  Intelligent reflecting surfaces (IRSs) have emerged as a promising technology
to improve the efficiency of wireless communication systems. However, passive
IRSs suffer from the ``multiplicative fading" effect, because the transmit
signal will go through two fading hops. With the ability to amplify and reflect
signals, active IRSs offer a potential way to tackle this issue, where the
amplification energy only experiences the second hop. However, the fundamental
limit and system design for active IRSs have not been fully understood,
especially for multiple-input multiple-output (MIMO) systems. In this work, we
consider the analysis and design for the large-scale active IRS-aided MIMO
system assuming only statistical channel state information (CSI) at the
transmitter and the IRS. The evaluation of the fundamental limit, i.e., ergodic
rate, turns out to be a very difficult problem. To this end, we leverage random
matrix theory (RMT) to derive the deterministic approximation (DA) for the
ergodic rate, and then design an algorithm to jointly optimize the transmit
covariance matrix at the transmitter and the reflection matrix at the active
IRS. Numerical results demonstrate the accuracy of the derived DA and the
effectiveness of the proposed optimization algorithm. The results in this work
reveal interesting physical insights with respect to the advantage of active
IRSs over their passive counterparts.
