Foundations of Discrete Harmonic Analysis

✍ Scribed by Vasily N. Malozemov, Sergey M. Masharsky

• Publisher: Birkhäuser
• Year: 2020
• Tongue: English
• Leaves: 257
• Series: Lecture Notes in Applied and Numerical Harmonic Analysis
• Edition: 1
• Category: Library

✦ Synopsis

This book provides an introduction to discrete harmonic analysis (DHA) with a view towards applications to digital signal processing. In a nutshell, DHA is used to determine the time-frequency structure of a digitized signal, providing a representation of the signal as a sum of spectral components that can then be analyzed.

The main methods of DHA are discrete Fourier transform and other discrete orthogonal transforms such as the Walsh and Haar transforms. Fast algorithms are used to process signals in real time, while additional options are provided by spline harmonic analysis. These topics are carefully covered in the book. With only modest prerequisites, some of which are recalled at the beginning, a profound mathematical theory is built almost from scratch. The 150 exercises included form an integral part of the text.

Based decades of teaching experience, this book provides a basis for lecture courses starting at the upper undergraduate level, and will also prove a valuable resource for mathematicians and engineers interested in digital signal processing.

✦ Table of Contents

LN-ANHA Series Preface
Preface
Contents
Acronyms
1 Preliminaries
1.1 Residuals
1.2 Greatest Common Divisor
1.3 Relative Primes
1.4 Permutations
1.5 Bitwise Summation
1.6 Complex Numbers
1.7 Roots of Unity
1.8 Finite Differences
2 Signal Transforms
2.1 Space of Signals
2.2 Discrete Fourier Transform
2.3 Parseval Equality
2.4 Sampling Theorem
2.5 Cyclic Convolution
2.6 Cyclic Correlation
2.7 Optimal Interpolation
2.8 Optimal Signal–Filter Pairs
2.9 Ensembles of Signals
2.10 Uncertainty Principle
3 Spline Subspaces
3.1 Periodic Bernoulli Functions
3.2 Periodic B-splines
3.3 Discrete Periodic Splines
3.4 Spline Interpolation
3.5 Smoothing of Discrete Periodic Data
3.6 Tangent Hyperbolas Method
3.7 Calculation of Discrete Spline's Values
3.8 Orthogonal Basis in a Space of Splines
3.9 Bases of Shifts
3.10 Wavelet Subspaces
3.11 First Limit Theorem
3.12 Second Limit Theorem
4 Fast Algorithms
4.1 Goertzel Algorithm
4.2 First Sequence of Orthogonal Bases
4.3 Fast Fourier Transform
4.4 Wavelet Bases
4.5 Haar Basis. Fast Haar Transform
4.6 Decimation in Frequency
4.7 Sampling Theorem in Haar Bases
4.8 Convolution Theorem in Haar Bases
4.9 Second Sequence of Orthogonal Bases
4.10 Fast Walsh Transform
4.11 Ordering of Walsh Functions
4.12 Sampling Theorem in Walsh Basis
4.13 Ahmed–Rao Bases
4.14 Calculation of DFT of Any Order
Appendix Solutions
References
Index

✦ Subjects

Signals, Sampling, Spline, DFT, FFT, Haar, Wavelet