Elsevier

Signal Processing

Volume 83, Issue 5, May 2003, Pages 973-981
Signal Processing

Analytical performance evaluation of association of active and passive tracks for airborne sensors

https://doi.org/10.1016/S0165-1684(02)00500-5Get rights and content

Abstract

An association algorithm for tracks generated by active and passive sensors was presented and its performance in terms of the probability of correct and false association was analytically evaluated by Farina and Miglioli (Signal Process. 69(3) (1998) 209). Based on the work in Farina and Miglioli, this paper concentrates on the derivation of simplified performance evaluation formula for deterministic target trajectories. The main contribution of this paper is: (a) the simplified formula of false association probability for a fixed value of correct association probability is presented, and (b) the simplified formula of correct association probability for a fixed value of false association probability is also derived. The advantage of the work is that the performance evaluation can be done with simple calculation instead of performing integral and using trial-and-error method. Numerical results and comparisons show that the presented methods are feasible and effective.

Section snippets

Problem formulation

In recent years, the association of active and passive tracks has been an active research area in multisensor data fusion and has been widely studied [2], [3], [5], [6], [7], [10], [11], [12], [13], [14], [16]. The association logic should decide whether the active and passive tracks have been generated by the same target or pertain to different objects. This paper studies the association performance evaluation for deterministic target trajectories. Let us consider the scenarios presented in [6]

Simplified formula of PFAS for given PC

Assume that Q is a non-central chi-square variable with four degrees of freedom and non-centrality parameter s, and its PDF, denoted as p(q), isp(q)=exp−s+q2∑k=0∞1k!s2kqk+122+kΓ(2+k).Assume λ(PC) is the threshold for a fixed value of PC. Then, we have the following lemma:

Lemma

For given PC, PFAS in Eq. (5) is equivalent toPFAS=Pr{Q⩽λ(PC)}=∫0λ(PC)p(q)dq.

Proof

Assume that the characteristic function of Q is φ(μ). Then we haveφ(μ)=E[ejμQ]=∫0∞ejμqp(q)dq=(1−2jμ)−2expjμs1−2jμandp(q)=12π∫−∞∞e−jμqφ(μ)dμ.

Simplified formula of PC for given PFAS

In some applications, the threshold may be determined such that the false association probability is below a certain value, i.e., the correct association probability for a fixed value of false association probability is required. In this case, the threshold is not only dependent on PFAS but also on s.

Since PFAS<0.5 is a reasonable requirement in this case, referring to [15], the inverse transformation of (12) can be written ast1=m11−29m1−ξ13,ifPFAS<0.5,whereξ1=−83ln(4PFAS(1−PFAS))225m1.

Test of goodness of fit for the approximations

In this section, the Kolmogorov test method is used to explain the reasonableness of the approximations. The first approximation is taken for example. The first approximation consists of two steps: first, approximating the non-central chi-square variable Q into X=̂c1Z1+b, and then approximating the cumulative distribution function of Z1 asFZ1(z)=Pr{Z1⩽z}≈0.5(1−1−e−K),ifz⩽m1−23,0.5(1+1−e−K),ifz>m1−23with K is similar to K1 in Eq. (14b) with z replacing t1.

Assuming that FQ(x) and FX(x) are the

Results and comparisons

In order to verify the effectiveness of the presented methods, the calculation of PFAS for a prescribed PC and the calculation of PC for a prescribed PFAS have been done. For notational convenience of comparisons, some notations are introduced. ΔPFAS is assumed to be the absolute differences of PFAS between the exact values obtained by using Eq. (5) and the approximate values obtained by using either Eq. (14) or Eq. (19). ΔPC is assumed to be the absolute differences of PC between the exact

Conclusions

The main contribution of this work is to derive the simplified closed-form formula of false association probability for a prescribed correct association probability and those of correct association probability for a prescribed false association probability in active and passive track association. The main results are represented by , , , . It should be emphasized, however, that the discussion in this paper is confined to deterministic target trajectories. The advantage of the work is that the

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