# Optimal Steady-State Range Prediction Filter for Tracking with LFM Waveforms
## 2 Problem Definition and Conventional Filter Design
### Conventional Filter Design Methods  
Steady-state performance analysis and filter design of the LFM \(\alpha\)-\(\beta\) filter have been investigated in some studies [8; 9; 10]. These conventional studies derive the optimal filter gains based on the Kalman filter equations because the \(\alpha\)-\(\beta\) filter is approximated as a steady-state Kalman filter. In the conventional filter design, the following relationship between \(\alpha\) and \(\beta\) in steady-state conditions is used [10]:  
\[\beta_{\pm}=\frac{2}{1+4c_{\rm RD}}\times\left(-\alpha(1+2c_{\rm RD})+2\pm 2 \sqrt{(1-\alpha c_{\rm RD})^{2}-\alpha}\right), \tag{7}\]  
where  
\[c_{\rm RD}=\Delta t/T \tag{8}\]  
is the normalized coefficient of range-Doppler coupling. To apply relation (7) in gain design, the following two methods are proposed. 1. Tracking-index-based method [8]: a well-known and useful design parameter is the tracking index, which is defined as [8; 14]: \[\Gamma=T^{2}\sigma_{\rm v}/\sigma_{\rm w}=\sqrt{\beta^{2}/(1-\alpha-\beta c_{ \rm RD})},\] (9) where \(\sigma_{\rm v}\) is the standard deviation of the process noise of the measurement model used in Kalman filter tracking [14]. Equation (9) is obtained by steady-state assumption for the Kalman filter using a random acceleration process noise [8]. In steady-state conditions, both the prediction and estimation error covariance matrices converge and the optimal gain relations are obtained using these fixed matrices in the Kalman filter equations. The value of \(\Gamma\) determines \(\alpha\) and \(\beta\) by Equations (7) and (9). In filter design, a suitable \(\Gamma\) is selected empirically based on the assumed degree of maneuvering of targets. 2. Maximum RMS error (RMSE)-based method [10]: the tracking index-based approach requires an empirical setting of \(\Gamma\). To resolve this problem, [PERSON] and [PERSON] [10] proposed the deterministic evaluation function and a design parameter to optimize the range estimation. Their proposed evaluation function corresponds to a maximum RMSE determined by a maximum target acceleration, which is defined as: \[\rm RMSE^{p}_{\rm max}\equiv\sqrt{E[(r^{\rm ca}_{t,k}-r_{\rm s,k})^{2}]}= \sigma_{\rm w}\sqrt{s_{11}/\sigma^{2}_{\rm w}+(L_{p}\Gamma_{\rm D})^{2}},\] (10) where \(E[\ ]\) denotes the mean, \(r^{\rm ca}_{t,k}\) the true range of the target undergoing constant acceleration, \(s_{11}\) the steady-state variance of the range estimation error assuming only measurement noise, \(L_{p}\Gamma^{2}_{\rm D}\) corresponds to the steady-state lag (bias error) for constant-acceleration target, and \(\Gamma_{\rm D}\) is called the deterministic tracking index defined using the maximum acceleration and \(\sigma_{\rm w}\); these are expressed as:\[s_{11} =\frac{2\alpha^{2}+\beta(2-3\alpha)-\beta^{2}c_{\rm RD}}{\alpha(4- \beta-2\alpha)-c_{\rm RD}(4\beta(\alpha-1)+\beta^{2}(2c_{\rm RD}+1))}\sigma_{\rm w }^{2}, \tag{11}\] \[L_{p} =\frac{2(\alpha-1)+(2\alpha+\beta)c_{\rm RD}}{2\beta},\] (12) \[\Gamma_{\rm D} =A_{\rm max}T^{2}/\sigma_{\rm w}, \tag{13}\]  
where \(A_{\rm max}\) is the maximum acceleration of the target. The optimal \(\alpha\) is determined by minimizing \(\rm RMSE_{\rm max}^{p}\) using Equation (7) and the known \(\Gamma_{\rm D}\) (i.e., \(A_{\rm max}\) and \(\sigma_{\rm w}\) are known). It is well known that the tracking index-based method is useful in the design of Kalman and \(\alpha\)-\(\beta\) filters. However, an empirical setting for \(\Gamma\) is required. The maximum RMSE-based approach automatically determines optimal \(\alpha\) and \(\beta\) from \(\Gamma_{\rm D}\). The setting of \(A_{\rm max}\) of \(\Gamma_{\rm D}\) is easily compared with \(\Gamma\) because its physical meaning is clear. However, the two above-dated methods assume only random acceleration process noise in the Kalman filter equations [8; 10]. The gain relation of Equation (7) is derived using this assumption.