*2.1. Operating Principles for FSK Radar*

An FSK radar measures the information of the target from the phase differences between two operating frequencies. The phase difference is obtained by comparing the transmitted and received signals at a single CW operating frequency. Accurate information can be obtained by increasing the measurement accuracy of the phase difference at each frequency. A block diagram of the FSK radar is shown in Figure 1. A quadrature architecture is adopted in the proposed FSK radar to avoid the null point problem in the CW Doppler radar [17,18]. Two discrete frequencies *f* <sup>1</sup> and *f* <sup>2</sup> are alternately transmitted for the switching time period of *T* with a duty cycle of 50% by a signal generator. A short time period is advantageous for increasing the measurement accuracy because the FSK radar assumes that the information of the target is constant during the period. However, a sufficient period for sampling baseband signals is needed to obtain the phase difference at each frequency. Thus, the time period should be optimized with consideration of the maximum velocity of the movement in the target and the maximum sampling frequency of the synchronous data-acquisition (DAQ) device.

**Figure 1.** Block diagram of the proposed frequency-shift keying (FSK) radar.

The transmitted signals *Tx*(*t*) in FSK radar can be expressed as follows:

$$T\_x(t) = \begin{cases} A\_{T1} \cdot \cos[2\pi f\_1 t + \varphi\_1(t)], & 0 < t \le \frac{T}{2} \\ A\_{T2} \cdot \cos[2\pi f\_2 t + \varphi\_2(t)], & \frac{T}{2} < t \le T \end{cases} \tag{1}$$

2 ∙ cos [2<sup>2</sup> + <sup>2</sup> ()], 2 < ≤ *φ φ* where *AT*<sup>1</sup> and *AT*<sup>2</sup> represent the amplitudes of the transmitted signals, and ϕ1(*t*) and ϕ2(*t*) represent the phase noises of the two transmitted frequencies in the signal generator, respectively. The receiving signals in the radar are modulated to the Doppler frequencies produced by the chest movements caused by the respiration and heartbeat [18]. The received signals *Rx*(*t*) in FSK radar can be expressed as follows:

$$R\_X(t) \approx \begin{cases} A\_{R1} \cdot \cos\left|2\pi f\_1 t - \frac{4\pi d\_0}{\lambda\_1} - \frac{4\pi x\_1(t)}{\lambda\_1} + \varphi\_1 \left(t - \frac{2d\_0}{c}\right)\right|, & 0 < t \le \frac{T}{2} \\\ A\_{R2} \cdot \cos\left|2\pi f\_2 t - \frac{4\pi d\_0}{\lambda\_2} - \frac{4\pi x\_2(t)}{\lambda\_2} + \varphi\_2 \left(t - \frac{2d\_0}{c}\right)\right|, & \frac{T}{2} < t \le T \end{cases} \tag{2}$$

 () ≈ { 1 ∙ cos [2<sup>1</sup> − 1 − 1 + <sup>1</sup> ( − )], 0 < ≤ 2 2 ∙ cos [2<sup>2</sup> − 4<sup>0</sup> 2 − 4<sup>2</sup> () 2 + <sup>2</sup> ( − 2<sup>0</sup> )], 2 < ≤ , *λ λ* where *AR*<sup>1</sup> and *AR*<sup>2</sup> represent the amplitudes of the received signals; *c* represents the velocity of light, λ<sup>1</sup> and λ<sup>2</sup> represent the wavelengths of the two frequencies, respectively; *d*<sup>0</sup> represents the fixed distance between the radar and the target; and *x*1(*t*) and *x*2(*t*) represents the displacements of the chest caused by the respiration and the heartbeat. The phase noise of the received signal is described with the time delay of 2*d*0/*c* by considering the time of the round trip of the signal at the distance. The in-phase (*I*<sup>1</sup>

+ Δ

()] + , = 1,2, and

4 () 

() =

∙ cos [

4<sup>0</sup> 

+

and *I*2) and quadrature (*Q*<sup>1</sup> and *Q*2) baseband signals, which are obtained from the down-conversion quadrature mixers, can be expressed as follows:

$$I\_k(t) = A\_I \cdot \cos\left[\frac{4\pi d\_0}{\lambda\_k} + \frac{4\pi \mathbf{x}\_k(t)}{\lambda\_k} + \Delta q\_k(t)\right] + D\mathbf{C}I\_k, \; k = 1, 2, \text{ and} \tag{3}$$

$$Q\_k(t) = A\_I A\_E \cdot \sin\left[\frac{4\pi d\_0}{\lambda\_k} + \frac{4\pi \text{x}\_k(t)}{\lambda\_k} + \Delta \phi\_k(t) + \phi\_E\right] + DCQ\_{k\prime} \text{ } k = 1, 2,\tag{4}$$

where *A<sup>I</sup>* represents the amplitude of the I-channel baseband signal; *A<sup>E</sup>* and φ*<sup>E</sup>* represent the errors of the amplitude and phase, respectively; *DCI<sup>k</sup>* and *DCQ<sup>k</sup>* represent the DC offset voltages in the *I* and *Q* channels, respectively; and ∆ϕ*<sup>k</sup>* represents the residual phase noise, which is the difference in the phase noise between the transmitted and received signals. The residual phase noise in the radar system can generally be neglected in the measurement of the distance and vital signs, owing to the range correlation effect [19]. The phase difference θ*<sup>k</sup>* between the transmitted and received signals at each frequency can be represented by using (3) and (4) as follows:

$$
\theta\_k \cong \frac{4\pi}{\lambda\_k} (d\_0 + \mathfrak{x}\_k(t)), \ k = 1, 2,\tag{5}
$$

which is identical to the difference of the CW Doppler radar. The detectable range in the phase difference, which is 0–2π, is determined by the characteristics of the trigonometric function, as indicated by (5). The distance *d*<sup>0</sup> can be measured from the subtraction in each phase difference using the two frequencies in the FSK radar, as follows:

$$d\_0 = \frac{c}{4\pi(f\_1 - f\_2)}(\theta\_1 - \theta\_2) - [\mathbf{x}\_1(t) - \mathbf{x}\_2(t)].\tag{6}$$

If the difference in the vital signs generated during the transmitting and receiving signals of each frequency can be neglected, the absolute distance can be obtained as follows:

$$d\_0 \cong \frac{c}{4\pi (f\_1 - f\_2)} (\theta\_1 - \theta\_2). \tag{7}$$

Thus, the error of the distance measurement in the FSK radar can show the cross-correlation of the vital signs at each operating frequency, and a low error corresponds to a high correlation rate between two vital signals. The periodic signals in the baseband can represent the vital signs included in the phase difference, because the FSK radar can be regarded as a CW Doppler radar with independent single-frequency operation. When the human motion does not have periodicity or is located outside of the frequency band of the vital signs, the vital signs can be detected through fast-Fourier transform (FFT) if the receiver is not saturated by the motion. The measured vital signs in the frequency band can be expressed as follows:

$$X(f) = \frac{\lambda\_k}{4\pi} \int\_{-\infty}^{\infty} \theta\_k(t) e^{-j2\pi f\_m t} dt. \tag{8}$$
