Estimation of Chaos Function for the Implementation of High-Resolution Measurement System

The use of chaotic maps as measurement system or as a signal quantisation unit in analogue to digital converters (ADC) is a fairly modern approach. Compared to existing ADC architectures, chaotic systems are advantageous because these are simple mathematical functions and can be implemented physically involving less components. Additionally, unique symbolic identities corresponding to an input value (initial condition) can be generated iteratively through the dynamics, thus simplifying the design complexities.

For the application of signal measurement system, the chaotic function tent map (TM) is found to be the suitable candidate, as dense distribution of points within the state-space can be realised from the dynamics. However,a significant issue that may arise while implementing the TM electronically is that, it is difficult to maintain the parameter of the map at the ideal value due to component imprecisions. When the map parameter is reduced, the dynamics is distorted from the ideal behaviour; hence estimating the initial condition from the symbols become difficult. If the knowledge of the non-ideal parameter is available, then the actual initial condition can be recovered. Hence, it becomes essential to determine the non-ideal parameter from the available dynamics.

In this work, two different approaches have been proposed for the parameter estimation of the TM. The first approach is realised from the symbolic dynamics of the TM in which the sequence corresponding to the map maximum is searched over a symbolic time series, and a difference equation is realised in terms of the map parameter. The second method is based on the identification of the map fixed-point through the noisy dynamics of the TM. It has been discovered that unique crossovers appear within the noisy samples and the information of the map fixed-point is preserved through those crossovers. The proposed methods have been broadly analysed through mathematical simulations and graphical results. The approaches deliver parameter estimates with errors below 1% and using short length trajectories as low as 200 iterations. This development can benefit accurate estimation of initial condition from the non-ideal dynamics and therefore maybe considered as a step forward in the development of chaos-based measurement systems and chaotic ADCs. The study and the proposed estimation methods can also be utilised in other areas of applications such as communication and encryption, where parameter estimation of the chaotic functions is one of the prime requirements.