The Estimation of “Transfer Functions” of Quadratic Systems

In recent years the use of stationary random inputs as a forcing function to experimentally determine the transfer function of linear systems has become widespread. The procedure involves the measurement of spectra and crossspectra between the input and output and the formation of the proper ratio. There are two basic reasons for using random testing functions. For many situations, particularly mechanical ones, these inputs are easier to generate than say steps or impulses and frequently they can be made to more closely approximate the in-service input. (The latter attribute is an advantage since the linearity of the system may not be complete but may be sufficiently so over the operating range of interest). A simple measure of linearity is immediately available i.e., the coherency. Although many experimenters try to generate a Gaussian stationary process for the forcing function this is not necessary from an expected value veiwpoint. Actually the probability structure plays no role in the general logic of the detailed procedure since only second moment characteristics of the process are relevant to the expected value calculations. However, a Gaussian input can be convenient since sometimes the evaluation of the variability of the estimates is simplified. In studying higher order systems by driving them with a random forcing function this use of a Gaussian process makes the calculations of the expected values considerably easier. In this paper we shall extend the spectral techniques of transfer function estimation of linear systems to time invariant quadratic systems when a stationary Gaussian process is used as a driving function.