Optimal Data and Training Symbol Ratio for Communication over Uncertain Channels

We consider the problem of determining the power ratio between the training symbols and data symbols in order to maximize the channel capacity for transmission over uncertain channels with a channel estimate available at both the transmitter and receiver. The receiver makes an estimate of the channel by using a known sequence of training symbols. This channel estimate is then transmitted back to the transmitter. The capacity that the transceiver maximizes is the worst case capacity, in the sense that given a noise covariance, the transceiver maximizes the minimal capacity over all distributions of the measurement noise under a fixed covariance matrix known at both the transmitter and receiver. We give an exact expression of the channel capacity as a function of the channel covariance matrix, and the number of training symbols used during a coherence time interval. This expression determines the number of training symbols that need to be used by finding the optimal integer number of training symbols that maximize the channel capacity. As a bi-product, we show that linear filters are optimal at both the transmitter and receiver.