In this section, the presented method for VS detection is analyzed based on the detailed steps as shown in Fig. 3.
Clutter suppression
The VS are usually covered by c[m] with strong amplitude, which can be estimated as:
$$ \Im =\frac{1}{M\times N}\sum \limits_{m=1}^M\sum \limits_{n=1}^N\Psi \left[m,n\right] $$
(8)
After removing (8), the resulting matrix Ω_{M × N} can be expressed as:
$$ \Omega =\Psi \Im $$
(9)
To remove a[m, n], the LTS algorithm is used. The result W_{M × N} is
$$ W={\Omega}^{\mathrm{T}}\hbox{} X{\left({X}^{\mathrm{T}}X\right)}^{1}{X}^{\mathrm{T}}{\Omega}^{\mathrm{T}} $$
(10)
where X = [x_{1}, x_{2}], x_{1} = [0, 1, . . …, N − 1]^{Τ} and \( {x}_2={\left[1,\kern0.5em 1,\kern0.5em ..\dots, 1\right]}_{N\times 1}^{\mathrm{T}} \).
Usually, the maximum SNR cannot be obtained only based on the matched filter, as the reflected pulses are different from the traditional radiated signals. Some other effective methods are required to improve SNR. In this paper, an infinite impulse response bandpass filter is used with the transfer function given by:
$$ {\leftH\left(\omega \right)\right}^2=\frac{1}{1+{\left(\omega /{\omega}_{\mathrm{c}}\right)}^{2{N}_{\mathrm{f}}}}. $$
(11)
where ω_{c} represents the cutoff frequency and N_{f} represents the filter order.
The bandpass filter is performed on (10) in the range direction for each slowtime index n; the outputs are:
$$ \Lambda \left[m,n\right]={\chi}_1W\left[m,n\right]+{\chi}_2W\left[m1,n\right]+\dots +{\chi}_{N_b+1}W\left[m{N}_b,n\right]{\kappa}_2W\left[m1,n\right]\dots {\kappa}_{N_a+1}W\left[m{N}_a,n\right] $$
(12)
where N_{
b
} = N_{
a
} = 5 and κ_{
i
} and χ_{
i
} are the filter coefficients.
Further, to improve SNR, an average extraction filter is applied in (12) in slowtime direction with λ = 7, and the result is given by:
$$ \Phi \left[k,n\right]=\frac{1}{7}\sum \limits_{m=7\lambda}^{8\lambda 1}\Lambda \left[m,n\right] $$
(13)
where k = 1, . …, ⌊M/λ⌋. ⌊M/λ⌋ represents the maximum integer less than M/λ.
After removing the various clutters as shown in (6), (7) can be described in the discrete form as:
$$ \Psi \left[m,n\right]={a}_vs\left(m{\delta}_T{\tau}_v\left({nT}_s\right)\right)={a}_vs\left(m{\delta}_Rv{\tau}_v\left({nT}_s\right)\right)=h\left[m,n\right]\kern0.5em $$
(14)
Range estimate
In this section, a new scheme for range estimate is developed by analyzing the skewness of VS.
The skewness for each range index m in (14) can be given by [43, 44]:
$$ Z(m)=E\left[{\left(\frac{\Psi \left[m,N\right]\mu }{\sigma}\right)}^3\right] $$
(15)
where Ε[•] represents the expectation and μ and σ are the mean and standard deviation.
In this paper, the skewness spectrum is analyzed to estimate the range between the radar antenna and human subject. To show the skewness of VS, one data acquired from one female subject is applied. The distance from the human subject to the radar antenna is 9 m outdoors, which will be introduced in Section 4.
As shown in Fig. 4a, the skewness in human subject area follows the periodicity approximately as show in Fig. 4b. To estimate the range, the DSFT is performed on (15), which has been widely applied in signal processing [44, 45]. The result is given by:
$$ K\left[o,p\right]=\sum \limits_{m=1}^MZ\left[m,1\right]\Xi \left[om\right]{e}^{j2 p\pi m/P}. $$
(16)
where P denotes the discrete frequency, which follows the uniform distribution. Ξ represents the Hamming window, which can be expressed as:
$$ \Xi (o)=\alpha \beta \cos \left(\frac{2\pi o}{O}\right),\kern1.5em o=0,1,..\dots O. $$
(17)
where α = 0.54, β = 0.46, and O = 512 is the width of the Hamming window [46].
The result K_{O × P} acquired by performing DSFT on (15) with human subject is shown in Fig. 5. When there is no any human subject in the detection environment, the calculated skewness is shown in Fig. 6a, and the corresponding timefrequency matrix is given in Fig. 6b.
It can be seen that the distance can be estimated as:
$$ \widehat{L}=v\widehat{\tau}\kern0.5em $$
(18)
where \( \widehat{\tau} \) represents the time estimate related to the maximum in (16).
Frequency estimate
The index of the time estimate in (13) can be given by:
$$ \Im =\widehat{\tau}/{\delta}_T $$
(19)
To acquire the frequency of human respiratory, the corresponding signal in slow time is chosen as the effective signal, which can be given by:
$$ \mathrm{O}=\Psi \left[\Im, n\right] $$
(20)
As an adaptive method, EEMD has been widely applied in analyzing nonstationary signals [45, 46], which overcomes the drawback in the traditional empirical mode decomposition [47]. Based on the EEMD technique, the nonstationary signal can be broken down into several intrinsic mode functions (IMFs) and a residual trend by employing the AWGN adaptively. The IMFs and residual trend can be used to reconstruct AWGN, which can improve SNR effectively.
To acquire the IMFs of (20), the steps for the EEMD method can be summarized as:

