Optimal choice of test signals for channel estimation

We examine the problem of determining the characteristics a slowly varying linear transmission channel by measuring the output when the input is a specially selected test signal. We assume the use of a cyclically repeating test signal and use of a linear estimator of the resulting signal to determine the echo channel. We find the optimal linear estimator and its associated test signal, which together minimises mean-squared error of the channel estimate. Ths optimization is carried out under the assumption that we know the autocovariance (or power spectrum) of the interfering noise and autocovariance of the echo channel. The test signal is also assumed to be constrained by a bound on total energy.