1618 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 4, APRIL 2009 Diversity Gains of Power Control With Noisy CSIT in MIMO Channels Tùng T. Kim, Member, IEEE, and Giuseppe Caire, Fellow, IEEE Abstract—A multiantenna channel with partial channel state in- formation at the transmitter (CSIT) is studied. The partial CSIT takes the form of the channel matrix corrupted by additive white Gaussian noise (AWGN) with a variance that is assumed to decay as a power of the signal-to-noise ratio (SNR). It is shown that under a long-term power constraint and in the regime of asymptotically high SNR, a large diversity gain over the channel can be achieved by using rarely a high power at the transmitter that compensates for bad channel realizations. Examples relating the diversity gain of the systems with the channel Doppler bandwidth are discussed. Index Terms—Diversity–multiplexing tradeoff, fading channels, large-deviation analysis, multiple-inpute multiple- output (MIMO) systems, power control. I. INTRODUCTION T HE diversity–multiplexing (D-M) tradeoff [1] elegantly characterizes the relationship between the reliability and throughput of multiple-input multiple-output (MIMO) systems in the regime of asymptotically high signal-to-noise ratios (SNR). While the original work [1] considers a system with no channel state information at the transmitter (CSIT), recent works have characterized the D-M tradeoff in practical scenarios where limited CSIT is available [2]–[6]. This work also considers a MIMO channel with partial CSIT. In sharp contrast to the explicitly quantized CSIT model in [4], which is a suitable model for frequency-division duplex (FDD) systems, in the current work we consider a noisy CSIT model, which perhaps better represents time-division duplex (TDD) systems, where reciprocity can be exploited to estimate the re- verse channel from the forward link. In both models, the trans- mitter exploits the knowledge of the conditional distribution of the channel matrix given the CSIT in order to improve reliability for fixed-rate (strictly delay-limited) transmission. However, a fundamental difference exists. With noiseless quantized feed- back [4], the transmitter knows exactly which quantization bin the current realization of the channel matrix belongs to. Hence, it can use power control in order to make sure that the desired rate can be reliably transmitted. Roughly speaking, in such a case the outage event consists of those channel states that cannot Manuscript received November 08, 2007; revised September 08, 2008. Cur- rent version published March 18, 2009. The work of G. Caire was supported in part by the National Science Foundation under Grant CCF-0635326. T. T. Kim was with the School of Electrical Engineering, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden. He is now with the Depart- ment of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: thanhkim@princeton.edu). G. Caire is with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail: caire@usc.edu). Communicated by A. J. Goldsmith, Associate Editor for Communications. Color version Figure 1 in this paper is available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TIT.2009.2013010 be compensated for by power control because of the average power constraint. With the noisy CSIT model explored in the present work, the channel matrix is conditionally Gaussian with given mean (the CSIT itself) and variance. Hence, an outage may occur for any given realization of the CSIT. The D-M tradeoff of this noisy CSIT model has been studied in [5], [6], where a particular rate-adaptation scheme under a short-term power constraint is shown to provide a constant di- versity gain at any multiplexing gain. It is known, however, that over slow-fading channels, temporal power control is greatly beneficial in improving the outage performance [7], [8]. In the current work, we relax the short-term power constraint imposed in [6] and allow the transmitter to adapt its power based on the noisy CSIT available. Since the technique used in [6] is very complicated to apply in this power control problem, we based the analysis on the elegant framework of integrals over the uni- tary group [9]. It turns out that the gain due to power control, even with impaired CSIT, is dramatic. This diversity gain is due to the use of high power levels when the channel is in an unlikely bad condition. It is argued that the dominant error events occur when some smallest square singular values of the channel matrix have the same order of magnitude of the CSIT noise variance. Limitations on the extra diversity gain when the noisy CSIT is obtained via prediction are finally discussed. II. SYSTEM MODEL Consider digital transmission over a MIMO wireless flat-fading channel. The transmitter uses antennas and the receiver has antennas. The complex-baseband received signal during fading block can be written as (1) The components of the channel matrix are indepen- dent and identically distributed (i.i.d.) complex Gaussian with zero mean and unit variance. We consider a quasi-static fading channel, i.e., the channel matrix is constant during a fading block of channel uses, but changes from one block to the other according to some block-wise ergodic and stationary Gaussian process. The components of the temporally and spatially white Gaussian noise matrix have zero mean and unit variance. The codeword is drawn from a codebook of rate bits per channel use. The rate is fixed for a given SNR, and is inde- pendent on the available CSIT. For brevity, we omit the block index whenever there is no ambiguity. Assume perfect knowledge of at the receiver. The trans- mitter has access to , a noisy version of , such that (2) 0018-9448/$25.00 © 2009 IEEE