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Discontinuous piecewise‐linear discrete‐time dynamics – maps with gaps in electronic systems

Orla Feely (School of Electrical, Electronic and Mechanical Engineering, University College Dublin, Dublin, Ireland)



Many important electronic systems are modelled by discrete‐time equations with nonlinearities that are discontinuous and piecewise‐linear, often arising as a result of quantization. Approximations based on linearization – the standard engineering response to nonlinearity – are often quite unhelpful in these systems, because of the form of the nonlinearity. Certain methods and results have been developed over a number of years for the analysis of discontinuous piecewise‐linear discrete‐time dynamics. The aim of this tutorial paper is to review that body of knowledge, and to show how it can be applied to representative electronic systems.


The paper uses an important electronic circuit – the ΣΔ modulator – as a central example, and considers the dynamical behaviour exhibited by this circuit and related circuits.


The circuits under investigation exhibit complex forms of behaviour that can be explained by the application of methods of nonlinear discrete‐time dynamics.


This paper is intended to provide a brief introduction to the body of research that exists into the behaviour of nonlinear discrete‐time circuits and systems with discontinuous piecewise‐linear nonlinearities.



Feely, O. (2011), "Discontinuous piecewise‐linear discrete‐time dynamics – maps with gaps in electronic systems", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 4, pp. 1296-1306.



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Copyright © 2011, Emerald Group Publishing Limited

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