Design of input signals for parameter estimation

The optimum energy-constrained and time-constrained input signal is obtained for estimating the parameters of a system. The output is corrupted by nonstationary, nonwhite additive observation noise, and the observation time is finite. The reproducing kernel Hilbert space formulation is used to obtain the parameter estimates and the error covariance matrix in terms of the input. The performance index, assumed to be a function of the error covariance matrix, is minimized by a variational procedure. A necessary condition for optimality is that the input satisfy a nonlinear Fredholm equation. An example estimates the gain of a single time constant system where the observation noise has an exponential autocorrelation function. For broadband noise, the optimum input is a portion of a sinusoid. For a noise bandwidth narrower than the system bandwidth, the optimum input switches sign as rapidly as possible, but near-optimum performance can be obtained with a relatively high frequency sinusoidal input.