MIMO MAC-BC Duality With Linear-Feedback Coding Schemes

We show that the rate regions achieved by linear-feedback coding schemes over dual multi-antenna Gaussian multi-access channels (MACs) and broadcast channels (BCs) with independent noises coincide. By dual here we mean: (1) the channel matrices of the MAC and the BC are transposes of each other and (2) the same total input-power constraint P is imposed on both the channels. We also present multi-letter expressions for the linear-feedback capacity regions of the two channels, i.e., for the set of all rates that are achievable with the linear-feedback coding schemes. We identify a sub-class of MAC and BC linear-feedback coding schemes that achieve the respective linear-feedback capacity regions, and within these subclasses, we identify pairs of MAC and BC coding schemes that achieve the same rate regions. In the two-user case, when the transmitters or the receiver are single-antenna, the capacity region for the Gaussian MAC is known [20], [15] and the capacity-achieving scheme is a linear-feedback coding scheme. With our results, we can thus determine the linear-feedback capacity region of the two-user Gaussian BC when either transmitter or receivers are single-antenna and we can identify the corresponding linear-feedback capacity-achieving coding schemes. Our results show that the control-theory inspired linear-feedback coding scheme by Elia [11], Wu et al. [30], and Ardestanizadeh et al. [1] is sumrate optimal among all the linear-feedback coding schemes for the symmetric single-antenna Gaussian BC with equal channel gains. More generally, we show that the linear-feedback sum-capacity of the scalar Gaussian BC with independent noises is achieved using a simple rearrangement of Ozarow's MAC encodings and decodings. In the K 3-user case, Kramer [16] and Ardestanizadeh et al. [2] determined the linear-feedback sum-capacity for the symmetric single-antenna Gaussian MAC with equal channel gains. Using our duality result, in this paper, we identify the linear-feedback sum-capacity for the K 3-user single-antenna Gaussian BC with equal channel gains. It is equal to the sum-rate achieved by Ardestanizadeh et al.'s linear-feedback coding scheme [1].