*2.2. Cross-Correlation Method*

The cross-correlation method can be used for increasing the power of a periodic signal in a noisy environment [20]. Respiration and heartbeat, which are vital signs obtained directly from the CW radar, are both periodic signals, but the heartbeat of a stationary subject has a relatively low SNR compared with the respiration of the subject. In the proposed FSK radar, the cross-correlation method is used for accurately extracting the heartbeat from the raw data. The cross-correlation between the

phase differences of signals at two operating frequencies can be mathematically expressed in the digital domain as follows:

$$\mathfrak{R}\_{\theta\_1 \theta\_2}[m] = \sum\_{n=0}^{N-m-1} \theta\_1[n+m]\theta\_2^\*[n], \ m = 1, 2, \dots, 2N - 1,\tag{9}$$

where *R* ˆ θ1θ2 represents the result of the raw correlation between the *n*th element of θ<sup>1</sup> and the *m*th element of θ2, and *N* represents the product of time and the sampling frequency [20]. The power levels of the elements simultaneously present in the two phase-different signals are increased using the cross-correlation method. The cross-correlated signal is expanded to double the length of the phase-difference signal in the time domain, as shown in Figure 2a. The frequency resolution of the signal is also increased in the frequency domain, as shown in Figure 2b, because the number of sampled data is increased at the same sampling frequency by the cross-correlation. Both white noise and noise signals caused by the hardware components that have different frequency characteristics are reduced by the cross-correlation. The proposed FSK radar can easily improve the vital-signs detection performance without additional signal processing, because the FSK radar uses the same hardware configuration to obtain the cross-correlated signal with the measured phase differences at the two operating frequencies. Thus, the peak detection accuracy of the heartbeat can be improved in the frequency domain because the SNR of the heartbeat in the proposed FSK radar is improved by the enhancement of the signal power level and the suppression of the noise level.

**Figure 2.** Cross-correlation method: (**a**) In the time domain; (**b**) in the frequency domain.
