Analog Filters, Introduction

Cuthbert Nyack

An ideal filter with cut off frequency of 1 has transfer function equal to 1 for frequencies from 0 to 1 and 0 for higher frequencies. One can also approach this by looking for a function f(x) which has gain of 0 for frequencies from 0 to 1 and very high gain for higher frequencies. A filter response can then be constructed from 1/(1 + f(x)).
The applet below shows some functions which can be used for f(x).
The red and pink curves show the simplest function f(x) = xⁿ.
Yellow and cyan show the chebyshev functions 1 + C_n and C_n respectively.

The applet below show the responses above converted to ideal filter approximations. Red and cyan show polynomial and Chebyshev approximations for low pass. Yellow and pink show high pass and band pass transformations.

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