizontal ground surface, the RCS can be evaluated using  
\[\sigma=\iint\limits_{A_{ill}}\sigma_{o}dA=\ \sigma_{o}A_{ill}, \tag{22}\]  
where \(\sigma_{o}\) is the backscattering cross-section per unit area normalized with respect to the area, \(A_{ill}\), of the ground patch illuminated by the radar antenna [52]. Here, \(\sigma_{o}\) is also referred to as the normalized radar cross-section (NRCS). The estimation of the NRCS falls in the realm of empirical research and well beyond the bounds of this article. A generic analytical model serves the scope of this work since the objective is an approximate estimation of the NRCS for a given terrain type. Such models do not adequately address the characteristics of land clutter for remote-sensing or surveillance use cases where more specific details of scatterers are required. For a pure look-down radar altimeter scenario of \(\psi=90^{\circ}\), the mean NRCS can be evaluated using the as a sum of the low grazing angle regime contribution, and the contribution due to specular scattering [53]:  
\[\overline{\sigma_{0}(dB)}=\overline{\sigma_{9.3G}}+10 log_{10}\Bigg{(}\frac{0.65 e^{-tan^{2}(90-\psi)/tan^{2}(0)}}{tan^{2}(0)}\Bigg{)}+10 log_{10}\bigg{(}\frac{f}{9.3}\bigg{)}+10 log_{10}(0.1\psi), \tag{23}\]  
where \(\overline{\sigma_{9.3G}}\) is the mean NRCS at frequency 9.3 GHz for \(\psi\ <10^{\circ}\), \(tan^{2}(\beta_{0})\) is the rms surface slope, and \(f\) is the operating frequency. The term \(10 log_{10}(0.1\psi)\) is added to cater to the effect of increasing \(\psi\) beyond \(10^{\circ}\). The reference of 9.3 GHz is evident in the mathematical expression since the specific experiments were conducted at this frequency at a very low grazing angle. The extrapolation beyond this grazing angle and operating frequency deems the model generic and, therefore, an approximation only. For an estimation of backscattering for a radar altimeter, such generic models provide sufficient substance for the estimation of performance metrics and parameters. (23) seems daunting from a mathematical perspective but it only signifies an increase in backscattering from the ground surface with increasing frequency owing to lesser absorption. Additionally, larger grazing angles cause more reflection. Interested readers are encouraged to refer to [52; 53; 54] for a detailed discourse and a more insightful understanding of clutter estimation from ground terrains. For the sake of completeness, it must be highlighted that the _landreflectivity_ function from MATLAB [55] utilizes the same model for clutter estimation at such high grazing angles and short wavelengths. There is some respite after all in the form of a high NRCS to somehow compensate for the high FSPL in the mmWave realm. The second term in the estimation of RCS is \(A_{ill}\).