The response of the transfer function of an alpha-beta filter to various measurement models

Abstract

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 ...

Fundamental Properties and Optimal Gains of a Steady-State Velocity Measured <i>α</i>-<i>β</i> Tracking Filter

“…Equation 14is the so-called Benedict-Bordner relation (BBR) [12], and this paper calls an α-β filter having this relation a BBR filter. This filter is stable when 0 < α ≤ 1.…”

“…An α-β filter is a simple and effective tracking filter that assumes constant velocity during the sampling interval [1] [6]- [12]. Because of its small computational load, the use of α-β filters has been proposed in various tracking systems.…”

General Solution to Constant Gain Tracking Filters

“…To make a comparison, one would substitute each one of these equations into Eq. (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and compare the plot for all r.…”

“…Typically, the noise can be written as a x = Rag, where ag is the sensor angular noise, which is a known parameter of the tracking system. Given that the range is in kilometers and the angle noise is in milliradians, the noise can be written as a function of the range (n) times a constant k. The tracking index r = apT 2 kn (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25) Specific system parameters are then plugged into Eq. (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25) and the a's are computed.…”

“…In this report, the general case will be solved first, and the a -ß filter [2] will be 1-2 NSWCDD/TR-01/68 solved as an illustrative example. Previously [9], the ^-transform or frequency domain method was used, but here the direct methods that have become more fashionable in recent years will be used. The solutions are independent of the particular relationship between a and ß that are discussed in [2] and [13].…”