Estimation of Input Variable as Initial Condition of a Chaos Based Analogue to Digital Converter

A realization of an analogue-to-digital converter(ADC) with improved conversion accuracy,using the chaotic behaviour of the tent map,is presented. In this approach, the analogue input signal to be measured, termed as the initial condition is applied to a chaotic map, and the symbolic dynamics resulting from the map evolution, is used to determine the initial condition in digital form. The unimodal piecewise linear tent map (TM) has been used for this purpose, because of its property of generating uniform distribution of points and robust chaos.

Through electronic implementation of the TMit is practically impossible to produce an ‘ideal’ TM behaviour with parameter values in the full range [0,1]. Due to component imprecision and various other factors, a non-ideal map with reduced height is observed. For such a map, converting the equivalent symbolic trajectory generated by TM iterations return erroneous results as the partitioning of the phase space embodied in the finite symbolic dynamics no longer has unique correspondence with the initial condition.

Two algorithmic solutions have been proposed to minimise the errors associated with a practical system. For one, it has been established that for a reduced-height map the partitioning will not remain of equal size. Considering that the height of the tent map used for this purpose is known from an independent but related research, a technique of partitioning the state space unevenly, depending on the map height has been proposed and has been shown that if the correct partitioning is used, the resulting symbolic dynamics again map uniquely to the initial condition.

Alternatively, it has been shown that the degree of deviation of the iterate values can be determined based on the parameter value, which in turn can be adjusted for depending on the symbolic sequence generated by the initial condition to determine the correct decimal equivalent values.

The both the approaches proved to be highly effective in obtaining a digital outcome corresponding to the initial condition using 8 symbolic iterations of the map in hardware domain, with the second approach outperforming the first in terms of accuracy, while the first method can easily be pipelined alongside generating the iterates and thus improve the speed. This development is promising because, in contrast to the commercially available ADCs, it places lower demand on the hardware resource and can be effectively implemented to give a real-time operation.