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In this paper, we present a design for a full-duplex near-echo-cancelling data modem based on a cambined "decision-directed" (DD) adaptive echo canceller and adaptive channel equalizer. This algorithm has the important feature that the interference to the canceller caused by the far-end signal is eliminated by subtracting an estimate of the far-end signal based on receiver decisions. This technique is extended to provide a new approach to full-duplex far-echo cancellation in which frequency offset of the far echo is adaptively estimated with little interference from the far-end signal and near echo. The approach uses receiver decisions on the far-end signal, but in addition, relies on a separate receiver structure devoted to joint estimation of the far-echo channel and the frequency offset in the far echo. The feasibility of both echo-cancelling modem algorithms is demonstrated by computer simulation with realistic channel distortions and with 4800 bps data transmission at which rate frequency offset in the far echo becomes important.