# Reduced-Complexity Tracking Algorithms on Chip for Real-Time Location Estimation
## 3 Alpha-Beta (\(\alpha\)-\(\beta\)) Filtering Algorithm  
In [31; 32], \(\alpha\)-\(\beta\) filtering algorithm is built on the steady state of KF, and uses \(\alpha\)-\(\beta\) filtering for position tracking. It does not require previous experience of state and measuring noise covariance. A two-dimensional model, using \(x\) as a position and \(s\) as a speed, the \(\alpha\)-\(\beta\) tracking filter can be organized as follows:  
(1) Prediction step (Time update phase):  
\[\widetilde{\mathbf{x}}_{k+1}=\hat{\mathbf{x}}_{k}+\Delta_{T}\hat{\mathbf{v}} _{k}, \tag{15}\]  
\[\widetilde{\mathbf{s}}_{k+1}=\hat{\mathbf{s}}_{k} \tag{16}\]  
(2) Estimation step (Measurement update phase):  
\[\hat{\mathbf{x}}_{k}=\widetilde{\mathbf{x}}_{k}\alpha(\mathbf{z}_{k}- \widetilde{\mathbf{x}}_{k}) \tag{17}\]  
\[\hat{\mathbf{s}}_{k}=\hat{\mathbf{s}}_{k-1}+(\beta/\Delta_{T})(\mathbf{z}_{k}- \widetilde{\mathbf{x}}_{k}), \tag{18}\]  
where \(\overset{\wedge}{\mathbf{x}}_{k}\), \(\hat{\mathbf{s}}_{k}\), \(\widetilde{\mathbf{x}}_{k}\), and \(\mathbf{z}_{k}\) are the position estimate matrix, speed estimate matrix, position prediction matrix, and the actual measurement equation, respectively. These equations are reduced from those KF algorithm as follows.  
\[\mathbf{H}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\end{bmatrix} \tag{19}\]  
\[\mathbf{K}=\begin{bmatrix}\alpha&0\\ 0&\alpha\\ \beta/\Delta_{T}&0\\ 0&\beta/\Delta_{T}\end{bmatrix}, \tag{20}\]  
where \(\alpha\) and \(\beta\) are between 0 and 1 for smoothing the estimated positions and speeds. If \(\alpha=\beta=0\), the measurement has no influence on the approach. If \(\alpha=\beta=1\), the history has no influence on the approach. As shown in Figure 2, first, position information enterstraining mode to iterate. KG will stabilize after repeated iterations, and then extract \(\alpha\) and \(\beta\) from KG to become \(\alpha\)-\(\beta\) filtering algorithm, by transporting the known training positioning information with a previous experience of measurement noise covariance and the state. It is mentioned in the literature that the training length of the KG coefficient can be based on iterations of the \(norm(\mathbf{P})\). After the training, the process takes regular intervals in time \(\Delta_{T}\), the coefficients of the \(\alpha\)-\(\beta\) tracker can be obtained from the matrix of KG, and then track the actual positioning information [32]. Decision mode is used in Figure 2 to replace the known of training mode. Finally, tracked by \(\alpha\) and \(\beta\) coefficients of the tracking method continuously [31]. Furthermore, on the basis of the result in Figure 3, the convergence rate of the KG of can be based on the number of iterations. In terms of convergence rate, the larger values the variances are, the more iterations the tracking approaches have. In this article, the training lengths are 17, 21, and 23 for the variances of 5, 10, and 15, respectively.  
Figure 3: The learning curves of the Kalman gain of the stable convergence in the light of the norm of the estimation error covariance with the iteration parameter.  
Figure 2: Flow diagram of \(\alpha\)-\(\beta\) filtering algorithm.