# Optimal Steady-State Range Prediction Filter for Tracking with LFM Waveforms
## 4 Performance Evaluation
### LFM Radar Simulation  
Finally, numerical simulation results are presented to demonstrate the effectiveness of the proposed method in range tracking using LFM radar. Figure 3 shows the simulation scenario. Although a two-dimensional problem is considered, we assume range measurements taken using a single LFM radar to demonstrate the validity of the proposed range tracking method for the target other than the constant-acceleration target assumed in the analysis. Figure 3a shows the true moving orbit of the target in the \(x-y\) plane and the radar position. The true motion of the target in \(x\) and \(y\) are \(x_{\mathrm{t,k}}=-5\cos((3\pi(N_{o}-k))/(2N_{o}))(80\sin((\pi kT)/20)-k^{2}T^{2} -60)\) and \(y_{\mathrm{t,k}}=-5\sin((3\pi(N_{o}-k))/(2N_{o}))(80\sin((\pi kT)/20)-k^{2}T^{2} -60)\), where \(N_{o}=200\) is the number of samples with respect to \(k\). In this setting, the true range in the simulation is \(r_{\mathrm{t,k}}^{\mathrm{sim}}\)=\(300+5(kT)^{2}+400\cos(\pi kT/20+\pi/2)\). The LFM radar is located at \((x,y)=(0,0)\) and its \(T\) is 0.1 s. Figure 3b shows the true acceleration of the target with respect to \(k\). Based on this target acceleration, it is assumed that the approximate target maximum acceleration \(A_{\mathrm{max}}=20\) m/s\({}^{2}\) has been obtained. Figure 3c shows the true and observed range data for \(c_{\mathrm{RD}}=\pm 0.5\) used as \(r_{\mathrm{o,k}}\). Two instances \(c_{\mathrm{RD}}=\pm 0.5\) are considered and their standard deviations for measurement noise are both \(\sigma_{\mathrm{w}}=0.2\) m. These settings give \(\Gamma_{\mathrm{D}}=1.0\) with Equation (13). We compared the RMS prediction errors of the conventional method of [10] and the proposed method for \(c_{\mathrm{RD}}=\pm 0.5\). As shown in Figure 3c, the range-offset dependence on \(c_{\mathrm{RD}}\) is confirmed, and their effects on the range prediction accuracy when applying the conventional and proposed methods were investigated. The RMS prediction error at each time \(kT\) is calculated from one thousand Monte Carlo simulations; that is,  
\[\mathrm{RMSE}_{k}=\sqrt{\frac{1}{1000}\sum_{m=1}^{1000}(r_{\mathrm{t,k}}^{ \mathrm{sim}}-r_{\mathrm{p,km}})^{2}}, \tag{19}\]  
where \(r_{\mathrm{p,km}}\) is the predicted range in the \(m\)-th Monte Carlo simulation. Figure 4 shows the simulation results for \(c_{\mathrm{RD}}=\pm 0.5\). For both instances, the proposed method achieves accurate tracking for steady states compared with the conventional method. In particular, the proposed method is effective around \(kT=10\) s because the filter gains are optimized for the maximum acceleration value. As shown in these results, the values of \(\mathrm{RMSE}_{k}\) for \(c_{\mathrm{RD}}=0.5\) is smaller than that for \(c_{\mathrm{RD}}=-0.5\). This is because that the positive \(c_{\mathrm{RD}}\) supports the prediction process in the tracking filtering as fully discussed in [11]. Note that the relationship between the range prediction error and \(c_{\mathrm{RD}}\) is also shown in Figure 2, and this also verifies the performance improvement when using positive \(c_{\mathrm{RD}}\). The performance difference between the conventional and proposed methods is relatively large for \(c_{\mathrm{RD}}=-0.5\). The reason is that the range prediction accuracy has been already improved through the effect of positive \(c_{\mathrm{RD}}\)[11] for \(c_{\mathrm{RD}}=0.5\). Thus, the likelihood of an improvement using the proposed  
method is small. In contrast, for \(c_{\rm RD}=-0.5\), the range prediction errors widen with negative \(c_{\rm RD}\). Therefore, using the proposed method seems to compensate for these errors effectively. Therefore, the performance improvement with this method becomes relatively large for \(c_{\rm RD}=-0.5\). In terms of \(c_{\rm RD}=0.5\), although smaller RMSE\({}_{k}\) values are achieved, the difference between the conventional and proposed methods is relatively small. However, the accuracy in steady-state tracking of the proposed method is always better than that of the conventional method.