Method of designing doubly terminated lossy ladder filters with increased element tolerances

A well established technique to increase the element tolerances in the design of selective passive filters, or to decrease the adverse effect of limited gain-bandwidth product of operational amplifiers is to use transfer function with a special type of magnitude response for which the passband ripple amplitude decreases with increasing frequency. Recently various techniques have been reported to construct this type of filtering function but they are either entirely numerical or, at least, require numerical optimisation of some relevant parameters. Two exact methods are proposed to find the explicit expressions for the predistorted characteristic functions of these filters applicable to both polynomial filters and filters with finite attenuation poles (real or complex) of any degree. The ratio of the magnitude of the first passband ripple to that of the last passband ripple can be prescribed in advance, according to the required specifications. Also a new method is presented of designing lossy ladder filters built with components having any dissipation factor. This method reduces to a simple algorithm suitable for computer implementation. Optimisation is performed in complex frequency domain, and partial derivatives in transfer function poles are determined in explicit form.