Joint Optimization of Detection and Tracking in Adaptive Radar Systems - Advances in Heuristic Signal Processing and Applications - page 123

as a tracking filter. The NSPP-based detector threshold optimization for the PDAF
case appeared in the works [ 5 , 20 , 22 , 48 ], and [ 2 ] which are all given in Fig. 6.2 .In
this section, we define the underlying optimization problems of these approaches.
The presentation is given as in Fig. 6.2 , i.e., in two parts, static and dynamic opti-
mization schemes. The section ends with a comparison of these approaches via a
simulation scenario.

6.3.1 Static Threshold Optimization (STOP)

The NSPP-based static threshold optimization problem is [ 2 ] to determine the opti-
mal P FA value such that

P FA f S [ P NSPP ]

P FA = arg min

(6.26)

subject to P D = f ROC (P FA ,ζ) and 0 ≤ P FA ≤ 1 ,

n x × n x

where f S : R is an appropriate scalar measure deduced from a matrix

(such as trace, determinant, or a matrix norm) and P NSPP is the steady-state covari-
ance matrix obtained by propagating one of the NSPP recursions to its steady-state,
i.e.,

→ R

P NSPP P NSPP (k | k),

lim
k →∞

(6.27)

where P NSPP (k k) corresponds to the output of either the HYCA or the MRE algo-

rithm at time step k . The equality constraint of the optimization problem is nothing
but an ROC curve relation which links P D to P FA , or vice-versa, through current
SNR ( ζ ), and the inequality constraint ensures that the resultant operating false
alarm value is a valid probability.

|

Remark 6.1 Note that this optimization is performed offline . The MRE and
HYCA recursions are initialized as explained in the Sects. 6.2.3 and 6.2.4 .In
the practical implementation, one cannot iterate the recursion given in ( 6.27 )
indefinitely. One should check whether the value of a suitably chosen norm of
the difference matrix between two consecutive covariance matrices is below
a chosen threshold to conclude that the recursion is converged, or whether a
maximum number of iterations is reached to conclude on its divergence.

The optimization problem given in ( 6.26 ) is a line search. Provided that the cost
function is unimodal , the global optimum point can be found directly applying well-
known numerical techniques, such as the Golden-Section or Fibonacci Search meth-
ods [ 11 ]. For each function evaluation at an arbitrary point P FA , one needs to obtain