On Achivable Rates in a Multi-Antenna Gaussian Broadcast Channel

| A Gaussian broadcast channel with r single-antenna receivers and t antennas at the transmitter is considered. Both transmitter and receivers have perfect knowledge of the channel. Despite apparent simplicity, this model is in general a non-degraded broadcast channel, for which the capacity region is not fully known. We propose a novel transmission scheme based on \ranked known interference" (RKI). In brief, the transmitter decomposes the channel into an ordered (or ranked) set of interference channels for which the interference signal of the i-th channel is generated as a linear combination of the signals transmitted in channels j < i. In this way, known techniques of coding for non-causally known interference can be applied to make the interference in each channel harmless without further power penalty. We show that the proposed scheme is throughputwise asymptotically optimal for both low and high SNR. We provide a modi cation of the basic RKI scheme which achieves optimal throughput for all SNRs in the special case of 2 antenna and 2 users. For independent Rayleigh fading closed-form throughput expressions are obtained in various cases of interest and numerical examples are provided for the in nite-dimensional Rayleigh channel. I. Problem statement The t 1 : : : r Gaussian boradcast channel (GBC) is described by y = Hx + z where the notation t 1 : : : r designates that the r receivers cannot cooperate. Here, x 2 C t denotes the vector of symbols transmitted in parallel from the t antennas in any channel use, H 2 C r t denotes the channel matrix, whose i; j-th element is the complex channel gain from antenna j to antenna i, y 2 C r denotes the vector of received signals at the r receivers and z 2 C r is the noise vector, N C (0; N0I). The channel matrix H is known to the transmitter and to all receivers and the input is constrained to satisfy trace(E[xx ]) E where E is the maximum allowed total transmit energy per channel use. We are interested in the channel throughput R (or ratesum), de ned as the sum of all achievable individual user rates. We shall consider the following scenarios: i) H is deterministic and xed; ii) H is xed during the transmission of each code word, but it is randomly selected according to a given probability distribution (composite channel). II. The \ranked known interference" scheme Let H = GQ be a QR-type decomposition of H where G 2 C r m is lower triangular andQ 2 C m t has orthonormal rows (we de ne m = minfr; tg). The transmitted signal is obtained as x = Qu. Let gi;j denote the (i; j)-th element of G and let di = jgi;ij . The original channel is turned into the set of interference channels yi = gi;iui + P j<i gi;juj + zi, for i = 1; : : : ;m, while no information is sent to usersm+1; : : : ; r. SinceQQ = I, trace(E[uu ]) = trace(E[xx ]). The signals ui are all generated by the transmitter: for each i-th channel ui is the wanted signal and P j<i gi;juj is an interference signal non-causally known to the transmitter. With some care [5], we can apply the results of [2] or (equivalently) the more recent results of [3], and show that by using an appropriate coding strategy, the capacity of each i-th channel is equal to that of the virtual interference-free channel yi = ui + zi: Proposition 1. The maximum achievable throughput of the RKI scheme is given byR max = Pm i=1[log( di)]+ where is the solution of P i=1 [ 1=di]+ = = E=N0.