John E. Gray

A derivation of an analytic expression for the tracking index for the alpha-beta-gamma filter

The authors derive an analytic expression for the alpha of an alpha-beta-gamma filter in terms of the tracking index. The result permits a direct calculation of alpha from a simple analytic expression instead of using numerical techniques. >

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

The response characteristics of the alpha-beta filter are used to quantify the filters performance against different measurement models representing a targets trajectory. The transfer functions for an alpha-beta filter are used to derive closed form (solutions) expressions for smoothed position and velocity outputs for various measurement models. The filters 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 parameters alpha and beta . Finally, the filters response to a sinusoidal input is determined.<<ETX>>

The Doppler spectrum for accelerating objects

The formalism for computing the Doppler spectrum for perfectly reflecting mirrors undergoing various types of accelerations is reviewed. This method is an amplification of work done by D. Censor and J. Cooper for one-dimensional waves. For sinusoidal waves, the formalism provides a computationally easy algorithm that enables determination of the Doppler spectrum. This method is exact and does not ignore the effects of motion on the amplitude, as is normally done. The mirror is an alternative means of determining the Doppler spectrum of point particles. From the exact result, an approximation method is derived that is of use to radar engineers. Extending these results to other commonly used radar waveforms is considered.<<ETX>>

Filter coefficient selection using design criteria

For tracking systems with a uniform data rate and stationary measurement noise, non-manoeuvring targets can be accurately tracked with a steady-state Kalman filter. The steady-state Kalman filter, which can be viewed as equivalent to an alpha-beta filter, has been widely applied to many different systems. A means of selecting the filter coefficients was proposed by Kalata (1984) using Kalman filter considerations. An alternative method based on noise reduction ratios is presented in this paper. Using a design criteria with the Kalata relation, optimal filter coefficients can be selected for specific applications. This method generalizes current methods for selecting the filter coefficients.

The solution to the Lyapunov equation in constant gain filtering and some of its applications

Presented in this paper is a solution to the Lyapunov equation in constant gain filtering. A specific method for deriving the noise reduction ratios for the alpha-beta and alpha-beta-gamma filters are explored using the Lyapunov equation. This enables us to simplify the computation drastically. For a 2 /spl times/ 2 matrix, one has to solve for one unknown instead of two. For a 3 /spl times/ 3 matrix, one has to solve for three unknowns instead of six and so on. Thus reduction in the number of variables has considerable advantage for higher dimensional filters such as the alpha-beta-gamma filter.

Characteristic functions in radar and sonar

We often encounter problems that involve either nonlinear or multi-dimensional combinations of random variables. One would like to be able to determine the probability density function of these combinations. The individual components that contribute to the combination are assumed known, but the combination is not known. Computations to determine this combined PDF axe tedious and difficult to accomplish in most cases. We propose a simpler means to bypass many of these difficulties. By using Fourier analysis and application of the definition of the Dirac delta function, the proper form the combined PDF is obtained with considerable simplification. The motivation for this is engineering applications that occur frequently in radar applications that involve combinations of probability density functions.

An alternative method to using plant noise as a means of selecting filter gains in a complex tracking environment

In this paper, we develop a methodology for comparing filter gain selection for several different filter coefficient relationships. When designing tracking filters, an important issue is the selection of the filter gains. The steady state solution of the Kalman filter leads to different filter coefficient relationships depending on what one assumes for the process noise model. Three commonly used relationships are the Benedict-Bordner relationship, the Kalata relationship, and the continuous white noise relationship. However, an analytic method for comparing these different relationships is needed. We develop a common methodology for comparing filter performance based upon cost functions in this paper and then discuss how the comparison might be used.

Solutions of matrix equations arising in track filter theory

The general solution of the equations that arise in constant coefficient filter theory are solved for arbitrary number of dimensions. Namely, we solve the first order matrix equation for the special case of scalar measurements. We also determine the covariance matrix for the general case. We then illustrate these results for the specific case of an alpha-beta filter.

Target tracking with explicit control of filter lag

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 for a step acceleration input. The standard predictor-corrector forms of the filter equations are algebraically rearranged into two uncoupled difference equations; one equation for the smoothed position and one for smoothed velocity. The equations are then solved for the smoothed estimates using the method of undetermined coefficients. The solution is shown to consist of the 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 preserves the nominal lags associated with the constant tracking interval regardless of how the update interval varies. An example demonstrates the improvement in performance from using this approach.

An interpretation of Woodward's ambiguity function and its generalization

We suggest a new interpretation of Woodwards ambiguity function as the expected value of an operator. The operator represents the physics of the interaction of the waveform with the object. This approach provides a new approach to understanding the return signal at the receiver and can reveal more detailed understanding of the underlying interactions within the return signal that are not usually brought out by standard signal processing techniques.