A Provably Convergent Block MM Algorithm for Range-based Multistatic Target Localization with Partial Transmitter Cooperation

This article focuses on locating a reflecting target using both bistatic and direct range measurements derived from the propagation delays of signals emitted by a transmitter with unknown position and received by spatially separated sensors with known positions. Addressing such a localization task is crucial for certain practical applications of multistatic systems, notably in the use of mobile units, hastily deployed nodes, or semi-cooperative illuminators of opportunity as sources of transmission. In the sense of maximum likelihood estimation, the problem lends itself to a formulation under the weighted nonlinear least squares (WNLS) framework. Beyond the challenge posed by nonconvexity, tackling the WNLS problem is inherently nontrivial as the squared residuals involve sums and differences of the Euclidean norm terms. To make the optimization more tractable, several approximations are employed, including a strategic two-block partitioning of variables and a series of proper upper-bounding functions for the corresponding subproblem objectives. They enable the development of an efficient block majorization–minimization algorithm, which admits closed-form updates at each iteration. Theoretically, we establish the convergence of the algorithm, confirming that every limit point of the generated iterates is a stationary point of the WNLS localization problem. We also present an analysis of the algorithmic complexity, and discuss how correlated noise may affect our approach. Simulations demonstrate the potential of the proposed method to attain higher positioning accuracy than the existing algebraic and semidefinite programming solutions, while being computationally lighter than the latter.

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