Second Order and Moderate Deviation Analysis of a Block Fading Channel with Deterministic and Energy Harvesting Power Constraints

Abstract: We consider a block fading additive white Gaussian noise (AWGN) channel with perfect channel state information (CSI) at the transmitter and the receiver. First, for a given codeword length and non-vanishing average probability of error, we obtain lower and upper bounds on the maximum transmission rate. We derive bounds for three kinds of power constraints inherent to a wireless transmitter. These include the canonical peak power constraint and average power constraint. In addition, we consider a time varying peak power constraint imposed by an \emph{energy harvesting device}-a mechanism that powers many modern-day wireless transmitters. The bounds characterize \emph{second order} deviation of finite blocklength coding rates from the channel capacity, which is in turn achieved by \textit{water-filling} power allocation across time. The bounds obtained also indicate the rate enhancement possible due to CSI at the transmitter in the finite blocklength regime. Next, we provide bounds on the optimal exponent with which error probability drops to zero when channel coding rate is simultaneously allowed to approach capacity \emph{at a certain rate}, as the codeword length increases. These bounds identify what is known as the moderate deviation regime of the block fading channel. We compare the bounds numerically to bring out the efficacy of our results.