In this paper, the source and relay transmit covariance matrices are jointly optimized for a fading multiple-antenna relay channel when the transmitters only have partial channel state information (CSI) in the form of covariance feedback. For both full-duplex and half-duplex transmissions, we evaluate lower and upper bounds on the ergodic channel capacity. These bounds require joint optimization over the source and relay transmit covariance matrices. The methods utilized in the previous literature cannot handle this joint optimization over the transmit covariance matrices for the system model considered in this paper. Therefore, we utilize matrix differential calculus and propose iterative algorithms that find the transmit covariance matrices to solve the joint optimization problem. In this method, there is no need to specify first the eigenvectors of the transmit covariance matrices. The algorithm updates both the eigenvectors and the eigenvalues at each iteration. Through simulations, we observe that lower and upper bounds are close to each other. However, the distance between the lower and upper bounds depends on the channel conditions. If the mutual information on the source-to-relay channel and the broadcast channel get closer to each other, the bounds on capacity also get closer.

C-11

Fading MIMO Relay Channels with Channel Estimation Error

Bengi Aygün, and Alkan Soysal

In IEEE International Conference on Communications (ICC), Jun 2013

This is the second part of a two-part paper on the joint channel estimation and resource allocation problem in MIMO systems with noisy channel estimation at the receiver side and partial CSI, in the form of covariance feedback, available at the transmitter side. We consider transmit-side correlated MIMO channels with block fading, where each block is divided into training and data transmission phases. In this paper, we extend the single-user results of Part I to the multiple access channel. For the data transmission phase, we propose an iterative algorithm to solve for the optimum system resources such as time, power and spatial dimensions. This algorithm updates the parameters of the users in a round-robin fashion. In particular, the algorithm updates the training and data transmission parameters of a user, when those of the rest of the users are fixed, in a way to maximize the achievable sum-rate in a multiple access channel; and iterates over users in a round-robin fashion. Finally, we provide a detailed numerical analysis to support the analytical results of both parts of this two-part paper.

J-3

Joint Channel Estimation and Resource Allocation for MIMO Systems-Part I: Single-User Analysis

Multiple antenna systems are known to provide very large data rates, when the perfect channel state information (CSI) is available at the receiver. However, this requires the receiver to perform a noise-free, multi-dimensional channel estimation, without using communication resources. In practice, any channel estimation is noisy and uses system resources. We shall examine the trade-off between improving channel estimation and increasing the achievable data rate. We consider transmit side correlated multi-input multi-output (MIMO) channels with block fading, where each block is divided into training and data transmission phases. The receiver has a noisy CSI that it obtains through a channel estimation process, while the transmitter has partial CSI in the form of covariance feedback. In Part I of this two-part paper, we consider the single-user case, and optimize the achievable rate jointly over parameters associated with the training phase and data transmission phase. In particular, we first choose the training signal to minimize the channel estimation error, and then, develop an iterative algorithm to solve for the optimum system resources such as time, power and spatial dimensions. Specifically, the algorithm finds the optimum training duration, the optimum allocation of power between training and data transmission phases, the optimum allocation of power over the antennas during the data transmission phase.

J-2

Optimality of Beamforming in Fading MIMO Multiple Access Channels

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.

C-6

MIMO Multiple Access Channels with Noisy Channel Estimation and Partial CSI Feedback

We consider both the single-user and the multi-user 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.

C-3

Optimum Power Allocation in Fading MIMO Multiple Access Channels with Partial CSI at the Transmitters