Wrapped phase and phase unwrapping in vital sign detection

Question

I am quite new to the world of radar signal processing and there is this issue that I can't figure out for the life of me.

In Remote Monitoring of Human Vital Signs Using mm-Wave FMCW Radar (Section II.B) it is said that

"After DC compensation, the phase of each column in the range-slow time matrix is calculated by using $tan^{−1}$ so that the output phases are wrapped in $\left[−\pi, \pi \right]$. In contrast, the phase can change beyond of $\pm \pi$ because $x \left( t \right)$, the physical displacement, can be greater than $\frac{\lambda_{max}}{4}$ (see (5))."

Question 1. Why does phase end up in the range $\left[- \pi, \pi \right]$ (The $\mathbf{arctan}$ function has a range of $\left[- \frac{\pi}{2}, \frac{\pi}{2} \right]$ so I initially thought the extracted phase would equally fall in this interval)

Question 2. How do we conclude from equation 5: $$4 \pi \cfrac{R_{0} + x \left( t_{0} \right)}{\lambda_{max}}$$

that the phase difference between two consecutive samples has to exceed $\pi$?

Answer 1

Question 1. How does it happen that the phase ends up in the range $\left[- \pi, \pi \right]$

You don't apply $\tan^{-1}$ blindly but use the sign of the imaginary parts to determine the correct quadrant. Almost all programming languages have a variant of atan() and it's called atan2(). In most formulas it's implied that you use the "correct quadrant" of the $\tan^{-1}$. Good reading on the topic https://en.wikipedia.org/wiki/Polar_coordinate_system#:~:text=In%20mathematics%2C%20the%20polar%20coordinate,angle%20from%20a%20reference%20direction.