IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 10, OCTOBER 2010 4793 Maximum Mutual Information Design for MIMO Systems With Imperfect Channel Knowledge Minhua Ding, Member, IEEE, and Steven D. Blostein, Senior Member, IEEE Abstract—New results on maximum mutual information design for multiple-input multiple-output (MIMO) systems are presented, assuming that both transmitter and receiver know only an estimate of the channel state as well as the transmit and receive correlation. Since an exact capacity expression is difficult to obtain for this case, a tight lower-bound on the mutual information between the input and the output of a MIMO channel has been previously formulated as a design criterion. However, in the previous literature, there has been no analytical expression of the optimum transmit covariance matrix for this lower-bound. Here it is shown that for the general case with channel correlation at both ends, there exists a unique and globally optimum transmit covariance matrix whose explicit expression can be conveniently determined. For the special case with transmit correlation only, the closed-form optimum transmit covariance matrix is presented. Interestingly, the optimal trans- mitters for the maximum mutual information design and the min- imum total mean-square error design share the same structure, as they do in the case with perfect channel state information. Simula- tion results are provided to demonstrate the effects of channel esti- mation errors and channel correlation on the mutual information. Index Terms—Channel state information (CSI), mean-square error (MSE), multiple-input multiple-output (MIMO), mutual information, optimization. I. INTRODUCTION M ULTIPLE-INPUT multiple-output (MIMO) systems are known to be capable of providing high data rates without increasing bandwidth in rich scattering wireless fading channels [1]. However, the capacity of a MIMO channel de- pends on the availability of channel state information (CSI) at both ends. Correspondingly, different transmit strategies should be used with different types of CSI. The case when the fading channel is perfectly known to both ends has been studied in [1]–[3] and [20]. More recently, optimal transmit strategies are obtained for the case when the CSI at the receiver (CSIR) is perfect and the CSI at the transmitter (CSIT) is the channel mean or covariance information [4]. The noncoherent case with Manuscript received November 04, 2008; revised June 26, 2009. Date of cur- rent version September 15, 2010. This work was performed while M. Ding was with the Department of Electrical and Computer Engineering, Queen’s Univer- sity, Kingston, ON, Canada, and was supported under the Natural Sciences and Engineering Research Council of Canada Discovery Grant 41731. M. Ding is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: minhua.ding@ieee.org). S. D. Blostein is with the Department of Electrical and Computer Engi- neering, Queen’s University at Kingston, Kingston, ON K7L 3N6 Canada (e-mail: steven.blostein@queensu.ca). Communicated by G. Taricco, Associate Editor for Communications. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2010.2059870 no instantaneous CSIT or CSIR has been studied in [5] and [6]. A comprehensive overview of the capacity results of MIMO systems can be found in [7]. In the above, either a perfect coherent system (perfect CSIR) or a noncoherent system (no instantaneous CSIR) has been assumed. In [8], a different MIMO channel scenario is considered, where the CSIR is obtained through channel estimation and contains estimation errors. The CSIT is assumed to be obtained from the receiver via a lossless feedback link and is the same as the CSIR. The CSI at both ends consists of the channel estimate and channel correlation information. Under this assumption of CSI, an exact capacity expression is hardly tractable. Instead, tight upper- and lower-bounds on capacity have been proposed for system design [8], which are generalizations from those for a single-input single-output (SISO) channel [9]. The case when the CSI at both ends consists of channel estimates and channel correlation has been studied in [10] and [11], where the upper- and lower-bounds are shown to be close and thus are both tight. In particular, the lower-bound on the ergodic capacity has been formulated and used as the design criterion [10], [11]. Unfortunately, so far, no expression for the optimum transmit covariance matrix has been obtained for the capacity lower-bound with channel mean (i.e., channel estimate) and channel correlation information at both ends. In this paper, we attempt to solve this problem. Our main contributions are listed as follows: • We show that a globally optimum transmit covariance ma- trix exists for the capacity lower-bound. We also present its expression, which clarifies the transmitter structure and can be conveniently determined. • The methodology employed to determine the optimum transmit covariance matrix enables us to determine the relationship between the maximum mutual information design and minimum total mean-square error (MSE) design with imperfect CSI. • In [11], due to the absence of the optimum covariance ma- trix for the capacity lower-bound, the effects of the same amount of transmit and receive correlation are found to be different (see [11, Figs. 4 and 5]). Based on the optimum transmit covariance matrix obtained here, we reassess the effects of transmit and receive correlation, and observe dif- ferent results from those in [11]. The differences between our work and that in the literature can be highlighted as follows: • In [12, Sec. VI], linear MIMO transceiver designs with im- perfect CSI at both ends have been considered. The authors have obtained results for the case with receive correlation only, which is mathematically equivalent to the perfect CSI 0018-9448/$26.00 © 2010 IEEE