research-article

Discrete Wavelet Transforms in the Large Time-Frequency Analysis Toolbox for MATLAB/GNU Octave

Authors:

ZdenĚK Průša,

Peter L. Søndergaard,

Pavel RajmicAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 42, Issue 4

Article No.: 32, Pages 1 - 23

https://doi.org/10.1145/2839298

Published: 14 June 2016 Publication History

Abstract

The discrete wavelet transform module is a recent addition to the Large Time-Frequency Analysis Toolbox (LTFAT). It provides implementations of various generalizations of Mallat's well-known algorithm (iterated filterbank) such that completely general filterbank trees, dual-tree complex wavelet transforms, and wavelet packets can be computed. The resulting transforms can be equivalently represented as filterbanks and analyzed as filterbank frames using fast algorithms.

References

[1]

A. Farras Abdelnour. 2007. Dense grid framelets with symmetric lowpass and bandpass filters. In 9th International Symposium on Signal Processing and Its Applications (ISSPA'07). Applied Mathematics and Computation 172, 1--4.

[2]

A. Farras Abdelnour. 2012. Symmetric wavelets dyadic sibling and dual frames. Signal Processing 92, 5, 1216--1229.

Digital Library

[3]

A. Farras Abdelnour and Ivan W. Selesnick. 2004. Symmetric nearly orthogonal and orthogonal nearly symmetric wavelets. The Arabian Journal for Science and Engineering 29, 2C, 3--16.

[4]

A. Farras Abdelnour and Ivan W. Selesnick. 2005. Symmetric nearly shift-invariant tight frame wavelets. IEEE Transactions on Signal Processing 53, 1, 231--239.

Digital Library

[5]

Sony Akkarakaran and P. P. Vaidyanathan. 2003. Nonuniform filter banks: New results and open problems. In Studies in Computational Mathematics: Beyond Wavelets, P. Monk, C. K. Chui, and L. Wuytack (Eds.). Vol. 10. Elsevier, 259--301.

[6]

O. Alkin and H. Caglar. 1995. Design of efficient M-band coders with linear-phase and perfect-reconstruction properties. IEEE Transactions on Signal Processing 43, 7, 1579--1590.

Digital Library

[7]

Peter Balazs. 2007. Basic definition and properties of Bessel multipliers. Journal of Mathematical Analysis and Applications 325, 1, 571--585.

[8]

Peter Balazs. 2008. Frames and finite dimensionality: Frame transformation, classification and algorithms. Applied Mathematical Sciences 2, 41--44, 2131--2144.

[9]

Peter Balazs, Monika Dörfler, Florent Jaillet, Nicki Holighaus, and Gino Angelo Velasco. 2011. Theory, implementation and applications of nonstationary Gabor frames. Journal of Computational and Applied Mathematics 236, 6, 1481--1496.

Digital Library

[10]

Ilker Bayram and Ivan W. Selesnick. 2008. On the dual-tree complex wavelet packet and M-band transforms. IEEE Transactions on Signal Processing 56, 6, 2298--2310.

Digital Library

[11]

John J. Benedetto and Shidong Li. 1998. The theory of multiresolution analysis frames and applications to filter banks. Applied and Computational Harmonic Analysis 5, 4, 389--427.

[12]

Helmut Bölcskei, Franz Hlawatsch, and Hans G. Feichtinger. 2002. Frame-theoretic analysis of oversampled filter banks. IEEE Transactions on Signal Processing 46, 12 3256--3268.

Digital Library

[13]

Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. 2011. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 3, 1, 1--122.

Digital Library

[14]

C. Sidney Burrus, Ramesh Gopinath, and Haitao Guo. 2013. Wavelets and wavelet transforms. OpenStax_CNX. Retrieved May 18, 2006 from http://cnx.org/content/col11454/1.5/.

[15]

Peter G. Casazza and Gitta Kutyniok (Eds.). 2013. Finite Frames: Theory and Applications. Birkhäuser, Basel, Switzerland.

