To detect the tiny chest displacement, a CW radar transmits a continuous wave signal that is directed toward the human subject. The phase shift of the reflected echo occurs due to the tiny movement of the chest surfaces (Harikesh et al., 2021). The phase shift of the reflected echo from the tiny movement of the chest surface can be described in Eq. 1 as: θ t ∝ 4 π λ x t (1)

Due to the presence of the wall, the phase-modulated reflected echo experiences attenuation that can be contributed by the depth of the wall, absorption, and reflection of the signal energy as it interacts with the wall material (Carr, 2000) shown in Figure 1. The received signal can be written as: R t = A R ⁡ cos 2 π f c t − t d + φ t − t d + θ 0 (2)

Where, t d is the round trip time delay and A R is the reflected signal amplitude = a × w × c ( a , w , a n d   c are the complex transmission coefficients for the air, wall, and chest respectively). Substituting t d = 2 d t − d t c c in Eq. 2 we get, R t ≈ A R ⁡ cos 2 π f c t − 4 π λ d 0 + x t + φ t − 2 d 0 c + θ 0 (3)

In Eq. 3 d 0 is the distance between the chest surface and the transmitter antenna ( d a is the distance covered in air, d w is the depth of the wall). Radar receiver consists of mixers which is non-linear (Carr, 2000; Islam et al., 2023). When there is a presence of two subjects in front of the radar, two reflected RF echoes produce an output as expressed in Eq. 4: M 2 t = ⁡ cos 2 π f c t * R 1 t + R 2 t (4) where, f c is the carrier frequency and R 1 t , R 2 t is the reflected echoes from the two subjects. M 2 t = ⁡ cos 2 π f c t * A 1 t cos 2 π f c t − 4 π λ d 0 , 1 + x 1 t + A 2 t cos 2 π f c t − 4 π λ d 0 , 2 + x 2 t ] = A 1 t 2 cos 4 π λ d 0 , 1 + x 1 t + ⁡ cos 4 π f c t − 4 π λ d 0 , 1 + x 1 t ] + A 2 t 2 cos 4 π λ d 0 , 2 + x 2 t + ⁡ cos 4 π f c t − d 0 , 2 + x 2 t ] (5) where A n t represents the amplitude modulation of the signal reflected by subject n, d 0 , n represents the nominal distance to subject n, and x n (t) stands for the n subject’s time-varying physiological displacement. For simplicity, the phase noise of the oscillator (φ) and constant phase shift ( θ 0 ) terms are ignored in Eq. 5. The output of the mixer carries modulated components at both sum and difference frequencies. The sum frequency can easily be rejected by a lowpass filter, leaving only the difference frequencies, which are at baseband and shown as B₂(t) in Eq. 6. B 2 t = A 1 t 2 cos 4 π λ d 0 , 1 + x 1 t + A 2 t 2 cos 4 π λ d 0 , 2 + x 2 t (6)

Due to the non-linear nature of the mixture it produces both the sum and difference frequencies that create an intermodulation tone (Carr, 2000; Islam et al., 2023). For the intermodulation tone, subharmonics would be dominant in the spectrum and would hinder the dominant peaks in the spectrum. Therefore, subharmonics compression is essential to extract the dominant fundamental frequency from the spectrum for multi-subject scenarios.

The radar-reflected echoes y t = B 2 (t) can be divided into two components using the MODWT method (Nguyen, , 2021; Shrifan et al., 2021). One is the detailed component and another is the approximation method shown below in the equation: y t = ∑ n c j , n ϕ j , n t + ∑ j = J ∞ ∑ n d j , n ψ j , n t (7)

In Eq. 7, the function ϕ(t) is the proportional function of two consecutive signals with the approximate function or the ratio coefficient { c j , n }, and ψ(t) denotes the wavelet function with the wavelet coefficient { d j , n }, represented Eq. 8 as: d j , n = 2 − j 2 ∫ − ∞ ∞ y t ψ j , n 2 − j t − n d t , c j , n = 2 − j 2 ∫ − ∞ ∞ y t ϕ j , n 2 − j t − n d t (8)

