Reciprocity for MIMO systems

We give reciprocity results for the channel capacity efficiency (i.e. the capacity/bandwidth – abbreviated here to capacity) of an N-transmit M-receive antenna wireless system, when the gain matrix is known to the receiver but not the transmitter. The channel may be non-Gaussian. Transmitter power or receiver noise may be variable. We consider the case when N or M is large but not both. Formulas are also given for the mean capacity when there is correlation at both ends. Capacity is approximately Gaussian. For a non-Gaussian channel the approximate mean capacity is not changed but the variance is increased by a certain factor. It is noted that reciprocity fails. The key to making it work is to introduce a ‘power factor’ that allows for the case when the power of each transmitter does not depend on N as well as for the usual case when the total power is fixed.