## 2 The \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) Filter
### Fundamental Properties Obtained in Conventional Studies
#### 2.2.4 Performance Analysis Method  
The performance of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is investigated using a simulation-based approach. The difference in the tracking accuracy between this and the \(\alpha\)-\(\beta\)-\(\gamma\) filters is determined via simulations and adaptive gain calculations [14; 15; 16; 17]. However, a theoretical analysis of the performance is not conducted and the appropriateness of the gain setting is not guaranteed. In the simulation-based gain settings presented in [15; 16], the considered gain values are limited (e.g., \(\alpha\) is set only 0.1, 0.7, 1.3, or 1.9). Thus, it seems that the performance and filter stability are only considered for these limited gain settings. Furthermore, the simulations only consider some situations, and the general advantages and disadvantages of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter are not clarified.  
In contrast, two theoretical performance indices are used for the performance analysis and optimal gain setting of the conventional \(\alpha\)-\(\beta\) and the \(\alpha\)-\(\beta\)-\(\gamma\) filters [8; 13]. One is the tracking performance index, which evaluates the performance for the prediction of maneuvering motions. The tracking performance is defined by the steady-state bias error as follows:  
\[e_{\text{fin}}=\lim_{k\to\infty}(x_{\text{tt},k}-x_{\text{p},k}), \tag{34}\]  
where \(x_{\text{tt},k}\) is the true motion of the target moving with a constant one-order-higher motion parameter compared with that in the filter's motion model. For example, \(x_{\text{tt},k}\) is the constant accelerating motion for the \(\alpha\)-\(\beta\) filter because this assumes the constant-velocity model for the prediction process. Similarly, \(x_{\text{tt},k}\) is the constant-jerk motion for the \(\alpha\)-\(\beta\)-\(\gamma\) filter. The other is the smoothing performance index, which evaluates the performance of noise reduction in the filtering process. The smoothing performance is defined by the steady-state error variance as follows:  
\[\sigma_{\text{p}}^{2}=\lim_{k\to\infty}E[(x_{\text{p},k}-x_{\text{t},k})^{2}], \tag{35}\]where \(E[\ ]\) denotes the mean with respect to \(k\) and \(x_{\text{ts,}k}\) is the true motion of the target moving with the same motion as that in the prediction model. Thus, \(x_{\text{ts,}k}\) is the constant-velocity motion for the \(\alpha\)-\(\beta\) filter, and \(x_{\text{ts,}k}\) is the constant acceleration motion for the \(\alpha\)-\(\beta\)-\(\gamma\) filter.  
These performance indices of the \(\alpha\)-\(\beta\) and the \(\alpha\)-\(\beta\)-\(\gamma\) filter have been reported in many studies [8; 9; 11; 12; 13] (also see Table 2 of this paper), and they have effectively been used for performance analysis and gain design. This is because that these indices are expressed in closed form and empirical simulation-based performance analysis and filter gain design are not required. However, these indices of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter have not been derived and hence, their theoretical performance has not been analyzed.  
In summary, the fundamental properties of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter, the stability conditions, and the tracking/smoothing performance have not been validated. Thus, in the rest of this paper, these are derived and the theoretical performance of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter based on the derived performance indices is determined to clarify the optimal performance and the difference in the performance between this filter and the conventional \(\alpha\)-\(\beta\) and the \(\alpha\)-\(\beta\)-\(\gamma\) filters.