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We consider wideband communication (e.g., using orthogonal frequency-division multiplexed (OFDM) systems) over a typical cellular "downlink," in which both the base station and the mobile may have multiple antennas, but the number of antennas at the mobile is assumed to be small. Implicit channel feedback can play a powerful role in such systems, especially for outdoor channels, which typically exhibit narrow spatial spreads. A summary of our findings is as follows: a) Implicit channel feedback regarding the covariance matrix for the downlink space-time channel can be obtained, without any power or bandwidth overhead, by suitably averaging uplink channel measurements across frequency. Since this approach relies on statistical reciprocity, it applies to both time-division duplex (TDD) and frequency-division duplex (FDD) systems. Using such covariance feedback yields significantly better performance at lower complexity than conventional space-time or space-frequency codes, which do not employ feedback; b) We provide guidelines for optimizing antenna spacing in systems with covariance feedback. Theoretical investigation of a hypothetical system with completely controllable channel eigenvalues shows that the optimal number of channel eigenmodes is roughly matched to the (small) number of receive antenna elements. Thus, while antenna elements in conventional systems without feedback should be spaced far apart in order to ensure uncorrelated responses, the optimal antenna spacing with covariance feedback is much smaller, thereby concentrating the channel energy into a small number of eigenmodes.