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This paper addresses issues related to the estimation and equalization of channels with a large delay spread and an impulse response with significant values only over small regions. For a time-varying environment, an adaptive matching pursuit (AMP) algorithm is considered which updates the channel estimate while exploiting the channel's sparsity. From the channel estimate, the equalizer taps are then derived. We show through simulations that using AMP to track the time-varying channel gives significant improvement in equalization performance over using the LMS or RLS algorithms. We also demonstrate how ANP may be modified to more quickly adapt to a fast time-varying channel.