The wavelet coefficient of the wavelet signal and its ratio coefficient are related in the following way: c j , n M = 1 2 j / 2 ∑ − ∞ ∞ g j , k y n − k m o d N , d j , n M = 1 2 j / 2 ∑ − ∞ ∞ h j , k y n − k m o d N (9) In Eq. 9, g_j,_k represents the low-pass filter, h_j,k is the high-pass filter, k is the discrete wavelet transform to the M^th levels, and N is the length of the signal sample. In general, the MODWT multiresolution analysis approach decomposes an original signal into detailed and approximate fields without sampling down in the filtering process to extract the fundamental frequency components and subharmonics or intermodulation-related components (Shrifan et al., 2021).

The proposed system consists of a 24-GHz single-channel on-shelf ST100 starter kit with the KLC1 CW radar module shown in Figure 2. The KLC1 radar transceiver has a single channel with a dual 4-patch antenna with a beam aperture of 80° and 34°, along with an output power of 15dBm (RFbeamMicrowave, 2024). The on-shelf ST100 starter kit with the KLC1 radar module has a graphical user interface (GUI) for data collection and it has a 16-bit low-noise analog-to-digital converter (ADC) that can process signals up to 16 KHz (RFbeamMicrowave, 2024). For this work, we used a sampling frequency of up to 1 KHz for data collection. The experiment was performed in a home environment (not an anechoic chamber) to resemble a life situation shown in Figure 3. We collected the data from ten different healthy subjects with an age group between 21 and 30 in different environments such as a room environment and rooftop of the floor. The participants also do not have any heart disease-related issues. This study was performed with the approval of the ethical review board of the Faculty of Biological Science, University of Dhaka, Dhaka, Bangladesh. During the data collection process, the participants were mostly in sedentary positions without any movements. We also used two different types of barriers such as brick walls and wood between the human subject and the radar system. The thickness of the brick wall is about 10 cm and the thickness of the wooden wall is approximately 5 cm. Each subject was positioned at five different distances ranging from 20 to 100 cm. A total of 35 data sets were collected for single-subject measurements, with 25 sets recorded in the presence of a brick wall and 10 sets with a wood wall. The data recording process was conducted for an approximate duration of 1 min each time. Therefore, 35 min of data, encompassing around 600 breathing and 2,500 heartbeat cycles, was collected. To further investigate, we also collected four data sets for concurrent measurements of two subjects positioned in front of the radar, with a subject-radar distance of 40 cm and a wood wall in between, as illustrated in Figure 3C.

For initial verification of the proposed system, we collected data just for single subject presence in front of the radar through the wall and wooden wall as well. The data was collected when the subjects were in seated and standing positions for a distance of 20 cm up to 100 cm Figure 4 illustrates the raw data that was collected when a single subject was in a seated position at a distance of 40 cm from the radar including the wall thickness. After capturing the data we removed the DC portion of the signal by deducting the mean of the signal and then filtered it with the bandpass filter with the cut-off frequency of 0.8–2 Hz for extracting the heartbeat waveform. Figures 4B, D illustrate the filtered heartbeat waveform. After filtering the heartbeat waveform we performed a Fast Fourier transform (FFT) to extract the fundamental frequency of the captured signal which is the heartbeat of the subject. During all the measurements we also collected data using the Biopac ECG acquisition module to compare the accuracy of our proposed system. Figure 5 shows that the radar-extracted heartbeat was 1.468 Hz and the Biopac ECG acquisition module captured heartbeat was around 1.47 Hz. This clearly illustrated that the radar signal extracted heartbeat waveform perfectly matches the gold standard reference system ECG module extracted heartbeat. It has been observed that when a single subject started crossing more than 50 cm the radar-captured signal strength started degrading and subharmonics became very dominant so that the peak of the FFT can not be determined as other subharmonics are very strong. Figure 6 clearly shows that the FFT of the radar-captured signal does not match the Biopac ECG acquisition module. By leveraging the MODWT method we could extract the heartbeat-related information even at distances of 80 and 100 cm. MODWT method decomposes the signal into four different levels based on the frequency range of the decomposition level (MathWorks, 2024). Additionally, it identifies the relative energy of the signal at multiple levels. Figure 7 illustrates that the radar-captured signal at a distance of 100 cm is being decomposed into four different levels and its associated relative energy has been calculated and shown in Figure 7. The highest relative energy decomposed signal was selected as the heartbeat signal and then its FFT was compared to the FFT of the Biopac ECG acquisition module. Figure 8 illustrates the time domain signal of the MODWT-decomposed best signal, the ECG signal, and then its associated FFT was also illustrated. From the FFT of the MODWT decomposed best signal it is visible that the subharmonics compression was performed accurately. For the subharmonics compression using the MODWT method, the FFT of the radar-captured signal showed 1.43 Hz whereas the ECG signal showed 1.47 Hz. That clearly illustrated the efficacy of the MODWT method.

