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

## TL;DR

This paper analyzes the finite blocklength performance of a block fading AWGN channel with perfect CSI, deriving bounds on transmission rates and error exponents under various power constraints, including energy harvesting scenarios.

## Contribution

It provides the first second order and moderate deviation bounds for block fading channels with deterministic and energy harvesting power constraints.

## Key findings

- Bounds on finite blocklength rates under different power constraints.
- Characterization of second order deviation from channel capacity.
- Bounds on error probability decay rate in the moderate deviation regime.

## 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.