Joint Optimization of Detection and Tracking in Adaptive Radar Systems - Advances in Heuristic Signal Processing and Applications - page 131

Fig. 6.7 STOP using TOC curves: ( a ) MRE case and ( b ) HYCA case. Note that the instability
region for the HYCA case has a slightly larger area than that of MRE. One can fit a line equation
for the optimum operating curves in both approaches. Then this equation, together with the ROC
curve relation, determines the STOP curve which can be used to find the optimum operating point
for an arbitrary SNR value

N of the HYCA algorithm is taken as 15. We run both recursions on a 500 500

regular P FA - P D grid. The borders of the grid are from 0 to 0 . 1for P FA and from 0
to1for P D . Both recursions are run over each point in this grid until convergence.
As mentioned before, the recursions do not converge to a steady-state covariance
for some of the grid points, due to non-existence of the limit given in ( 6.27 ). This
causes an instability region [ 20 ] as illustrated in Fig. 6.7 .
We define the scalar performance function ( f S [·]

×

) as the steady-state position

estimation error, i.e.,

f S [ P NSPP ] σ ss

P 1 NSPP + P 3 NSPP

POS =

(6.40)

where P i NSPP is the i th diagonal element of P NSPP . The TOC curves are obtained as
the contours of the corresponding performance measure surface. The superimposi-
tion of these curves onto the ROC curves is shown in Fig. 6.7 where the functional
form of the ROC curves is given by 14

P D = P 1 /( 1 + ζ)

. (6.41)

FA

Note that the collection of optimum operating points for different SNR values, con-
sisting of the tangential points of TOC and ROC curves, are well-behaved. A line
fitting works quite well and results in approximations

P D =− 9 . 523 P FA + 1 . 002 for MRE, (6.42)

P D =− 5 . 943 P FA + 0 . 979 for HYCA. (6.43)

14 This is valid for a special case of a NP detector under HOG SQL
I

assumption.