Finite Blocklength Rates over a Fading Channel with CSIT and CSIR

In this work, we obtain lower and upper bounds on the maximal {transmission} rate at a given codeword length $n$, average probability of error $\epsilon$ and power constraint $\bar{P}$, over a finite valued, block fading additive white Gaussian noise (AWGN) channel with channel state information (CSI) at the transmitter {and the receiver}. {These bounds characterize} deviation of the finite blocklength coding rates from the channel capacity which is in turn achieved by the \textit{water filling} power allocation across time. {The bounds obtained also} characterize the rate enhancement possible due to the {CSI} at the transmitter in the finite blocklength regime. The results are further elucidated via numerical examples.