Analog to digital conversion in physical measurements

There exist measuring devices where an analog input is converted into a digital output. Such converters can have a nonlinear internal dynamics. We show how measurements with such converting devices can be understood using concepts from symbolic dynamics. Our approach is based on a nonlinear one-to-one mapping between the analog input and the digital output of the device. We analyze the Bernoulli shift and the tent map which are realized in speci®c analog/digital (A/D) converters. Furthermore, we discuss the sources of errors that are inevitable in physical realizations of such systems and suggest methods for error reduction. Ó 2000 Elsevier Science Ltd. All rights reserved. Progress in physics is based on the intimate relationship between experiment and theory. Experiments involve measurements in which the precision of the measured quantities depends essentially on the particular characteristics of the measuring devices. Often this process of measurement involves a conversion of the analog measured quantity into a digital number by means of a converter. For such analog/digital (A/D) converters, there exists an internal relationship between the quantity to be measured (analog input), and the result of the measurement process (digital output). We assume that the conversion of the analog signal into a digital number is the sole process in such converters and we neglect errors associated with the input analog signal itself. Thus, the dynamics of the input±output relationship determines the whole conversion process and it is responsible for the precision of the conversion itself. This problem is especially relevant for converters based on nonlinear processes, for which it is known that the errors grow exponentially fast. In this paper, we analyze the properties of a such relationship in order to identify and understand the sources of possible errors involved in the conversion process. As a paradigm of a device having a nonlinear internal dynamics, we consider the algorithmic A/D converters [1±5]. The internal dynamics of such converters is described by well-known one-dimensional chaotic maps [6], the Bernoulli shift xn‡1 ˆ G1…xn† ˆ 2xn jmod 1 …1† and the tent map www.elsevier.nl/locate/chaos Chaos, Solitons and Fractals 11 (2000) 1247±1251 * Corresponding author. Technical University of Lodz, Division of Dynamics, Stefanowskiego 1/15, 90-924 Lodz, Poland. Tel.: +4842-332231; fax: +4842-365646. E-mail address: [email protected] (T. Kapitaniak). 0960-0779/00/$ see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 0 7 7 9 ( 9 9 ) 0 0 0 0 3 X xn‡1 ˆ G2…xn† ˆ 1ÿ 2 xn ÿ 12 : …2† Both maps are characterized by a positive Lyapunov exponent k ˆ ln 2 and they often serve as model examples in mathematical work devoted to the theory of dynamical systems (see e.g. [7]). The Bernoulli shift is realized by the circuit shown in Fig. 1(a) [6]. This circuit represents a K-bit algorithmic A/D converter, which is recycling multistage converter with one bit stage. The converter achieves K-bit resolution by converting the input once and then circulating the residue function G1…:† through the state …K ÿ 1† times. The tent map is realized as a Grey-code algorithmic A/D converter [6] with a folding circuit having a transfer characteristic given by Eq. (2) as shown in Fig. 1(b). To analyze the input±output relationship of such A/D converters from the point of view of the theory of dynamical systems, we apply concepts from symbolic dynamics. In analyzing the conversion process, we use the sensitive dependence on the initial conditions, a fundamental property of Eqs.(1) and (2), to argue that precise A/D conversion is possible. We identify the sources of errors in such converters and present a simple method of error reduction. Suppose we need to measure the quantity X, and let v denote the set of all possible values of X. Let us consider an unknown value x1 of X , which is the analog input of the converter, as an initial condition that is iterated internally using the chaotic map G generating the sequence fxkg ˆ x1x2; . . . ; xL with xk ˆ Gkÿ1…x1† and k ˆ 1; 2; . . . ; L. To describe such an orbit in terms of a symbolic sequence, we need to choose a partition by coarse-graining the space v. In the case corresponding to the maps (1) and (2), we split the space v ˆ ‰0; 1Š into two cells whose partition is the critical point xc ˆ 1=2. With any given orbit fxkg one can associate its symbol sequence fakg using a two-letter alphabet de®ned as ak ˆ 0 for xk 6 xc and ak ˆ 1 for xk > xc. The sequence fakg is called the itinerary of x1. Let O be a set of all possible symbolic sequences. The process of conversion is pregnant with meaning if there exists a one-to-one mapping between v and O. Once the itinerary characterizing the orbit is obtained. We get a result in form of a digital output