# Wrapped phase and phase unwrapping in vital sign detection

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 $$\pi$$?

• I'm also quite new to world of radar signal processing. I don't understand why we have to make phase unwrapping process. Could you light in me about it? It would be nice conversation speak about it. I would appreciate if you share useful resources, I am stuck on this. I have read 4-5 articles about detection of human vital signs but Phase unwrapping process was not clearly stated in any of them. Thanks in advance. Commented Dec 8, 2022 at 7:05

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.