Joint Optimization of Detection and Tracking in Adaptive Radar Systems - Advances in Heuristic Signal Processing and Applications - page 126

Proof The cost function given in ( 6.29 ) can be rewritten as

E tr x(k) k)

Z k − 1

2

J (k,P FA ) = −ˆ x(k |

k) Z k − 1 E tr x(k) k) T x(k)

= −ˆ x(k | −ˆ x(k |

k) T Z k − 1

E tr x(k) k) x(k)

= −ˆ x(k | −ˆ x(k |

k) T Z k − 1 tr E x(k) k) x(k)

= −ˆ x(k | −ˆ x(k |

tr E E x(k) k) x(k) k) T Z k Z k − 1 = −ˆ x(k | −ˆ x(k |

tr E P(k Z k − 1 = | k) |

tr P MRE (k | k)

≈

tr P(k 1 ) − q 2 λ(k)V(k),P D tr W(k)S(k)W T (k) ,

= | k −

where the first equality is due to the property that the trace of a scalar is itself, the
third one is due the property that tr { AB }= tr { BA } , the fourth one is due to linearity

of tr operators, and the fifth one follows from the smoothing property [ 9 ]

of expectations. Note that W(k)S(k)W T (k)

{·} and E [·]

W(k)S(k)W T (k)

0,
and q 2 (λ(k)V(k),P D ) is the only term that depends on P FA . Hence the minimiza-
tion of

≥ 0 implies tr { }≥

(k,P FA ) can be achieved by maximizing q 2 (λ(k)V(k),P D ) over P FA ,
which completes the proof.

J

Remark 6.2 We experimentally observe that choosing any other scalar mea-
sures for the function f S [·] results

in the same optimization problem given in ( 6.30 ) where the elements of the
set are the determinant, 1-norm (the largest column sum), 2-norm (the largest
singular value),

from the set {|·| , · 1 , · 2 , · ∞ , · F }

∞ -norm (the largest row sum) and Frobenius-norm of a ma-

trix, respectively.

Due to mathematical intractability, the problem given in ( 6.30 ) was solved by uti-
lizing some line search algorithms that require only the evaluation of the cost func-
tion (e.g., Golden-Section or Fibonacci Search methods) [ 22 ]. We call this scheme
as DYNAMIC-MRE-LS in Fig. 6.2 .

Lemma 6.2 (A Closed-Form Solution [ 5 ]) An approximate closed-form solution
for the MRE-based dynamic threshold optimization can be found for a special type