# Towards mmWave Altimetry for UAS: Exploring the Potential of 77 GHz Automotive Radars
## 4 Theoretical Basis of Performance Metrics
### Supplementary Performance Metrics  
The performance metrics elaborated in the preceding text directly govern the waveform specifications of an FMCW radar employed for drone altimetry. However, for the sake of completeness, it is imperative to provide a succinct summary of other metrics as well. Section 6 briefly discusses the potential directions for future work which includes an experimental verification in dynamic conditions. Nevertheless, one of the concerns in range estimation is the rapid rate of descent (RoD). Moreover, the vertical velocity component becomes more relevant if RoD estimation is necessitated. Moreover, with the widespread use of AoA estimation in emerging applications, it is essential to briefly summarize the associated performance metrics as well. Given the scope of the existing work, an abridged discussion on velocity and angular estimation has been offered in the proceeding text. Readers are encouraged to consult references [21; 22; 28; 29; 30; 31; 34] for detailed derivations. Velocity estimation in an FMCW radar requires at least two chirps and estimates the range corresponding to each chirp. According to [PERSON]'s second law of motion, the measured phase difference corresponds to the motion \(\Delta d\) of the target given as  
\[\Delta d=vT_{C}. \tag{32}\]  
Plugging \(\Delta d\) from (32) into (8), the phase difference \(\Delta\varphi_{IF}\) can be expressed as  
\[\Delta\varphi_{IF}=\frac{4\pi vT_{c}}{\lambda}. \tag{33}\]  
Rearranging (33), the final expression for velocity estimation comes out to be  
\[v=\frac{\lambda\Delta\varphi_{IF}}{4\pi T_{c}}. \tag{34}\]  
Figure 5: FMCW chirp frame [28]. The phase difference can only be increased to a limit \(|\Delta\varphi_{IF}|<\pi\) beyond which the estimation becomes ambiguous. Accordingly, the upper bound on the maximum unambiguous velocity is given by  
\[v_{max}=\frac{\lambda}{4T_{c}}. \tag{35}\]  
The two-chirp velocity measurement method fails when multiple objects with different velocities align at the same radar distance, generating identical IF frequencies and resulting in a single peak in the range-FFT. To address this, the radar system transmits a chirp frame of \(N\) equally spaced chirps for an accurate velocity measurement as already illustrated in Figure 5. After the range-FFT, another FFT of order \(N\) is performed in a second dimension to distribute the phase changes into Doppler bins. According to the properties of FFT,  
\[\Delta\varphi_{IF}>\frac{2\pi}{N}. \tag{36}\]  
The final expression for velocity resolution, \(\Delta v\), can then be obtained by using (36) and plugging (27) into (31) as  
\[\Delta v>\frac{\lambda}{2T_{f}}. \tag{37}\]  
Per the application requirements concerning the localization of obstacles and pedestrians in the radar FOV, contemporary automotive radar platforms pack multiple antennae for accurate AoA estimation. The platform under consideration, IWR1843 BOOST, offers antenna arrays for AoA estimation in the azimuth, as well as the elevation, planes. Appropriately, AoA is an important performance metric similar to range and velocity. The following discourse covers the basics of this metric considering only two antennae in a single dimension. Practically more than two antennae in multiple dimensions are realized for spatial localization and high AoA accuracy. The angular estimation relies on the observation that a slight deviation in an object position from the boresight of the radar causes a phase shift in the range-FFT or Doppler-FFT peak. By performing an FFT operation along the antenna dimension, this phase shift can be estimated. Utilizing a minimum of two RX antennae as shown in Figure 6, a phase shift in the FFT peak due to reflected waves arriving at different times facilitates the AoA estimation. Equation (8) is based on the RTOF where the AoA works on the concept of transmission of the EM wave from a single Tx antenna and reception by multiple Rx antennae after reflection from the target.