Alpha-beta filter

An alpha-beta filter (also fg filter or gh filter) is an observer for estimating and smoothing quantities, which is related to the Kalman filter . The alpha-beta filter is used in radar trackers, for example .

Formulas

Result of smoothing a signal (blue) using an alpha-beta filter (red) with alpha = 0.85 and beta = 0.005.

The alpha-beta filter can be applied to systems in which one quantity (for example a position) results from another quantity (for example speed) through integration. ${\displaystyle {\textbf {x}}}$ ${\displaystyle {\textbf {v}}}$

Based on the previous (possibly initial) estimate of the sizes at the point in time , the predicted value for the size is obtained from the formula ${\displaystyle {\hat {\textbf {x}}}_{k-1},{\hat {\textbf {v}}}_{k-1}}$ ${\displaystyle k-1}$

${\displaystyle {\text{(1)}}\quad {\hat {\textbf {x}}}_{k}\leftarrow {\hat {\textbf {x}}}_{k-1}+\Delta {\textrm {T}}\ {\textbf {}}{\hat {\textbf {v}}}_{k-1}}$,

where is the time difference between the two times. ${\displaystyle \Delta T}$

If the system is not driven, the estimate of the size remains received: . Otherwise, adjustments are possible in this formula. ${\displaystyle {\textbf {v}}}$ ${\displaystyle {\text{(2)}}\quad {\hat {\textbf {v}}}_{k}\leftarrow {\hat {\textbf {v}}}_{k-1}}$

The residual between the new reading and the current predicted value is ${\displaystyle k}$

${\displaystyle {\text{(3)}}\quad {\hat {\textbf {r}}}_{k}\leftarrow {\textbf {x}}_{k}-{\hat {\textbf {x}}}_{k}}$

The estimate of the two quantities is updated using the alpha and beta values.

${\displaystyle {\text{(4)}}\quad {\hat {\textbf {x}}}_{k}\leftarrow {\hat {\textbf {x}}}_{k}+(\alpha )\ {\hat {\textbf {r}}}_{k}}$

${\displaystyle {\text{(5)}}\quad {\hat {\textbf {v}}}_{k}\leftarrow {\hat {\textbf {v}}}_{k}+(\beta /[\Delta {\textrm {T}}])\ {\hat {\textbf {r}}}_{k}}$,

here and are to be chosen. ${\displaystyle \alpha }$ ${\displaystyle \beta }$

For convergence and stability, alpha and beta should be small positive numbers:

${\displaystyle \quad 0<\alpha <1}$

${\displaystyle \quad 0<\beta \leq 2}$

${\displaystyle \quad 0<4-2\alpha -\beta }$

Individual evidence

1. ^ David Lee Hall, Sonya AH McMullen: Mathematical Techniques in Multisensor Data Fusion . Artech House, 2004, ISBN 978-1-58053-335-5 ( limited preview in Google Book Search [accessed July 1, 2020]).
2. ^ C. Frank Asquith: Weight selection in first-order linear filters. Technical report. Army Intertial Guidance and Control Laboratory Center, Redstone Arsenal, Alabama, 1969. https://doi.org/10.21236/ad0859332