The variance of non-parametric errors-in-variables estimates

Frequency response functions (FRF) measured by taking the ratio of the output to the input Fourier coefficients of the steady state response of the system to a periodic excitation are considered. Under assumptions of additive Gaussian noise on both the inputs and outputs the variance of such measurements is infinite. If an exclusion zone is applied the variance can be rendered finite. An expression for the variance, which includes the case where the input and output noise sources are correlated, is given. The expression is useful for the estimate without exclusion zone over a wide range of input signal to noise ratio.