Quantized linear systems on integer lattices: a frequency-based approach

The roundoff errors in computer simulations of continuous dynamical systems, which are caused by finiteness of machine arithmetic, can lead to qualitative discrepancies between phase portraits of the resulting spatially discretized systems and the original systems. These effects can be modelled on a multidimensional integer lattice by using a dynamical system obtained by composing the transition operator of the original system with a quantizer representing the computer discretization procedure. Such model systems manifest pseudorandomness which can be studied using a rigorous probability theoretic approach. To this end, the lattice Zn is endowed with a class of frequency measurable subsets and a spatial frequency functional as a finitely additive probability measure describing the limit fractions of such sets in large rectangular fragments of the lattice. Using a multivariate version of Weyl’s equidistribution criterion and a related nonresonance condition, we introduce an algebra of frequency measurable quasiperiodic subsets of the lattice. The frequencybased approach is applied to quantized linear systems with the transition operator R ◦ L, where L is a nonsingular matrix of the original linear system in Rn, and R : Rn → Zn is a quantizer (in an idealized fixed-point arithmetic with no overflow) which commutes with the additive group of translations of the lattice. It is shown that, for almost every L, the events associated with the deviation of trajectories of the quantized and original systems are frequency measurable quasiperiodic subsets of the lattice whose frequencies are amenable to computation involving geometric probabilities on finite-dimensional tori. Using the skew products of measure preserving toral automorphisms, we prove mutual independence and uniform distribution of the quantization errors and investigate statistical properties of invertibility loss for the quantized linear system, thus extending the results obtained by V.V.Voevodin in the 1960s. In the case when L is similar to an orthogonal matrix, we establish a functional central limit theorem for the deviations of trajectories of the quantized and original systems. As an example, these results are demonstrated for rounded-off planar rotations. This work is a slightly edited version of two research reports: I.Vladimirov, “Quantized linear systems on integer lattices: frequency-based approach”, Parts I, II, Centre for Applied Dynamical Systems and Environmental Modelling, CADSEM Reports 96–032, 96–033, October 1996, Deakin University, Geelong, Victoria, Australia, which were issued while the author was with the Institute for Information Transmission Problems, the Russian Academy of Sciences, Moscow, 127994, GSP–4, Bolshoi Karetny Lane 19. None of the original results have been removed, nor have new results been added in the present version except for a numerical example on p. 58 of the last section. Present address: UNSW Canberra, ACT 2600, Australia. E-mail: [email protected].