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We consider the Gaussian multiple-input multiple-output (MIMO) broadcastchannel with common and private messages. We obtain the degrees of freedom(DoF) region of this channel. We first show that a parallel Gaussian broadcastchannel with unmatched sub-channels can be constructed from any given GaussianMIMO broadcast channel by using the generalized singular value decomposition(GSVD) and a relaxation on the power constraint for the channel input, in a waythat the capacity region of the constructed parallel channel provides an outerbound for the capacity region of the original channel. The capacity region ofthe parallel Gaussian broadcast channel with unmatched sub-channels is known,using which we obtain an explicit outer bound for the DoF region of theGaussian MIMO broadcast channel. We finally show that this outer bound for theDoF region can be attained both by the achievable scheme that uses a classicalGaussian coding for the common message and dirty-paper coding (DPC) for theprivate messages, as well as by a variation of the zero-forcing (ZF) scheme.

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