Noncontact Monitoring of Respiration and Heartbeat Based on Two-Wave Model Using a Millimeter-Wave MIMO FM-CW Radar

Abstract

This paper deals with the non-contact measurement of heartbeat and respiration using a millimeter-wave multiple-input–multiple-output (MIMO) frequency-modulated continuous-wave (FM-CW) radar. Monitoring heartbeat and respiration is useful for detecting cardiac diseases and understanding stress levels. Contact sensors are not suitable for these sorts of long-term measurements due to the discomfort and skin irritation they cause. Therefore, the use of non-contact sensors, such as radars, is desirable. In this study, we obtained heartbeat and respiration information from phase data measured using a millimeter-wave MIMO FM-CW radar. We propose a two-wave model based on a Fourier series expansion and extract respiration and heartbeat information as a minimization problem. This model makes it possible to produce respiration and heartbeat waveforms. The produced heartbeat waveform can be used for estimating the interbeat interval (IBI). Experiments were conducted to confirm the usefulness of the proposed method. Moreover, the estimated results were compared with the contact sensor’s results. The results for both types of sensors were in good agreement.

1. Introduction

Understanding the vital information of every living body is highly useful for detecting signs of illness and knowing its state of health. This information is generally obtained from the heartbeat, respiration, blood pressure, and body temperature. In particular, heartbeat and respiration can be used to detect signs of heart disease and understand stress levels. Contact sensors, including electrocardiograms (ECGs) and photoplethysmograms, can measure these vital signals accurately, but they are not suitable for long-term measurements in everyday life because of the discomfort and skin irritation they cause. To overcome this problem, the development of non-contact sensors is required. One potential candidate for non-contact sensors is radar, which uses microwaves or millimeter waves. Radar can measure small skin displacements caused by the breathing and heartbeats of clothed targets without contact. However, radar does not directly measure electrical signals from the heart as electrocardiograms do. Furthermore, small skin displacements on the chest are affected by respiration and the heartbeat. Thus, they must be separated for data analysis.

In recent years, collision-prevention radars that use millimeter waves have become widespread in automotive cars. By using the same millimeter-wave radars, many experiments attempting to measure the vital signals of humans have been conducted. Doppler radars [1,2,3,4,5] and frequency-modulated continuous wave (FM-CW) radars [6,7,8,9,10,11] are the types mainly used in these experiments. Since an FM-CW radar can measure the distance of a target, it has the advantage of being able to monitor multiple targets in range and measure signals with a good signal-to-noise ratio (SNR) compared with Doppler radar [10]. In addition, an FM-CW radar with multiple-input–multiple-output (MIMO) technology makes it possible to measure multiple targets distributed in the azimuth direction [4,12]. A vital-sensing radar mainly collects phase [1,2,3,4,5,6,7,8,9,10,11] and amplitude [13] information. The amplitude shows the fluctuation of the waveform with respect to distance. However, this does not represent a physical measurement related to the distance of a small displacement. On the other hand, the phase change does provide a physical measurement, as it corresponds directly to the variation in the relative distance of a small displacement. However, if there is movement of the target itself, it is necessary to distinguish between target movement and the small displacements seen due to heartbeat and respiration. This makes the analysis more complex. In this study, the target was assumed to be in a resting state while being measured. Data analysis was carried out by measuring phase data. First, it is necessary to separate the respiration and heartbeat waveforms from the phase data. In general, these waveforms are often separated using a filter [14] or the arctangent method [15,16]. In addition, by using the statistical properties of the data, they can be separated using methods such as the projection matrix method [8] or variational mode decomposition [9]. The separated heartbeat waveform often has a complex shape due to residuals from the respiration waveform which cannot be completely eliminated. Therefore, various methods have been proposed to distinguish each heartbeat component, such as the topology method [17,18], the quasi-arctangent method [16], artificial intelligence (AI) [14], and template matching [19]. As mentioned above, the procedures for analyzing heartbeat and respiration waveforms are highly complicated. In this study, to solve this problem, a two-wave model based on a Fourier series expansion is proposed. For example, in reference [4], the displacements of skin on the chest due to respiration and heartbeat are modeled using sinusoidal signals to represent the skin’s periodic vibration. However, the shape of the actual observed waveform, especially for respiration, is close to a square wave or triangular wave. Periodic functions such as these waves are expressed by a combination of harmonics in Fourier series expansions. Based on this model, we analyzed heartbeat and respiration parameters as a minimization problem and separated their waveforms. The modeling of these waveforms, especially heartbeats [4,19,20], has been used to understand their shape. However, these modeling techniques have not been used for the parameter estimation of heartbeats and respiration. We used our method to produce heartbeat and respiration waveforms from the measured phase data. This allowed for an estimation of the interbeat interval (IBI), which is necessary to determine variability in the heartbeat rate. In order to verify the model’s basic measurement capabilities, we conducted experiments in which a single target was lying down. Moreover, the estimated results were compared with a contact sensor’s measurements. The results for both types of sensors showed good agreement. Section 2 describes the theoretical background of the FM-CW MIMO radar and the methodology of our FM-CW signal and MIMO array processing techniques. Additionally, Section 3 and Section 4 describe a measurement method based on a two-wave model that is used for the heartbeat and respiration parameters, which includes a method for reconstructing heartbeat and respiration waveforms. In Section 5, the experimental results are presented. Finally, Section 6 concludes and presents the final outcomes of our study.

