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We consider pilot-aided transmission (PAT) for a general class of systems encompassing linear modulation and a linear time-varying channel. For these systems, and given a pilot energy constraint, we derive a tight lower bound on the mean squared error (MSE) of pilot-aided channel estimates as well as necessary and sufficient conditions on PAT to attain this bound. We then apply these results to the design of single-antenna PAT for doubly selective channels and arrive at novel MSE-optimal PAT schemes. In this application, we assume a block-based cyclic-prefix PAT and a basis expansion model for the channel.