## 1 Introduction  
In monitoring systems for aircraft, robots, and intelligent cars using remote sensors such as radar and cameras, the accurate tracking of moving objects is essential. For such applications, Kalman filter- [1; 2; 3; 4] and Bayesian filter-based trackers [5] are commonly used to track the position, velocity, and acceleration of the targets. However, in recent years, the importance of jerk, which is the derivative of acceleration with respect to time, has been realized to address tracking problems [4; 6; 7]. In [4; 6], it was proved that the use of a high-dimensional parameter such as jerk can improve the performance of the tracking filters for highly maneuvering targets. Thus, the consideration of jerk in tracking filters is promising. However, commonly used and well-analyzed conventional tracking filters do not consider tracking jerk. A steady-state second-order Kalman filter, known as an \(\alpha\)-\(\beta\) filter [8; 9; 10], tracks position and velocity only, and a steady-state third-order tracking filter, known as an \(\alpha\)-\(\beta\)-\(\gamma\) filter [6; 11; 12; 13], tracks these in addition to acceleration.  
To estimate and use the jerk in tracking filters, [PERSON] et al. [14] proposed a steady-state fourth-order Kalman filter considering jerk tracking, which is called an \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter, and they analyzed its performance by determining its transfer functions and stability conditions [14]. However, their correctness has not been proved and validated. Furthermore, they also proposed a simulation-based optimal design and performance analysis of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter using the Taguchi method, evolution-programming, and genetic- algorithm [14; 15; 16], but the theoretical performance of the filter has not been analyzed. Furthermore, these studies have investigated some limited setting of the filter parameters (filter gains).  
[PERSON] et al. also performed optimal design using a fading memory filter algorithm for the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter [17]. However, its optimal quality and effectiveness in general situations are not guaranteed. Thus, the performance of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter has not been analyzed and validated theoretically, and its advantages over the traditional \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters are unknown. In other words, the selection criteria for the above filters for practical use have not been established.  
In this paper, we theoretically analyze the steady-state performance of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter by deriving its closed-form performance indices. First, we show an example in which the filter is unstable even when the stability conditions derived by [PERSON] et al. [14] are satisfied, and then, we derive the correct conditions to ensure convergence of the steady state for the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter. Then, the steady-state errors for targets moving with constant jerk or snap (time-derivative of the jerk) are derived in closed form as the analytical performance indices of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter. Finally, to investigate the advantages and disadvantages of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter, the performance with optimal gains that are obtained with the derived indices are compared with those of traditional \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters using theoretical analyses and numerical simulations.  
Table 1 lists the main symbols used in this paper.