## 2 The \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) Filter  
### Problem Definition  
We investigate the position prediction performance using a filter called the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter [14]. For simplicity, one-dimensional tracking along the \(x\)-axis is considered. The tracking process of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is as follows. Similar to other tracking filters, the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter invokes an iterative prediction and smoothing process to estimate the state of the target. The state of the target includes the position, velocity, acceleration, and jerk (fourth-order tracking). The prediction process is conducted under the assumption that the target's jerk is constant during a sampling interval, which is expressed as: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Description** & **Unit** \\ \hline \(T\) & Sampling interval & [s] \\ \(k\) & Discrete sampling index & Dimensionless \\ \(()_{k}\) & Parameter at index \(k\) & \\ \(x_{\rm P}\) & Predicted position & [m] \\ \(v_{\rm P}\) & Predicted velocity & [m/s] \\ \(a_{\rm P}\) & Predicted acceleration & [m/s\({}^{2}\)] \\ \(j_{\rm P}\) & Predicted jerk & [m/s\({}^{3}\)] \\ \(x_{\rm s}\) & Smoothed (estimated) position & [m] \\ \(v_{\rm s}\) & Smoothed (estimated) velocity & [m/s] \\ \(a_{\rm s}\) & Smoothed (estimated) acceleration & [m/s\({}^{2}\)] \\ \(j_{\rm s}\) & Smoothed (estimated) jerk & [m/s\({}^{3}\)] \\ \(x_{\rm o}\) & Observed (measured) position & [m] \\ \(\alpha\) & Filter gain for \(x_{\rm s}\) & Dimensionless \\ \(\beta\) & Filter gain for \(v_{\rm s}\) & Dimensionless \\ \(\gamma\) & Filter gain for \(a_{\rm s}\) & Dimensionless \\ \(\delta\) & Filter gain for \(j_{\rm s}\) & Dimensionless \\ \(B_{\rm x}\) & Error variance of \(x_{\rm o}\) & [m\({}^{2}\)] \\ E() & Mean with respect to \(k\) & \\ \(\sigma_{\rm P}^{2}\) & Smoothing performance index & [m\({}^{2}\)] \\ \(e_{\rm fin}\) & Tracking performance index & [m] \\ \(s_{\rm c}\) & Snap assumed in the derivation of \(e_{\rm fin}\) & [m/s\({}^{4}\)] \\ \(e\) & RMS prediction error of Monte Carlo simulations & [m] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of main symbols. \[x_{\rm p,k} =x_{\rm s,k-1}+Tv_{\rm s,k-1}+\frac{T^{2}}{2}a_{\rm s,k-1}+\frac{T^{ 3}}{6}j_{\rm s,k-1}, \tag{1}\] \[v_{\rm p,k} =v_{\rm s,k-1}+Ta_{\rm s,k-1}+\frac{T^{2}}{2}j_{\rm s,k-1},\] (2) \[a_{\rm p,k} =a_{\rm s,k-1}+Tj_{\rm s,k-1},\] (3) \[j_{\rm p,k} =j_{\rm s,k-1}, \tag{4}\]  
where \(x_{\rm p,k}\) is the predicted position at time \(kT\), \(T\) is the sampling interval, \(x_{\rm s,k}\) is the smoothed target position, \(v_{\rm p,k}\) is the predicted velocity, \(v_{\rm s,k}\) is the smoothed velocity, \(a_{\rm p,k}\) is the predicted acceleration, \(a_{\rm s,k}\) is the smoothed acceleration, \(j_{\rm p,k}\) is the predicted jerk, and \(j_{\rm s,k}\) is the smoothed jerk. The smoothing process is defined as:  
\[x_{\rm s,k} =x_{\rm p,k}+\alpha(x_{\rm o,k}-x_{\rm p,k}), \tag{5}\] \[v_{\rm s,k} =v_{\rm p,k}+\frac{\beta}{T}(x_{\rm o,k}-x_{\rm p,k}),\] (6) \[a_{\rm s,k} =a_{\rm p,k}+\frac{\gamma}{T^{2}}(x_{\rm o,k}-x_{\rm p,k}),\] (7) \[j_{\rm s,k} =j_{\rm p,k}+\frac{\delta}{T^{3}}(x_{\rm o,k}-x_{\rm p,k}), \tag{8}\]  
where \(x_{\rm o,k}\) is the measured position, and \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are dimensionless filter gains. In the steady state, the filter gains are converged to a fixed value. Thus, the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter uses fixed gains. The definition of the smoothing process in this paper is slightly different from that in [PERSON]'s study [14; 15; 16]. Their study sets gains corresponding to the acceleration and jerk as \(\gamma/2\) and \(\delta/6\), respectively. However, this study uses the above definition to improve the mathematical generality. Equations (1)-(8) are the definitions of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter.