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Transmission of a time-discrete message over a time-continuous channel is considered. The channel is assumed to be stationary and Gaussian. The following results are shown. 1) An optimum receiver will always contain a matched matrix filter followed by a sampling unit. 2) Jointly optimized transmitting and receiving filters will always be strictly band limited to a set of Nyquist domains. It is shown that both these results are true for any kind of message and under any measure of performance. On the basis of these results a general time-discrete matrix channel equivalent to the original time-continuous scalar channel is derived. The significance of the results is that any optimization of transmitter and receiver is reduced from a time-continuous problem to a time-discrete one.