Search for chaotic output from a sigma-delta modulator: an archetypal retarded nonlinear system

David Thomas Hughes; S. McLaughlin; Gary Ushaw; Bernard Mulgrew; David S. Broomhead

doi:10.1117/12.162673

18 November 1993 Search for chaotic output from a sigma-delta modulator: an archetypal retarded nonlinear system

David Thomas Hughes, S. McLaughlin, Gary Ushaw, Bernard Mulgrew, David S. Broomhead

Proceedings Volume 2038, Chaos in Communications; (1993) https://doi.org/10.1117/12.162673
Event: SPIE's 1993 International Symposium on Optics, Imaging, and Instrumentation, 1993, San Diego, CA, United States

Abstract

One of the most common methods by which a non-linear physical system can be determined to be stable, globally unstable, or in fact chaotic is by an investigation of its Lyapunov exponents. When the system contains a delay equation the correct co-ordinate space for the system is infinite dimensional. In this case special techniques must be used to make the problem solvable by numerical methods. We have investigated a practical example of such a retarded non-linear system: the continuous Sigma-Delta modulator (CSDM). The CSDM is an adaptation of the widely used digital Sigma-Delta modulator in which, now, the input and output are continuous functions of time, the quantizer has been replaced with a sharp hyperbolic tangent function and the feedback delay time can be varied continuously. This allows the system dynamics to be modeled by delay ordinary-differential equations. We have simulated such a device for various feedback configurations and delay times and have been able to show how a full calculation of the Lyapunov exponent spectrum allows a detailed analysis of stability conditions for the CSDM. Furthermore, we discuss the implications of this approach for retarded non-linear systems in general and a range of signal processing applications in particular.

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David Thomas Hughes, S. McLaughlin, Gary Ushaw, Bernard Mulgrew, and David S. Broomhead "Search for chaotic output from a sigma-delta modulator: an archetypal retarded nonlinear system", Proc. SPIE 2038, Chaos in Communications, (18 November 1993); https://doi.org/10.1117/12.162673

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