Beta encoders: Symbolic dynamics and electronic implementation

Tohru Kohda, Yoshihiko Horio, Yoichiro Takahashi, Kazuyuki Aihara

Research output: Contribution to journalReview articlepeer-review

6 Citations (Scopus)


A new class of analog-to-digital (A/D) and digital-to-analog (D/A) converters that uses a flaky quantizer, known as a β-encoder, has been shown to have exponential bit rate accuracy and a self-correcting property for fluctuations of the amplifier factor β and quantizer threshold ν. The probabilistic behavior of this flaky quantizer is explained by the deterministic dynamics of a multivalued RényiParry map on the middle interval, as defined here. This map is eventually locally onto map of [ν - 1, ν], which is topologically conjugate to Parry's (β, α)-map with α = (β - 1)(ν - 1). This viewpoint allows us to obtain a decoded sample, which is equal to the midpoint of the subinterval, and its associated characteristic equation for recovering β, which improves the quantization error by more than 3 dB when β > 1.5. The invariant subinterval under the RényiParry map shows that ν should be set to around the midpoint of its associated greedy and lazy values. Furthermore, a new A/D converter referred to as the negative β-encoder is introduced, and shown to further improve the quantization error of the β-encoder. Then, a switched-capacitor (SC) electronic circuit technique is proposed for implementing A/D converter circuits based on several types of β-encoders. Electric circuit experiments were used to verify the validity of these circuits against deviations and mismatches of the circuit parameters. Finally, we demonstrate that chaotic attractors can be observed experimentally from these β-encoder circuits.

Original languageEnglish
Article number1230031
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Issue number9
Publication statusPublished - 2012 Sept


  • Analog-to-digital (A/D)conversion
  • Chaotic dynamics
  • Discrete-time integrator
  • Ergodic theory
  • Flaky quantizer
  • Number theory
  • Switched-capacitor circuit
  • β-encoder
  • β-expansion
  • β-shift
  • β-transformation


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