Complex Curve Fitting Research Articles

Multidimensional bilinear transformations

The bilinear transformation is extended to transform multi-variable polynomials, using discrete convolution and the Kronecker product. This approach is very simple, easy for computer implementation, and useful in complex curve fitting and stability studies of discrete systems and the design of digital filters etc.

On Levy's identification, its generalization and applications

Levy's complex curve fitting is re-visited and re-solved using a vector-matrix approach. The problem is formulated in a general minimum-norm setting, and a unified geometric optimization method is used to result in a more comprehensive solution procedure and more programmable algorithms. The basic problem solved in the real-domain is then extended to allow complex-domain operation, and to allow the use of only the gain-frequency or the phase-frequency response. Extension is made to the identification of two-dimensional systems. Using this approach, sequential computation is made possible, and flexibility of accepting general weighting functions is provided. Potential applications beyond identification, including filter design and stable model-reduction, are discussed.

Relaxation models for moist soils suitable at microwave frequencies

Relaxation models based on modifications of the Debye equation have been used in analysis of the complex dielectric constant results on various types of soils at microwave frequencies. A computer-based algorithm has been developed to provide accurate analysis through complex curve fitting. Results of this study suggest that Bergmann modification of the Debye equation seems to fit experimental data on 12% moisture content soils. Below 1 GHz one may need to consider phenomena such as residual surface effects and intermediate forms of bound water.

Transfer Functions from Sampled Impulse Responses

A mathematical formulation is developed for obtaining the frequency responses from known sampled impulse responses of linear dynamical systems. This is done by assuming a straight line approximation from one sampled point to the next and performing the usual Fourier Transformation for this interval. This is repeated for all sampled points and the results are summed together. In practice sampled impulse responses can be obtained using correlation techniques. The frequency responses as obtained above are then processed by a generalised method, developed by the authors, to determine the transfer functions. The unique feature of this method is that the transfer functions can be obtained from the frequency responses without any idea about the actual order of the system and its poles and zeroes. However, the actual principle employed, that of complex curve fitting, is already a well established technique. Finally, many examples are presented which show the validity of the methods where impulse responses are exact and also corrupted by errors. The treatment is restricted to systems which have a finite dc gain. The numerical calculations are processed on an ICT 1905 computer.

Complex-curve fitting

The mathematical analysis of linear dynamic systems, based on experimental test results, often requires that the frequency response of the system be fitted by an algebraic expression. The form in which this expression is usually desired is that of a ratio of two frequency-dependent polynomials. In this paper, a method of evaluation of the polynomial coefficients is presented. It is based on the minimization of the weighted sum of the squares of the errors between the absolute magnitudes of the actual function and the polynomial ratio, taken at various values of frequency (the independent variable). The problem of the evaluation of the unknown coefficients is reduced to that of the numerical solution of certain determinants. The elements of these determinants are functions of the amplitude ratio and phase shift, taken at various values of frequency. This form of solution is particularly adaptable to digital computing methods, be-because of the simplicity in the required programming. The treatment is restricted to systems which have no poles on the imaginary axis; i.e., to systems having a finite, steady-state (zero frequency) magnitude.