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We adopt a basis expansion approach to model the time and frequency (i.e., doubly) selective fading channel and employ a superimposed training technique to acquire the channel state information. Specifically, we add a known periodic sequence onto, instead of multiplexing it into, the information sequence. There is no reduction in transmission rate, and it is possible to estimate the time-varying channel without requiring receiver diversity. We propose a novel Doppler frequency estimator and describe two equalizers for symbol recovery. Bit error rate performance is demonstrated for the iterative joint data-channel estimator. Tradeoffs regarding power allocation to the pilot and to the symbols are also explored.