Design of Lossy Ladder Filters by Digital Computer

Nonavailability of ideal elements is a major drawback in the development of filters having some prescribed characteristic. In this paper, a method for designing lossy filters built with elements having unequal dissipation factors using a high-speed digital computer as the main tool is presented along with the results of a preliminary study of the method. The magnitude function, within the first order, is shown to be a multilinear function of the element values for very small change in the element values. The basic idea is to perturb the element values of the lossy filter with the aim of making the magnitude function of the lossy filter proportional to that of the ideal filter. An error function is defined as the sum of squared differences between the magnitude characteristic of the lossy filter and the same of the ideal filter (multiplied by a suitable constant factor) at some discrete frequencies in the passband and the stop band. The error function is then minimized by the steepest descent method of minimization. Results of using the suggested method in designing a lossy Tchebycheff filter of degree 9 are included. It is found that the minimization of the pass-band ripple is associated with a decrease in the stop-band attenuation. The paper also includes details on programming methods.