FMCW radar periodically transmits frequency signals that increase linearly over time. When the target is stationary, the millimeter-wave radar’s range resolution is on the order of centimeters, while the amplitude of chest or heart movements is on the millimeter scale. Therefore, the human body can be approximated as a static target, enabling static target localization.

The periodically transmitted linearly increasing frequency signals, known as chirp signals, can be expressed as:

S T X ( t ) = A T X exp ( j ( 2 π f c t + π γ t 2 + φ ( t ) ) ) (1)

where f c is the starting frequency, B is the transmission signal bandwidth, T c is the pulse duration, γ = B / T c is the linear frequency modulation slope, and φ is the initial phase. When the human target is stationary, the radial distance from the radar to the human body is R c , 0 ≤ t ≤ T C . The echo signal received by the antenna after a time delay of τ = 2 R c / c (where c is the speed of light) is given by:

S R X ( t ) = A R X exp ( j ( 2 π f c ( t − τ ) + π γ ( t − τ ) 2 + φ ( t − t d ) ) ) (2)

The mixer combines the transmitted signal with the echo signal, and after passing through a low-pass filter, the intermediate frequency (IF) signal is obtained. The phase of the IF signal corresponds to the phase difference between the transmitted signal and the echo signal, which can be expressed as:

S I F ( t ) = A T A R exp ( j ( 2 π γ τ t + 2 π f c τ + π γ t 2 + Δ φ ( t ) ) ) ≈ A T A R exp ( j ( 2 π γ τ t + 2 π f c τ ) ) A T A R exp ( j 2 π ( f b t + ϕ b ) ) (3)

Since π γ t 2 is very small, it can be approximated Δ φ ( t ) as negligible. Based on the range-related effect, the term can also be ignored. Thus, the frequency and phase can be expressed as: f b = 2 γ R c / c ,#Math_17#. By calculating the FFT of the intermediate frequency (IF) signal, its frequency spectrum can be expressed as:

S I F ( f ) = A T A R T c exp ( j ϕ b ) s i n c ( T C ( f − f b ) ) (4)

By using incoherent accumulation, the distance corresponding to the maximum energy point can be determined. The frequency f b of the intermediate frequency (IF) signal is obtained by finding the peak index of the frequency spectrum. Based on the formula: R c = c f b / 2 γ .

The MUSIC algorithm performs eigenvalue decomposition on the covariance matrix of the data received by the array antenna. From the matrix of eigenvectors corresponding to different eigenvalues, it distinguishes between the signal subspace and the noise subspace. The eigenvectors corresponding to the larger eigenvalues form the signal subspace, while those corresponding to the smaller eigenvalues constitute the noise subspace. Due to the orthogonality between the signal and noise subspaces, the spatial spectrum (or beamforming spectrum) can be calculated. By searching for peaks in the spectrum, the direction of the target can be determined. Therefore, by applying the MUSIC algorithm to process the target phase signal from range bin, the direction information of the target can be obtained.

After determining the target range bin, the phase information for that range bin is obtained using the arctangent function:

φ b = tan − 1 i m a g ( d a t a ) r e a l ( d a t a ) (5)

Since the phase obtained by the arctangent function lies within the range of [ − π , π ] , any phase values exceeding this range will be wrapped back into it, causing phase ambiguity. To resolve this, phase unwrapping is applied to restore the phase to its original form, resulting in phase changes that contain the target’s vital sign information, as shown in the (6), where Δ ϕ ( t ) represents the difference between two adjacent phase values:

Δ ϕ ( t ) = { ϕ ( t ) − ϕ ( t − 1 ) , | ϕ ( t ) − ϕ ( t − 1 ) | ≤ π ϕ ( t ) − ϕ ( t − 1 ) + 2 π , ϕ ( t ) − ϕ ( t − 1 ) < − π ϕ ( t ) − ϕ ( t − 1 ) − 2 π , ϕ ( t ) − ϕ ( t − 1 ) > π (6)

Since heartbeats cause weaker fluctuations compared to chest movements, this paper implements phase differencing in the signal processing process by subtracting consecutive phase values from the unwrapped phase. This approach effectively eliminates baseline phase drift and suppresses low-frequency components, such as the respiratory signal. As a result, high-frequency components, such as the heartbeat signal, are enhanced, facilitating better identification of the heartbeat signal.

The chest displacement caused by human breathing ranges from 1 to 12 mm, while the heart displacement due to heartbeat is much smaller, ranging from 0.1 to 0.5 mm. The human respiratory rate is between 16 to 20 breaths per minute, while the heart rate typically ranges from 60 to 100 bpm. These two frequencies are significantly different from each other. A bandpass filter can extract signals within a specific frequency range. Therefore, a bandpass filter can be used to initially separate the respiratory and heartbeat signals from the radar data.

Due to varying scenarios, the processing methods also differ. When targets are at different range bins, the phase signals of different range bins can be processed, and the bandpass filter can separate the respiratory and heartbeat signals for each target. However, when multiple targets are located at the same range bin, after using the bandpass filter to separate the respiratory and heartbeat signals, the VMD algorithm is employed to decompose the breathing or heartbeat signals of each individual target.

The VMD algorithm iteratively searches for the optimal solution of the model to determine the center frequency and bandwidth of each component. After the modalities are demodulated into baseband signals, they become smooth signals. Moreover, the VMD algorithm can determine the number of modes to be decomposed, exhibiting strong adaptive characteristics. It overcomes the modal aliasing issue present in the Empirical Mode Decomposition (EMD) algorithm, making VMD more suitable for the practical needs of this study.

After obtaining the vital sign signals of each target, the heart rate and respiratory rate of each target can be estimated by calculating the frequency peak with the maximum amplitude in the spectrum, as shown in the following formula:

V h r = f h r ∗ 60 (7)

where V h r represents the heart rate value, and f h r is the frequency peak in the spectrum corresponding to the heart rate.

