Capacity of nearly decomposable Markovian fading channels under asymmetric receiver-sender side information

We investigate the following issue: if fast fades are Markovian and known at the receiver, while the transmitter has only a coarse quantization of the fading process, what capacity penalty comes from having the transmitter act on the current coarse quantization alone? For time-varying channels which experience rapid time variations, sender and receiver typically have asymmetric channel side information. To avoid the expense of providing, through feedback, detailed channel side information to the sender, the receiver offers the sender only a coarse, generally time-averaged, representation of the state of the channel, which we term slow variations. Thus, the receiver tracks the fast variations of the channel (and the slow ones perforce) while the sender receives feedback only about the slow variations. While the fast variations (micro-states) remain Markovian, the slow variations (macro-states) are not. We compute an approximate channel capacity in the following sense: each rate smaller than the "approximate" capacity, computed using results by Caire and Shamai, can be achieved for sufficiently large separation between the time scales for the slow and fast fades. The difference between the true capacity and the approximate capacity is O(εlog²(ε)log(-log(ε))), where ε is the ratio between the speed of variation of the channel in the macro- and micro-states. The approximate capacity is computed by power allocation between the slowly varying states using appropriate water filling.