## Introduction

In recent years, the detection of living persons behind obstacles impulse ultra-wideband (UWB) radar has been investigated1,2,3,4,5. UWB radar has been used for the detection of humans6,7, moving subjects8,9, imaging in through-wall conditions10,11, search and rescue12,13, positioning indoors14,15, and public order and security16 because of its high range resolution and penetrability17,18,19,20. It can be used to detect human vital sign signals such as respiration and heart rates, but this can be difficult as thorax movement is usually only several millimeters, and the signal attenuation can be severe.

Previous researches on human vital sign detection have focused on suppressing stationary and/or nonstationary clutters, estimating the respiration frequency and heart rate, the analysis of signal characteristics, and other related problems21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42. The characteristics of human respiration signals were analyzed in both time and frequency domain24,25 using the Hilbert-Huang transform (HHT) and fast Fourier transform (FFT). In27, clutter similar to respiration signals was suppressed using adaptive clutter cancellation. In28, the arctangent demodulation (AD) technique was employed in a UWB pulse radar system to accurately extract vital sign signals over long distances and in and through-wall conditions. However, this technique is complex and decreases the computational efficiency. A maximum likelihood estimator was considered in29 to determine the period of vital sign signals under considering the additive-white-Gaussian-noise (AWGN). A post-processing algorithm was developed for respiration motion detection. The stationary and nonstationary clutters were removed by employing the singular-value-decomposition (SVD) algorithm when the signal-to-noise ratio (SNR) is low30. A tracing technique was considered to extract the respiration impulse response, but this approach is effective only when the SNR is high and over small distances35. LTS i.e. linear trend subtraction method was employed in30 to reduce the linear trend. In37, a higher order cumulants (HOC) technique was applied to extract vital sign signals considering that the fourth order cumulants of AWGN is zero. The classic EEMD i.e. ensemble empirical mode decomposition technique was used in38 to estimate the heart rate of a living person by improving the SNR and removing clutter. An improved arctangent demodulation (AD) algorithm was proposed to increase the accuracies of human heart rate estimate40. In42, a state-space method (SSM) was employed to extract vital sign signals including human respiration and heartbeat signals.

Most detection techniques are not effective over long distances and in through-wall conditions. To solve these problems, an algorithm based on UWB radar is developed here to calculate accurately vital sign signals in challenging environments. This is achieved by suppressing stationary, non-stationary and other clutter as well as the linear trend. The products of the vital sign signals and their harmonics are removed by employing a CSD-based frequency accumulation (FA) method. This is shown to perform better than several well-known techniques. The range information between human subject and radar is estimated considering the characteristics of vital sign signals and a discrete short-time Fourier transform (DSFT). Experimental results obtained using a UWB radar system at the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences (IECAS), are used to evaluate the performance of the proposed technique.

The following parts of the paper are arranged as follows. The model of signal detection is given in Section 2, and the proposed method for vital sign detection is presented in Section 3. Section 4 provides performance results to evaluate the detection method, and the conclusions are presented in Section 5.

## Signal Detection

### Vital Sign Signal Model

Vital sign signals may be obtained due to the time delay changes of a transmitted UWB pulse. The distance between human chest and radar is32

$$d(t)={d}_{0}+r(t)={d}_{0}+{A}_{r}\,\sin (2\pi {f}_{r}t)+{A}_{h}\,\sin (2\pi {f}_{h}t),$$
(1)

where the range between the radar and the center of human chest is given by d030, fr and Ar are the frequency and amplitude of human respiratory movement, respectively. $${f}_{h}$$ and $${A}_{h}$$ are the heart rate and heartbeat amplitude, respectively32.

With only one human subject present in the detection environment and all other objects are considered as static, the impulse responses are

$$h(\tau ,\,t)={a}_{v}\delta (\tau -{\tau }_{v}(t))+\sum _{i}{a}_{i}\delta (\tau -{\tau }_{i}),$$
(2)

where $$\sum _{i}{a}_{i}\delta (\tau -{\tau }_{i})$$ represent the responses from all these static objects with time delay τi and vibration amplitude ai30, and $${a}_{v}\delta (\tau -{\tau }_{v}(t))$$ represents the impulse response of the vital signs with amplitude av and time delay $${\tau }_{v}(t)$$32. This delay is then

$${\tau }_{v}(t)=\frac{2d(t)}{v}={\tau }_{0}+{\tau }_{r}\,\sin (2\pi {f}_{r}t)+{\tau }_{h}\,\sin (2\pi {f}_{h}t),$$
(3)

where $$v=3\times {10}^{8}$$ m/s, $${\tau }_{0}=2{d}_{0}/v$$,$${\tau }_{r}=2{A}_{r}/v$$, and $${\tau }_{h}=2{A}_{h}/v$$.

