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The joint design of transmitter and receiver for multichannel data transmission over dispersive channels is considered. The design criterion is the minimization of the mean squared error (MSE) at the receiver output under the constraint of a fixed transmit power. The focus is on the practically important case where the transmitter employs finite impulse response (FIR) filters, and the channel impulse response has arbitrary length. The proposed algorithm allows a straightforward transmitter design and generally yields near-optimal solutions for the transmit filters. Under certain conditions, the exact solutions for optimum block transmission, as known from the literature, are obtained.