Rational modeling from frequency/wavenumber samples

The signal processing community is, by now, very familiar with such time domain rational modeling techniques as Prony's method and Pade approximation. These methods allow the engineer to construct rational models of systems from time domain data (usually impulse response samples, autocorrelation samples, etc.). There has been a recent interest in frequency domain techniques in a number of various applications. Such disciplines as stellar imaging, seismic analysis and sonar engineering use spatial frequency data routinely.;The goal of the thesis is to make useful contributions to the following general problem. Given a set of frequencies, spatial or temporal, and the prescribed response values of an unknown multivariable or multidimensional linear system at those frequencies, construct a rational model for the unknown process. A natural question which arises in an interpolation problem is "What is the order of the unknown system?" It is well known that the Hankel matrix of time-domain Markov parameters is able to uncover the order of an unknown rational function. An important intent of the thesis is to develop corresponding matrix structures to be used with frequency/wavenumber samples. A useful structure which naturally occurs in this context is the Lowner matrix structure, named after the late Czech mathematician and Stanford professor Charles Lowner. In the thesis, it will be shown that the Lowner matrix is a frequency domain generalization of the Hankel matrix. Like the Hankel matrix, the Lowner matrix uncovers the order of our unknown rational model, provides information on multivariable systems and is also used to construct the model itself.;The thesis will show the feasibility of obtaining a rational model in a twofold manner: First, uncover the order of the frequency/wavenumber data; then follow with a parameter identification procedure. The techniques have been used to completely solve the scalar and matrix 1-D rational interpolation problem. These 1-D methods are then used to solve the 2-D rational interpolation problem. A few multidimensional generalizations a well as a preliminary definition for a 2-D Lowner matrix is also given.