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Chaotic behaviours of stable second‐order digital filters with two's complement arithmetic
Author(s) -
Ling Bingo WingKuen,
Hung WaiFung,
Tam Peter KwongShun
Publication year - 2003
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.243
Subject(s) - mathematics , complement (music) , ellipse , plane (geometry) , fixed point , aperiodic graph , fractal , phase plane , fixed point arithmetic , unit circle , chaotic , set (abstract data type) , mathematical analysis , arithmetic , combinatorics , geometry , computer science , physics , biochemistry , chemistry , nonlinear system , quantum mechanics , complementation , artificial intelligence , programming language , gene , phenotype
In this paper, the behaviours of stable second‐order digital filters with two's complement arithmetic are investigated. It is found that even though the poles are inside the unit circle and the trajectory converges to a fixed point on the phase plane, that fixed point is not necessarily the origin. That fixed point is found and the set of initial conditions corresponding to such trajectories is determined. This set of initial conditions is a set of polygons inside the unit square, whereas it is an ellipse for the marginally stable case. Also, it is found that the occurrence of limit cycles and chaotic fractal pattern on the phase plane can be characterized by the periodic and aperiodic behaviours of the symbolic sequences, respectively. The fractal pattern is polygonal, whereas it is elliptical for the marginally stable case. Copyright © 2003 John Wiley & Sons, Ltd.