2. MIMO FM-CW Radar

The principle of the MIMO FM-CW radar used for data acquisition is explained. It is assumed that the target lies on his back on the floor. The FM-CW radar is positioned above the target, and its electromagnetic wave illuminates the target’s chest. Figure 1 shows the relationship between frequency and time in the FM-CW radar. The FM-CW radar [21] uses a frequency-modulated continuous wave as its transmitting wave.

$$S_t\left(t\right)=A\exp \left[j2\pi \left(f_0 t+{\displaystyle \frac{M_{fm}}{2}}{t}^2\right)\right]$$

(1)

$A$ is the amplitude of the transmitting wave, $f_0$ is the center frequency of the transmitting signal, $M_{fm}=\mathrm{\Delta }f/\mathrm{\Delta }t$ is the modulation rate, and $\mathrm{\Delta }f$ and $\mathrm{\Delta }t$ are the sweep frequency width and sweep time.

If the transmitting and receiving antennas’ positions are at the origin and the distance between the radar and its target is $R_0$, a wave received from the target is written as:

$$S_r\left(t\right)=A^{\prime }\exp \left[j2\pi \left(f_0 \left(t-\tau \right)+{\displaystyle \frac{M_{fm}}{2}}{\left(t-\tau \right)}^2\right)\right],$$

(2)

where $A^{\prime }$ is the amplitude of the receiving wave and $\tau =2R_0/c$, in which c is the speed of light. A beat signal is yielded by the square-law detection of the transmitting and receiving waves and is written as follows:

$$S_b\left(t\right)=A^{″}\exp \left[j2\pi \left(f_0\tau +M_{fm}\tau t-{\displaystyle \frac{M_{fm}}{2}}{\tau }^2\right)\right],$$

(3)

where $A^{″}$ is the amplitude of the beat signal. ${\displaystyle \frac{M_{fm}}{2}}{\tau }^2$ is typically very small and can be neglected. After applying the Fourier transform to this signal, a beat spectrum is obtained, as follows:

$$S_b\left(f\right)=A^{‴}\exp \left(j2k_0{R}_0\right){\displaystyle \frac{\sin \left[\pi \left(f-f_b\right)\right]}{\pi \left(f-f_b\right)}}$$

(4)

where $A^{‴}$ is the amplitude of the beat spectrum. $k_0=2\pi /{\lambda }_0$ is the wave number and ${\lambda }_0$ is the wavelength of $f_0$. The beat frequency is as in Equation (3): $f_b=M_{fm}\tau =2R_0{M}_{fm}/c$, in which $f_b$ is proportional to the distance $R_0$.