Digital Library

[16]

Ole Christensen and Say Song Goh. 2014. From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa. Applied and Computational Harmonic Analysis 36, 2, 198--214.

[17]

Ronald R. Coifman and Mladen Victor Wickerhauser. 1992. Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory 38, 2, 713--718.

Digital Library

[18]

Patrick L. Combettes and Jean-Christophe Pesquet. 2011. Proximal splitting methods in signal processing. In Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Heinz H. Bauschke, Regina S. Burachik, Patrick L. Combettes, Veit Elser, D. Russell Luke, and Henry Wolkowicz (Eds.). Springer, New York, NY, 185--212.

[19]

Z. Cvetkovic and M. Vetterli. 1998. Oversampled filter banks. IEEE Transactions on Signal Processing 46, 5, 1245--1255.

Digital Library

[20]

Ingrid Daubechies. 1992. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, PA.

Digital Library

[21]

Ingrid Daubechies, Michel Defrise, and Christine De Mol. 2004. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics 57, 11, 1413--1457.

[22]

Ingrid Daubechies, A. Grossmann, and Y. Meyer. 1986. Painless nonorthogonal expansions. Journal of Mathematical Physics 27, 5, 1271--1283.

[23]

Ingrid Daubechies, Bin Han, Amos Ron, and Zuowei Shen. 2003. Framelets: MRA-based constructions of wavelet frames. Applied and Computational Harmonic Analysis 14, 1, 1--46.

[24]

Bogdan Dumitrescu, Ilker Bayram, and Ivan W. Selesnick. 2008. Optimization of symmetric self-Hilbertian filters for the dual-tree complex wavelet transform. IEEE Signal Processing Letters 15, 146--149.

[25]

Martin Ehler. 2007. The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes. Ph.D. Dissertation. Philipps--Universität Marburg, Marburg/Lahn, Germany.

[26]

Matthew Fickus, Melody L. Massar, and Dustin G. Mixon. 2013. Finite frames and filter banks. In Finite Frames: Theory and Applications, Peter G. Casazza and Gitta Kutyniok (Eds.). Birkhäuser, Basel, Switzerland, 337--380.

[27]

James E. Fowler. 2005. The redundant discrete wavelet transform and additive noise. IEEE Signal Processing Letters 12, 9, 629--632.

[28]

Ramesh A. Gopinath and C. Sidney Burrus. 1995. On cosine-modulated wavelet orthonormal bases. IEEE Transactions on Image Processing 4, 2, 162--176.

Digital Library

[29]

D. Griffin and J. Lim. 1984. Signal estimation from modified short-time Fourier transform. IEEE Transactions on Acoustics, Speech and Signal Processing 32, 2, 236--243.

[30]

Karlheinz Gröchenig. 1993. Acceleration of the frame algorithm. IEEE Transactions on Signal Processing 41, 12, 3331--3340.

Digital Library

[31]

Nicki Holighaus, Zdeněk Průša, and Christoph Wiesmeyr. 2015. Designing tight filter bank frames for nonlinear frequency scales. Sampling Theory and Applications 2015.

[32]

M. Holschneider, R. Kronland-Martinet, J. Morlet, and Ph. Tchamitchian. 1990. A real-time algorithm for signal analysis with the help of the wavelet transform. In Wavelets, Jean-Michel Combes, Alexander Grossmann, and Philippe Tchamitchian (Eds.). Springer, Berlin, 286--297.

[33]

Nick G. Kingsbury. 2000. A dual-tree complex wavelet transform with improved orthogonality and symmetry properties. In ICIP. 375--378.

[34]

Nick G. Kingsbury. 2001. Complex wavelets for shift invariant analysis and filtering of signals. Applied and Computational Harmonic Analysis 10, 3, 234--253.

[35]

Nick G. Kingsbury. 2003. Design of Q-shift complex wavelets for image processing using frequency domain energy minimization. In Proceedings of the 2003 International Conference on Image Processing (ICIP'03). Vol. 1. I--1013--16.

