The Topography of a Third Order IIR Digital Filter Zeros and Poles in the z-Plane Discretized due to the Quantization of the Direct Form Coefficients

Abstract

Earlier it was found that the zeros and poles of IIR digital filters with finite word length are elements of the set of algebraic numbers. Therefore, not every point of the unit circle of the z-plane can be a zero and / or a pole of such digital filters. The position of admissible positions for zeros and poles depends on the degree of algebraic numbers and the length of the fractional part of the coefficients of the equivalent canonical structure of the corresponding order. The formation of the corresponding configurations is considered as a $\boldsymbol{z}$-plane discretization due to the quantization of the filter coefficients. The z-plane discretization for second-order filters has been well studied. The geometric locus of the corresponding algebraic numbers is a system of concentric circles, that is, plane algebraic curves of the second degree. This paper presents the results of studying the geometrical place of the third order IIR filter zeros and poles.