Optimum allpole filters with Chebyshev passband magnitude response

Abstract

A new kind of the allpole lowpass filter functions with Chebyshev behaviour in the passband is derived by minimizing the passband insertion loss. The ripple parameter ε, commonly used to control the attenuation at the cutoff frequency, in proposed approximation is used as a degree of freedom by which the passband attenuation can be controlled, while keeping the attenuation at the cutoff frequency equal to 3 dB. The approximation method relies on solving the nonlinear equation such that the area under the filter characteristic function in passband is minimized. Obtained filters, called Optimum Chebyshev filters, exhibit minimum insertion loss and maximum cutoff slope. Filter polynomial coefficients and the elements values for the LC ladder implementation are tabulated for filter degrees from 3 to 11.Comparison of the Optimum Chebyshev and Least Square Monotonic filters is also discussed in the paper, since both filters are developed subject to the minimum reflected power in the passband; the first with the equiripple and the second with the staircase (monotonic) behavior in the passband. It is also revealed that the Optimum Chebyshev approximation can be considered as the optimal solution among the class of filters whose passband response is bound to be a staircase.