$$R(\tau ,\,t)=s(\tau )\ast h(t,\,\tau )={a}_{v}s(\tau -{\tau }_{v}(t))+\sum _{i}{a}_{i}s(\tau -{\tau }_{i}),$$
(4)

where $$s(\tau )$$ represents the transmitted pulse. $$R(\tau ,t)$$ for a respiration signal is shown in Fig. 1. Slow-time represents the received pulse number and fast-time represents the range. In the signal model, t is slow-time with frequency components denoted by f, and τ is fast-time with frequency components denoted by υ. The average human chest location is shown as a dashed line.

The discrete signal corresponding to (4) is

$$\begin{array}{rcl}R[m,n] & = & {a}_{v}s(m{\delta }_{T}-{\tau }_{v}(n{T}_{s}))+\sum _{i}{a}_{i}s(m{\delta }_{T}-{\tau }_{i})\\ & = & {a}_{v}s(m{\delta }_{R}-v{\tau }_{v}(n{T}_{s}))+\sum _{i}{a}_{i}s(m{\delta }_{R}-v{\tau }_{i}/2)\\ & = & h[m,\,n]+c[m],\end{array}$$
(5)

where $$1/{T}_{s}$$ represents the pulse repetition frequency (PRF), $$t=n{T}_{s}$$, $$n=0,\ldots ,\,N-1$$ denote t samples, δT represents the sampling time in τ, $${\delta }_{R}=v{\delta }_{T}/2$$ $$m=0,\ldots ,\,M-1$$ denote τ samples, $$h[m,n]$$ represent vital signs, and $$c[m]$$ represent static clutters. To avoid ambiguities in range and frequency aliasing, Ts should satisfy

$$1/{T}_{s}\ge 2(\max ({f}_{r},\,{f}_{h})).$$
(6)

In a real environment, (5) becomes

$${R}_{M\times N}={h}_{M\times N}+{c}_{M\times N}+{a}_{M\times N}+{w}_{M\times N}+{q}_{M\times N}+{g}_{M\times N}+{z}_{M\times N},$$
(7)

where $${a}_{M\times N}$$ represents linear trend term, $${w}_{M\times N}$$ represents AWGN, $${q}_{M\times N}$$ represents non-stationary clutters, $${g}_{M\times N}$$ represents unknown clutters, and $${z}_{M\times N}$$ represents the clutters introduced by moving objects. A reflected pulse must be acquired within one pulse duration so that

$${T}_{w}+\,\max \,\{{\tau }_{v}(t)\}-\,\min \,\{{\tau }_{v}(t)\} < {T}_{s},$$
(8)

where Tw is the −6 dB pulse width.

### Vital Sign Signal Detection

In static environments, the received pulses are shown in (4). Usually, the pulses without $$c[m]$$ are

$$\tilde{R}(\tau ,\,t)={a}_{v}s(\tau -{\tau }_{v}(t)).$$
(9)

The frequencies of the vital sign signals are obtained by performing FT on $$\tilde{R}({\rm{m}}{\delta }_{T},t)$$ in t given by

$$Y(m{\delta }_{T},\,f)={\int }_{-\infty }^{+\infty }\tilde{R}(m{\delta }_{T},\,t){e}^{-j2\pi ft}dt.$$
(10)

This can be acquired using the two-dimension FT of $$Y(\upsilon ,f)$$ given by

$$Y(m{\delta }_{T},\,f)={\int }_{-\infty }^{+\infty }Y(\upsilon ,\,f){e}^{j2\pi \upsilon \tau }d\upsilon ,$$
(11)

where

$$Y(\upsilon ,\,f)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\tilde{R}(m{\delta }_{T},\,t){e}^{-j2\pi ft}{e}^{-j2\pi \upsilon \tau }dtd\tau ,$$
(12)
$$\begin{array}{rcl}Y(\upsilon ,f) & = & {\int }_{-\infty }^{+\infty }{a}_{v}S(\upsilon ){e}^{-j2\pi ft}{e}^{-j2\pi \upsilon {\tau }_{v}(t)}dt,\\ & = & {a}_{v}S(\upsilon ){e}^{-j2\pi \upsilon {\tau }_{0}}{\int }_{-\infty }^{+\infty }{e}^{-j2\pi \upsilon {m}_{b}\sin (2\pi {f}_{r}t)}{e}^{-j2\pi \upsilon {m}_{h}\sin (2\pi {f}_{h}t)}{e}^{-j2\pi ft}dt,\end{array}$$
(13)

