Cauer filter

Cauer filters or elliptical filters are frequency filters that are designed for a very steep transition in the frequency response from the pass band to the stop band . They are named after Wilhelm Cauer . In contrast to the similarly structured Chebyshev filters , Cauer filters have an oscillating course of the transfer function in both the passband and the stopband .

For the design of a Cauer filter, use is made of the rational elliptic functions , from which the name of this filter type is derived. In contrast to other filters such as Chebyshev filters or Butterworth filters , a given amplitude tolerance scheme with given constant guaranteed attenuation in the stop band and given ripple in the pass band as well as given transition frequencies can be implemented with a system of minimal order. This means that the circuitry is less complex than with other filter types. However, this advantage is bought at the cost of strong phase distortions in the transfer function. Excessively strong phase distortions are undesirable in some filter applications, so that, despite the advantages of elliptical filters, preference is given to the Chebyshev or Butterworth filter and the increased circuit complexity in certain applications. However, an all-pass filter can also be used to correct the phase, also at the expense of increased circuit complexity. ${\displaystyle R_{n}(\xi ,\omega )}$

Transfer function

Transfer function of a 4th order Cauer filter with ε = 0.5 and ξ = 1.05

The transfer function of a Cauer low pass of the order is given by: ${\displaystyle n}$

${\displaystyle |G_{n}(\mathrm {j} \omega )|={\frac {1}{\sqrt {1+\epsilon ^{2}R_{n}^{2}(\xi ,\omega )}}}}$

and the square of the magnitude of the frequency curve is:

${\displaystyle |G_{n}(\mathrm {j} \omega )|^{2}={\frac {1}{1+\epsilon ^{2}R_{n}^{2}(\xi ,\omega )}}}$

each with the rational elliptic functions of the order . The factor is a parameter that primarily influences the ripple of the transfer function. The parameter influences the selectivity of the filter. The figure on the right shows the transfer function of a 4th order Cauer filter with the parameters ε = 0.5 and ξ = 1.05 with the abbreviation ${\displaystyle R_{n}(\xi ,\omega )}$ ${\displaystyle n}$ ${\displaystyle \epsilon }$ ${\displaystyle \xi }$

${\displaystyle L_{n}=R_{n}(\xi ,\omega )}$

shown.

For the practical application and dimensioning of Cauer filters, filter tables or corresponding software packages such as GNU Octave or MATLAB are used . The required component values ​​for a filter can be read directly from these tables up to a certain filter order.

literature

• Wilhelm Cauer : filter circuits . VDI-Verlag, Berlin, 1931.
• Wilhelm Cauer: Theory of the linear alternating current circuits. Volume 1. Academic Publishing Society Becker and Erler, Leipzig, 1941.
• Wilhelm Cauer: Theory of the linear alternating current circuits. Volume 2. Edited from the estate by Ernst Glowatzki. Akademie-Verlag, Berlin 1960.
• Anatol I. Zverev: Handbook of Filter Synthesis. John Wiley & Sons, New York NY et al. 1967, ISBN 0-471-74942-7 (also: Wiley-Interscience, Hoboken NJ 2005).
• Leland B. Jackson: Digital Filters and Signal Processing. Kluwer, Boston MA et al. 1986, ISBN 0-89838-174-6 .