Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Sound and Fourier Series Read chapter Read first chapter

2019 | OriginalPaper | Chapter

5. The Filter Representation of Wavelets

Author : Øyvind Ryan

Published in: Linear Algebra, Signal Processing, and Wavelets - A Unified Approach

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We saw in the previous chapter that the wavelet kernels G and H had some repeating structure in the rows and columns, similar to the circular Toeplitz structure in filters. Clearly the matrices are not filters, however. Nevertheless, we will now prove that wavelet kernels can be implemented easily in terms of filters. but that several filters are needed in the computation. Each of these filters will have an interpretation in terms of how the wavelet transform treats different frequencies. Much has been done in establishing efficient implementations of filters, and by expressing a wavelet transform in terms of filters we can take advantage of this.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Motivation for Wavelets and Some Simple Examples
next chapter Constructing Interesting Wavelets
Footnotes
1
The name windowing is a bit strange. Here this does not correspond to applying a window to the sound samples as we explained in Exercise 3.​9. We will see that it rather corresponds to applying a filter coefficient to a sound sample.
 
2
It seems strange to use the name matrixing, for something which obviously is matrix multiplication. The reason must be that the procedure has been established outside a linear algebra framework.
 
Literature
2.
T. Barnwell, An experimental study of sub-band coder design incorporating recursive quadrature filters and optimum ADPCM, in Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP, pp. 808–811 (1981)
9.
A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45(5), 485–560 (1992)MathSciNetCrossRef
11.
A. Croisier, D. Esteban, C. Galand, Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques, in International Conference on Information Sciences and Systems, pp. 443–446 (1976)
36.
T.D. Lookabaugh, M.G. Perkins, Application of the Princen-Bradley filter bank to speech and image compression. IEEE Trans. Acoust. Speech Signal Process. 38(11), 1914–1926 (1990)CrossRef
55.
D. Pan, A tutorial on MPEG/audio compression. IEEE Multimedia 2, 60–74 (1995)CrossRef
61.
T.A. Ramstad, S.O. Aase, J.H. Husøy, Subband Compression of Images: Principles and Examples, vol. 6 (Elsevier, Amsterdam, 1995)
63.
J.H. Rothweiler, Polyphase quadrature filters – a new subband coding technique, in ICASSP 83, Boston, pp. 1280–1283 (1983)
67.
M.J.T. Smith, T.P. Barnwell, A new filter bank theory for time-frequency representation. IEEE Trans. Acoust. Speech Signal Process. 35(3), 314–327 (1987)CrossRef
76.
M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, Upper Saddler River, 1995)
Metadata
Title
The Filter Representation of Wavelets
Author
Øyvind Ryan
Copyright Year
2019
Book
Linear Algebra, Signal Processing, and Wavelets - A Unified Approach

Print ISBN: 978-3-030-01811-5

Electronic ISBN: 978-3-030-01812-2

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-01812-2_5

Premium Partner

    Image Credits