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This paper considers the joint design of the transmitter and the receiver for linear multiple-input multiple-output (MIMO) channels when the channel state information (CSI) is available at both sides of the link. Assuming that the data symbols are mapped into points of fixed signal constellations, a transceiver structure employing widely linear (WL) filters, rather than linear ones, is proposed in order to exploit the statistical redundancy of some constellations, i.e., the correlation between the signals and their conjugate version. WL filters linearly and independently process both the in-phase and the quadrature components of the input signals and, with a limited increase in the computational complexity, allows one to improve the system performances. It is shown that, by representing the basic system model in terms of in-phase and quadrature components, the proposed structure can be derived by utilizing the already existing procedure for the design of the linear transceiver according to the minimum-mean-square-error (MMSE) criterion. The performances of the WL and linear transceivers are compared in terms of mean-square error, symbol error rate, and achieved multiplexing gain.