Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-13T09:08:47.341Z Has data issue: false hasContentIssue false
  • English
  • Français

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 Special Collection in honour of Professor Bernd Krauskopf

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
*
ka.shinohara@r.hit-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e14
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Daubechies, I., DeVore, R., Gunturk, C. S. and Vaishampayan, V. A., “Beta expansions: a new approach to digitally corrected A/D conversion”, IEEE 2002 Int. Sympo. Circ. and Syst. ISCAS (2002) II; doi:10.1109/ISCAS.2002.1011470.CrossRefGoogle Scholar
Daubechies, I. and Yilmaz, O., “Robust and practical analog-to-digital conversion with exponential precision”, IEEE Trans. Inform. Theory 52 (2006) 3533–3545; doi:10.1109/TIT.2006.878220.CrossRefGoogle Scholar
Kohda, T., Horio, Y., Takahashi, Y. and Aihara, K., “Beta encoders: symbolic dynamics and electronic implementation”, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012) Article ID: 1230031; doi:10.1142/S0218127412300315.CrossRefGoogle Scholar
Makino, T., Iwata, Y., Shinohara, K., Jitsumatsu, Y., Hotta, M., San, H. and Aihara, K., “Rigorous estimates of quantization error for A/D converters based on Beta-map”, NOLTA 6 (2015) 99–111; doi:10.1587/nolta.6.99.CrossRefGoogle Scholar
Mori, M., “Fredholm determinant for piecewise linear transformations”, Osaka J. Math. 27 (1990) 81–116; doi:10.18910/10418.Google Scholar
San, H., Kato, T., Maruyama, T., Aihara, K. and Hotta, M., “Non-binary pipeline analog-to-digital converter based on $\beta$ -expansion”, IEICE Trans. Fundam. Electron Commun. Comput. Sci. 96 (2013) 415–421; doi:10.1587/transfun.E96.A.415.CrossRefGoogle Scholar
Shinohara, K. and Kobayashi, K., “Estimation of mean squared errors of non-binary A/D encoders through Fredholm determinants of piecewise-linear transformations”, NOLTA 9 (2018) 243–258; doi:10.1587/nolta.9.243.CrossRefGoogle Scholar
Suzuki, R., Maruyama, T., San, H., Aihara, K. and Hotta, M., “Robust cyclic ADC architecture based on $\beta$ -expansion”, IEICE Trans. Electron. 96 (2013) 553–559; doi:10.1587/transele.E96.C.553.CrossRefGoogle Scholar