Principles of Modern Radar - Volume 2 1891121537

15.2 Multitarget Tracking 679track i in hypothesis l. The multiplicative factor, γ , in (15.20) is found withγ = φ!v!μ φ (φ) μ v (v) V −φ−v(c)m(k)!(15.21)where φ is the number of false alarms at t k , v is the number of new targets at t k , m(k) is thenumber of measurements at t k , μ φ (φ) is the probability mass function (pmf) of φ, μ v (v)is the pmf of v, V is the track gate volume, and c is a normalization constant. Hence,(15.20) provides the complete probability for hypothesis l at time t k .If features are not included, then f i, j is equal to the likelihood whose log is shown in(15.4). If the observed SNR is rigorously incorporated into f i, j as a feature [14], then thisbecomes[f i, j z j (k) ] ( )τ j= exp τ ji, j p ( ) τR 0 j |H 1 , R i , R 0 j ≥Rjth× p ( R 0 j |H 0 , R i = 0, R 0 j ≥R th) 1−τ j(15.22)where R 0 j is the observed SNR of measurement j, R th is the SNR detection threshold, R iis the (average) SNR of track i, H 1 is the hypothesis that measurement j originates fromthe target tracked by track i, and H 0 is the hypothesis that measurement j is a false alarm.Hence, if hypothesis l includes assignment of measurement j to track i, then τ j is equalto one and (15.22) reduces to[f i, j z j (k) ] ( )τ j= exp τ ji, j p ( )R 0 j |H 1 , R i , R 0 j ≥R th (15.23)If hypothesis l instead treats measurement j as a false alarm, then τ j is equal to zero and(15.22) reduces to[f i, j z j (k) ] ( )τ j= exp τ ji, j p ( )R 0 j |H 0 , R i = 0, R 0 j ≥R th (15.24)Note that this formulation treats the kinematics and features as though they are independent.This greatly simplifies the math, since finding a closed-form expression for the SNRbasedlikelihoods as a function of target orientation would be extremely challenging (if notimpossible). The resulting multiplication of their likelihoods in (15.22) is hence somewhatad hoc and can allow the feature terms to disproportionately drive assignment results. Ata minimum, using the complete hypothesis cost from (15.19) appears to help. Furtherdiscussion on this point is found in [14, 15].If the target’s radar cross section (RCS) statistics are well-modeled as Rayleighdistributed(e.g., Swerling II), then [11, 14] documents that the likelihood that measurementj originates from target i, conditioned on the fact that the detection exceeded the detectionthreshold, is given byp ( R 0 j |H 1 , R i , R 0 j ≥R th) =( 1R i + 1)exp[ ]Rth −R 0 jR i + 1(15.25)This likelihood has also been derived for more complex targets (e.g., whose RCS statisticsinclude a Rician-distributed mix of Rayleigh and fixed-amplitude parts) in the literature[15]. Regardless of the assumption of the threat’s RCS statistics, the likelihood that themeasurement is a false alarm is typically modeled with the exponential distribution,p ( R 0 j |H 0 , R i = 0, R 0 j ≥R th) = exp( −R0 j)(15.26)