Dynamical behaviour of digital filters subject to 2's complement arithmetic nonlinearity

The thesis investigates a two-dimensional nonlinear map describing a second-order direct form digital filter constrained to operate with 2’s complement arithmetic. Attention is primarily directed towards the lossless case in which the filter parameters are on the lower boundary of the linear stability region. An effective tool for studying the dynamics of the map is the association of an infinite symbolic sequence with each orbit. Identification of the admissible periodic sequences is a fundamental theme. A comprehensive answer is given for when string fragments +0... 0+, and related patterns, appear within some admissible sequence. Infinite families of admissible periodic sequences are exhibited as analytic solutions to the trigonometric formulation of Chua and Lin’s admissibility criterion. The role of the filter parameter a = 2cos θ in determining admissibility is emphasised. Examples demonstrate that a periodic sequence may be admissible over several disjoint parameter intervals. \ud Strategies are developed that permit powerful computer searches for admissible periodic sequences. Tables are included listing the periodic sequences admissible for fixed values of the parameter, up to period 200. Parameter values for the tabulation progress by steps of 0.05 across the entire range (0,π). A complete list, covering all parameter values, of the admissible periodic sequences up to period 20 is presented; associated with each sequence is its interval of admissibility. A classification is undertaken of all periodic sequences with an interval of admissibility having left end-point θ = 0. For odd period sequences it is definitive; in the even case a partial classification is achieved, definitive when the period is twice a prime. An investigation of comparable scope is presented for admissibility around θ = π/2 The topological dynamics of a one-dimensional reduction of the filter map is explored. Orbits are attracted to an invariant set, itself a disjoint union of intervals, that becomes increasingly fragmented as the parameter a approaches 1