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.
Table of Contents
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
"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