On quantization effects in state-variable filter implementations

Studying the effects of roundoff errors in digital filters requires specialized study of each implementation of each of various filter structures. Even the study of these special structures, however, is fraught with difficulties; different implementations of the same structure can have different roundoff behavior, because rounding is done at different points in the structure. The limitations of practical VLSI architectures suggest two models of computation that accurately reflect the vast majority of filter implementations. Such implementations can be accurately described in a factored state variable form that represents the actual computations in the implementations. The quantization noise behavior of different filter structures can be studied under this unified framework. Necessary and sufficient conditions for the optimality of filter realizations expressed in the factored state variable form are derived, as a simple extension of important earlier work with the usual state variable form.