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Birkhäuser Boston eBooks

Uncertainty Principle and Wavelet Transforms

Alexander I. Saichev, Wojbor A. Woyczyński

Published Jan 1, 1997• 2 citations
Publisher

Abstract

The method of wavelet transforms , which provides a decomposition of functions in terms of a fixed family of functions of constant shape but varying scales and locations, recently acquired broad significance in the analysis of signals and of experimental data from various physical phenomena. It is clear that the potential of this method has not yet been fully tapped. Nevertheless, its value for the whole spectrum of problems in many areas of science and engineering, including the study of electromagnetic and turbulent hydrodynamic fields, image reconstruction algorithms, prediction of earthquakes and tsunami waves, and statistical analysis of economic data, is by now quite obvious. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-4158-4_7

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The method of wavelet transforms , which provides a decomposition of functions in terms of a fixed family of functions of constant shape but varying scales and locations, recently acquired broad significance in the analysis of signals and of experimental data from various physical phenomena. It is clear that the potential of this method has not yet been fully tapped. Nevertheless, its value for the whole spectrum of problems in many areas of science and engineering, including the study of electromagnetic and turbulent hydrodynamic fields, image reconstruction algorithms, prediction of earthquakes and tsunami waves, and statistical analysis of economic data, is by now quite obvious. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.
Fourier transform has been shown to be a powerful tool in many area of science. However, there is another class of unitary transforms, the wavelet transforms, which are as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and very useful for image processing and data compressiyon. In this paper, we construct quantum algorithms for Haar wavelet transforms and show its application in analyzing the multi-scale structure of the dynamical system by the Logistic Map (x! x(1 x)). 1
We present a general method for calculating relevant moments of the wavelet transform.Explicit results are given for the time, frequency, and scale moments.Using the results obtained, we show that the wavelet transform has unique characteristics that are not possessed by other methods.We discuss whether these unusual characteristics are physically or mathematically important.Exactly solvable examples are given, and the results are contrasted to those of the standard methods such as the spectrogram and the Wigner distribution.
A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation is insensitive to fine structures of the interference pattern generated by a Mach-Zehnder interferometer. Actually Niels Bohr’s indeterminacy principle cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relation of quantum electrodynamics. The geometric quantization approach makes the heuristic arguments concerning quantum parallelism rigorous by considering wave packets as symplectic spinors on the hologram plane. It includes the standard uncertainty inequality as a special case. As an application to the area of quantum computers, we study synchronized neural networks and their photonic implementations as suggested by recent experiments in brain research. As another result of the geometric quantization approach to holography, the uncertainty minimizing Gabor wavelets arise which provide useful wavelet expansions for image analysis, segmentation, and compression. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.