β -expansion attractors observed in A/D converters

• Published in: Chaos (Woodbury, N.Y.) N.Y.), 2012-12, Vol.22 (4), p.047512-047512
• Main Authors: Kohda, Tohru, Horio, Yoshihiko, Aihara, Kazuyuki
• Format: Article
• Language: English
• ISSN: 1054-1500; 1089-7682
• DOI: 10.1063/1.4758813

The recently proposed β -encoders, analog-to-digital converters using an amplifier with a factor β and a flaky quantizer with threshold ν , have proven to be explained by the deterministic dynamics of multi-valued Rényi-Parry maps. Such a map is locally eventually onto [ ν − 1 ,   ν ) , which is topologically conjugate to Parry's ( β , α ) -map with α = ( β − 1 ) ( ν − 1 ) . This implies that β -encoders have a closed subinterval [ ν − 1 , ν ) , which includes an attractor. Thus, the iteration of the multi-valued Rényi-Parry map performs the β -expansion of x while quantization errors in β -encoders behave chaotically and do not converge to a fixed point. This β -expansion attractor is relatively simpler than previously reported attractors. The object of this paper is twofold: to observe the embedded attractors in the β -encoder and to identify attractors that are useful for spread-spectrum codes and optimization techniques using pseudo-random numbers.