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An assignment-based solution for the data association problem in synchronous passive multisensor (Type 3) tracking systems involves two steps: first measurement-to-measurement or static association is solved using a multidimensional (S-dimensional or S-D with S sensors) assignment, and then measurement-to-track association is solved using a 2-D assignment. This solution is computationally very expensive and, to rectify an efficient (S+1)-D assignment algorithm has been proposed in the literature. Two new assignment-based algorithms are proposed that use prior track information (i.e., predicted state and covariance) which result in improved tracking performance compared with the existing solutions, while requiring considerably less computations. One of the proposed algorithms, the gated assignment, is similar to the two-step solution mentioned above except that it uses prior track information and avoids the need to consider all possible association hypotheses in the static association step. The second algorithm, the gated (S+1)-D assignment, combines the gated assignment and the (S+1)-D algorithms. An approximation to the (S+1)-D algorithm is also derived when sensor measurements are independent, which results in an extremely fast solution. Simulation results confirm that the proposed algorithms show improved tracking performance and faster execution times.