A digital filter design procedure with application program in APL

This study develops a fully documented computer program in APL for the synthesis of transfer functions and state-space realizations of digital filters configured in cascaded second-order subsections. The program designs Butterworth, Chebyshev, and Elliptic filters of the low-pass, high-pass, band-pass, and band-stop types. The realizations are designed to be computationally efficient with low roundoff noise.

The study essentially spans the complete development of a cascade digital filter by working through the following steps:

1. design parameters,

2. normalized analog transfer function,

3. denormalized analog transfer function,

4. digital transfer function,

5. realization coefficients.

Each step is examined in detail and at each point in the mathematical development, the program functions that perform the calculations at that point are identified. In addition, program function flow charts, detailed logic flow charts, and a glossary of terms are provided to complete the program documentation.

The mathematical development begins with descriptions and definitions of the Butterworth, Chebyshev, and Elliptic normalized analog low-pass filters. A standard form, for use throughout the study, is then established for the transfer functions of these filters. Next, procedures are established for calculating the coefficients of the filters. A review is then made of computation parameters that can be used in the transformation from analog to digital transfer functions. These parameters, and the bilinear transformation, are then employed to generate low-pass and high-pass digital transfer functions with the desired cutoff frequencies and attenuation levels. The transformations to digital band-pass and band-stop transfer functions are handled in a similar manner. The transfer function development concludes with a general equation consisting of a constant times a ratio of products of second-order polynomials with real-valued coefficients.

The realization portion of the mathematical development begins with a brief review of the state-space concept. This is followed by procedures for applying the standard form equations in this study to second-order filter structures that employ state-space equations. Two of these structures have appeared only recently in the literature. The procedure used for selecting the appropriate filter structure for each second-order subsection of the transfer function is also relatively new and comes from the same source.

Numerical examples are provided to demonstrate the operation of the program. These examples are taken from the sources used in the mathematical development. Detailed comparisons are made between the results here and those of the original sources.