The Expanded State Model for Digital Filter Representation: Theory and Applications

The common mathematical relationships that describe digital filters are particularly useful in investigating the filter properties if implemented in (of course, non-obtainable) infinite precision. In this case, any computational algorithms that satisfy the mathematical relationships (e.g., transfer functions, state equations) would produce satisfactory (and easily predictable) results. In the actual case of limited precision arithmetic, the specific computational procedure used is critical in its effects on roundoff errors, computational complexity, etc. In earlier works, a number of equation-based and schematic models have been presented for the representation of digital filters. It has been found that either the techniques do not qualify as satisfactory models or are very cumbersome and lacking in intuitive appeal. This paper presents a new model (the expanded state model ) for obtaining equations describing a digital filter given a block diagram, signal flow graph, transfer function or state equations. The model uniquely describes the diagram and is thus particularly useful in obtaining exact filter simulations. Transformations to and from the standard state variable representation are quite straightforward and thus allow convenient application of familiar stability, controllability and observability results. Also implied is an organized method for generation of a suitable model which satisfies a given transfer function or set of state equations. The model equations are simple to write by inspection, and transformations between the equations and the block diagram are readily accomplished. The form of the model allows insight into and straightforward solution of the problems of computability, solvability and the arrangement of network equations in computable form. Equivalent graphical techniques which allow unique insight into these problems are also presented. The particular way in which the model equations are defined and written allows the inclusion of error terms at the exact location of each mathematical operation. This leads to new, precise results to the problem of determining the noise variance for a general digital filter, network, system or computational algorithm due to errors generated in performing mathematical computations in the filter. Other results of importance allow a convenient computation of output sensitivity and eigenvalue sensitivity to inaccuracies in realizing multiplier coefficients. All results depend on the model matrices and require no analysis by the designer. They are thus perfectly suited to computer implementation and can be performed in a computer with no user derivation of any network properties (such as impulse responses, etc). All information required for the various computations to proceed is contained in the model matrices. Computer algorithms, which perform the trans formations from the expanded state model to state variable form and vice-versa, determine computability and computable form, and calculate normalized noise variance for a general digital filter, are included in an appendix. Throughout the development, suitable examples emphasize the value, importance and simplicity of the model and analysis techniques