Outage diversity of MIMO block-fading channels with causal channel state information

We study the outage diversity of the multiple-input multiple-output (MIMO) Rayleigh block-fading channel when causal channel state information (CSI) is available at the transmitter (CSIT). Within this setting, we consider the optimal power allocation for blocks b = 1, ..., B given perfect CSIT for blocks 1, ..., b-u only, subject to a long-term power constraint. The parameter 0 ≤ u ≤ B is a fixed arbitrary integer that determines the delay in acquiring perfect knowledge of the CSI at the transmitter. Without explicitly solving the optimal power allocation problem, we derive the outage diversity of the system. For general 0 ≤ u ≤ B, we derive a simple recursive expression for computing the outage diversity. For the special case u = 0, it is shown that the outage diversity is infinite, coinciding with previously known results. For 1 ≤ u ≤ B, the outage diversity becomes finite and for the special case of u = 1, 2 it can be expressed in simple closed form.