## 3 Respiration Detection
### Windowed Fourier Transform  
Once the clutter and linear trend have been suppressed, AWGN given by \(\mathbf{w}[m,n]\) is the main signal component that degrades detection performance [42]. To reduce the effect of AWGN, the slow time kurtosis is employed which is given by [43]  
\[k_{m}=\frac{\mathrm{E}\Bigl{[}\left(\mathbf{F}[m,n]\right)^{4}\Bigr{]}}{ \left\{\mathrm{E}\Bigl{[}\left(\mathbf{F}[m,n]\right)^{2}\Bigr{]}\right\}^{2}}, \tag{19}\]  
where \(\mathrm{E}[]\) denotes expectation with respect to \(n\). For a Gaussian distributed random variable, the kurtosis is three. Thus, the excess kurtosis is used here which is [44; 45; 46]  
\[\hat{k}_{m}=k_{m}-3. \tag{20}\]  
Excess kurtosis is denoted as kurtosis in the remainder of the paper. To illustrate the characteristics of the kurtosis, experiments were conducted outdoors with and without a female subject. The experimental setup is explained in detail in Section 4. The kurtosis \(\mathbf{Z}_{M\times 1}\) is shown in Figure 5. The red lines indicate the kurtosis with a subject and the black lines without a subject. The distances between the antenna and test subject are 3 m, 6 m, 9 m, and 11 m. These results indicate that the kurtosis is negative when a subject is present, and near zero without a subject. An FFT is usually performed on \(\mathbf{F}_{M\times N}\) in each slow time dimension to extract the respiration information. However, it cannot also be used to provide the range [47]. The windowed FFT (WFT) has been used to determine time variations [48; 49]. Thus, in this paper, the range of the human subject is obtained from the discrete WFT (DWFT)  
\[\mathbf{K}[o,p]=\sum_{m=1}^{M}\mathbf{Z}[m]\boldsymbol{\lambda}[o-m]e^{-\beta p \tau m/P}. \tag{21}\]  
where \(p\) denotes the \(p\)th discrete frequency, \(p=0,1,\ldots,P-1\), and \(\boldsymbol{\lambda}\) is the Hamming window given by  
\[\boldsymbol{\lambda}(o)=\alpha-\beta\cos\left(\frac{2\pi o}{O}\right),\ \ o=0,1,\cdots,O-1, \tag{22}\]  
where \(\alpha=0.54\), \(\beta=0.46\), and \(O=512\) corresponds to the width of the window [50]. The result \(\mathbf{K}_{O\times P}\) without a human subject is given in Figure 6, while the result with a subject is shown in Figure 7. The vertical axis is frequency and the horizontal axis is fast time. Comparing these results shows that the signal is periodic when a subject is present. The time-frequency characteristics of this periodic signal are shown in Figure 8.  
Figure 5: The kurtosis of the received signal with (red) and without (black) a human subject. The distances between the antenna and subject are (**a**) 300 cm, (**b**) 600 cm, (**c**) 900 cm, and (**d**) 1100 cm. To obtain the range of the subject from \(\mathbf{K}_{\mathcal{O}\times\mathcal{P}}\), a frequency window of width 0.6 to 1.2 GHz is employed. If \(\hat{\tau}\) corresponds to the maximum amplitude index, then the estimated range is  
\[\overline{\overline{L}}=v\times\frac{\hat{\tau}}{2}\,. \tag{23}\]  
Figure 6: The time-frequency matrix without a test subject after the DWFT. \[\hat{k}_{m}=k_{m}-3. \tag{20}\]0 The time-frequency matrices obtained using the DWFT. The distances between the antenna and subject are (**a**) 300 cm, (**b**) 600 cm, (**c**) 900 cm, and (**d**) 1100 cm. To improve the accuracy of the range estimate given by (23), a frequency accumulator is employed to eliminate the harmonics of the respiration signal. The frequency of this signal is typically in the range 0.2 Hz to 0.4 Hz [23; 24]. After an FFT is performed on \(\mathbf{A}_{M\times N}\) to obtain \(\mathbf{F}_{M\times N}\), a frequency window of width 0.1 Hz to 0.8 Hz is used. A typical output is shown in Figure 9a. The resulting matrix is \(\mathbf{U}_{M\times N^{\prime}}\) where \(N\prime=\gamma_{2}-\gamma_{1}+1\) is the number of values between 0.1 to 0.8 Hz, \(\gamma_{2}\) is the index corresponding to 0.8 Hz, and \(\gamma_{1}\) is the index corresponding to 0.1 Hz. Then the following operation is performed [51]  
\[\widetilde{\mathbf{U}}[m,i]=\left\{