and $$S(\upsilon )$$ is the FT of the transmitted UWB pulse. This can be represented using Bessel functions as

$$\begin{array}{rcl}Y(\upsilon ,\,f) & = & {a}_{v}S(\upsilon ){e}^{-j2\pi \upsilon {\tau }_{0}}\\ & & \times \,{\int }_{-\infty }^{+\infty }(\sum _{k=-\infty }^{+\infty }{J}_{k}({\beta }_{r}\upsilon ){e}^{-j2\pi k{f}_{r}t})(\sum _{l=-\infty }^{+\infty }{J}_{l}({\beta }_{h}\upsilon ){e}^{-j2\pi l{f}_{b}t}){e}^{-j2\pi ft}dt.\end{array}$$
(14)
$${e}^{-jz\sin (2\pi {f}_{0}t)}=\sum _{k=-\infty }^{+\infty }{J}_{k}(z){e}^{-j2\pi k{f}_{0}t}.$$
(15)

where $${\beta }_{r}=2\pi {A}_{r}$$ and $${\beta }_{h}=2\pi {A}_{h}$$.

Then, (10) is rewritten as

$$Y(m{\delta }_{T},\,f)={a}_{v}\sum _{k=-\infty }^{+\infty }\sum _{l=-\infty }^{+\infty }{G}_{kl}(\tau )\delta (f-k{f}_{r}-l{f}_{h}).$$
(16)

where

$${G}_{kl}(\tau )={\int }_{-\infty }^{+\infty }S(\upsilon ){J}_{k}({\beta }_{r}\upsilon ){J}_{l}({\beta }_{h}\upsilon ){e}^{j2\pi \upsilon (\tau -{\tau }_{0})}d\upsilon .$$
(17)

The maximum value of (17) can be obtained as

$${C}_{kl}={G}_{kl}({\tau }_{0})={\int }_{-\infty }^{+\infty }S(\upsilon ){J}_{k}({\beta }_{r}\upsilon ){J}_{l}({\beta }_{h}\upsilon )d\upsilon$$
(18)

and setting $$l=0$$ in

$$Y({\tau }_{0},\,f)={a}_{v}\sum _{k=-\infty }^{+\infty }\sum _{l=-\infty }^{+\infty }{C}_{kl}\delta (f-k{f}_{r}-l{f}_{h}),$$
(19)

gives the respiration signal

$${C}_{k{\rm{0}}}={\int }_{-\infty }^{+\infty }S(\upsilon ){J}_{k}({\beta }_{r}\upsilon ){J}_{{\rm{0}}}({\beta }_{h}\upsilon )d\upsilon .$$
(20)

setting k = 0 in (19), we can acquire the heartbeat signals as

$${C}_{0l}={\int }_{-\infty }^{+\infty }S(\upsilon ){J}_{0}({\beta }_{b}\upsilon ){J}_{l}({\beta }_{h}\upsilon )d\upsilon \approx {\int }_{-\infty }^{+\infty }S(\upsilon ){J}_{l}({\beta }_{h}\upsilon )d\upsilon .$$
(21)

The experimental data was obtained using a UWB radar with parameters given in32. The radar center frequency is 400 MHz and the PRF is 600 kHz. The UWB pulses were acquired form six segments simultaneously with a time window of 124 ns, Ms = 682 samples per segment, and M = 4092 fast-time samples. NA samples are averaged to improve the SNR. Thus, a set of data is obtained every MsNA/PRF = 34.1 ms, and it takes 17.6 s to receive N = 512 pulses in slow-time. A combination of equivalent-time and real-time sampling34 is employed as it provides better performance than equivalent-time sampling only35.

Figure 2 shows the matrix R obtained using the UWB radar with a human subject at a distance of 9 m from the receiver in an outdoor environment which is described in Section IV. The vital sign signals are not evident due to the significant signal attenuation caused by the through-wall and long distance conditions. Thus, obtaining these signals in real environments is challenging, which motivates the development of a new method for vital sign detection.

