On the Capacity of the Vector MAC and BC with Feedback

We determine the feedback capacity region of a two user Gaussian multiple access channel (MAC) with multiple antennas at the base station and a single antenna at each user. The vector MAC and broadcast channels (BC) with a single antenna at the base station and multiple antennas at each user are shown to be equivalent to scalar MACs and BCs, respectively. We also determine the capacity enhancement due to feedback at high SNR for the vector MAC and BC. The capacity benefits of multiple antenna systems are highly dependent on the amount of channel knowledge at the transmitter and receiver [1] [2]. In practice, the channel state is learned at the receiver from a pilot signal and conveyed back to the transmitter through a feedback channel. The existence of a feedback channel opens up several interesting possibilities in terms of the information that can be conveyed from the receiver to the transmitter to enhance the forward channel capacity. Channel state feedback is just one of these possibilities. In this work we explore the potential capacity benefits of a feedback channel beyond providing channel state information. Therefore we assume perfect channel knowledge at all transmitters and receivers. We consider the feedback model that information theorists have traditionally been interested in: a feedback channel that makes the received signal available to the transmitter instantaneously. Feeding back the channel output to the transmitter establishes the absolute limit of how much the forward channel capacity can be enhanced by relaying information besides channel state information on a feedback channel. In particular, we explore the feedback capacity of a two user vector Gaussian MAC and BC with either multiple antennas at the base station and a single antenna at each mobile or with multiple antennas at each mobile and a single antenna at the base station. For a single user discrete memoryless channel it is well known that feedback does not increase the channel capacity [3]. However, for multiuser channels feedback can increase the channel capacity. The feedback capacity of a two user scalar Gaussian MAC is found by Ozarow in [4]. Ozarow also shows that feedback increases the capacity of a scalar Gaussian BC [5]. However, the capacity region of even the scalar Gaussian BC with feedback remains unknown. 1 The System Model We begin with the channel model for the vector Gaussian MAC. Figure 1 shows two kinds