Then, it is assumed that the radar antennas’ configuration is a MIMO array with mm transmitting and nn receiving antennas. If a Uniform Linear Array (ULA) with N elements consisting of mm by nn along the x-axis with a uniform spacing of $d={\lambda }_0/2$ can be constructed from the MIMO array, it is possible to estimate a target direction ${\theta }_0$ at the slant range $R_0$. If the target is located at $\left(x_0,z_0\right)=\left(R_0\cos {\theta }_0,R_0\sin {\theta }_0\right)$, as shown in Figure 2, on the condition that the target area is in the far field, a signal vector $s\left(R_0\right)$ consisting of N elements can be given as:

$$\begin{gathered} s\left(R_0\right)=A^{‴}\exp \left(j2k_0{R}_0\right){\left[s_1 s_2 \cdots s_n \cdots s_N \right]}^{ T}, \\ \mathrm{with}{s}_n=\exp \left[j{k}_0\left(n-1\right)d\sin {\theta }_0\right], \end{gathered}$$

(5)

where T is a transpose of $s\left(R_0\right)$. The beamforming conducted using the signal vector $s\left(R_0\right)$ is considered in the following equation:

$$\begin{array}{c}{S}_{BF}\left(\theta ,R_0\right)=A^{‴}\exp \left(j2k_0{R}_0\right){\displaystyle \sum _{n=1}^N{s}_n\exp \left[-j{k}_{0}d\left(n-1\right) \sin \theta \right]}\hfill \\ =A^{‴}\exp \left(j2k_0{R}_0\right){\displaystyle \frac{\sin \left[\frac{k_{0}d}{2}N\left(\sin \theta -\sin {\theta }_0\right)\right]}{\sin \left[\frac{k_{0}d}{2}\left(\sin \theta -\sin {\theta }_0\right)\right]}}\exp \left[-j{k}_0{\displaystyle \frac{d\left(n-1\right)}{2}} \left(\sin \theta -\sin {\theta }_0\right)\right]\hfill \end{array}$$

(6)

This also allows us to estimate the azimuth angle ${\theta }_0$ of the target. From Equations (4) and (6), we can calculate both the distance $R_0$ and the azimuth ${\theta }_0$ of the target. Moreover, if $\theta$ is equal to ${\theta }_0$ in Equation (6), an exponential function $\exp \left(j2k_0{R}_0\right)$ that contains information on the distance between the radar and target is retained. The phase term of the exponential function is derived as follows:

$${\theta }_{ph}=2k_0{R}_0 \mathrm{i}\mathrm{n} \mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{s}.$$

(7)

${\theta }_{ph}$ is constrained to the interval (−π, +π). This phenomenon is called the wrapped phase [22]. The radar repeats the measurement for each PRI (Pulse Repetition Interval). If the target makes a small movement, the phase ${\theta }_{ph}$ is changed with each PRI. When an unwrapping process used to reconstruct an original phase from a wrapped phase is applied to the measured phase, ${\theta }_{ph}^{mes}\left(t\right)=2k_0{R}_0\left(t\right)$ represents the slant range motion of the target. In this research, the phase of the target’s chest echo is measured. It is assumed that the target remains stationary, and since their chest skin vibrates due to respiration and their heartbeat, this vital information can be measured using the MIMO FM-CW radar.

3. The Principle of Measuring Heartbeat and Respiration Parameters

This section describes a method used for estimating a person’s heartbeat rate and respiration rate using a MIMO FM-CW radar. The ideal phase ${\theta }_{ph}^{ideal}\left(t\right)$ consists of three components, which are as follows:

$${\theta }_{ph}^{ideal}\left(t\right)={\theta }_r\left(t\right)+{\theta }_h\left(t\right)+n\left(t\right).$$

(8)

In this research, the body movement of the target is neglected. ${\theta }_r\left(t\right)$ and ${\theta }_h\left(t\right)$ are the phase terms of the respiration and heartbeat of the target, respectively, as a function of time $t$, and $n\left(t\right)$ is a noise term. In order to consider ${\theta }_r\left(t\right)$ and ${\theta }_h\left(t\right)$, a number of displacement models based on respiration and heartbeat have been proposed [4,19,20,23]. This research represents ${\theta }_r\left(t\right)$ and ${\theta }_h\left(t\right)$ as a finite sum of the sine and cosine functions in a period $T$ whose center time is $t_0$ based on the trigonometric Fourier series. They are expressed as:

$${\theta }_r\left(p_r\left(t_0\right),t\right)=a_{0,r}+{\displaystyle \sum _{q=1}^3\left\{a_{q,r}\cos \left[2\pi q f_r\left(t-t_0\right)\right]+b_{q,r}\sin \left[2\pi q f_r\left(t-t_0\right)\right]\right\}},$$

