Estimation of transfer function parameters with output Fourier transform sensitivity vectors

We derive a technique for estimating a small number of parameters of a spatially or temporally varying transfer function for which a parametric model is known. Such systems occur in transmission lines with faults or mismatches, ultrasonic imaging, semiconductor layering and tomography in various applications. We assume that it is important not only to know that a boundary exists but also to estimate the spatially parameters of the medium across the boundary. The method does not require that the input need be known, but it must be applied to the diverse paths simultaneously, such as with a plane wave hitting a plane surface. The output signals at the various points are time synchronous unless delay is the varying parameter. The transfer function model may be nonlinear in the parameters, and we may also have to estimate the nominal values around which the parameters are locally varying. The procedure is constructed in the context of estimating at any point in space the parameters of the system by using the eigenstructure of the covariance matrix of vectors whose elements are Fourier transform values of the responses. We require that the available responses in the spatial neighborhood are independent enough to yield a good estimate of the covariance matrix. As an example we examine the case of an unknown parameter in a thin layer between 2 semi-infinite layers