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

We solve the transmitter optimization problem and determine a necessary and sucient condition under which beamforming achieves Shannon capacity in a narrowband point to point communication system employing multiple transmit and receive antennas. We assume perfect channel state information at the receiver (CSIR) and imperfect channel state feedback from the receiver to the transmitter. We consider the cases of mean and covariance feedback. The channel is modeled at the transmitter as a matrix of complex jointly Gaussian random variables with either a zero mean and a known covariance matrix (covariance feedback), or a non-zero mean and a white covariance matrix (mean feedback). For both cases we develop a necessary and sucient condition for when the Shannon capacity is achieved through beamforming, i.e. the channel can be treated like a scalar channel and one-dimensional codes can be used to achieve capacity. We also provide a waterpouring interpretation of our results and nd that less channel uncertainty not only increases the system capacity but may also allow this higher capacity to be achieved with scalar codes which involves signi cantly less complexity in practice than vector coding.

Abstract—The analysis of the multiple-antenna capacity in the high- regime has hitherto focused on the high- slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of prominent channel features since, for a majority of channels, the slope equals the minimum of the number of transmit and receive antennas. Furthermore, a characterization based solely on the slope captures only the scaling but it has no notion of the power required for a certain capacity. This paper advocates a more refined characterization whereby, as a function of �f, the high- capacity is expanded as an affine function where the impact of channel features such as antenna correlation, unfaded components, etc., resides in the zero-order term or power offset. The power offset, for which we find insightful closed-form expressions, is shown to play a chief role for levels of practical interest. Index Terms—Antenna correlation, channel capacity, coherent communication, fading channels, high- analysis, multiantenna arrays, Ricean channels.

This paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels. The analysis is not restricted to the popular separable correlation model, but rather it embraces a more general representation that subsumes most of the channel models that have been treated in the literature. For arbitrary signal-to-noise ratios @ A, the characterization is conducted in the regime of large numbers of antennas. For the low- and high- regions, in turn, we uncover compact capacity expansions that are valid for arbitrary numbers of antennas and that shed insight on how antenna correlation impacts the tradeoffs among power, bandwidth, and rate.

Abstract—We consider the sum capacity of a multi-input multi-output (MIMO) multiple access channel (MAC) where the receiver has the perfect channel state information (CSI), while the transmitters have either no or partial CSI. When the transmitters have partial CSI, it is in the form of either the covariance matrix of the channel or the mean matrix of the channel. For the covariance feedback case, we mainly consider physical models that result in single-sided correlation structures. For the mean feedback case, we consider physical models that result in in-phase received signals. Under these assumptions, we analyze the MIMO-MAC from three different viewpoints. First, we consider a finite-sized system. We show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices, in the covariance and mean feedback models, respectively. Also, we find the conditions under which beamforming is optimal for all users. Second, in the covariance feedback case, we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system. In the mean feedback case, we show through simulations that this is not necessarily true. Third, we consider the asymptotic case where the number of users is large. We show that in both no and partial CSI cases, beamforming is asymptotically optimal. In particular, in the case of no CSI, we show that a simple form of beamforming, which may be characterized as an arbitrary antenna selection scheme, achieves the sum capacity. In the case of partial CSI, we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity. Finally, we generalize our covariance feedback results to double-sided correlation structures in the Appendix. Index Terms—Multi-user MIMO, MIMO multiple access channel, partial CSI, covariance feedback, mean feedback, optimality of beamforming, large system analysis. I.

Abstract — We consider both the single-user and the multiuser power allocation problems in MIMO systems, where the receiver side has the perfect channel state information (CSI), and the transmitter side has partial CSI, which is in the form of covariance feedback. In a single-user MIMO system, we consider an iterative algorithm that solves for the eigenvalues of the optimum transmit covariance matrix that maximizes the rate. The algorithm is based on enforcing the Karush-Kuhn-Tucker (KKT) optimality conditions of the optimization problem at each iteration. We prove that this algorithm converges to the unique global optimum power allocation when initiated at an arbitrary point. We, then, consider the multi-user generalization of the problem, which is to find the eigenvalues of the optimum transmit covariance matrices of all users that maximize the sum rate of the MIMO multiple access channel (MIMO-MAC). For this problem, we propose an algorithm that finds the unique optimum power allocation policies of all users. At a given iteration, the multi-user algorithm updates the power allocation of one user, given the power allocations of the rest of the users, and iterates over all users in a round-robin fashion. Finally, we make several suggestions that significantly improve the convergence rate of the proposed algorithms. Index Terms — Multi-user MIMO, MIMO multiple access channel, partial CSI, covariance feedback, optimum power allocation.

This paper proposes linear precoder designs exploiting statistical channel knowledge at the transmitter in a multiple-input multiple-output (MIMO) wireless system. The paper focuses on channel statistics, since obtaining real-time channel state information at the transmitter can be difficult due to channel dynamics. The considered channel statistics consist of the channel mean and transmit antenna correlation. The receiver is assumed to know the instantaneous channel precisely. The precoder operates along with a space–time block code (STBC) and aims to minimize the Chernoff bound on the pairwise error probability (PEP) between a pair of block codewords, averaged over channel fading statistics. Two PEP design criteria are studied—minimum distance and average distance. The optimal precoder with an orthogonal STBC is established, using a convex optimization framework. Different relaxations then extend the solution to systems with nonorthogonal STBCs. In both cases, the precoder is a function of both the channel mean and the transmit correlation. A linear precoder acts as a combination of a multimode beamformer and an input shaping matrix, matching each side to the channel and to the input signal structure, respectively. Both the optimal beam direction and the power of each mode, obtained via a dynamic water-filling process, depend on the signal-to-noise ratio (SNR). Asymptotic analyses of the results reveal that, for all STBCs, the precoder approaches a single-mode beamformer on the dominant right singular vector of the channel mean as the channel factor increases. On the other hand, as the SNR increases, it approaches an equipower multiple-mode beamformer, matched to the eigenvectors of the transmit correlation. Design examples and numerical simulation results for both orthogonal and nonorthogonal STBC precoding solutions are provided, illustrating the precoding array gain.

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The benefits of using multiple antennas at both the transmitter and the receiver in a wireless system are well established. Multiple-input multiple-output (MIMO) systems enable a growth in transmission rate linear in the minimum of the number of antennas at either end [1][2]. MIMO techniques also enhance link reliability and

In this paper, a novel transmission technique for the multiple-input multiple-output (MIMO) broad-cast channel is proposed that allows simultaneous transmission to multiple users with limited feedback from each user. During a training phase, the base station modulates a training sequence on multiple sets of randomly chosen orthogonal beamforming vectors. Each user sends the index of the best beamforming vector and the corresponding signal-to-interfence-plus-noise ratio for that set of orthogonal vectors back to the base station. The base station opportunistically determines the users and corresponding orthogonal vectors that maximize the sum capacity. Based on the capacity expressions, the optimal amount of training to maximize the sum capacity is derived as a function of the system parameters. The main advantage of the proposed system is that it provides throughput gains for the MIMO broadcast channel with a small feedback overhead, and provides these gains even with a small number of active users. Numerical simulations show that a 20 % gain in sum capacity is achieved (for a small number of users) over conventional opportunistic space division multiple access, and a 100 % gain (for a large number of users) over conventional opportunistic beamforming when the number of transmit antennas is four.