(9)

$${\theta }_h\left(p_h\left(t_0\right),t\right)=a_{0,h}+a_{1,h}\cos \left[2\pi f_h\left(t-t_0\right)\right]+b_{1,h}\sin \left[2\pi f_h\left(t-t_0\right)\right],$$

(10)

where $p_r\left(t_0\right)=\left[a_{0,r}, a_{1,r} , a_{2,r}, a_{3,r}, b_{1,r} , b_{2,r}, b_{3,r} , f_r\right]$ and $p_h\left(t_0\right)=\left[a_{0,h}, a_{1,h} , b_{1,h}, f_h\right]$ are the weighting parameters and fundamental frequency of the respiration and heartbeat phases. The parameters $a_{0,r}$ and $a_{0,h}$ are the direct current (DC) components of respiration and the heartbeat and include information about the distance between the target and the antenna. ${\theta }_r\left(p_r\left(t_0\right),t\right)$ consists of three harmonics, and ${\theta }_h\left(p_h\left(t_0\right),t\right)$ contains one harmonic. We call this analysis method the two-wave model. The difference between both models is their contribution to the measured phase signal ${\theta }_{ph}^{mes}\left(t\right)$. The average range of the chest motion caused by respiration varies from 1 mm to 2 mm. The average chest displacement that is due to the heartbeat is from 0.2 mm to 0.4 mm. This means that the contribution of respiration ${\theta }_r\left(p_r\left(t_0\right),t\right)$ is greater than that of the heartbeat ${\theta }_h\left(p_h\left(t_0\right),t\right)$ in Equation (8). In addition, although a high SNR can be achieved from the effects of the time and space integrations (Fourier transform and beamforming) of the MIMO FM-CW radar, the noise term cannot be neglected, and restrictions thus arise in parameter estimation. Therefore, we assume that there are eight unknowns and four unknowns describing the components of the respiration and heartbeat waveforms.

We next demonstrate how the 12 unknowns of the heartbeat and respiration can be estimated at the specified time $t_0$. Using the measured phases ${\theta }_{ph}^{mes}\left(t\right)$ and the two-wave model ${\theta }_{ph}^{ideal}\left(t\right)$, the following cost function is considered:

$$\begin{array}{l}\Phi \left(t_0, p_r\left(t_0\right),p_h \left(t_0\right), \mathrm{\Delta }T,T_w\right)=\\ {\displaystyle \underset{0}{\overset{T_{mes}}{\int }}{\left|w\left(t_0,\mathrm{\Delta }T,T_w,t\right)\left\{{\theta }_{ph}^{mes}\left(t\right)-\left[{\theta }_r\left(p_r\left(t_0\right),t\right)+\lambda {\theta }_{sw}{\lambda }_h\left(p_r\left(t_0\right),t\right)\right]\right\}\right|}^{2}dt}\end{array},$$

(11)

where $T_{mes}$ is the time duration of the measurement and $n\left(t\right)$ is not considered in Equation (11). λ_sw is a parameter that can be 0 or 1, and its original setting is 1. $w\left(t_0,\mathrm{\Delta }T,T_w,t\right)$ is a non-negative window function. Regarding the window function $w$, $t_0$ is a center position and $T_w$ and $\mathrm{\Delta }T$ are the shape parameters. If $t_0$ = 0, the window function can be written as follows:

