  In this paper, pilot-assisted transmission over time-selective flat fading
channels is studied. It is assumed that noncausal and causal Wiener filters are
employed at the receiver to perform channel estimation with the aid of training
symbols sent periodically by the transmitter. For both filters, the variances
of estimate errors are obtained from the Doppler power spectrum of the channel.
Subsequently, achievable rate expressions are provided. The training period,
and data and training power allocations are jointly optimized by maximizing the
achievable rate expressions. Numerical results are obtained by modeling the
fading as a Gauss-Markov process. The achievable rates of causal and noncausal
filtering approaches are compared. For the particular ranges of parameters
considered in the paper, the performance loss incurred by using a causal filter
as opposed to a noncausal filter is shown to be small. The impact of aliasing
that occurs in the undersampled version of the channel Doppler spectrum due to
fast fading is analyzed. Finally, energy-per-bit requirements are investigated
in the presence of noncausal and causal Wiener filters.
