We consider space-time communication over a typical cellular "downlink" based on Orthogonal Frequency Division Multiplexing (OFDM), in which the base station may have several antenna elements, while the mobile has 1 or 2 antenna elements. 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. The covariance feedback can be used to obtain better performance on the downlink, at lower encoding and decoding complexity, compared to standard space-time coding (which does not employ channel feedback). (b) The conventional design without channel feedback is to space the transmit antennas far enough apart so as to ensure uncorrelated responses. However, when implicit feedback is available, much better performance is obtained with significantly smaller antenna spacing, optimized such that the number of dominant eigenmodes of the channel matches the number of receive antenna elements.