  In this paper, the bit energy requirements of training-based transmission
over block Rayleigh fading channels are studied. Pilot signals are employed to
obtain the minimum mean-square-error (MMSE) estimate of the channel fading
coefficients. Energy efficiency is analyzed in the worst case scenario where
the channel estimate is assumed to be perfect and the error in the estimate is
considered as another source of additive Gaussian noise. It is shown that bit
energy requirement grows without bound as the 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
schemes are analyzed and shown to improve the energy efficiency in the low-snr
regime. Energy efficiency analysis is also carried out when peak power
constraints are imposed on pilot signals.