## Detection Algorithm

The new method for detection of the vital sign is designed in this section. A flowchart of this algorithm is given in Fig. 3. Two volunteers participated in this research. They were informed of the risks associated with participating in the experiments. This research was approved by both Ocean University of China and the Chinese Academy of Sciences. All experiments were performed in accordance with the relevant international guidelines and regulations.

### Clutter Suppression

Respiration signals are typically corrupted by strong static clutters which are approximately slow-time invariant30. The DC component can be estimated as

$$\Im =\frac{1}{M\times N}\sum _{m=1}^{M}\sum _{n=1}^{N}R[m,\,n]$$
(22)

resulting in the matrix31

$${{\rm{\Omega }}}_{M\times N}={{\rm{\Psi }}}_{M\times N}-\Im$$
(23)

To further remove static clutter, range profile subtraction is employed which is32

$$\widehat{{\rm{\Omega }}}[m,n]={\rm{\Omega }}[m,n]-{\rm{\Omega }}[m,\,n-1]$$
(24)

and then adaptive background subtraction is applied using a weighting coefficient $$\lambda$$ to obtain

$$p[m,\,n]=\lambda p[m,\,n-1]+(1-\lambda ){\rm{\Omega }}[m,\,n]$$
(25)

which gives

$$T[m,\,n]={\rm{\Omega }}[m,\,n]-p[m,\,n]$$
(26)

where $$\lambda =0.9$$ and $$p[m,1]={\rm{\Omega }}[m,1]$$43.

The LTS method estimates the linear trend $${a}_{M\times N}$$ and removes it from the received pulses. A least-squares fit is used to eliminate $${a}_{M\times N}$$ from the received pulses in slow-time by assuming that the starting point is a linear model for the received pulses. This can be expressed in matrix form as

$$T=X\beta$$
(27)

where $$X=[\frac{k}{N}\,{1}_{N}]$$, $$k=0,1,\ldots ,N-1,$$ and $$\beta$$ denotes the unknown coefficient matrix which can be obtained by multiplying (27) by $${X}^{{\rm{T}}}$$

$${X}^{{\rm{T}}}T=({X}^{{\rm{T}}}X)\beta$$
(28)

Rearranging gives

$${({X}^{{\rm{T}}}X)}^{-1}{X}^{{\rm{T}}}T=\beta$$
(29)

Using (28) and (29), the linear trend can be calculated as

$$a=X\beta =X{({X}^{{\rm{T}}}X)}^{-1}{X}^{{\rm{T}}}T$$
(30)

As a result, we can acquire the results given by

$$W=T-X{({X}^{{\rm{T}}}X)}^{-1}{X}^{{\rm{T}}}T$$
(31)

In high SNR conditions, vital signs can easily be obtained even under $${q}_{M\times N}$$ conditions. However, this clutter can significantly degrade signal estimation. SVD divides $$W$$ into complementary subspaces which represent independent features of the received signal. This can be used to reduce the static/non-static clutter and increase the SNR. This decomposition is given by

$$W=US{V}^{{\rm{T}}}$$
(32)

where $$S$$ is a diagonal matrix, $$U$$ is a $$M\times M$$ unitary matrix and $$V$$ is a $$N\times N$$ unitary matrix. The elements of $$S$$ are the singular values $${\sigma }_{i}$$ which are ranked in descending order, i.e. $$\sigma 1\ge \sigma 2\ge \sigma 3\ge \cdot \cdot \cdot \ge {\sigma }_{r}\ge 0$$. The columns of $$U$$ are the left singular vectors and the columns of $$V$$ are the right singular vectors. The eigenvectors of $$W{W}^{{\rm{T}}}$$ are U and the eigenvectors of $${W}^{{\rm{T}}}W$$ are V.

The SVD in (32) can be expressed as

$$\begin{array}{rcl}W & = & {\sigma }_{1}(\begin{array}{c}\cdot \\ \cdot \\ {\mu }_{1}\\ \cdot \\ \cdot \end{array})({\upsilon }_{1}\ldots )+{\sigma }_{k}(\begin{array}{c}\cdot \\ \cdot \\ {\mu }_{k}\\ \cdot \\ \cdot \end{array})(\ldots {\upsilon }_{k}\ldots )+\cdot \cdot \cdot +{\sigma }_{N}(\begin{array}{c}\cdot \\ \cdot \\ {\mu }_{N}\\ \cdot \\ \cdot \end{array})(\ldots {\upsilon }_{N}\ldots )\\ & = & {M}_{1}+{M}_{2}+\cdot \cdot \cdot +{M}_{k}+\cdot \cdot \cdot +{M}_{G}\end{array}$$
(33)

