On the Capacity and Energy Efficiency of Training-Based Transmissions Over Fading Channels

In this paper, the capacity and energy efficiency of training-based communication schemes employed for transmission over
a

priori
unknown Rayleigh block-fading channels are studied. Initially, the case in which the product of the estimate error and transmitted signal is assumed to be Gaussian noise is considered. In this case, it is shown that bit energy requirements grow without bound as the signal-to-noise ratio (SNR) goes to zero, and the minimum bit energy is achieved at a nonzero SNR value below which one should not operate. The effect of the block length on both the minimum bit energy and the SNR value at which the minimum is achieved is investigated. Flash training and transmission schemes are analyzed and shown to improve the energy efficiency in the low-SNR regime. In the second part of this paper, the capacity and energy efficiency of training-based schemes are investigated when the channel input vector in each coherence block is subject to peak power constraints. The capacity-achieving input structure is characterized and the magnitude distribution of the optimal input is shown to be discrete with a finite number of mass points. The capacity, bit energy requirements, and optimal resource allocation strategies are obtained through numerical analysis. The improvements in energy efficiency when
on-off
keying (OOK) with fixed peak power and vanishing duty cycle is employed are studied.