## 2 The \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) Filter
### Fundamental Properties Obtained in Conventional Studies
#### 2.2.2 Relationship to Kalman Filter  
The \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is equivalent to a steady-state Kalman filter with a constant-jerk model [14; 15], which is similar to the well-known conventional \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters [8; 12]. The equations for the Kalman filter are:  
\[\tilde{\mathbf{x}}_{k} =F\mathbf{\hat{x}}_{k-1}, \tag{15}\] \[\tilde{\mathbf{P}}_{k} =F\mathbf{\hat{P}}_{k-1}F^{\mathrm{T}}+\mathbf{Q},\] (16) \[K_{k} =\tilde{\mathbf{P}}_{k}H^{\mathrm{T}}(H\tilde{\mathbf{P}}_{k}H^{\mathrm{ T}}+\mathbf{B})^{-1},\] (17) \[\hat{\mathbf{x}}_{k} =\tilde{\mathbf{x}}_{k}+K_{k}(\mathbf{z}_{k}-\mathbf{H}\tilde{\mathbf{x}}_{k}),\] (18) \[\tilde{\mathbf{P}}_{k} =\tilde{\mathbf{P}}_{k}-K_{k}H\tilde{\mathbf{P}}_{k}, \tag{19}\]  
where \(\mathbf{x}\) is a state vector, forecasts and estimates are denoted by tildes and hats, respectively, superscript \"T\" and \"\(-1\)\" denote transpositions and inversions, \(\mathbf{z}\) is a measurement vector, \(\mathbf{F}\) is the transition matrix, \(\mathbf{P}_{k}\) is the error covariance matrix at time \(kT\), \(\mathbf{Q}\) is the covariance matrix for the process noise, \(\mathbf{K}_{k}\) is the optimal gain (Kalman gain) at time \(kT\), and \(\mathbf{B}\) is the covariance matrix for the measurement noise. The \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is obtained by substituting the following into (15) and (19): Vectors \(\mathbf{\hat{x}}_{k}=(x_{\mathrm{p},k}\ v_{\mathrm{p},k}\ a_{\mathrm{p},k}\ j_{ \mathrm{p},k})^{\mathrm{T}},\mathbf{\hat{x}}_{k}=(x_{\mathrm{s},k}\ v_{\mathrm{s}, k}\ a_{\mathrm{s},k}\ j_{\mathrm{s},k})^{\mathrm{T}}\), \(\mathbf{z}_{k}=x_{\mathrm{o},k}\), and matrices:  
\[\mathbf{F} = \begin{pmatrix}1&T&T^{2}/2&T^{3}/6\\ 0&1&T&T^{2}/2\\ 0&0&1&T\\ 0&0&0&1\end{pmatrix} , \tag{20}\] \[\mathbf{H} = \begin{pmatrix}1&0&0&0\end{pmatrix} ,\] (21) \[\mathbf{K}_{k} = \begin{pmatrix}\alpha_{k}&\beta_{k}/T&\gamma_{k}/T^{2}&\delta_{k }/T^{3}\end{pmatrix} ,\] (22) \[\mathbf{B} =B_{\mathbf{x}}, \tag{23}\]  
By assuming the steady state with \(k\to\infty\), the gains in (22) become constant and the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter of definitions (1)-(8) is obtained. Thus, the investigation of the properties of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is equivalent to that of the steady-state properties of the fourth-order Kalman filter tracker. 2.3 Relationship to the \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) Filters  
As indicated above, the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is a fourth-order filter. Traditionally, the steady-state second- and third-order tracking filters are well known as \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters, respectively. These filters are popularly used because of their simplicity, utility in real-time applications, and usefulness in the design of complicated tracking systems. The \(\alpha\)-\(\beta\) filter is a steady-state second-order Kalman filter using the constant-velocity model that tracks the position and velocity of moving objects. Its filtering process is as follows [8]\[x_{\text{p},k} =x_{\text{s},k-1}+Tv_{\text{s},k-1}, \tag{24}\] \[v_{\text{p},k} =v_{\text{s},k-1},\] (25) \[x_{\text{s},k} =x_{\text{p},k}+\alpha(x_{\text{o},k}-x_{\text{p},k}),\] (26) \[v_{\text{s},k} =v_{\text{p},k}+\frac{\beta}{T}(x_{\text{o},k}-x_{\text{p},k}). \tag{27}\]  
Please note that the smoothing processes of (26) and (27) are the same as those of (5) and (6). Similarly, the \(\alpha\)-\(\beta\)-\(\gamma\) filter is a steady-state third-order Kalman filter using the constant acceleration model; thus, it tracks the position, velocity, and acceleration of moving objects.