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We present a novel channel partitioning and modulation technique for linear time-varying (LTV) channels using adaptive bases of localized complex exponentials. We show that localized complex exponentials are approximate eigenfunctions of underspread LTV channels. A basis of localized complex exponentials that approximately diagonalizes the LTV channel is selected adaptively by the receiver during a training period. The basis selection process is equivalent to matching the support intervals of the localized complex exponentials to the rate of the channel time variation. The receiver sends information regarding the selected basis to the transmitter which modulates the subsequent data stream in this basis. The adaptive modulation technique performs significantly better than conventional orthogonal frequency-division multiplexing systems for rapidly varying LTV channels such as time-frequency-selective mobile radio channels.