where $${M}_{k}$$ is the kth intrinsic image with the same dimensions as $$W$$. The intrinsic images are arranged in descending order. M1 is the first intrinsic image and is considered as $${q}_{M\times N}$$. The images M2, M3, …, Mk typically contain the majority of the vital sign signals, while Mk+1,…, MG mainly contain noise. Therefore, (33) can be rewritten as

$$W={M}_{C}+{M}_{T}+{M}_{N}$$
(34)

where $${M}_{C}$$ denotes $${q}_{M\times N}$$, i.e. the first intrinsic image, $${M}_{T}$$ is the images containing the vital sign signals and $${M}_{N}$$ is the remaining images. SVD can be used to remove $${q}_{M\times N}$$ which gives

$${\rm{\Phi }}=US{V}^{{\rm{T}}}=\sum _{i=2}^{G}{u}_{i}{\sigma }_{ii}{v}_{i}^{{\rm{T}}}$$
(35)

Note that it is difficult to estimate k so it is not possible to suppress the noise using this technique.

### SNR Improvement

The detection environment including the signal frequency can have a significant effect on the received signals37. As a result, the traditional matched filter maybe invalid for vital signs detection. Instead, a Butterworth bandpass filter (BPF) is employed which has transfer function44

$${|H(\omega )|}^{2}=\frac{1}{1+{(\omega /{\omega }_{c})}^{2{N}_{f}}}$$
(36)

where $${N}_{f}=5$$ is the filter order and $${\omega }_{c}$$ is the cutoff frequency. This filter is composed of a Butterworth low pass filter (LPF) and a Butterworth high pass filter (HPF). This filter is used on (35) in the fast-time which gives

$$\begin{array}{rcl}{\rm{\Lambda }}[m,\,n] & = & {\chi }_{1}{\rm{\Phi }}[m,\,n]+{\chi }_{2}{\rm{\Phi }}[m-1,\,n]+\mathrm{...}+{\chi }_{{N}_{b}+1}{\rm{\Phi }}[m-{N}_{b},\,n]\\ & & -\,{\kappa }_{2}{\rm{\Phi }}[m-1,\,n]-\cdots -{\kappa }_{{N}_{a}+1}{\rm{\Phi }}[m-{N}_{a},\,n]\end{array}$$
(37)

where $${N}_{b}={N}_{a}=5$$, $${\kappa }_{i}$$ and $${\chi }_{i}$$ are the filter coefficients.

An average extraction filter is now used to eliminate low- and high-frequency clutter. The output is

$${\rm{\Psi }}[k,\,n]=\frac{1}{\beta }\sum _{m=\beta k}^{\beta (k+1)-1}{\rm{\Lambda }}[m,\,n],$$
(38)

where $$k=1,\,\ldots ,\,\lfloor M/\beta \rfloor$$, $$\beta =7$$ is the number of filter coefficients, $$\lfloor M/\beta \rfloor$$ is the maximum integer less than $$M/\beta$$, and $${{\boldsymbol{\Lambda }}}_{M\times N}=0$$ when $$M > \lfloor M/\beta \rfloor$$. Equations (2238) are used to suppress the components $$c[m]$$, $${a}_{M\times N}$$, $${q}_{M\times N}$$, and $${g}_{M\times N}$$. $${z}_{M\times N}$$ can be removed when there are no motions in the detection area. The resulting ideal signal is then

$$\begin{array}{rcl}{\rm{\Psi }}[m,\,n] & = & {a}_{v}s(m{\delta }_{T}-{\tau }_{v}(n{T}_{s}))\\ & = & {a}_{v}s(m{\delta }_{R}-v{\tau }_{v}(n{T}_{s}))=h[m,\,n]\end{array}$$
(39)

### Range Determination

The standard deviation (SD) of the received pulses is used for range estimation of human subject. The SD for fast-time index m is45

$${\rm{\Gamma }}[m]=\sqrt{\frac{\sum _{n=1}^{N}{({\rm{\Psi }}[m,n]-{\mu }_{m})}^{2}}{N-1}}$$
(40)

where $${\mu }_{m}$$ is the mean. The data from a female volunteer at 9 m far from the radar were used for SD calculation in through-wall conditions, and Fig. 4 gives the results. This figure shows that the SD is large when a subject is present, which indicates that it can be used for the detection of vital sign signals. Figure 5 shows the corresponding spectrums obtained from the SD values in the target area using an FT, which indicates the SD is periodic, and thus can be used for range estimation.

