Degrees of Freedom Region of the Gaussian MIMO Broadcast Channel with Common and Private Messages

We obtain the degrees of freedom region of the Gaussian multiple-input multiple-output (MIMO) broadcast channel with common and private messages. We first show that a parallel Gaussian broadcast channel with unmatched sub-channels can be constructed from any given Gaussian MIMO broadcast channel by using the generalized singular value decomposition (GSVD) and a relaxation on the power constraint for the channel input, in a way that the capacity region of the constructed parallel channel provides an outer bound for the capacity region of the original channel. The capacity region of the parallel Gaussian broadcast channel with unmatched sub-channels is known, using which we obtain an explicit outer bound for the degrees of freedom region of the Gaussian MIMO broadcast channel. We finally show that this outer bound for the degrees of freedom region can be attained by the achievable scheme that uses a classical Gaussian coding for the common message and dirty-paper coding (DPC) for the private messages.