# A New Model for Human Running Micro-Doppler FMCW Radar Features
## 3 Radar Echo Model of Human Running  
Most existing research methods use simple single-carrier frequency continuous waves in studying human motion [24]. However, these waves can only be used to acquire the velocity of the subject and not its spatial location. FMCW radar is used for human target detection and motion information extraction. Firstly, an FMCW radar point target echo model is constructed. Then, the FMCW radar echo model of human targets is established according to the rigid body scattering characteristic. The plural form of the FMCW radar signal is:  
\[m(t)=e^{i2\pi(f_{0}+0.5 Kt^{2})}, \tag{4}\]  
where \(K\) is the slope, determined by the bandwidth and period, and \(f_{0}\) is the starting frequency of the chirp signal. The echo signal can be expressed as:  
\[r(t)=\sigma m(t-\tau)=\sigma e^{i2\pi[f_{0}(t-\tau)+0.5K(t-\tau)^{2}]}, \tag{5}\]  
where \(\tau\) is the two-way delay of electromagnetic wave, and \(\sigma\) is the amplitude attenuation factor. The received signal is mixed with the local oscillating signal to obtain the intermediate frequency (IF) signal:  
\[s(t)=e^{i2\pi(f_{0}\tau+K\tau t-0.5K\tau^{2})}. \tag{6}\]  
Figure 2: Space trajectory. (**a**) Positional changes in vertical translation and (**b**) angle changes in hip flexion and left and right rotation. For FMCW radar, the frequency of the IF signal can be used to estimate the target distance. A one-dimensional fast Fourier transform (FFT) is carried out on the IF signal. The distance amplitude is obtained after phase-reference accumulation. The phase information on each distance unit responds to target phase change information of the Doppler effect. Therefore, without loss of generality, when modeling the radar echo model of the human body, the radar signal can be directly restored from the phase of the signal to a single-frequency signal. The human body model is constructed from geometric bodies such as spheres and ellipsoids. First, it is necessary to discuss the radar scattering cross-sectional area of ellipsoids and spheres. The radar cross section (RCS) of each part can be determined by the relationship between the location of the target and the location of the radar, as well as the radii of geometry. The relationship between the ellipsoid and the radar is shown in Figure 3. The x-y-z coordinate system is the global coordinate system. It belongs to the same coordinate system as the coordinate system in Figure 1 and has the same definition. The radar cross-section (\(RCS\)) of an ellipsoid, which is given by:  
\[RCS_{ellip}=\frac{\pi a^{2}b^{2}c^{2}}{\left(a^{2}sin^{2}\theta cos^{2}\varphi+b ^{2}sin^{2}\theta sin^{2}\varphi+c^{2}cos^{2}\theta\right)^{2}}, \tag{7}\]  
where the ellipsoid's three semi-axes lengths in the \(x\), \(y\), and \(z\) directions are denoted by \(a\), \(b\), and \(c\), respectively. \(\theta\) and \(\varphi\) denote the pitch and azimuth angles; when \(a=b=c\), the above equation simplifies to the sphere \(RCS\) formula. In order to calculate the FMCW radar echo of the human body, the distance of each body component relative to the radar needs to be calculated based on the spatial coordinates of the ellipsoid of different parts. Using the \(RCS\) of different body parts, the echo attenuation coefficients are calculated. Finally, the echo arithmetic of the human body model is calculated by weighted summation. Assume the center coordinate of the \(i\)th part is expressed as the distance from the radar as \(R_{i}\). In the simulation process, since the duration of each chirp signal is short, usually in the order of ms, the human body moves at a limited speed during a chirp pulse. It can be approximated that the radial distance \(R_{i}\) does not change significantly during a chirp pulse. Then the radar echo from one body part is:  
\[S_{i}(t)=e^{\mid 2\pi f_{0}(t-\frac{2R_{i}}{c})}. \tag{8}\]  
Figure 3: Geometric relationship between the radar and one part of the person.