Modern mathematical tools for analysing nonlinear e ects in digital

| A number of mathematical tools are used for analysing nonlinear over-ow eeects in digital lters. In our investigations we found if useful to use the probabilis-tic approach and the circle-map theory in the analysis of complex behaviour in second-order digital lter. In the rst part we introduce some deenition (rotation number, nonwander-ing and chain recurrent points, mixing, exact-ness) and present several theorems. In the second part we describe how to apply the theories described to the dynamical systems deened by digital lters. 1 Mathematical tools 1.1 Circle-map theory Let us recall some important deenitions and theorems from the circle-map theory. Let f : S 1 7 ?! S 1 be a continuous map of S 1 to itself. The map : IR 7 ?! S 1 deened by (t) = (cos(2t); sin(2t)) is continuous and onto. Let F : IR 7 ?!: IR be a lift of f, i.e. F is continuous, F = f , and for each x 2 IR, F(x + 1) = F(x) + k, where k is an integer constant. The integer k is unique for a given continuous map f and is called the degree of f. In this paper we are interested in degree-one maps only. An important concept in the study of degree-one maps is the rotation number. Let f be a degree-one map, and let F be a lift of f. If x 2 IR, then the rotation number of x under F is deened by: We say that a map of a circle is non-decreasing if its lift is non-decreasing.