On the capacity region of the vector fading broadcast channel with no CSIT

We develop an upperbound on the capacity region of an isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel. Using this upperbound we prove the optimality of the Alamouti scheme [1] in a broadcast setting and extend the recent results [2] on the capacity region of the fading scalar non-degraded broadcast channel to fading vector non-degraded broadcast channels. The upperbound is fundamental in that it makes no assumption regarding the distribution of the users\´ channel magnitudes, the distribution of the additive noise, or the amount of channel information available at the receiver. The scalar upperbound explicitly characterizes the loss of degrees of freedom in a vector broadcast channel when the transmitter has no information about the "direction" of the users\´ channel vectors.