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ESTIMATION OF MEAN SQUARED ERRORS OF NON-BINARY A/D ENCODERS THROUGH FREDHOLM DETERMINANTS OF PIECEWISE-LINEAR TRANSFORMATIONS II: GENERAL CASE

Part of: Applications - Dynamical systems and ergodic theory Circuits, networks

Published online by Cambridge University Press:  19 March 2025

KATSUTOSHI SHINOHARA Open the ORCID record for KATSUTOSHI SHINOHARA [Opens in a new window]
KATSUTOSHI SHINOHARA*
Affiliation:
Hitotsubashi University, Graduate School of Commerce and Management, Naka 2-1, Kunitachi, Tokyo, Japan
*
[email protected]
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Abstract

We conduct a theoretical analysis of the performance of $\beta $-encoders. The $\beta $-encoders are A/D (analogue-to-digital) encoders, the design of which is based on the expansion of real numbers with noninteger radix. For the practical use of such encoders, it is important to have theoretical upper bounds of their errors. We investigate the generating function of the Perron–Frobenius operator of the corresponding one-dimensional map and deduce the invariant measure of it. Using this, we derive an approximate value of the upper bound of the mean squared error of the quantization process of such encoders. We also discuss the results from a numerical viewpoint.

Keywords

β-expansionPerron–Frobenius operatoranalogue-to-digital converter

MSC classification

Primary: 94C05: Analytic circuit theory
Secondary: 37N99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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