# Optimal Steady-State Range Prediction Filter for Tracking with LFM Waveforms
## 3 Proposed Optimal Range Prediction Filter Design
### RMS-Index for LFM \(\alpha\)-\(\beta\) Filter  
The RMS-index approach minimizes the RMSE of the prediction error of range (or position). In this section, the RMS-index is defined and derived for the LFM \(\alpha\)-\(\beta\) filter. Similar to Equation (10), the RMS-index for the range prediction is defined as [13]:  
\[\rm RMSE_{pred}\equiv\sqrt{E[(r_{\rm t,k}^{\alpha}-r_{\rm p,k})^{2}]}=\sqrt {\sigma_{\rm r}^{2}+(L_{\rm rp}A_{\rm max}T^{2})^{2}}=\sigma_{\rm w}\sqrt{ \sigma_{\rm r}^{2}/\sigma_{\rm w}^{2}+(L_{\rm rp})^{2}\Gamma_{\rm D}^{2}}, \tag{14}\]  
where \(\sigma_{\rm r}^{2}\) is the steady-state variance of the range prediction errors assuming only sensor noise and \(L_{\rm rp}A_{\rm max}T^{2}\) is the steady-state bias error of the range prediction assuming a target moving with constant acceleration \(A_{\rm max}\). The RMS-index expresses the maximum RMS error for the prediction of the range (or position) in steady-state conditions. Because the LFM \(\alpha\)-\(\beta\) filter is using the constant-velocity model as indicated in Equations (1) and (2), there is a bias error in the steady state associated with the accelerating target, and the filter diverges for the target moving with constant jerk. Thus, the maximum (worst-case) error for the steady-state LFM \(\alpha\)-\(\beta\) filter is calculated by assuming the target moving with \(A_{\rm max}\)[10].  
To derive \(\rm RMSE_{pred}\), \(\sigma_{\rm r}^{2}\) (steady-state variance of random errors) and \(L_{\rm rp}A_{\rm max}T^{2}\) (steady-state bias error) are calculated one by one. First, \(\sigma_{\rm r}^{2}\) of the LFM \(\alpha\)-\(\beta\) filter is calculated. There is no bias error associated with targets moving with constant velocity because the \(\alpha\)-\(\beta\) filter uses the constant-velocity model. Hence, in the constant-velocity target tracking, there are only random errors arising from the radar measurement noise. These mean that \(\sigma_{\rm r}^{2}\) is calculated by the mean square error for the constant-velocity target, and is calculated as \[\sigma_{\rm r}^{2}\equiv E[(r_{\rm t,k}^{\rm cv}-r_{\rm p,k})^{2}]=\frac{2\alpha^{2 }+2\beta+\alpha\beta-\beta^{2}c_{\rm RD}}{\alpha(4-\beta-2\alpha)-c_{\rm RD}(4 \beta(\alpha-1)+\beta^{2}(2c_{\rm RD}+1))}\sigma_{\rm w}^{2}, \tag{15}\]  
where \(r_{\rm t,k}^{\rm cv}\) is the true range of the target moving with constant velocity. Equation (15) contains the effects on sensor noise only and does not contain the errors resulting from differences between the model and the motion. The derivation of Equation (15) is given in Appendix A.  
Next, \(L_{\rm rp}A_{\rm max}T^{2}\) is calculated as the bias error for the target for which the acceleration is \(A_{\rm max}\). The bias error is the mean range prediction error in the steady-state, and is calculated from  
\[\lim_{k\to\infty}E[r_{\rm t,k}^{\rm ca}-r_{\rm p,k}]=\left[\frac{1}{\beta}- \left(\frac{1}{2}+\frac{\alpha}{\beta}\right)c_{\rm RD}\right]A_{\rm max}T^{ 2}\equiv L_{\rm rp}A_{\rm max}T^{2}. \tag{16}\]  
The derivation of Equation (16) is given in Appendix B. Using the derived results of Equations (15) and (16), the RMS-index \(\rm RMSE_{\rm pred}\) is calculated using Equation (14).