# Towards mmWave Altimetry for UAS: Exploring the Potential of 77 GHz Automotive Radars
## 4 Theoretical Basis of Performance Metrics
### A Complex Baseband FMCW Radar  
Figure 1 depicts a system-level diagram of an FMCW radar employing an in-phase and quadrature (I/Q) complex baseband architecture. One of the most widely utilized waveforms in said radars is linear frequency modulation (LFM). The FMCW generator routes an LFM-modulated continuous wave for amplification at the power amplifier (PA) stage. Following the reflection from the target, the wave undergoes low noise amplification (LNA) at the receiver. Afterward, it is multiplied by a time-delayed and a time-delayed with a 90-degree phase offset version of the transmitted wave in the respective paths. Subsequently, both signals pass through low-pass IF filters to eliminate high-frequency elements. This yields the I and Q components, which are then sampled by analog-to-digital converters (ADCs) and prepared for further signal processing. Power dividers (PDs) are employed at various stages of the process, given the parallel processing inherent in complex baseband architecture. The transmitted signal, \(S_{Tx}\), can then be given as  
\[S_{Tx}=A_{Tx}cos(2\pi f(t)t+\varphi_{Tx}(t)), \tag{1}\]  
where \(\varphi_{Tx}(t)\) and \(A_{Tx}\) are the initial phase and the amplitude of the transmitted signal and \(f(t)\) is the instantaneous frequency of the FMCW generator, given by  
\[f(t)=f_{c}+\text{St}, \tag{2}\]  
where \(f_{c}\) is the initial carrier frequency and \(S\) is the slope of the LFM waveform, the rate of change of frequency sweep with respect to time. This signal is also known as chirp. Figure 2 shows an LFM chirp with BW \(B\), given by the equation  
\[B=ST_{c}. \tag{3}\]Similarly, the received signal \(S_{Rx}\) after reflection from the target, can be expressed as  
\[S_{Rx}=A_{Rx}cos(2\pi f_{c}t+\varphi_{Rx}(t)), \tag{4}\]  
where \(\varphi_{Rx}(t)\) and \(A_{Rx}\) are the initial phase and the amplitude of the received signal. As the received signal is subject to multiplication with the transmitted signal, a duration \(\tau\) is elapsed, known as the roundtrip time of flight (RTOF) expressed using [PERSON]'s second law of motion as  
\[\tau=\frac{2R}{c}, \tag{5}\]  
where \(R\) is the range of the target from the radar and \(c\) is the speed of light. Using the formula for the product to sum trigonometric functions yields the following signal at the output of the mixer:  
\[S_{Tx}S_{Rx}=\frac{1}{2}cos(2\pi S\tau t+\varphi_{Tx}(t)-\varphi_{Rx}(t))+ \frac{1}{2}cos(2\pi S\tau t+4\pi f_{c}t+\varphi_{Tx}(t)+\varphi_{Rx}(t)). \tag{6}\]  
Afterward, the low-pass filter suppresses the high-frequency component containing \(4\pi f_{c}t\). The difference of frequencies comes out to be \(f_{IF}\) and the phase difference \(\varphi_{Tx}(t)-\varphi_{Rx}(t)\), is the initial phase of the resultant I and Q components of the IF signal, expressed as  
\[\varphi_{IF}=2\pi f_{c}\tau. \tag{7}\]  
Plugging (5) into (8) and expressing (7) in terms of wavelength, \(\lambda\), gives us the following:  
\[\varphi_{IF}=\frac{4\pi R}{\lambda}. \tag{8}\]  
Equation (6) can then be used to express \(I_{IF}\) as  
\[I_{IF}=\frac{1}{2}A_{IF}cos(2\pi f_{IF}t+\varphi_{IF}), \tag{9}\]  
Figure 1: System-level diagram of an FMCW radar [31].  
Figure 2: (**a**) Slope of \(f(t)\), and (**b**) FMCW chirp in time domain [28].  
where \(A_{IF}\) is the amplitude of the signal. Similarly, \(Q_{IF}\) can be given as  
\[Q_{IF}=\frac{1}{2}A_{IF}sin(2\pi f_{IF}t+\varphi_{IF}). \tag{10}\]  
Subsequently, the IF signal, \(S_{IF}\) can expressed as  
\[S_{IF} =I_{IF}+jQ_{IF}, \tag{11}\]  
which depicts the IF signal as a sum of I and Q components which is subject to signal processing in the next stage for range, velocity, and angle of arrival estimation.