On the Capacity of MIMO Systems with Magnitude Knowledge and Phase Uncertainty Miquel Payaro Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), Spain mpayaro@cttc.es Jinhong Yuan Department of Electrical Engineering, University of New South Wales, Australia jinhong@ee.unsw.edu.au Miguel A. Lagunas Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), Spain m.a.lagunas@cttc.es Abstract- In this work, we study different capacity formula- tions in multi-input multi-output channels where the transmitter is only informed with the magnitude of the complex channel matrix coefficients. Perfect channel knowledge is assumed at the receiver side. First, we give the expressions for the ergodic, compound, and outage capacities for our particular model of channel state information. Next, focusing on the compound for- mulation, we find that the optimal transmitter strategy consists in independent signaling through the transmit dimensions. Finally, we present some results on the optimal power allocation for the maximization of the outage mutual information for a simple MIMO system where the transmitter is equipped with only two antennas. I. INTRODUCTION The transceiver design in multi-input multi-output (MIMO) communication systems depends not only on the chosen figure of merit (MSE, SINR, BER, or capacity), but also on the quantity and the quality of the channel state information (CSI) that is made available at the transmitter side. As far as capacity is concerned, it is well known [1], that the optimum signaling strategy for MIMO channels with Gaussian noise consists in transmitting random Gaussian vectors with zero mean. Since a Gaussian process is fully characterized by its mean and covariance, the only remaining degree of freedom to optimize the mutual information is through the covariance matrix of these vectors. In addition to the quantity and the quality of the CSI at the transmitter side, the optimal design of the covariance matrix also depends on the model utilized to characterize the channel. On one hand, many authors consider the MIMO channel as a stochastic matrix, whose entries are commonly modeled as circularly symmetric complex Gaussian random variables, with a given mean and covariance. Within this context, two different kind of approaches can be considered depending on the time-varying characteristics of the channel. If the channel fluctuations are fast enough so that, during the transmission of a message, its long term average properties are unveiled, then the optimal covariance design aims at maximizing the ergodic mutual information, because it is the measure that controls the rate at which reliable communication is possible [2]. On the contrary, in a slow fading scenario, the maximum rate achievable with a certain probability is dictated by the outage M. Payaro is a Visiting Research Associate at the Department of Electrical Engineering, University of New South Wales (UNSW), Australia mutual information [3], and thus the covariance matrix should be optimized according to this criterion. On the other hand, much attention is recently being paid to models that describe the channel assuming that it is a deterministic (thus fixed) quantity belonging to a certain set, which takes into account the possible effects of lack of knowledge about the channel state that the transmitter may be experiencing. Within this framework, the design of the covariance matrix is focused at maximizing the worst case mutual information, whose supremum is defined in [4] as the compound capacity. Concerning the stochastic models for the channel, the er- godic capacity of multiantenna systems with partial channel state information at the transmitter side has been studied, for example, in [5] and [6], in which a single receiver architecture is considered, and it is assumed that the transmitter has access only to either the mean value or the covariance of the channel vector. In the latter case, capacity can be achieved by a covariance matrix that has the same eigenvectors as the true channel covariance matrix. The outage formulation has been largely less studied. In [2], it was conjectured that for Rayleigh channels the optimal strategy consists in performing uniform power allocation among a subset of the transmit dimensions. More recently, the outage capacity has been studied in [7] for the case of a single receive dimension. As far as the compound capacity is concerned, the optimality of the uniform power allocation scheme has been proven in [8] using game-theoretic justifications under a mild assumption on the channel isotropy. In addition, in [9] the authors proved that beamforming maximizes the compound capacity in rank one Ricean linear vector Gaussian channels. For the sake of completeness, see further [10] for a tutorial on capacity under different assumptions of the knowledge at the transmitter side. In this work, we study the ergodic, compound and outage capacity formulations for a different model of incomplete knowledge of the channel matrix at the transmitting end. We assume that the transmitter has perfect knowledge of the magnitude of the complex channel coefficients but a complete lack of knowledge of the channel phases (see further [11], [12]). Many interesting practical situations fit in this model. In time division duplex (TDD) mobile communication systems, for example, if electromagnetic reciprocity between uplink and downlink channels is exploited, then perfect knowledge of the 7th Australian Communications Theory Workshop 1-4244-0214-X/06/$20.00 (.)2006 IEEE 43