$$w\left(t_0=0,\mathrm{\Delta }T,T_w,t\right)=\left\{\begin{array}{c}\begin{array}{cc}0.5\left\{1-\cos \left[2\pi {\displaystyle \frac{t+\left(\mathrm{\Delta }T+\frac{T_W}{2}\right)}{2\mathrm{\Delta }T}}\right]\right\}& -\mathrm{\Delta }T-{\displaystyle \frac{T_W}{2}}\le t\le -{\displaystyle \frac{T_W}{2}}\end{array}\\ \begin{array}{cc}1& -{\displaystyle \frac{T_W}{2}}\le t\le {\displaystyle \frac{T_W}{2}}\end{array}\\ \begin{array}{cc}0.5\left\{1-\cos \left[2\pi {\displaystyle \frac{t-\left(\frac{T_W}{2}+\mathrm{\Delta }t\right)}{2\mathrm{\Delta }T}}\right]\right\}& {\displaystyle \frac{T_W}{2}}\le t\le {\displaystyle \frac{T_W}{2}}+\mathrm{\Delta }t.\end{array}\end{array}\right.$$

(12)

An example of $w$ is shown in Figure 3. This window function specifies the time $t_0$ and the data estimation period $T_w$ for evaluating the unknowns. If the cost function reaches a minimum, the unknowns related to the heartbeat and respiration are accurately estimated. To minimize the cost function of Equation (11), particle swarm optimization (PSO) [24] is used, which is just one of the global optimization methods used in this research. However, it is difficult to estimate all parameters simultaneously due to an imbalance in the contributions of the respiration and heartbeat waveforms. Nevertheless, the parameters are calculated using the following two-step estimation:

(1)

λ_sw becomes 0 and only the respiration parameters $p_r\left(t_0\right)$ are estimated.

(2)

λ_sw becomes 1 and the respiration parameters estimated in the first step are used to represent ${\theta }_r\left(p_r\left(t_0\right),t\right)$. Then, the heartbeat parameters $p_h\left(t_0\right)$ are estimated.

A calculation flow chart of this two-step estimation is shown in Figure 4. By multiplying the fundamental frequencies $f_r$ and $f_h$ by 60, the heartbeat rate and respiration rate can be calculated.

4. Reconstruction of Waveform

This section introduces a reconstruction method for the waveforms of the heartbeats and respirations included in the measured phase ${\theta }_{ph}^{mes}\left(t\right)$. The 12 parameters estimated by Equation (11) are the unknowns in Equations (9) and (10) and contain information pertaining to the phase waveforms of the heartbeat and respiration. Thus, they can be used to reconstruct the time waveforms of the heartbeat and respiration. For example, when the respiration parameters are estimated at $t_0$, this waveform ${\theta }_r\left(p_r\left(t_0\right),t\right)$ is valid around $t_0$. However, the waveform is not applicable to other time points. Then, when the waveform estimated at $t_1$, which is slightly delayed from $t_0$, ${\theta }_r\left(p_r\left(t_1\right),t\right)$ is valid around $t_1$. The respiration waveform between $t_0$ and $t_1$ is expressed as follows:

$${\theta }_r\left(t\right)={\displaystyle \frac{t_1-t}{t_1-t_0}}{\theta }_r\left(p_r\left(t_0\right),t\right)+{\displaystyle \frac{t-t_0}{t_1-t_0}}{\theta }_r\left( p_r\left(t_1\right),t\right).$$

(13)

This calculation is based on the crossover of the real Genetic Algorithm (GA) [25], which is similar to PSO. The same procedure can be repeated for the heartbeat waveform ${\theta }_h\left(t\right)$. This allows the measured phase ${\theta }_{ph}^{mes}\left(t\right)$ to be separated into a heartbeat phase ${\theta }_h\left(t\right)$ and respiration phase ${\theta }_r\left(t\right)$, respectively.

5. Experimental Results

In this article, we conducted vital-sensing experiments using a millimeter-wave MIMO FM-CW radar (TI-AW1243). The radar specifications are shown in

Table 1. Radar specification.

Radar Type	FM-CW
Center Frequency	79 GHz
Bandwidth	3 GHz
Sweep Time	100 µs
PRI	30 ms
MIMO Transmitting Antenna	2
MIMO Receiving Antenna	4

. The measurement setup is shown in Figure 5.

The experiment included two healthy participants: two males, aged twenty and fifty. The subjects remained in a lying down position for measurement conducted using the MIMO FM-CW radar. The radar was placed above the participant. At the same time, a Piezoelectric Respiration (PZT) sensor and Blood Volume Pulse (BVP) sensor were attached to their chest and fingertip to measure their respiration and heartbeat rates.

