## 7 Application to UWB Doppler Radar Simulation
### Tracking of Medium Maneuvering Target  
#### 7.1.1 Simulation Setup  
First, we show the application examples for a maneuvering target that assumes near-field radar remote sensing for surveillance and robot monitoring systems. We simulate the UWB Doppler radar tracking [23; 24; 25] of a maneuvering target and compare the tracking errors of the filters assumed in the previous section and the PVM Kalman filter. For one-dimensional tracking assumed in the previous sections, the steady state performances of the PVM Kalman and the proposed filters are the same. However, for two-dimensional tracking assumed in this section, a difference in their tracking accuracy occurs because the Kalman filter considers correlations between the \(x-y\) positions and velocities. Therefore, a comparison between the \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filters and the PVM Kalman filter is also necessary.  
Figure 3 shows the simulation scenario and the true acceleration of a target. The true target position is \((x_{\rm t,k},y_{\rm t,k})=(0.5+0.3 kT\sin(2\pi kT/10),\,1.5+0.1(kT)^{1.2}\cos( 2\pi kT/12))\). Two-dimensional tracking in the \(x\)-\(y\) plane of the point target is assumed. We consider two Doppler radars located at \((x,y)=(0.5\;\;{\rm m},0)\) and \((1.0\;\;{\rm m},0)\). The sampling interval \(T\) is 100 ms, and the observation time is 4 s. The transmitted signal is a UWB pulse with a center frequency of 26.4 GHz and a bandwidth of 500 MHz. The received radar signals are calculated using ray tracing with the addition of Gaussian white noise. The radars measure position using ranging results [24] and measure velocity using the Doppler shift with the method presented in [25]. We determine a standard derivation for this noise to set \(B_{\rm x}=0.030^{2}\) m\({}^{2}\) and \(B_{\rm v}=0.10^{2}\) m\({}^{2}\)/s\({}^{2}\). These values are the averages along the two axes (\(x\) and \(y\)and are set based on the experimental results in [23; 24]. Thus, the \(R_{\text{xv}}\) of the assumed radar system is 9.0. For simplicity, we use this value for both axes. In addition, to evaluate the performance for smaller \(R_{\text{xv}}\), the case \(R_{\text{xv}}=1.0\) is generated by adding Gaussian white noise to the measured velocity data. \(B_{\text{v}}\) in this instance is \(0.30^{2}\) m\({}^{2}\)/s\({}^{2}\).  
The \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filters are implemented for each axis, for which we use the same gain. The implementation of the PVM Kalman filter is the same as in [27], and its covariance matrix for process noise is calculated from the optimal \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filter gains designed with the proposed strategy using Equations (34)-(36). The initial values of the state vectors and the error covariance matrix of the PVM Kalman filter are all zero. Using the RMS prediction error calculated from 1000 Monte Carlo simulations, the performance is defined as:  
\[\epsilon_{k}=\sqrt{\frac{1}{1000}\sum_{m=1}^{1000}\{(x_{\text{t,k}}-x_{\text{p,m,k}})^{2}+(y_{\text{t,k}}-y_{\text{p,m,k}})^{2}\}}, \tag{37}\]  
where \(x_{\text{p,m,k}}\) and \(y_{\text{p,m,k}}\) are the predicted positions in the \(m\)-th Monte Carlo simulation.