On the Performance of Optimal Input Signals for Frequency Response Estimation

We consider the problem of minimum-variance excitation design for frequency response estimation based on finite impulse response (FIR) and output error (OE) models. The objective is to minimize the power of the input signal to be used in the system identification experiment subject to a model accuracy constraint. For FIR and OE models this leads to a finite dimensional semi-definite programming optimization problem. We study, in detail, how to apply this approach to the estimation of the frequency response at a given frequency,. The first case concerns minimizing the asymptotic variance of the estimated frequency response based on an FIR model estimate. We compare the optimal input signal with a sinusoidal signal with frequency that gives the same model accuracy, and show that the input power can, at best, be reduced by a factor of two when using the optimal input signal. Conditions are given under which the sinusoidal signal is optimal, and it is shown that this is a common case for higher order FIR models. Next, we study FIR model based estimation of the absolute value and phase of the frequency response at a given frequency,. We derive the corresponding optimal input signals and compare their performances with that of a sinusoidal input signal with frequency. The relative reduction of input power when using the optimal solution is at best a factor of two. Finally, we discuss how to extend the FIR results to OE system identification by using an input parametrization proposed by Stoica and Soderstrom (1982).