We provide an overview of the extensive recent results on the Shannon capacity of single-user and multiuser multiple-input multiple-output (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about the underlying time-varying channel model and how well it can be tracked at the receiver, as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. For time-varying MIMO channels there are multiple Shannon theoretic capacity definitions and, for each definition, different correlation models and channel information assumptions that we consider. We first provide a comprehensive summary of ergodic and capacity versus outage results for single-user MIMO channels. These results indicate that the capacity gain obtained from multiple antennas heavily depends

The use of space-division multiple access (SDMA) in the downlink of a multiuser multiple-input, multiple-output (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering the co-channel interference of other users. Typical optimization problems of interest include the capacity problem—maximizing the sum information rate subject to a power constraint—or the power control problem—minimizing transmitted power such that a certain quality-of-service metric for each user is met. Neither of these problems possess closed-form solutions for the general multiuser MIMO channel, but the imposition of certain constraints can lead to closed-form solutions. This paper presents two such constrained solutions. The first, referred to as “block-diagonalization,” is a generalization of channel inversion when there are multiple antennas at each receiver. It is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high SNR. The second, known as “successive optimization, ” is an alternative method for solving the power minimization problem one user at a time, and it yields superior results in some (e.g., low SNR) situations. Both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined. In order to accommodate more general scenarios, we also propose a framework for coordinated transmitter-receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas. While the proposed algorithms are suboptimal, they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity.

This paper characterizes the sum capacity of a class of non-degraded Gaussian vectB broadcast channels where a singletransmitter with multiple transmit terminals sends independent information to multiple receivers. Coordinat+[ is allowed among the transmit teminals, but not among the different receivers. The sum capacity is shown t be a saddlepoint of a Gaussian mu al informat]R game, where a signal player chooses a tansmit covariance matrix to maximize the mutual information, and a noise player chooses a fictitious noise correlation to minimize the mutual information. This result holds fort he class of Gaussian channels whose saddle-point satisfies a full rank condition. Furt her,t he sum capacity is achieved using a precoding method for Gaussian channels with additive side information non-causally known at the transmitter. The optimal precoding structure is shown t correspond to a decision-feedback equalizer that decomposes t e broadcast channel into a series of single-user channels with intk ference pre-subtract] at the transmiter.

This paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the base-station. In contrast to the conventional sum-power constraint on the transmit antennas, this paper adopts a more realistic per-antenna power constraint, because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier. Assuming perfect channel knowledge at the transmitter, this paper investigates two different transmission schemes under the per-antenna power constraint: a minimum-power beamforming design for downlink channels with a single antenna at each remote user and a capacity-achieving transmitter design for downlink channels with multiple antennas at each remote user. It is shown that in both cases, the per-antenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise. This generalizes previous uplink–downlink duality results and transforms the per-antenna transmitter optimization problem into an equivalent minimax optimization problem. Further, it is shown that various notions of uplink–downlink duality may be unified under a Lagrangian duality framework. This new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink per-antenna transmitter optimization problem.

Abstract—The sum capacity of a Gaussian vector broadcast channel is the saddle point of a minimax Gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint. This sum capacity result has been proved using two different methods, one based on decision-feedback equalization and the other based on a duality between uplink and downlink channels. This paper illustrates the connection between the two approaches by establishing that uplink–downlink duality is equivalent to Lagrangian duality in minimax optimization. This minimax Lagrangian duality relation allows the optimal transmit covariance and the least-favorable-noise covariance matrices in a Gaussian vector broadcast channel to be characterized in terms of the dual variables. In particular, it reveals that the least favorable noise is not unique. Further, the new Lagrangian interpretation of uplink–downlink duality allows the duality relation to be generalized to Gaussian vector broadcast channels with arbitrary linear constraints. However, duality depends critically on the linearity of input constraints. Duality breaks down when the input constraint is an arbitrary convex constraint. This shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink–downlink duality representation. Index Terms—Broadcast channel, Lagrangian duality, minimax optimization, multiple-input multiple-output (MIMO), multipleaccess

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Convex optimization methods are widely used in the

Abstract—A precoding scheme for multiuser broadcast communications is described, which fills the gap between the low-complexity Tomlinson–Harashima precoding and the sphere decoderbased system of Peel et al. Simulation results show that, replacing the closest-point search with the Babai approximation, the full diversity order supported by the channel is available to each user, as in the system of Peel et al., and unlike Tomlinson–Harashima precoding, which suffers some diversity penalty. The complexity of the scheme is similar to that of Tomlinson–Harashima precoding. Index Terms—Lattice reduction, multiple-input multiple-output (MIMO) broadcast channels, MIMO precoding.

We identify the common underlying form of the capacity expression that is applicable to both cases where causal or non-causal side information is made available to the transmitter. Using this common form we find that for the single user channel, the multiple access channel, the degraded broadcast channel, and the degraded relay channel, the sum capacity with causal and non-causal side information are identical when all the transmitter side information is also made available to all the receivers. A genie-aided outerbound is developed that states that when a genie provides n bits of side information to a receiver the resulting capacity improvement can not be more than n bits. Combining these two results we are able to bound the relative capacity advantage of non-causal side information over causal side information for both single user as well as various multiple user communication scenarios. Applications of these capacity bounds are demonstrated through examples of random access channels. Interestingly, the capacity results indicate that the excessive MAC layer overheads common in present wireless systems may be avoided through coding across multiple access blocks. It is also shown that even one bit of side information at the transmitter can result in unbounded capacity improvement.

The capacity region of the multiple antenna (transmit and receive) broadcast channel is considered. We propose an outer bound to the capacity region by converting this nondegraded broadcast channel into a degraded one with users privy to the signals of users ordered below them. We extend our proof techniques in the characterization of the sum capacity of the multiple antenna broadcast channel to evaluate this outer bound with Gaussian inputs. Our main result is the observation that if Gaussian inputs are optimal to the constructed degraded channel, then the capacity region of the multiple antenna broadcast channel is characterized.