Foundations of discrete harmonic analysis

Vasily N. Malozemov, Sergey M. Masharsky

Research output: Book/Report/AnthologyBookResearchpeer-review


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.
Original languageEnglish
Place of PublicationSwitzerland
PublisherSpringer Nature
Number of pages243
ISBN (Electronic)978-3-030-47048-7
ISBN (Print)978-3-030-47047-0
StatePublished - Aug 2020

Publication series

NameApplied and Numerical Harmonic Analysis
ISSN (Print)2296-5009

Scopus subject areas

  • Applied Mathematics


  • discrete harmonic analysis
  • discrete periodic signals
  • Discrete Fourier Transform
  • convolution theorem
  • correlation theorem
  • discrete Walsh transform
  • discrete Haar transforms
  • discrete Ahmed-Rao bases
  • discrete periodic splines
  • wavelet expansions
  • wavelet packets
  • fast transforms


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