The effects of quantization errors on rotated filters

Abstract

A digital filter that has been designed by rotation of the frequency response of a one-dimensional continuous filter, and then bilinearly transformed into a two-dimensional digital filter is called a rotated filter. Other useful filters such as circularly symmetric low-pass, high-pass, or bandpass filters can be obtained by parallel or cascade connection of rotated filters. These filters can be used in image processing and geophysics. Rotated filters are marginally stable if the rotation angle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta</tex> satisfies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">270°&lt;\beta&lt;360°</tex> , when designed from a stable one-dimensional continuous filter. A slight change in the coefficients has a significant effect on the stability of rotated filters. In this paper, the effects of quantization errors on the stability of rotated filters are investigated. A method to predict the stability state of rotated filters after coefficient quantization is given. A coefficient perturbation technique is discussed and used to stabilize a filter if found to be unstable. For real coefficients, a region of guaranteed stability is defined for some forms of fixed-point and floating-point arithmetic. The effects of coefficient quantization and coefficient perturbation on the frequency response also are discussed.