A Derivation of an Analytic Expression for the

Tracking Index for the Alpha-Beta-Gamma Filter

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 a re 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 selection of gains is based on optimal relationships between the filter coefficients that balance the twin goals of good noise reduction and rapid maneuver response. The first optimal relation between filter coefficients was derived by Benedict and Bordner [3]. The so called Benedict-Bordner relationship fixed beta in terms of alpha. The correct choice of alpha remained an open question until Kalata's introduction of a parameter called the tracking index [4]. Using the Kalman update equations in the steady state, a relation between the filter gains and the ratio of target maneuverability uncertainty to position measurement uncertainty was derived. This ratio is referred to as the tracking index and may be used for the determination of the free parameter alpha.

The tracking index r is defined as

(1)

which is a function of the assumed target maneuverability variance O'� (deviation from modeled behavior) and the radar measurement noise variance 0';. The relationship between the tracking index and the filter coefficients for an alpha-beta filter is

Kalata relation, the tracking index expressed in tet1t1.S of, is

r = _2(,---1_-_ '),--- 2 , (3)

which can be readily solved for , in terms of r to give

(4 + r) - (8r + r2)1/2 , = 4 . (4)

Once r is known, the value of , can be determined using (4) and the values of alpha and beta are fixed for a specified ,. Thbles of the values of a and f3 as a function of the range and the tracking index can be computed for storage in a real time computer program. This enables near-optimal tracking for targets of all ranges.

The optimal Kalata relations, in the sense of measurement reduction and maneuver following capability, between the coefficients for the alpha-beta-gamma filter a re

f3 = 2(2 - a) - 4';1- a f32

,,= 2a while the tracking index is

2 r2 = -"-. I- a

(5)

(6)

(7)

Making the substitution a = 1- s2, transforms (5) into

f3 = 2(1- sf· Solving (7) for " and substituting it and (8) into (5) yields

sr (1- si "2 = (1 +s)

which is equivalent to the equation

where

s3 + bs2 + cs + d = 0

r b= -- 3 2 ' r c=Z+3, d =-1.

(8)

(9)

(10)

(11)

A convenient way to express relationships involving a and f3 is to introduce a damping parameter, = (1- a)1/2 that is discussed further in [5]. For the

(2) The substitution s = y - b /3 reduces (10) to

l+ py+q =0 (12)

Manuscript received August 21, 1992; revised December 23, 1992.

IEEE Log No. T-AE St29/3/08913.

u.s. Government work. u.s. copyright does not apply.

0018-9251/93/$3.00 © 1993 IE EE

where

b2 p =c-3,

The substitution y = z - pj(3z) reduces (13) to an equivalent quadratic equation which has a solution

(13)

(14)

1064 IEEE TRANSACTIONS ON AEROSPACE A ND ELECTRONIC SYSTEMS VOL. 29, NO.3 JULY 1993

alphas for two filters 0.9

O.B

0.7

0.6

0.5 1 0.

'0 0.4

0.3

log of tracking index

Fig. 1.

TABLE 1 Matlab Program for Computing Alphas as Function of 1taclting

Index

calculates alpha as function of tracking index gam=input('tracking index less than 1'); r = (4 + gam - sqrt(8*gam + gam."2»/4; alpha = 1- r. "2 beta = 2. (1- r)."2 b = gam/2- 3; c = gam/2+3; d = -1; P = c - b."2/3; q = (2. b."3)/27 - (b .• c )/3 + d; v = sqrt(q."2 + (4. p."3)/27); z = -Ceq + v/2)." (1/3); s = z - p./(3*Z) - b/3; alpha1 = 1 - s."2 beta1 = 2. (1- s)."2 gamma1 = (beta1."2)./(2.alpha1) end;

For the normal range of values of the filter parameters (0 < a < 1), the term inside the square root is positive, so there is one real solution for

z. The parameter s

expressed in terms of z is

P b s = z - 3z - 3' (15)

Note that the same tracking index can produce a significantly different alpha for the two filters. Fig. 1 shows alpha versus tracking index for the alpha-beta and alpha-beta-gamma filter. Thble I is Matlab [6] code for computing alpha from the tracking index.

The advantage of this change of variables to compute the tracking index is that it is simple. It avoids using polynomial root solving programs to solve the sixth-order polynomial equation that results without this change of variable. At the same time, it avoids the usual complex root checking which is usually necessary when solving cubic equations. Thus, this method is useful for real time applications.

CORRESPONDENCE

ACKNOWLEDGMENTS

We wish to thank Dr. Bar-Shalom for his comments on the contents of this paper.

JOHN E. GRAY WILLIAM MURRAY Code B32 N BVa) SurfKe Warfare Center Dablgren, VA 22448

REFERENCES

[1]

[2]

[3]

[4]

[5]

[6]

Bogler, P. L. (1990) Radar Principles with Application to Tracking Systems. New York: Wiley, 1990.

Chui, C. K., and Chen, G. (1990) Kalman Filtering with Real Time Applications (2nd ed.). New York: Springer, 1990.

Benedict, R. T., and Bordner, G. W. (1962) Synthesis of an optimal set of radar track-while scan smoothing equations. IRE Transactions on Automatic Control, Af'..-7 (1962), 27-32.

Kalata, P. R. (1984) The tracking index: A generalized parameter for a -(j and a-f3-T target trackers. IEEE Transactions on Aerospace and Electronic Systems, AES-20 (Mar. 1984), 174-182

Gray, J. E., and Murray, W. J. (1990) Analysis of alpha-beta filters for potential applications in anti-air kill evaluation improvements and as acceleration detectors. 1l:chnical report 89-269, Naval Surface Warfare Center, Dahlgren, VA, Feb. 1990.

Matlab (1989) Matlab Jor the MS-Dos Personal Computers-User's Guide. The Math Works Inc., South Natick, MA: 1989.

ISAR Using Thomson's Multiwindowed Adaptive Spectrum Estimation Method

We compare two of Thomson's muitiwindowed spectral estimation methods to conventional windowed discrete Fourier transform methods for generating inverse synthetic aperture radar (ISAR) images. The compared images are generated from synthetic data and data obtained from the IPIX research radar. The images show that Thomson's adaptive multiwindowed spectral estimation method is superior to the windowed fast Fourier transform (FFT) in generating ISAR images.

Manuscript received February 6, 1992.

IEEE Log No. T-AES/29/3/08914.

0018-9251/931$3.00 © 1993 IEEE

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