The DSFT of (40) is used for analyzing signal characteristics in time-frequency domain46,47. Windowing is also employed, which gives

$$K[o,\,p]=\sum _{m=1}^{M}{\rm{\Gamma }}[m]{\rm{\Xi }}[o-m]{e}^{-j2p\pi m/{\bf{P}}},$$
(41)

where p represents frequency and $${\rm{\Xi }}$$ represents the used Hamming window which is expressed as

$${\rm{\Xi }}(o)=\alpha -\beta \,\cos (\frac{2\pi o}{O}),\,o=0,\,1,\,\cdots ,\,O,$$
(42)

where α = 0.54 and β = 0.46. The DSFT window length must be chosen carefully. If it is too small, there will be excessive, and the result tends to zero length increases. A window of O = 512 was chosen based on extensive simulation results. The DSFT is shown in Fig. 6 and has a range error of only 0.104 m. Figure 7 shows the calculated SD values and the corresponding spectrums using DSFT without a human subject in the environment. Compared Figs 6 with 7(b), result indicates that the range information can be acquired by employing the DSFT technique, which is given by

$$\widehat{L}=\frac{v\widehat{\tau }}{{\rm{2}}}$$
(43)

where $$\widehat{\tau }$$ denotes the time delay, i.e. the peak value from (39).

### Frequency Estimation

The time index for the delay estimate $$\widehat{\tau }$$ is

$$\Im =\widehat{\tau }/{\delta }_{T}.$$
(44)

To estimate the vital sign signal frequencies, an FT is performed on (39), which gives

$$Y(\upsilon ,\,{\rm{t}})={a}_{v}S(\upsilon ){e}^{-j2\pi \upsilon {\tau }_{v}(t)}.$$
(45)

Using the time index in (44), (45) can be given by

$$Y(\Im ,\,{\rm{t}})={a}_{v}S(\frac{\Im }{{\delta }_{{\rm{T}}}}){e}^{-j2\pi \frac{\Im }{{\delta }_{{\rm{T}}}}{\tau }_{v}(t)},$$
(46)

and taking the logarithm gives4

$$ln[Y(\Im ,\,t)]=ln({a}_{v})+ln[S(\frac{\Im }{{\delta }_{{\rm{T}}}})]-j2\pi \frac{\Im }{{\delta }_{{\rm{T}}}}{\tau }_{v}(t)+j2k\pi ,$$
(47)

where the term j2 represents the phase variation due to any non-static objects and k is an integer. The unwrapping method can be used to remove this term20, but differentiation provides a simpler solution16. The imaginary part of (47) includes the phase variation of the vital signs $$j2\pi \frac{\Im }{{\delta }_{{\rm{T}}}}{\tau }_{v}(t)$$. Compared with the approach in Section II, the products of the vital signs and the harmonics are not as significant when only the phase is considered. Further, differentiating the logarithm term is simple and straightforward, making this method suitable for real-time vital sign detection16. Using (3), the result is

$$\lambda \text{'}=4{\pi }^{2}\frac{\Im }{{\delta }_{{\rm{T}}}}[{f}_{r}{\tau }_{r}\,\cos (2\pi {f}_{r}t)-{f}_{h}{\tau }_{h}\,\sin (2\pi {f}_{h}t)]$$
(48)

where $$\text{'}$$ denotes differentiation, and the corresponding discrete form as

$$\lambda \text{'}[{\rm{n}}]=4{\pi }^{2}\frac{\Im }{{\delta }_{{\rm{T}}}}[{f}_{r}{\tau }_{r}\,\cos (2\pi {f}_{r}n{T}_{s})-{f}_{h}{\tau }_{h}\,\sin (2\pi {f}_{h}n{T}_{s})]$$
(49)

The respiration frequency is typically 0.2 Hz to 0.5 Hz with an amplitude of 5 to 15 mm, and the heart rate is 0.8 Hz to 2.5 Hz with an amplitude of 2 to 3 mm38. The SNCR can thus be improved using a window for these frequencies. A rectangular window with length $$\kappa$$ in the frequency domain is employed for each slow-time dimension which gives

$${\boldsymbol{\Sigma }}[n]=\chi [n]\odot {\rm{FFT}}\{\lambda \text{'}[n]\},\,n={k}^{\ast },\,{k}^{\ast }+1,\,\ldots ,\,{k}^{\ast }+\kappa -1$$
(50)

where $$\odot$$ is the window operator and $${k}^{\ast }$$ is the minimum index. The FA method proposed in48 is employed to suppress any remaining clutters existing in vital sign signals in the same frequency band. This gives

$$\mathop{{\bf{a}}}\limits^{{\boldsymbol{a}}}[n]=l[n]+jl[n]$$
(51)

where

$$l[n]=\{\begin{array}{ll}2{\rm{\Sigma }}[n], & n > 0\\ {\rm{\Sigma }}[n], & n=0\\ 0, & n < 0\end{array}$$
(52)

