ABSTRACT

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 Schroedinger 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.

chapter Chapter 1|19 pages

Introduction

chapter Chapter 2|3 pages

Auto-correlation functions

chapter Chapter 3|4 pages

The time-independent Schrödinger equation

chapter Chapter 4|4 pages

The time-dependent Schrödinger equation

chapter Chapter 7|12 pages

Difference equations and the harmonic inversion

chapter Chapter 9|7 pages

Dimensionality reduction in the frequency domain

chapter Chapter 10|6 pages

Dimensionality reduction in the time domain

chapter Chapter 11|8 pages

The basic features of the Padé approximant (PA)

chapter Chapter 12|5 pages

Gaussian quadratures and the Padé approximant

chapter Chapter 13|16 pages

Padé–Schur approximant (PSA) with no spurious roots

chapter Chapter 15|16 pages

Exact iterative solution of a system of linear equations

chapter Chapter 16|2 pages

Relaxation methods and sequence accelerations

chapter Chapter 17|7 pages

Quantification: harmonic inversion in the time domain

chapter Chapter 20|2 pages

The base-transient concept in signal processing

chapter Chapter 23|6 pages

The Lanczos orthogonal polynomials Pn (u) and Qn (u)

chapter Chapter 24|2 pages

Recursions for derivatives of the Lanczos polynomials

chapter Chapter 25|4 pages

The secular equation and the characteristic polynomial

chapter Chapter 27|2 pages

The Wronskian for the Lanczos polynomials

chapter Chapter 28|3 pages

Finding accurate zeros of high-degree polynomials

chapter Chapter 29|5 pages

Recursions for sums involving Lanczos polynomials

chapter Chapter 31|6 pages

Completeness proof for the Lanczos polynomials

chapter Chapter 32|2 pages

Duality: the states | ψn ) and polynomials Qn (u)

chapter Chapter 34|3 pages

The explicit Lanczos algorithm

chapter Chapter 35|3 pages

The explicit Lanczos wave packet propagation

chapter Chapter 36|5 pages

The Padé–Lanczos approximant (PLA)

chapter Chapter 38|5 pages

The exact spectrum via the Green function series

chapter Chapter 39|3 pages

Uniqueness of the amplitudes {dk }

chapter Chapter 41|3 pages

The Lanczos continued fractions (LCF)

chapter Chapter 42|2 pages

Equations for eigenvalues uk via continued fractions

chapter Chapter 49|5 pages

Auto-correlation functions at large times

chapter Chapter 50|8 pages

The power moment problem in celestial mechanics

chapter Chapter 51|4 pages

Mass positivity and the power moment problem

chapter Chapter 55|6 pages

The modified moment problem

chapter Chapter 56|6 pages

The mixed moment problem

chapter Chapter 58|2 pages

Mapping from monomials un to polynomials Qn (u)

chapter Chapter 59|4 pages

Mapping of state vectors: Schrödinger ↔ Lanczos

chapter Chapter 62|3 pages

A variational principle for a quadratic Hankel form

chapter Chapter 64|9 pages

Tridiagonal inhomogeneous systems of linear equations

chapter Chapter 65|6 pages

Delayed time series

chapter Chapter 66|2 pages

Delayed Green function

chapter Chapter 67|4 pages

The quotient-difference (QD) recursive algorithm

chapter Chapter 68|6 pages

The product-difference (PD) recursive algorithm

chapter Chapter 69|7 pages

Delayed Lanczos continued fractions

chapter Chapter 70|4 pages

Delayed Pad é–Lanczos approximant

chapter Chapter 73|8 pages

Illustrations

chapter Chapter 74|2 pages

The uncertainty principle and resolution improvement

chapter Chapter 75|2 pages

Prediction/extrapolation for resolution improvement

chapter Chapter 76|9 pages

Main advantages of the Padé-based spectral analysis

chapter Chapter 77|5 pages

Conclusions