Adaptive Allocation of Pilot and Data Power for Time-Selective Fading Channels with Feedback

We consider data transmission through a time-selective (correlated) flat Rayleigh fading channel under an average power constraint. The channel is estimated at the receiver with a pilot signal, and the estimate is fed back to the transmitter. The estimate is used for coherent demodulation, and to adapt the data and pilot powers. We start with a block fading channel in which the channel gain changes according to a Gauss-Markov process. The channel estimate is updated during each coherence block with a Kalman filter, and optimizing the data and pilot powers is formulated as a dynamic program. We then study a continuous limit in which the coherence time tends to zero, and the correlation between successive channel gains tends to one, so that the channel process becomes a diffusion process. In this limit it is shown that the optimal pilot power control policy is "bang-bang", i.e., depending on the current system state (channel estimate and associated error variance) the pilot power is either the maximum allowable, or zero. The associated regions of the state space are illustrated numerically for specific system values. This example shows that the achievable rate with the optimized training policy provides substantial gains relative to constant training power at low SNRs