Application of the Principle of Analytic Continuation to Interpolate/Extrapolate System Responses Resulting in Reduced Computations—Part A: Parametric Methods

Abstract

This review paper is a sequel to our earlier paper entitled “Application of the principle of analytic continuation to interpolate/extrapolate system responses resulting in reduced computations—Part A: Parametric methods” dealing with parametric methods in the context of the principle of analytic continu-ation and providing its relationship to reduced rank modeling using the total least-squares-based singular value decomposition methodology. The problem with a parametric method is that the quality of the solution is determined by the choice of the basis functions, and the use of bad basis functions generates bad solutions. A priori, it is quite difficult to recognize what are good basis functions and what are bad basis functions, even though methodologies exist in theory on how to choose good ones. The advantage of the nonparametric methods is that no such choices of the basis functions need to be made, as the solution procedure itself develops the nature of the solution and no a priori information is necessary. This is accomplished through the use of the Hilbert transform, which exploits one of the fundamental properties of nature, i.e., causality. The Hilbert transform illustrates that the real and imaginary parts of any nonminimum-phase transfer function from a causal system satisfy this relationship. In addition, some parameterization can also be made of this procedure, which can enable one to generate a nonminimum-phase function from its amplitude response and from that generate the phase response and, thereby, can compute the time-domain data for the amplitude-only case except for a delay in the response. This uncertainty is removed in holography, as in such a procedure, amplitude and phase information is measured for a specific look angle, thus eliminating the phase ambiguity. An overview of the technique along with examples is presented to illustrate this methodology.