# Towards mmWave Altimetry for UAS: Exploring the Potential of 77 GHz Automotive Radars
## 4 Theoretical Basis of Performance Metrics
### Link Budget for Radar Altimeter
Figure 4 illustrates a horizontal elliptical area illuminated by the radar antenna on the ground surface. It is governed by the half-power beamwidth (HPBW) of the antenna in both the azimuth (H-plane) and elevation (E-plane), as well as the altitude [56]. The higher the altitude, the longer the axes of the ellipse. \(A_{ill}\) can be expressed using the area of the antenna beam, \(A_{beam}\), and angle of incidence, \(\theta_{i}\), as  
\[A_{ill}=\frac{A_{beam}}{cos(\theta_{i})}. \tag{24}\]Assuming a non-symmetrical area of the elliptical antenna beamwidth, (24) can be expanded as  
\[A_{ill}=\pi\frac{\theta_{e}}{2}\frac{\theta_{a}}{2}R^{2}sec(\theta_{i}), \tag{25}\]  
where \(\theta_{a}\) and \(\theta_{e}\) are the HPBW in radians of the antenna in the azimuth and elevation planes, respectively, and \(R\) is the altitude of the platform AGL. The grazing angle, \(\psi=90^{\circ}-\theta_{i}\), for a pure look-down radar altimeter is \(90^{\circ}\). Accordingly, the effect of \(sec(\theta_{i})\) can be ignored in (25). The RCS can now be calculated using (22). Before proceeding to the expression for maximum range, it is crucial to estimate the noise power for the subsequent evaluation of the SNR. The expression of the noise power, \(P_{n}\), in watts is given as  
\[P_{n}=k_{B}TB_{n}F, \tag{26}\]  
where \(k_{B}\) is the Boltzmann constant, \(T\) is the temperature in Kelvin degrees, \(B_{n}\) is the noise BW, and \(F\) is the noise figure of the receiver. It is appropriate to offer some disambiguation and insight into the estimation of the noise BW. This term is often mistakenly approximated in the literature and warrants careful consideration. The only important aspect to understand here is the identification of the exact point where the received signal shall be subject to detection in the receive chain. As shown in Figure 1, after the beat frequencies pass through the IF filter, the signals are sampled in two independent I and Q paths. The time-domain signals I and Q are sampled by the analog-to-digital converter (ADC) and subject to fast Fourier transform (FFT). Subsequently, the noise power becomes split into frequency bins where the BW of each bin is equal to the inverse of the chirp duration, \(T_{c}\)[28; 58]. Furthermore, it must be noted here that an FMCW radar generates a series of chips for the estimation of velocity. This aspect was not covered in the theoretical discussion on FMCW radars since velocity estimation is not covered in this work. Nevertheless, there is no restriction on orchestrating a chirp frame with multiple chirps to extract benefits in terms of the increased SNR. Figure 5 shows an \(N\) chirp frame. The observation time for the estimation of noise can be extended to the entire frame instead of just one chirp. This is termed the integration time or frame time, \(T_{f}\), given as  
\[T_{f}=NT_{c}. \tag{27}\]  
The noise BW in a frequency bin can then be given as  
\[B_{n}=\frac{1}{NT_{c}}. \tag{28}\]  
Plugging (28) into (26), the modified expression for \(P_{n}\) becomes  
\[P_{n}=\frac{k_{B}TF}{NT_{c}}. \tag{29}\]  
It is pertinent to understand that the modified expression for the noise power is consistent with the zero-mean Gaussian distribution of its power spectral density (PSD). It implies that the noise tends to average itself out as the observation interval is increased. To evaluate the maximum range, \(P_{min}\) is the threshold for the minimum power that can  
Figure 4: Area illuminated by radar antenna [57]. be used by the radar with an acceptable value of the probability of false alarm (PFA). An estimation of the PFA for the minimum value, \(SNR_{min}\), is beyond the scope of this work and, therefore, the value specified by the manufacturer serves the purpose [28].