The Problem of Aliasing and Folding Effects in Spectrum of Sampled Signals in View of Information Theory

Authors

Abstract

In this paper, the problem of aliasing and folding effects in spectrum of sampled signals in view of Information Theory is discussed. To this end, the information content of deterministic continuous time signals, which are continuous functions, is formulated first. Then, this notion is extended to the sampled versions of these signals. In connection with it, new signal objects that are partly functions but partly not are introduced. It is shown that they allow to interpret correctly what the Whittaker–Shannon reconstruction formula in fact does. With help of this tool, the spectrum of the sampled signal is correctly calculated. The result achieved demonstrates that no aliasing and folding effects occur in the latter. Finally, it is shown that a Banach–Tarski-like paradox can be observed on the occasion of signal sampling.

References

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing. New Jersey: Prentice Hall, 1998.

R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory, New York: Springer-Verlag, 1991.

R. N. Bracewell, The Fourier Transform and Its Applications, New York: McGraw-Hill, 2000.

A. Borys, “Further discussion on modeling of measuring process via sampling of signals,” Intl Journal of Electronics and Telecommu-nications, vol. 66, no. 3, pp. 507-513, 2020.

A. Borys, “Spectrum aliasing does not occur in case of ideal signal sampling,” Intl Journal of Electronics and Telecommunications, vol. 67, no. 1, pp. 71-77, 2021.

A. Borys, “Extended definitions of spectrum of a sampled signal,” Intl Journal of Electronics and Telecommunications, vol. 67, no. 3, pp. 395-401, 2021.

R. F. Hoskins, Delta Functions: Introduction to Generalised Functions. Oxford: Woodhead Publishing, 2010.

A. Borys, “Some topological aspects of sampling theorem and reconstruction formula,” Intl Journal of Electronics and Telecommu-nications, vol. 66, no. 2, pp. 301-307, 2020.

A. Borys, “Measuring process via sampling of signals, and functions with attributes,” Intl Journal of Electronics and Telecommunications, vol. 66, no. 2, pp. 309-314, 2020.

A. Borys, “Filtering property of signal sampling in general and under-sampling as a specific operation of filtering connected with signal shaping at the same time,” Intl Journal of Electronics and Telecommu-nications, vol. 66, no. 3, pp. 589-594, 2020.

G. Tomkowicz, S. Wagon, The Banach-Tarski Paradox. Cambridge: Cambridge University Press, 2019.

Downloads

Published

2024-04-19

Issue

Section

Signals, Circuits, Systems