I)
Add the AWGN to (20);

II)
Based on the empirical mode decomposition, (20) can be decomposed into IMFs as:

1)
Let v = 0, which is used to indicate the vth IMF;

2)
To acquire all these local maximum and minimum values of (20);

3)
To get the envelopes including the upper envelope r_{
u
}(t) and the lower envelope r_{
l
}(t)of (20) based on the values from 2) by employing the cubic spline function;

4)
To generate the average envelope from 3), which can be given by [48]:
$$ m(t)=\frac{r_u(t)+{r}_l(t)}{2} $$
(21)

5)
Subtracting (21) from (20), which can be given by:
$$ h(t)={\tilde{\mathrm{O}}}_{1\times \mathrm{N}}m(t) $$
(22)

6)
Go to step (II), (22) is processed until an IMF meets the stoppage criteria;

7)
v = v + 1, IMF_{
v
}(t) = h(t). The residue trend can be given by [49]:
$$ {q}_v(t)={\tilde{\mathrm{O}}}_{1\times \mathrm{N}}{\mathrm{IMF}}_v(t) $$
(23)

8)
Go to step (II) and (23) is processed. The whole decomposition stops until the amplitude of IMF_{
v
}(t) is small enough.
Based on the empirical mode decomposition method, (20) can be broken into several IMFs, and the residue signal given by:
$$ {\tilde{\mathrm{O}}}_{1\times \mathrm{N}}=\sum \limits_{v=1}^{M_v}{\mathrm{IMF}}_v(t)+{q}_{M_v}(t) $$
(24)

III)
For each added AWGN, (20) is processed by employing the steps (I)–(II) repeatedly;

IV)
The average values of the IMFs are considered as the final result, which are given by:
$$ {\tilde{\mathrm{O}}}_{1\times \mathrm{N}}=\sum \limits_{v=1}^{M_v}{\mathrm{IMF}}_v^{\mathrm{mean}}(t)+{q}_{M_v}^{\mathrm{mean}}(t) $$
(25)
where
$$ {IMF}_v^{\mathrm{mean}}(t)=\frac{1}{N_v}\sum \limits_{v=1}^{N_v}{\mathrm{IMF}}_v(t) $$
(26)
$$ {q}_{M_v}^{\mathrm{mean}}(t)=\frac{1}{N_v}\sum \limits_{v=1}^{N_v}{q}_v(t) $$
(27)
and N_{
v
} is the times of the added AWGN.
For the EEMD method, two key parameters are required to be determined such as the amplitude and the times of the added AWGN. Usually, the relationship between the amplitude of (20) and the added AWGN can be given by:
$$ {\varepsilon}_n=\frac{\varepsilon }{\sqrt{N_v}} $$
(28)
where ε is the standard deviation of the added AWGN and ε_{
n
} is the error between (20) and the ideal signal reconstructed based on the chosen IMFs.
Figure 7 shows the timefrequency matrix of (20) and the acquired IMFs based on EEMD, and Fig. 8 shows the corresponding welch power spectral density of the IMFs. As known, the frequency of human respiratory movement is usually within 0.2–0.5 Hz with the amplitude of 0.5–1.5 cm. As a result, the corresponding IMFs with the power being in the range of 0.1–0.8 Hz are chosen to reconstruct the signal, which can be given by:
$$ {\tilde{\mathrm{O}}}_{1\times \mathrm{N}}\kern0.5em =\kern0.5em \sum \limits_{v=4}^5{\mathrm{IMF}}_v^{\mathrm{mean}}(t) $$
(29)
A rectangular window χ is performed on the frequency components of (29), which gives
$$ \Omega \left[n\right]=\chi \left[n\right]\left\{\mathrm{DFT}\left\{{\tilde{\mathrm{O}}}_{1\times N}\right\}\right\}\kern2.12em n\in {K}^{\ast };{K}^{\ast }=\left\{{k}^{\ast },{k}^{\ast }+1,\dots, {k}^{\ast }+\kappa 1\right\} $$
(30)
where \( \mathrm{DFT}\left\{{\tilde{\mathrm{O}}}_{1\times N}\right\} \) is the discrete fast Fourier transform (DFT) of (29) and k^{∗} corresponds to the index of the lowest retained frequency component.
The frequency of human respiratory movement can be acquired as:
$$ {f}_{\mathrm{r}}=w\left({\mu}_{\mathrm{r}}\right)\kern0.5em $$
(31)
where μ_{r} corresponds to the index of the maximum value in (30), and w ∈ (0.1, 0.8).
As known, the harmonics are the major factor affecting the frequency estimate. To suppress the harmonics, an accumulation method is proposed [50], which gives:
$$ \delta \left[n\right]=l\left[n\right]+ jl\left[n\right] $$
(32)
where
$$ l\left[n\right]=\left\{\begin{array}{l}2\Omega \left[n\right],\kern4.5em \kappa >0\\ {}\Omega \left[n\right],\kern5.5em \kappa =0\\ {}0,\kern8em \kappa <0\end{array}\right. $$
(33)