  Energy efficiency of fixed-rate transmissions is studied in the presence of
queueing constraints and channel uncertainty. It is assumed that neither the
transmitter nor the receiver has channel side information prior to
transmission. The channel coefficients are estimated at the receiver via
minimum mean-square-error (MMSE) estimation with the aid of training symbols.
It is further assumed that the system operates under statistical queueing
constraints in the form of limitations on buffer violation probabilities. The
optimal fraction of power allocated to training is identified. Spectral
efficiency--bit energy tradeoff is analyzed in the low-power and wideband
regimes by employing the effective capacity formulation. In particular, it is
shown that the bit energy increases without bound in the low-power regime as
the average power vanishes. A similar conclusion is reached in the wideband
regime if the number of noninteracting subchannels grow without bound with
increasing bandwidth. On the other hand, it is proven that if the number of
resolvable independent paths and hence the number of noninteracting subchannels
remain bounded as the available bandwidth increases, the bit energy diminishes
to its minimum value in the wideband regime. For this case, expressions for the
minimum bit energy and wideband slope are derived. Overall, energy costs of
channel uncertainty and queueing constraints are identified, and the impact of
multipath richness and sparsity is determined.