5.1. Data Analysis

Firstly, target 1, who was in his twenties, was measured. A radar detection image of target 1, obtained using the MIMO FM-CW radar, is shown in Figure 6. The echo in the white circle is the target. The other echoes are the baseline. The complex signal of the pixel with the maximum power in the target echo was recorded for 1 min, and the data, excluding the first and last 10 s, were used for analysis. Moreover, arctangent demodulation [15] and phase unwrapping [22] were applied to these data. Since the IQ imbalance of the AD converter was calibrated and the two-wave model estimates the DC components of Equations (9) and (10), we did not consider DC offset removal. Figure 7a shows the unwrapped phase signal of target 1. A frequency spectrum of this waveform is presented in Figure 7b. The respiration waveform exhibits periodicity, and its spectrum is composed of multiple harmonics with a fundamental frequency of 0.26 Hz. At this wavelength, a phase change of 1 radian corresponds to a distance change of 0.95 mm.

The heartbeat waveform tends to be overshadowed by the respiration waveform, and its spectrum is confirmed around 1.62 Hz. Both spectra are given as average values over 40 s, and the time variability of the heartbeat and respiration cannot be confirmed. Our proposed method was applied to the measured phase data in order to estimate the respiration and heartbeat rates per second. In the first step, PSO searched the signal with a fundamental frequency $f_r$ between 0.15 and 0.7, which is the frequency range of respiration. In the second step, PSO looked for a signal with a fundamental frequency $f_h$ between 0.8 and 2.0, which is the frequency range of heartbeats. The parameters for the width of the window function $w$ were $T_w$ = 5 s and $\mathrm{\Delta }T$ = 1 s for the first step and $T_w$ = 2 s and $\mathrm{\Delta }T$ = 1 s for the second step. This difference in width $T_w$ was caused by the different periods of respiration and a heartbeat.

For example, the 12 parameters of the respiration and heartbeat waveforms measured at t = 10 s, as estimated by PSO and two-step estimation, are shown in

Table 2. The estimated parameters of the respiration waveform at 10 s.

$a_{0,r}$	$a_{1,r}$	$a_{2,r}$	$a_{3,r}$	$b_{1,r}$	$b_{2,r}$	$b_{3,r}$	$f_r$
−23.46	0.06	−1.43	−0.99	−0.99	0.24	0.31	0.17

and

Table 3. The estimated parameters of the heartbeat waveform at 10 s.

$a_{0,h}$	$a_{1,h}$	$b_{1,h}$	$f_h$
0.04	0.27	0.07	1.71

. Figure 8a shows the waveform of Equation (9) using the parameters listed in Table 2. The parameters listed in Table 3 were derived from the difference between the original signal and the estimated signal in Figure 8a. A waveform of the combination of Equations (9) and (10) is shown in Figure 8b.

The waveform calculated using the parameters estimated at 10 s is very similar to the measured phase around 10 s. The heartbeat and respiration rates per second, calculated from their fundamental frequencies, are shown in Figure 9. It is possible to confirm how the respiration and heartbeat rates fluctuate over time. The target’s average respiration rate and heartbeat rate are 15.1 and 95.1 (bpm), respectively.

Moreover, we reconstructed the target’s respiration and heartbeat waveforms from the measured phase data. Both waveforms were calculated using Equation (13), and their appearance for 40 s (respiration) and 10 s (heartbeat) is shown in Figure 10. The heartbeat and respiration waveforms obtained from a PZT sensor attached to the chest and a BVP sensor attached to the fingertip are presented in the secondary vertical axis in Figure 10. The radar and contact sensors provide similar time fluctuation results. In the heartbeat waveform, a time difference between the radar and the BVP can be seen due to the difference in the observation locations. It can be observed that the results of the BVP sensor, which was placed on the fingertip, were delayed compared with the results of the radar observing the chest. The IBI (interbeat interval) of the heartbeat can be accurately calculated from radar signals instead of the contact sensor. Figure 11 shows the heartbeat’s IBI, calculated from the peak positions of the BVP and radar heartbeat waveforms. The results were roughly consistent. The average IBIs obtained from both sensors are 0.63 (s).