## Performance Results

Compared with several well-known techniques, the detection capability of the presented algorithm is tested in this section. This is achieved using experimental results obtained in both indoor and outdoor environments.

### Experimental Setup

Figure 8 shows the UWB radar experimental setup. The wall has a thickness of 100 cm and is composed of brick (30 cm), reinforced concrete (40 cm), and wood (30 cm). The four experiments described below were conducted to obtain data for performance evaluation.

1. 1.

This experiment was carried out at IECAS outdoors as shown in Fig. 9(a). A female subject stood breathing normally and facing the radar. The subject stood at 11 m, 9 m, 6 m, and 3 m far from the radar successively. The radar was placed on a table at a height of 1.5 m.

2. 2.

This experiment was conducted indoors at the China National Fire Equipment Quality Supervision Center (CNFEQSC) as shown in Fig. 9(b). A male subject stood 12 m, 10 m, 7 m, and 4 m far from the radar. The radar was placed on a table at a height of 1.3 m.

An actuator was used instead of a human subject in experiments 3 and 4. The actuator moves at a frequency of 0.3333 Hz with an amplitude of 3 mm.

3. 3.

This experiment was carried out at IECAS outdoors as shown in Fig. 10(a). The actuator was placed on a table at a height of 1.3 m at a distance of 11 m from the radar.

4. 4.

This experiment was conducted at CNFEQSC indoors as shown in Fig. 10(b). The actuator was placed 12 m, 10 m, 7 m, and 4 m far from the radar.

The performance of the proposed, FFT, constant false alarm ratio (CFAR)34, and MHOC37 methods as well as the advanced method (AM)36, the FA method with different times, and the CSD42 and logarithm CSD (LCSD) techniques are evaluated in the following sections using the experimental data.

### Initial Detection Performance

In this subsection, based on the data from experiment 1, the performance of the clutter suppression steps is considered with the subject 9 m from the radar. The results after removing the DC component and static clutter are given in Fig. 11(a,b), respectively, and Fig. 11(c) presents the results after removing the linear trend. These figures show that human respiration signals are quite weak and as a consequence, the oscillations are challenging to determine. The results after SVD and fast-time filtering are given in Fig. 11(d,e), respectively. These show a further reduction in the clutter which improves the respiration signal. Figure 11(f) shows the results after slow-time filtering and indicates that the respiration signal is significantly improved compared to the received signal in Fig. 2.

### Human Subject Detection Outdoors

In this subsection, the data from experiment 1 is used for evaluating the performance of the detection method. Usually, the SNR of vital signs increases with decreasing distance between the radar and human subject due to large-scale attenuation32. The improvement in SNR can thus be used to evaluate the performance, which can be obtained as

$$SNR=20\,{\mathrm{log}}_{10}(\frac{\sum _{n={\mu }_{r}-1}^{{\mu }_{r}+1}|\mathop{{\bf{a}}}\limits^{{\boldsymbol{a}}}[n]|}{\sum _{n={\gamma }_{1}}^{{\mu }_{r}-2}|\mathop{{\bf{a}}}\limits^{{\boldsymbol{a}}}[n]|+\sum _{n={\mu }_{r}+2}^{{\gamma }_{2}}|\mathop{{\bf{a}}}\limits^{{\boldsymbol{a}}}[n]|}),$$
(53)

where $${\mu }_{r}$$ is the frequency index corresponding to the target, and $${\gamma }_{1}$$ and $${\gamma }_{2}$$ are the boundaries of the frequency window.

The data sets acquired outdoors were used to evaluate four detection methods. Figure 12 gives the calculated SD results and the corresponding spectrums using the DSFT technique are shown in Fig. 13. The errors in the estimations of range information are 26 cm, 11 cm, 5 cm, and 5 cm at ranges of 11 m, 9 m, 6 m and 3 m, respectively.

