A New Z Domain Continued Fraction Expansion

Abstract

A new <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</tex> domain continued fraction expansion is presented which proceeds in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z - 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 - z^{-1}</tex> factors. It is proved to be always convergent for polynomials whose roots all lie within the unit circle. The procedure involves a unique decomposition of the given polynomial into a mirror image polynomial (MIP) and an antimirror image polynomial (AMIP) whose degrees differ by unity. The ratio of these polynomials is shown to possess the properties of a digital reactance function. The continued fraction expansion developed here-due to the fact that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z - 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 - z^{-1}</tex> are the inverse transmittances of digital accumulators, as well as of switched capacitor integrators-has application to the synthesis of digital and switched capacitor ladder filters.