  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. In these schemes, periodically transmitted
training symbols are used at the receiver to obtain the minimum
mean-square-error (MMSE) estimate of the channel fading coefficients.
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 the paper, the capacity and energy efficiency of
training-based schemes are investigated when the channel input 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 bit energy is again shown to grow without bound as SNR decreases to zero
due to the presence of peakedness constraints. The improvements in energy
efficiency when on-off keying with fixed peak power and vanishing duty cycle is
employed are studied. Comparisons of the performances of training-based and
noncoherent transmission schemes are provided.
