Quantum-Mechanical Signal Processing and Spectral Analysis

by Dzevad Belkic

Overview

Quantum-Mechanical Signal Processing and Spectral Analysis describes the novel application of quantum mechanical methods to signal processing across a range of interdisciplinary research fields. Conventionally, signal processing is viewed as an engineering discipline with its own specific scope, methods, concerns and priorities, not usually encompassing quantum mechanics. However, the dynamics of systems that generate time signals can be successfully described by the general principles and methods of quantum physics, especially within the Schrödinger framework. Most time signals that are measured experimentally are mathematically equivalent to quantum-mechanical auto-correlation functions built from the evolution operator and wavefunctions. This fact allows us to apply the rich conceptual strategies and mathematical apparatus of quantum mechanics to signal processing.

Among the leading quantum-mechanical signal processing methods, this book emphasizes the role of Pade approximant and the Lanczos algorithm, highlighting the major benefits of their combination. These two methods are carefully incorporated within a unified framework of scattering and spectroscopy, developing an algorithmic power that can be exported to other disciplines. The novelty of the author's approach to key signal processing problems, the harmonic inversion and the moment problem, is in establishing the Pade approximant and Lanczos algorithm as entirely algerbraic spectral estimators. This is of paramount theoretical and practical importance, as now spectral analysis can be carried out from closed analytical expressions. This overrides the notorious mathematical ill-conditioning problems with round-off errors that plague inverse reconstructions in those fields that rely upon signal processing.

Quantum-Mechanical Signal Processing and Spectral Analysis will be an invaluable resource for researchers involved in signal processing across a wide range of disciplines.

Product Details

ISBN-13:	9780750310192
Publisher:	Taylor & Francis
Publication date:	12/31/2004
Series:	Series in Atomic Molecular Physics
Pages:	478
Product dimensions:	6.12(w) x 9.19(h) x (d)

Table of Contents

Preface, General Introduction, Direct Link of Spectral Analysis with Eigenvalue Problems in Quantum Physics, Ubiquitous Padé Approximant across Quantum Physics and Signal processing, Versatile Lanczos Methods for Nearest Neighbour Interactions, Synergistic Combination of the Padé and Lanczos Methodologies, First Exact analytical Padé Method and continued Fractions, Algorithmic Strengths of Methods of Moments in Physics and Mathematics, Proof-of-Principle Illustrations of Quantum-mechanical Signal Processing, General Conclusions,Outlook for Quantum-mechanical Processing of generic time signals, Appendices, References

Editorial Reviews

"The special importance of this book, in my opinion, is that it links different investigations from various branches of science: physics, mathematics, chemistry, medicine, and numerical analysis. Moreover, the book also addresses similar problems from engineering. It is clearly shown that a number of phenomena that seemed not to be directly connected, actually are, and this realization is expected to spur progress in many areas. Most importantly, the applicability and superiority of the presented spectral methods are demonstrated. The book is superbly written, and it is self-contained, so that it can be treated as a fundamental source of information for researchers and students in a wide range of fields. Therefore, I strongly recommend this book to colleagues and students."
-Professor Ivan Mancev

"The present book concerns one of the most beautiful and useful developments in mathematics. It takes the reader from the classical moment problem of Stieltjes via Pade's rational representations, using Hankel determinants and Frobenius normal forms, through the Krylov-Lanczos development to modern algorithmic techniques and codes. The deeper algebraic connections between signal processing and quantum mechanics are implicit and leaven important concrete applications typical of the interdisciplinary faculties of physics and medicine."
-Professor Erkki Brandas

From the Publisher

Reviewer: Andy (Yuan-Guang) Xu, PhD (Columbia University Medical Center)
Description: This book presents the application of a mathematical model, the Padé approximation in quantum mechanics and spectral analysis. It describes the development of the fast Padé transformation method for signal processing, based on the Padé-Lanczos approximation from the theories of quantum mechanical scattering and spectroscopy. The eigenvalue problem in quantum mechanics and the inverse reconstruction problem in spectral analysis can both be tackled using the same mathematical formulation.
Purpose: The major purpose is to outline the mathematical framework of a robust spectral analysis algorithm that can be used to replace or supplement the Fast-Fourier transformation method, which has been widely used in many commercial signal-processing software packages. As was observed in magnetic resonance imaging (MRI) and single photon emission computerized tomography (SPECT) studies, the use of the Fast-Fourier transformation for signal processing is usually accompanied by long data acquisition time, insufficient spatial resolution, and/or spurious spectrum distribution. These issues could cause considerable inconvenience to medical imaging in routine clinical practice.
Audience: The method described in this book opens an avenue for implementation of accurate and stable signal processing software that might be of great interest to the medical imaging community.
Features: The 77 short chapters cover problems associated with spectral analysis using the Fast-Fourier transformation; the unified-theory for quantum mechanical scattering and spectroscopy based on the Padé-Lanczos approximation; the Padé-Lanczos approximation as applied to spectral analysis, i.e., the fast Padé transformation; techniques for improving the accuracy and stability of the Fast Padé approximation: the constrained root reflection technique, the Padé-Schur approximant; methods for solving the eigenvalue problem or the harmonic inversion problem following the Padé-Lanczos approximation; and illustrates the application of the fast Padé transformation to MRI.
Assessment: The book is well written and well organized. It is a useful mathematical tool for those who are involved in theoretical research on image reconstruction and quantum scattering and spectroscopy. The book is expected to be less appealing to medical physics researchers due to its highly mathematical and theoretical nature.

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