Next, target 2, who was in his fifties, was measured. Figure 12a shows the unwrapped phase waveform of target 2. Figure 12b shows the spectrum of the phase waveform. As shown in Figure 12b, two peaks can be clearly confirmed. The peak around 0.52 Hz is the target’s respiration, and the peak around 1.53 Hz is his heartbeat. In this case, the respiration waveform does not have harmonics, and the heartbeat frequency is three times the respiration frequency. This means that the fundamental frequency of the heartbeat is interpreted as the third harmonic of respiration.

Therefore, regarding the heartbeat’s parameter estimation, we stopped using the three harmonics in Equation (9) and changed them to one harmonic (q = 2, 3 were neglected). Although it is very important to adjust the harmonics for each target, we leave this subject for future work. Next, the respiration and heartbeat rates per second were calculated. These results are shown in Figure 13. The respiration and heartbeat rates were constant over time, and the average heartbeat rate and respiration rate were 30.6 and 90.7 (bpm), respectively. The respiration and heartbeat waveforms were reconstructed and are shown in Figure 14, together with the results of the PZT and BVP sensors. The results obtained using the radar are very similar to those obtained using the PZT and BVP sensors. In addition, the IBIs of the heartbeat were calculated from the BVP and radar waveforms. The results are shown in Figure 15. The two IBIs are almost identical in both their time variation and value, and the average IBIs obtained from both sensors are 0.66 (s).

5.2. Discussion

In the above section, the proposed two-step estimation method estimated the respiration and heartbeat rates of two targets and reconstructed their waveforms. Although the IBI was estimated from a short period, from 40 s, the total ME (mean error) and MSE (mean square error) of target 1 and target 2 were −0.0015 (s) and 0.027 (s), calculated via the following equations:

$$\mathrm{Mean} \mathrm{error}: ME={\displaystyle \frac{{\displaystyle {\sum }_{m=0}^{m=M-1}\left\{IB{I}_{radar}\left(m\right)-IB{I}_{BVP}\left(m\right)\right\}}}{M}},$$

(14)

$$\mathrm{Mean} \mathrm{square} \mathrm{error}: RMSE=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{m=0}^{m=M-1}{\left\{IB{I}_{radar}\left(m\right)-IB{I}_{BVP}\left(m\right)\right\}}^2}}{M}}},$$

(15)

where $IB{I}_{radar}\left(m\right)$ is the m-th IBI sample detected by radar and $IB{I}_{BVP}\left(m\right)$ is the corresponding sample from the BVP sensor. M is the total number of heartbeats. In addition, we evaluated the accuracy of the proposed algorithm using the following equations:

$$\mathrm{Mean} \mathrm{accuracy}: MA=100\times \left[1-{\displaystyle \frac{1}{M}}{\displaystyle {\sum }_{m=0}^{m=M-1}\left|\frac{IB{I}_{radar}\left(m\right)-IB{I}_{BVP}\left(m\right)}{IB{I}_{BVP}\left(m\right)}\right|}\right] [\%]$$

(16)

Pearson Correlation Coefficient:

$$PCC={\displaystyle \frac{{\displaystyle {\sum }_{m=0}^{m=M-1}\left\{IB{I}_{radar}\left(m\right)-I\overline{B}{I}_{rdar}\right\}\left\{IB{I}_{BVP}\left(m\right)-I\overline{B}{I}_{BVP}\right\}}}{\sqrt{{\displaystyle {\sum }_{m=0}^{m=M-1}{\left\{IB{I}_{radar}\left(m\right)-I\overline{B}{I}_{radar}\right\}}^2}}{\displaystyle {\sum }_{m=0}^{m=M-1}{\left\{IB{I}_{BVP}\left(m\right)-I\overline{B}{I}_{BVP}\right\}}^2}}}$$

(17)

