cases are given.

    Energy efficiency is further analyzed in the presence of channel uncertainties. The scenario in which *a priori* unknown fading coefficients are estimated at the receiver via minimum mean-square-error (MMSE) estimation with the aid of training symbols, is considered. The optimal fraction of power allocated to training is identified under QoS constraints. It is proven that the minimum bit energy in the low-power regime is attained at a certain nonzero power level below which bit energy increases without bound with vanishing power. Hence, it is shown that it is extremely energy inefficient to operate at very low power levels when the channel is only imperfectly known.

    *Index Terms:* bit energy, channel estimation, effective capacity, energy efficiency, fading channels, fixed-rate transmission, imperfect channel knowledge, low-power regime, minimum bit energy, QoS constraints, spectral efficiency, wideband regime, wideband slope.
author:
- '[^1] [^2]'
title: 'The Impact of QoS Constraints on the Energy Efficiency of Fixed-Rate Wireless Transmissions'
---

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Introduction
============

The two key characteristics of wireless communications that most greatly impact system design and performance are 1) the randomly-varying channel conditions and 2) limited energy resources. In wireless systems, the power of the received signal fluctuates randomly over time due 