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We present a derivation of an analyUc expression for the alpha of an alpha-beta-gamma filter in terms of tbe tracking index. Tbe result permits a dtrect calculation of alpha from a simple analytic expression instead of using numerical techniques to determine alpha.Both the alpha-beta and alpha-beta-gamma filters are tracking filters commonly used in radar systems [1]_ These filters are steady state versions of the Kalman filter [2] which find usage in target intensive tracking environments, where computational restrictions limit the complexity of the filters that can be implemented.

The response characteristics of the alpha-beta filter are used to quantify the filter's performance against different measurement models rcprcsenting a target's trajectory.The transfer functions for an alphabcta filter are used to derive closed form (solutions) expressions for smoothed position and velocity outputs for various measurement models. The filter's response to constant velocity targets is found to be the input plus a sinusoidal transient.Constant acceleration measurement models, in addition, yield a steady state bias that is a function of the filter parametersa and R. Finally, the filter's response to a sinusoidal input is determined. Part I. IntroductionWith a tracking radar, it is possible to measure the position of the target directly, but it is not possible to measure the velocity of the target directly [l]. A means of estimating the future positions and vclocities of the target is needed. Some of the earliest filters used in tracking radars are the alpha-beta filter and the alpha-beta-gamma filter. The variable gain alpha-beta filter combines elements of the Kalman filter [2] and the alpha-beta filter and has found application when large numbers of objects are being tracked. The information derivcd from the filter is used not only to arrive at pointing orders for radars, but also for other applications including prediction of intercept points, guidance laws for missile orders, computation of decelerations, etc. It is useful to know the response of the filter to different measurement models other than the constant velocity model. To calculate the deterministic responses of the alpha-beta filter to different measurement models, a factorization of the transfer functions is presented that leads to manageable computations of the response function of the alpha-beta filter.The equations for tracking a one dimensional target with a alphabcta filter assume that the tracking model is constant velocity. Given this assumption, the mean squared error in the filtered velocity and position estimates is minimized to get optimal estimates. This is done by choosing two weighting coefficients that weight the differences of the predicted and measured position and combines them with the present prediction to arrive at new estimates. The tracking equations for the alpha-beta filter are given bywhere 15 (k) = smoothed position at the k-th interval, 16 (k) = predicted position at the k-th interval, (k) = measured position at the k-th interval, vs (k) = smoothed velocity at the k-th interval, vp (k) = predicted velocity at the k-th interval T = radar update interval or period, a , 6 = filter weighing coefficients.These equations are one-dimensional, but can be extended to three dimensions by substituting successively y and z for x in equations (1-4). The filter equations are usually analyzed in one dimension and the resulting analysis is usually extended to three dimensions with the assumption that this gives similar results.The first major advance in using the tracking equations was to find a means to o timally select a ...

Dahlgren, VA 22448 wmurray Qnswc.navy.mil jgray@nswc.navy.mil +4 bst ract This paper presents a method of gain adjustment for an alpha-beta filter when data points are lost or when the tracking interval changes. The steady state position and velocity lags are first derived f o r a step acceleration input. The standard predictor-corrector form of the filter equations are algebraically rearranged into two uncoupled difference equations; one equation f o r the smoothed position and one f o r smoothed velocity. The equations are then solved f o r the smoothed estimates using the method of undetermined coeficients. The solution is shown to consist of input acceleration, transient terms and steady state lags. The transient terms counteract the effects of the steady state lags until the time determined by the filters lag time. The steady-state lags are used for optimal adjustment of filter gains for aperiodic track conditions. For a varying track update interval, the filter gains which preserve a nominal periodic filter lag are derived. Such gain selection will preserve the nominal lags associated with the constant tracking interval regardless of how the update interval varies. A n example demonstrates the improvement in performance from using this approach.

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