Synthesis of 2-D state-space fixed-point digital-filter structures with minimum roundoff noise

Abstract

Based on a roundoff-noise analysis, a general synthesis procedure is developed which leads to an optimal local state-space 2-D digital-filter realization that minimizes the output-noise power due to roundoff subject to a scaling condition on the state variables. The outputnoise power and the signal scaling condition are closely related to two positive-definite matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> . These matrices provide two sets of invariants, called the 2-D second-order modes of the filter, which play a crucial role in the minimization of the output-noise power. With the availability of matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> , the 2-D similarity transformation that yields an optimal state-space realization can be obtained by solving separately two 1-D optimization problems so that the well-developed techniques for minimizing roundoff noise in 1-D state-space digital filters can also be used for minimizing roundoff noise in 2-D state-space digital filters.