This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.
Bibliographical noteFunding Information:
Manuscript received March 13, 2010; accepted September 26, 2010. Date of publication October 14, 2010; date of current version December 17, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dominic K. C. Ho. The work of D. P. Palomar and Y. Huang was supported by the Hong Kong RGC 618709 research grant. The work of S. Zhang was supported by National Natural Science Foundation of China (Grant 10771133).
- Constrained maximum likelihood steering direction estimation
- fractional QCQP
- radar applications