Method of undetermined coefficients for circuits and filters using Legendre functions

This article presents a new way to implement matching networks and filters using the method of undetermined coefficients. A method is proposed for approximating the transmission coefficient of the synthesized filter, taking into account the required amplitude-frequency characteristics. To synthesize the filter, an approximating function (AF) was used using orthogonal Legendre polynomials, which is a mathematical description using a system of equations. Filter properties whose implementation is based on modified Legendre approximating functions usually depend on the interval on which they are defined and have the property that they are orthogonal on this interval. An example of seventh order filter synthesis using modified Legendre approximating functions is given. The filter circuit is implemented, the elements of the filter circuit are calculated based on the selected approximating modified function. The criteria used were minimization of the unevenness of the group delay time (GDT) and minimization of the complex approximation error for given values of the AF parameters. As a result, the number of filter elements, the group delay value and the complex approximation error are significantly reduced.