# Short-Range Noncontact Sensors for Healthcare and Other Emerging Applications: A Review
## 2 Working Principle of CW Doppler Radar
### Demodulation Techniques
The \(I/Q\) signals before ADC are as follows [61]:  
\[B_{I}\left(t\right)=A_{I}\cdot cos\left[\theta_{o}+4\pi x\left(t\right)/\lambda +\Delta\theta\left(t\right)\right]+DC_{I} \tag{7}\]  
\[B_{Q}\left(t\right)=A_{Q}\cdot sin\left[\theta_{o}+4\pi x\left(t\right)/ \lambda+\Delta\theta\left(t\right)\right]+DC_{Q} \tag{8}\]  
where \(A_{I}\) and \(A_{Q}\) are the amplitude of \(I/Q\) channels, and \(DC_{I}\) and \(DC_{Q}\) are the DC offsets. The DC offset is from the reflections from the surrounding stationary objects and the circuit imperfections like mixer self-mixing [66]. It is seen that, if we can calibrate \(I/Q\) signals to remove the DC offsets \(DC_{I}\) and \(DC_{Q}\) and normalize the amplitude factors \(A_{I}\) and \(A_{Q}\), the \(I/Q\) signals will become (5) and (6) which will fit with the unit circle in the \(I/Q\) plane as shown in Figure 4.  
By using arctangent demodulation, the subject's movement \(x\left(t\right)\) can be recovered:  
\[\theta_{o}+4\pi x\left(t\right)/\lambda+\Delta\theta\left(t\right)=arctan \left(\frac{B_{Q}\left(t\right)}{B_{I}\left(t\right)}\right)=arctan\left( \frac{sin\left[\theta_{o}+4\pi x\left(t\right)/\lambda+\Delta\theta\left(t \right)\right]}{cos\left[\theta_{o}+4\pi x\left(t\right)/\lambda+\Delta \theta\left(t\right)\right]}\right) \tag{9}\]  
In a coherent Doppler CW radar, \(\Delta\theta\left(t\right)\) is negligible due to range correlation effect [8]. \(\theta_{o}\) is dependent on the fixed distance to the subject so it is a DC value which can be easily subtracted. Therefore, as long as the carrier wave length \(\lambda\) is known, \(x\left(t\right)\) can be easily recovered from Equation (9). Figure 3: Quadrature Doppler radar receiver with \(I/Q\) channels. As discussed, DC calibration is required for effective arctangent demodulation. Since DC offset values are unpredictable, the calibration is challenging. The conventional method is by manually adjusting the \(DC_{I}/DC_{Q}\) values as well as the amplitude factors \(A_{I}/A_{Q}\) to fit the \(I/Q\) signals on the unit circle in the \(I/Q\) plane [67]. However, the manual circle fitting method is inefficient, sensitive to environmental changes, and sometimes inaccurate. Moreover, the estimation of calibration parameters is a subjective approach, which may vary from one person to another so that errors are inevitable. To have a robust DC calibration approach, researchers have proposed an algorithm based on compressed sensing using \(\ell_{1}\) minimization [56]. Based on the input \(I/Q\) signals, the algorithm would automatically calculate the DC offset values \(DC_{I}/DC_{Q}\) and a factor \(A\) to normalize the amplitudes. In this method, it assumes the there is no \(I/Q\) mismatch. The algorithm has been proved to be effective in arctangent demodulation and robust towards large interference [56]. Another challenge in arctangent demodulation is the phase discontinuity. Mathematically, an arctangent function only allows a native codomain range of (\(-\pi/2\), \(+\pi/2\)). If the target motion displacement is relatively large compared to the carrier wavelength, the demodulation may exceed this range so that a phase discontinuity will occur. Figure 5a shows the demodulation of a 0.2 Hz 16 mm displacement movement in 2.4 GHz Doppler radar using the regular arctangent demodulation. It is seen that large discontinuity happens. Such discontinuity can be theoretically compensated in digital signal processing by shifting an integer multiple of \(\pi\), which is a procedure called phase unwrapping [22]. However, in the real use case, it is actually not easy for a hardware or software to automatically make a judicious choice on which point needs a shift. It is extremely difficult when the vibration amplitude is large because it would cause severe discontinuity issues and the discontinuous data points may need to be compensated by different integer multiples of \(\pi\). The other drawback of compensating by an integer multiple of \(\pi\) is the uncertainty of robustness to noise.