Trade-off between Capacity and Diversity for Block Fading Sub-Channels

We consider the trade-off between multiplexing gain and diversity gain for the block-fading sub-channel model. The channel vector may be rank deficient and has arbitrary covariance structure. We derive this trade-off by considering the scaling law of the ergodic capacity, which represents the multiplexing gain, and the error probability, which determines the diversity gain at high SNR. With a fixed multiplexing gain, we give an upper bound and a lower bound on the maximum diversity gain, and give an optimization procedure to get the exact maximum diversity gain.