848 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 Isotropic Fading Vector Broadcast Channels: The Scalar Upper Bound and Loss in Degrees of Freedom Syed Ali Jafar, Member, IEEE, and Andrea J. Goldsmith, Fellow, IEEE Abstract—We propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel. The scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users’ channel magnitudes, the distribution of the additive noise experienced by each user, or the amount of channel knowledge available at the receiver. Using this upper bound, we prove the optimality of the Alamouti scheme in a broadcast setting, extend the recent results on the capacity of nondegraded, fading scalar broadcast channels to nondegraded fading vector broadcast channels, and determine the capacity region of a fading vector Gaussian broadcast channel with channel magnitude feedback. We also provide an example of a Rayleigh-fading broadcast channel with no channel state information available to the receiver (CSIR), where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose, because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver. A tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of CSIR before applying the scalar upper bound. Index Terms—Broadcast channel, channel capacity, channel state information, degrees of freedom, fading channels, multiple antennas. I. INTRODUCTION I N this paper, we consider the capacity region of a fading vector broadcast channel with antennas at the transmitter and users with a single receive antenna at each user. We as- sume that, conditioned on the side information available to the transmitter, the channel fading vector may be modeled as an er- godic isotropic [1]–[3] stochastic process. In other words, all spatial directions are equivalent from the transmitter’s stand- point. The isotropic fading assumption captures many practical scenarios where no directional information is available to the transmitter. We do not make any assumptions on the distribu- tion of additive noise, the amount of channel state information available to the receiver, the amount of channel magnitude infor- mation available to the transmitter, the distribution of the users’ Manuscript received March 19, 2004; revised October 15, 2004. The material in this paper was presented in part at the 2004 IEEE International Conference on Communications, Paris, France, June 2004. S. A. Jafar is with the Department of Electrical Engineering and Com- puter Science, University of California Irvine, Irvine, CA 92697-2625 USA (e-mail:syed@ece.uci.edu). A. J. Goldsmith is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail:andrea@wsl.stanford.edu). Communicated by A. Lapidoth, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2004.842621 channel magnitudes, or even the availability of feedback as long as the feedback does not provide any noncausal directional in- formation for future channel realizations. We allow channel fade processes that may not be stationary or memoryless. For this broad class of isotropic fading vector broadcast channels we show that the capacity region is bounded above by the capacity region of a scalar broadcast channel. We will refer to this upper bound as the “scalar upper bound.” Like many fundamental concepts, the scalar upper bound af- fords a simple intuitive explanation. The transmit antennas provide spatial dimensions. However, with a single receive antenna, each user’s channel occupies only one spatial dimen- sion. While the transmitter would like to concentrate its power only along the direction corresponding to the users’ channels, isotropic fading, by its definition, implies that the transmitter is unable to discriminate between various directions. Thus, the transmit power is spread uniformly over all spatial dimensions. Each users’ channel occupies only one of these dimensions, and therefore each user is able to collect only a fraction of the transmit power. Thus, although there are spatial di- mensions available, the inability of the transmitter to track the users’ channel directions reduces the system to a scalar broad- cast channel with the transmit power reduced by a factor of . The scalar upper bound is a valuable tool for proving capacity results for isotropically fading vector broadcast channels. By re- ducing the vector broadcast channel to a scalar channel we are able to make use of the known capacity results for scalar chan- nels. Several applications of the scalar upper bound are provided in this paper. II. SYSTEM MODEL DEFINITIONS The following definitions of the vector (BC-V) and scalar (BC-S) fading broadcast channel models are used throughout this paper. A. Broadcast Channel Model BC-V The fading vector broadcast channel model BC-V consists of transmit antennas at the base station and users with a single receive antenna at each user and is given by the input/output relationship . . . (1) 0018-9448/$20.00 © 2005 IEEE