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By applying the rate-distortion theory to multiple-input-multiple-output (MIMO) systems, we first present the mathematical relation between feedback rate and closed-loop capacity. The lower and upper bounds on rate are then presented. Finally, we perform simulations of the bounds for various SNRs and numbers of antennas, and draw relevant conclusions. The proposed concepts and expressions of the bounds, the conclusions and numerical results, as well as the method that employs the rate-distortion theory to study feedback of MIMO systems, are crucial to researches and designs based on a global consideration of SNR, antenna number, feedback rate and capacity.