$I\overline{B}{I}_{radar}$ and $I\overline{B}{I}_{BVP}$ are averaged values. The MA and PCC are 97.09% and 0.93. These results demonstrate that the proposed method has basic measurement capabilities for vital sensing. These values can be compared to a number of references [4,26]. Although the accuracy of the proposed method was evaluated, the amount of data generated was small. The computational efficiency of the proposed method is relatively low because it uses PSO for its parameter estimation. Recently, parallel PSO calculations using GPUs [27] have been proposed, and it is expected that their calculation speed will be improved. In the data analysis of target 2, it was confirmed that respiration harmonics affect the parameter estimation of the heartbeat. In such cases, individualized approaches are needed. Moreover, the search frequency ranges for respiration and heartbeat were specified according to a two-step estimation based on PSO. Therefore, when the actual parameters are outside the specified range due to heart disease, etc., this parameter estimation may become difficult. In the future, we will need to consider countermeasures to deal with cases where it is hard to estimate these parameters. The important thing in these cases is judging that something unexpected has happened. Then, we would use the proposed method to measure health conditions such as stress, where the heartbeat rate and respiration rate do not change greatly. It is also necessary to examine the measurements from different distances and angles, as well as use multiple targets. In our preliminary experiments, we found that when observing from the side or back of the target, the SNR was low. These are issues for future work. Finally, we discuss the influence of random body movement (RBM) [5,11]. In this experiment, we assumed that the target was stationary. Due to our two-step estimation, $a_{0,r}$ contained phase information related to the distance between the target and the radar. Figure 16 shows the time change of $a_{0,r}$. We were able to confirm that target 1 moved slowly, by a change of about 3 radians over a period of 40 s. It seems that the position of the chest skin varied during the measurement period due to changes in the target’s state of tension.

In reference [11], using FM-CW radar, a Kalman filter was used to handle RBM (0–0.5 m/s). It is expected that, if the DC component $a_{0,r}$ in Equation (9) is extended to a linear relation such as $a_{0,r1}+a_{0,r2}t$, where $a_{0,r1}$ and $a_{0,r2}$ are constant numbers, it may be possible to evaluate the linear motion of the target within the estimation period $T_w$. In

Table 4. Comparison of non-contact monitoring methods of respiration and heartbeat.

Reference	Signal	Freq. (GHz)	Key Technology	Vital Signs	% of Targets	Range (m)	Movement	Accuracy
[10]	FM-CW	76.4	ZA-SEFLMS	RR/HR	1	0.5–3.0	Static	Maximum mean error is less than 0.74 bpm (HR). Mean error is within 0.25 bpm (RR)
[12]	IR-UWB	4.3	LCMV	RR	3	3	Static	Respiration rate error: 2%
[4]	CW	2.4	DeepMining	RR/HR	3	1	Static	85.3%
[5]	CW	24/5.8	Polynomial fitting + Matched filter	RR/HR	1	1	RBMs	-
[11]	FM-CW	77–81	Kalman filter	RR/HR	1	0.5–2.0	RBMs	Still: Measured errors of RR and HR were, respectively, less than 2 bpm and 3 bpm. Moving: Measured errors of RR and HR were less than 5 bpm.
This work	FM-CW	79	Two-wave model	RR/HR	1	0.7	Slow RBM	97.09%

ZA-SEFLMS: Zero attracting sign exponentially forgetting least mean square; IR-UWB: impulse-response ultra-wideband; LCMV: linearly constrained minimum variance beamformer; RR: respiration rate; HR: heartbeat rate.

, the results of our proposed method are summarized with other non-contact monitoring methods. As described above, it is currently difficult to handle measurements obtained via the proposed method under a variety of conditions and to compare them with the results of other radar-based methods. We plan to investigate these issues in the future.

6. Conclusions

We examined the non-contact measurement of heartbeat and respiration using a millimeter-wave MIMO FM-CW radar. We proposed a two-wave model based on the Fourier series expansion and extracted respiration and heartbeat information as a minimization problem. Experiments that included two subjects were carried out. Each target was measured while lying on the floor. The proposed method measured the time variation of the heartbeat and respiration rates and reconstructed their waveforms to estimate the interbeat interval (IBI). In addition, these results were compared with measured results obtained from contact sensors (PZT and BVP). Both sets of results demonstrated good agreement. In the future, we plan to develop a two-wave model that incorporates the physical characteristics of the measured targets into the calculation process and measures the respiration and heartbeat rates of moving targets.