β-expansion attractors observed in A/D converters

Area of Science:

• * Digital signal processing
• * Dynamical systems theory
• * Information theory

Background:

• * β-encoders, a type of analog-to-digital converter, utilize amplifiers with factor β and quantizers with threshold ν.
• * Their behavior is mathematically described by multi-valued Rényi-Parry maps, which are topologically conjugate to Parry's (β,α)-maps.

Purpose of the Study:

• * To analyze the embedded attractors within the dynamics of β-encoders.
• * To identify attractors suitable for applications in spread-spectrum coding.
• * To explore the use of these attractors in optimization techniques leveraging pseudo-random numbers.

Main Methods:

• * Analysis of deterministic dynamics of multi-valued Rényi-Parry maps.
• * Topological conjugacy between multi-valued Rényi-Parry maps and Parry's (β,α)-maps.
• * Observation of attractors within the [ν-1, ν) interval.

Main Results:

• * β-encoders exhibit a closed subinterval [ν-1, ν) containing a deterministic attractor.
• * Quantization errors in β-encoders display chaotic behavior, not converging to a fixed point.
• * The identified β-expansion attractors are simpler than previously documented ones.

Conclusions:

• * The deterministic dynamics of multi-valued Rényi-Parry maps provide a robust explanation for β-encoder behavior.
• * The discovered simpler attractors offer potential for advanced applications in signal processing and random number generation.
• * Further research into these attractors can advance spread-spectrum techniques and optimization algorithms.