The Synthesis of Recursive Digital Filters

Abstract

The circumstances and methods of the synthesis of linear digital recursive filters which are both stable and physically realizable are described. It is shown that any amplitude requency transfer function expressible as an even trigonometric rational polynomial can be synthesized by a real stable linear digital recursive filter. The degree of the corresponding difference equation is twice the degree of the denominator of the rational trigonometric polynomial. A class of even rational trigonometric functions which exhibit pointwise convergence to the deal rectangular low or high-pass amplitude frequency transfer function is chosen. A member of this class is shown to approximate more closely the ideal rectangular filter than does the corresponding classical continuous Butterworth filter. This class of filters is then mechanized. The phase and unit-impulse response functions are calculated for the corresponding difference equations of degrees 2 and 4.