Principles of Modern Radar - Volume 2 1891121537

680 CHAPTER 15 Multitarget, Multisensor TrackingHence, by making a few assumptions about the target RCS statistics, the likely number offalse alarms, and the likely number of new targets, observed SNR can be incorporated intothe rigorous hypothesis costs from (15.19) to potentially improve measurement-to-trackdata association when the kinematics are ambiguous.Although this technique has merit, it is not a panacea. The probability of error basedon observed SNRs can be derived in closed form for Swerling II targets. This analysisshows that SNR-assisted data association may not be sufficient in the context of SwerlingII targets to achieve desired performance unless the observed SNRs are accurate andthe differences between the observed SNRs for closely-spaced tracks are substantiallydifferent.To illustrate this point, consider the simple case with two targets in track, the second ofwhich has a larger observed SNR [15]. The SNR of the second target can thus be written asR 2 =R 1 + R R > 0 (15.27)where R 1 is the SNR of the first track and R 2 is the SNR of the second track, both ofwhich are in linear units (not dB). Using only the features (e.g., if the kinematic costswere completely ambiguous), an error would occur if the SNR of the measurement fromthe first object, R 01 , is larger than the observed SNR of the measurement from the secondobject, R 02 . It follows that the probability of error is given by∫ ∞[∫ ∞RP E =02p (R 01 |H 1 , R 1 , R 02 ) dR 01∫ ∞]p (R 02 |H 1 , R 2 , R 02 ≥R th ) dR 02 (15.28)RR thp (R 01 |H 1 , R 1 ) dR 01thwhich reduces to1P E =2 + RR 1 +1(15.29)for Swerling II targets. This makes intuitive sense. If the two targets have identical observedSNRs, then R is equal to zero and the probability of error is 0.5; when the observed SNRsare identical, feature-based association is completely ambiguous.The degree to which the observed SNRs must be different to achieve a desirably lowP E (assuming Swerling II targets) may come as a surprise. Inspection of (15.29) revealsthat even if the second target’s observed SNR is twice as large as that of the first (in linearunits, not dB), then the probability of error is 0.33. The probability of error (based solelyon the SNR term, as in (15.29)) is plotted in Figure 15-4a for Swerling II targets. In thisplot, the independent axis is the SNR of the weaker target and the dependent axis is theprobability of error from (15.29). A family of curves is presented for varying levels ofSNR differences between the weaker and stronger targets. For Swerling II targets, thedifference between the weaker and stronger observed SNRs must be large (relative to theSNR of the weaker target) before P E reaches a desirably low level.The primary reason for this disappointing result is that the pdfs of observed SNR forSwerling II targets overlap substantially, as shown in [15]. If the target RCS statistics wereinstead well-modeled as fixed-amplitude, then (15.25) would change and the correspondingpdfs of the observed SNRs would be bell-shaped, rather than exponential-looking.As a result, a satisfactory P E could be attained via feature-assisted tracking for a muchsmaller difference in observed SNRs. To illustrate this, Figure 15-4b plots (15.28) assumingfixed-amplitude targets, using the same family of curves as in Figure 15-4a. The P E