# Elliptic filter

Cauer filter or elliptic filters are frequency filters that are designed for a very steep transition of the frequency response from the passband to the stopband. They are named after Wilhelm Cauer. In contrast to the similarly constructed Chebyshev filters have Cauer filter in both the passband and stopband on an oscillating shape of the transfer function.

For the design of a Cauer filter is made use of the rational elliptic functions, which is also the name of this type of filter is derived. Unlike other filters such as Chebyshev or Butterworth filter a predetermined amplitude tolerance scheme with a given constant Warranty attenuation in the stop band and a given ripple in the passband and given transition frequencies can be realized with a system of minimal order in the filter design. This means a lower circuit complexity than with other types of filters. This advantage is paid for you, however, by strong phase distortions of the transfer function. Overly strong phase distortions are undesirable in some filtering applications, so that despite the advantages of elliptic filters the Chebyshev or Butterworth filter and the increased circuit complexity is given in certain applications the preference. It can also be an all-pass filter used to correct the phase, also paid for by increased circuit complexity.

## Transfer function

The transfer function is at Cauer filters of order given by:

And the magnitude frequency response:

With each of the rational elliptic functions of order. The factor is a parameter which affects primarily the ripple of the transfer function. The parameter influences the sensitivity of the filter. In the adjacent figure, the transfer function of a Cauer filter is 4th order with the parameters ε = 0.5 and ξ = 1.05 with the abbreviation

Shown.

For practical applications, and dimensioning of Cauer filters are working with filter tables or relevant software packages such as GNU Octave or MATLAB. From these tables, the required component values can be read directly on a filter to a certain filter order.