In addition, when multiple targets are located at the same range bin, the issue of matching the targets with their corresponding vital sign signals arises. By sorting the spectral amplitudes obtained from the MUSIC algorithm along with the amplitudes of the vital sign signals, the correspondence between each target and its respective vital sign signal can be established. The flowchart of the proposed method is shown in Figure 1.

In this study, radar echo data and gold standard data were collected using a radar module, a fingertip oximeter, and a respiratory monitoring belt. The radar module consists of a radar board and a data acquisition board. The radar board is the IWR6843ISK-ODS from Texas Instruments (TI), and the data acquisition board is the DCA1000EVM from TI. The radar board is tuned to a frequency range of 60 - 64 GHz and includes 4 receiving antennas and 3 transmitting antennas. Radar echo data is collected via the Samtec interface and stored in binary files. The fingertip oximeter is the CMS50E module from CONTEC, with data stored in CSV format at a sampling frequency of 1 sample per second, meeting the experimental requirements. The respiratory monitoring belt is a product developed by Renhe Technology, with monitoring results transmitted via Bluetooth. The equipment diagram is shown in Figure 2. In the diagram, the radar board and data acquisition board are shown in Figure 2(a), the fingertip oximeter is shown in Figure 2(b), and the respiratory monitoring belt is shown in Figure 2(c).

The data acquisition scenario in this study involves three targets sitting in front of the radar board, each wearing a fingertip oximeter and a respiratory monitoring belt to record ground truth values. The relative positions of multiple targets are shown in Figure 3. In Figure 3(a), all three targets are located at the same range bin, with the distance between the radar board and the human body set to 0.8 meters. In Figure 3(b), the three targets are located at different range bins, with the radar board distances to the individuals being 0.6 meters, 1.4 meters, and 1.9 meters, respectively. During data collection, the radar board sampling rate was set

to 4 MHz, with 200 sampling points, 2 chirps, and 1024 frames, resulting in a total collection duration of 1 minute.

The radar echo data is stored in two channels, real and imaginary, resulting in a data length of 6,553,600 points. The radar echo signal is shown in Figure 4. Before processing the data, the real and imaginary channel data are combined into a complex form. Since this study aims to detect vital sign signals from multiple targets, static clutter removal is applied prior to signal processing to enhance the vital sign signals.

The echo signals are arranged into a matrix form with fast time and slow time dimensions. The fast time dimension, representing data from different range bins, is subjected to a Range FFT, completing the conversion from the time domain to the frequency domain. This process yields a Range FFT image, as shown in Figure 5. Figure 5(a) shows the image when all three targets are located at the same range bin, while Figure 5(b) shows the image when the targets are located at different range bins. In both cases, the Range FFT images clearly display the data.

When multiple targets are located at the same range bin, it is necessary to distinguish them based on the direction of targets. Validation was conducted using data where multiple targets are located at different range bins. The MUSIC algorithm was employed to determine the direction of targets, and the results are shown in Figure 6. Specifically, Figure 6(a) illustrates the scenario where multiple targets are in the same range bin, while Figure 6(b) depicts the scenario where

targets are in different range bins. For targets in different range bins, the target closer to the radar board exhibits a larger signal amplitude, with the strongest energy corresponding to an echo signal from 66.5˚. Additionally, the amplitudes of three signals decrease progressively, further verifying the effectiveness of the MUSIC algorithm. In this study, the number of signal sources for the MUSIC algorithm was set to 3, with 4 array elements and 1024 sampling points. This approach successfully enables the localization of multiple targets within the same range bin.

After determining the target’s range information, the phase signal corresponding to the identified range cell is extracted. Due to the presence of phase wrapping, phase unwrapping is required to restore the signal to its original form. Phase differencing is then applied to enhance the target’s heartbeat signal and eliminate phase drift, completing the data preprocessing.

The band-pass filter for the respiratory signal is set to a frequency range of 0.1 - 0.6 Hz, while for the heartbeat signal, the filter is set to a frequency range of 0.8 - 3 Hz. These band-pass filters effectively separate the respiration and heartbeat signals. The signal waveforms before and after filtering are shown in Figure 7. Specifically, Figure 7(a) displays the signal prior to bandpass filtering, while Figure 7(b) illustrates the respiration signal after bandpass filtering, demonstrating the effect of the filtering process.

When multiple targets are located at the same range bin, the VMD algorithm is used to separate the vital sign signals of these targets. During the implementation of the VMD algorithm, a secondary penalty factor needs to be determined. To avoid mode mixing and excessive computational load, the alpha parameter is set to 50,000, which corresponds to the length of the input signal. To reduce noise interference, the fidelity coefficient τ is set to 0, and the convergence condition ε is set to 10⁻⁷. After obtaining the vital sign signals, the breathing rate and heart

rate of each target are determined by identifying the peak frequency in the signal’s spectrum. As shown in Figure 8, when the peak frequency is 1.07 Hz, the corresponding heart rate is 64.2 bpm.

The detection results for the three individuals at different range bins are 0.69 meters, 1.37 meters, and 1.93 meters, achieving a tracking accuracy of 94%. The vital sign signals are shown in Figure 9. The mean absolute error (MAE) represents the average absolute difference between the true physiological measurements of the targets and the monitoring results obtained using the proposed method. For the case where the three individuals are at the same range bin, the MAE for the breathing rate is 3 bpm, and the MAE for the heart rate is 5 bpm.

When the individuals are at different range bins, the MAE for the breathing rate is 2 bpm, and the MAE for the heart rate is 4 bpm. These results demonstrate that the proposed method effectively addresses the separation of vital sign signals and the positioning of multiple targets within the same range bin.