A new formulation of the approximation problem

This paper presents a novel formulation of the approximation problem for fitting a rational transfer function to a piecewise connected straight-line representation of the filter attenuation. It is shown that it is possible to generate starting functions in a noniterative fashion using properties attributable to the associated phase function of the attenuation requirement. The phase functions thus determined are used to find a minimum order for realization of the rational function. It is then shown that it is convenient to introduce the concept of a separating function to develop individual polynomial phase requirements for strictly Hurwitz polynomials of equal degree. Three examples are illustrated to show the results obtainable for what are referred to as simple or complicated filter requirements. The results show that cusp requirements are to be avoided if possible, however, results are obtainable if desired in such cases. The starting functions are iterated to final results using a parameter vector formed from the Hurwitz parameters of the polynomials. These results are obtained for a squared error sum without a requirement for either weighting functions or accelerating factors. Hence it is conjectured that the initial functions are in the vicinity of a global minimum. Data supporting this conjecture are shown for each case by walking in small steps along the initial gradient of the starting function. The examples treated here are for nontabulated filter requirements. Useful results are obtained in each example with passband errors resulting on the order of 0.1 dB or less in most cases.