The IIR Transfer Function H(z) From A General Laplace H(s): Improved Procedure For Computing Coefficients
J. Bombardieri, Harvey A. Cohen · 2005
An improved method is derived and demonstrated for the design of Infinite Impulse Response (IIR) Digital Filters or Discrete Control Systems from Analogue Prototypes. The dynamics and structure of these prototypes being initially specified in the Laplace Domain. The method depends on the calculation presented here of a closed form result for the application of the bilinear transform s=2/T(1-x/sup -1//1+x/sup -1/ ) to a general Laplace transfer function pocessing both numerator and denominator polynomials. Previous workers have utilized a closed form in the pole only case (denominator only), this closed form being expressed as the Q matrix transforming a polynomial in a to a polynomial in a. The Bilinear transform has been previously applied to compute the equivalent discrete recursive system coefficients associated with general Laplace transfer functions. However for the general form of arbitrary order this procedure has previosly required prior factorization of the transfer functions into low order units. The use of our closed form result offers several advantages compared to othe methods involving the use of the bilinear trans form. Firstly the transform may be efficiently computed for filters of arbitrary order and structure. Secondly, the requirement of factorization is eliminated wherever a direct form or state realization is applicable. Thirdly, where it is necessary to utilize a factorized realization, the factorization may be deferred until after the biliear transform. This avoids the propagation of error incurred by numerically locating the roots of the Laplace polynomials through the bilinear transform stage. The properties of this result relating to all pole and pole/zero prototypes are elaborated and the method is verified by applying it to previosly transformed systems whose coefficients an available in the literature.