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This paper investigates important properties of acquisition receivers that employ commonly used serial-search strategies. In particular, we focus on the properties of the mean acquisition time (MAT) for wide bandwidth signals in dense multipath channels. We show that a lower bound of the MAT over all possible search strategies is the solution to an integer programming problem with a convex objective function. We also give an upper bound expression for the MAT over all possible search strategies. We demonstrate that the MAT of the fixed-step serial search (FSSS) does not depend on the timing delay of the first resolvable path, thereby simplifying the evaluation of the MAT of the FSSS. The results in this paper can be applied to design and analysis of fast acquisition systems in various wideband scenarios.