Table 1 gives the estimations of human respiration frequency, range errors, and SNR values from (53) for four detection algorithms. This shows that the presented method supplies the most accurate range estimates and high SNR, particularly for long distances. At a distance of 11 m, the SNR with the proposed method is 3.25 dB while the SNR is only −15.3 dB with the CFAR method, a difference of 18.6 dB. The respiration frequency estimates using the proposed method are 0.26 Hz at 11 m, 0.29 Hz at 9 m, 0.26 Hz at 6 m, and 0.32 Hz at 3 m, respectively, which are considerably more accurate than with the other methods. Figure 14 gives the results acquired from the CFAR technique, which indicate that the range information cannot be estimated accurately with this method over long distances. Figure 14(b–d) show that there remains significant clutter at distances of 6 m or more.

To evaluate the heart rate estimation performance, the data with a male subject outdoors was used. A typical resting heart rate is 75 to 81 beats per minute, corresponding to frequencies of 1.25 to 1.35 Hz. Figure 15 shows that the heart rate estimates for the proposed method are 1.34 Hz, 1.34 Hz, and 1.45 Hz, respectively. It was not possible to obtain estimates using the CFAR method. The corresponding SNR values for the proposed and CFAR methods are given in Table 2. This shows that the proposed method provides a significant SNR improvement over the CFAR method as the smallest difference is 9.43 dB at a distance of 6 m. The very low SNR values for the CFAR method are the reason the heart rate could not be estimated.

### Human Subject Detection Indoors

The data from experiment 2 is now used for evaluating the detection performance. Figure 16 shows the calculated SD values with the proposed method and the DSFT results are given in Fig. 17. The range estimates are 11.8 m, 9.89 m, 6.95 m, and 4.06 m for distances of 12 m, 10 m, 7 m and 4 m, respectively, and show that the range information can be estimated more accurately indoors. This is because it is a controlled environment which lacks factors such as wind. The respiration frequency estimation results are shown in Fig. 18. The estimates of for the four distances are 0.26 Hz, 0.29 Hz, 0.26 Hz, and 0.32 Hz.

### Clutter Suppression

The capability of removing clutters was evaluated using the data from experiment 1 at 6 m. The harmonics and the products of the vital sign signals were suppressed using the FA method and the results are given in Figs 19 and 20. This shows that the four-FA method can supply a better SNR improvement than the FFT, two-FA, and one-FA methods. Table 3 gives the SNR values and shows that the additional improvement is minimal when the FA method is performed six times. The SNR for the FFT, CSD, logarithm CSD (LCSD) and differentiated LCSD (DLCSD) methods is also given in Table 4. The corresponding frequency domain results for the CSD, LCSD, and DLCSD methods are given in Fig. 21(b–d), respectively. The range estimation results in Fig. 14 show that these methods do not effectively suppress the clutter compared to the proposed method.

### Actuator Experiment Results

The data from the conducted actuator experiments were used for testing the presented algorithm. The actuator was used to imitate human respiration with the amplitude of 3 mm and a frequency of 0.3333 Hz and. Figure 22 shows the SD values for the proposed method using the data from experiment 4 and the corresponding range and frequency estimation results are given in Figs 23 and 24, respectively. The frequency estimate from Fig. 24(a) is 0.353 Hz. The frequency estimate using the CSD method from Fig. 24(b) is 0.11 Hz and the estimate using the LCSD method from Fig. 24(c) is 0.56 Hz. The FFT method provides the worst performance as the frequency estimate from Fig. 24(d) is 0.12 Hz. Compared with the actual frequency of 0.3333 Hz, the proposed method provides the best estimate with an error of only 0.019 Hz. The frequency estimates using the outdoor data from experiment iii is shown in Fig. 25. The frequency estimates and corresponding deviations are given in Table 4 for three trials of the experiment. All these results indicate that the presented algorithm has the smallest deviation and significantly outperforms the other algorithms.

## Conclusion

Vital sign detection was considered in this paper for applications such as after a natural disaster. The respiration and heartbeat frequencies were estimated using an impulse UWB radar. The CSD-based FA method was shown to be effective in suppressing signal products and harmonics. The discrete short-time Fourier transform (DSFT) of the calculated standard deviation (SD) values was used to estimate the range information of the volunteer. Experimental data obtained using a UWB radar were used to evaluate the performance of several techniques. The results obtained indicate that the proposed method can more effectively remove clutter and improve the SNR than the other methods. Further, it provides better range and frequency estimates49,50,51,52,53.