We also experimented with the presence of the two subjects in front of the radar through the wood as well. The distance of the subject from the radar was approximately 40 cm from the human and the shoulder-to-shoulder distance between the subjects was almost 10 cm Figure 9 illustrates that when there is a presence of two subjects through the wall, radar receives multiple reflections and echoes from the target and therefore, isolating the independent respiratory patterns from the combined mixture is always a challenging task (Islam et al., 2022a). The FFT of the mixture signal shows multiple peaks and the fundamental frequency can not be determined due to the presence of the other dominant peaks shown in Figure 9. Figures 9C, E illustrates the ECG signal of individual subjects 1, and 2 and its associated FFTs (d) and (f). Then the MODWT method was applied to the mixture of the signal to isolate the independent respiratory patterns. Figure 10 illustrates the two best-decomposed MODWT signals that were selected based on the relative energy of the signal. After decomposition and selecting the best two signals, we performed the FFT. The FFT peak of the best two decomposed signals is 1.648 and 1.43 Hz which quite accurately matches with the ECG signal.

To test the reproducibility and repeatability of the proposed system, we collected a dataset from seven participants for single-subject and eight participants for two-subject concurrent measurements, having distances from 20 cm to up to 100 cm. We also performed the typical FFT method for extraction of the heart rates through the brick wall and the wooden barrier as well. Table 1 illustrates the overall accuracy investigation for the single subject and Table 2 shows the accuracy investigation for two-subject concurrent measurement. The direct FFT method demonstrated reasonable accuracy up to 60 cm (92.25% at 60 cm) but experienced a decline, dropping to 86.6% at 100 cm distance. On the other hand, the proposed MODWT method showed a better accuracy of 95.25% at 60 cm and 93.45% at a distance of 100 cm, which is reasonable for applications like post-disaster and rescue operations. For the post-disaster search and rescue applications the proposed technology can be the key for saving the lives of trapped and injured people. The direct FFT method can not be implemented for two subjects’ concurrent measurements as it can not isolate the vital signs information for two subjects. However, the MODWT method can also be used for two-subject measurement as it can isolate the heart rate of individual subjects from the multiple echoes. The overall accuracy for two-subject measurements was approximately 97.04%. It is important to highlight that, for two-subject concurrent measurements, four trials were conducted with the radar positioned 40 cm from the human subjects. The accuracy achieved in this scenario closely matches that of the single-subject scenario (96.84% at 40 cm). Figure 11 illustrates the comparative average accuracy of the direct FFT and MODWT methods. The results indicated the efficacy of the proposed MODWT method both for single-subject and two-subject measurements. Using the MODWT method, we could only extract the heart rates of two individuals from their mixed signals. Moreover, the system can detect vital signs up to 1 m accurately, facing challenges beyond this distance in accurately extracting heart rates.