This is a kalman filter to estimate 1D location using acceleration data

$$\hat{x}^- = F\cdot \hat{x} + \Delta t * \begin{bmatrix} speed\\ acc \end{bmatrix}$$ $$P_{k}^- = F\cdot P_{k-1}\cdot F^T + Q$$ $$K_{k} = P_{k}^- \cdot H^T \cdot (H\cdot P_{k}^- \cdot H^T +R)^{-1}$$ $$P_{k} = (I - K_{k} \cdot H) \cdot P_{k}^-$$ $$\hat{z} = \hat{x} + \Delta t \cdot \begin{bmatrix} speed\\acc \end{bmatrix}$$ $$\hat{x} = \hat{x}^- + K \cdot (z - H \cdot \hat{x}^-)$$ $$F = \begin{bmatrix} 1 & \Delta t \\\ 0 & 1 \end{bmatrix}$$ $$H = \begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}$$