Optimal Linear Fusion for Distributed Detection Via Semidefinite Programming

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Consider the problem of signal detection via multiple distributed noisy sensors. We study a linear decision fusion rule of [Z. Quan, S. Cui, and A. H. Sayed, “Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks,” <emphasis emphasistype="boldital">IEEE J. Sel. Topics Signal Process.</emphasis>, vol. 2, no. 1, pp. 28–40, Feb. 2008] to combine the local statistics from individual sensors into a global statistic for binary hypothesis testing. The objective is to maximize the probability of detection subject to an upper limit on the probability of false alarm. We propose a more efficient solution that employs a divide-and-conquer strategy to divide the decision optimization problem into two subproblems. Each subproblem is a nonconvex program with a quadratic constraint. Through a judicious reformulation and by employing a special matrix decomposition technique, we show that the two nonconvex subproblems can be solved by semidefinite programs in a globally optimal fashion. Hence, we can obtain the optimal linear fusion rule for the distributed detection problem. Compared with the likelihood-ratio test approach, optimal linear fusion can achieve comparable performance with considerable design flexibility and reduced complexity. </para>