Block fading channels with limited channel state information

In this paper, we provide a lower bound to maximum expected rate for block fading channels with limited Channel State Information (CSI). We assume a channel fading block accommodates at least one codeword so that for block Rayleigh fading, the ergodic channel capacity is zero when no CSI is available at the transmitter. The CSI fed back to the transmitter is subject to a cardinality constraint. We derive a counting function to describe an efficient use of the available cardinality by formulating the capacity loss due to limited cardinality as a variational calculus problem. The feedback schemes with different cardinality constraints can be easily derived by scaling the counting function to fit the cardinality constraints. With Rayleigh fading channels, the feedback scheme and the corresponding lower bound of maximum expected rate can be solved in closed form. According to our results, the expected rate achieved by using the counting function is tight.