[36]

Jelena Kovačević and Amina Chebira. 2008. An introduction to frames. Foundations and Trends in Signal Processing 2, 1, 1--94.

Digital Library

[37]

Tony Lin, Shufang Xu, Qingyun Shi, and Pengwei Hao. 2006. An algebraic construction of orthonormal M-band wavelets with perfect reconstruction. Applied Mathematics and Computation 172, 2, 717--730.

Digital Library

[38]

Stéphane G. Mallat. 1989. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 7, 674--693.

Digital Library

[39]

Nuria González Prelcic, Oscar W. Márquez, and Santiago González. 1996. Uvi wave, the ultimate toolbox for wavelet transforms and filter banks. In Proceedings of the 4th Bayona Workshop on Intelligent Methods in Signal Processing and Communications. Bayona, Spain, 224--227.

[40]

Zdeněk Průša, Peter L. Søndergaard, Nicki Holighaus, Christoph Wiesmeyr, and Peter Balazs. 2014. The large time-frequency analysis toolbox 2.0. In Sound, Music, and Motion, Mitsuko Aramaki, Olivier Derrien, Richard Kronland-Martinet, and Sølvi Ystad (Eds.). Springer International Publishing, 419--442.

[41]

Pavel Rajmic and Zdeněk Průša. 2014. Discrete wavelet transform of finite signals: Detailed study of the algorithm. International Journal of Wavelets, Multiresolution and Information Processing 12, 01, 1450001.

[42]

Olivier Rioul. 1993. A discrete-time multiresolution theory. IEEE Transactions on Signal Processing 41, 8, 2591--2606.

Digital Library

[43]

Amos Ron and Zuowei Shen. 1997. Affine systems in L²(ℝ^d): The analysis of the analysis operator. Journal of Functional Analysis 148, 2, 408--447.

[44]

Ivan W. Selesnick. 2001. The double density DWT. In Wavelets in Signal and Image Analysis. Springer, 39--66.

[45]

Ivan W. Selesnick. 2004. The double-density dual-tree DWT. IEEE Transactions on Signal Processing 52, 5, 1304--1314.

Digital Library

[46]

Ivan W. Selesnick. 2006. A higher density discrete wavelet transform. IEEE Transactions on Signal Processing 54, 8, 3039--3048.

Digital Library

[47]

Ivan W. Selesnick and A. Farras Abdelnour. 2004. Symmetric wavelet tight frames with two generators. Applied and Computational Harmonic Analysis 17, 2, 211--225.

[48]

Ivan W. Selesnick, Richard G. Baraniuk, and Nick G. Kingsbury. 2005. The dual-tree complex wavelet transform. IEEE Signal Processing Magazine 22, 6, 123--151.

[49]

M. Shensa. 1992. The discrete wavelet transform: Wedding the à trous and Mallat algorithms. IEEE Transactions on Signal Processing 40, 10, 2464--2482.

Digital Library

[50]

Peter L. Søndergaard, Bruno Torrésani, and Peter Balazs. 2012. The linear time frequency analysis toolbox. International Journal of Wavelets, Multiresolution Analysis and Information Processing 10, 4.

[51]

David Stanhill and Yehoshua Y. Zeevi. 1996. Frame analysis of wavelet type filter banks. In Digital Signal Processing Workshop Proceedings, IEEE. 435--438.

[52]

Peter Steffen, Peter N. Heller, Ramesh A. Gopinath, and C. Sidney Burrus. 1993. Theory of regular M-band wavelet bases. IEEE Transactions on Signal Processing 41, 12, 3497--3511.

Digital Library

[53]

Gilbert Strang and T. Nguyen. 1997. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA.

[54]

Wim Sweldens. 1996. The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis. 3, 2, 186--200.

[55]

Carl Taswell. 1994. Near-best basis selection algorithms with non-additive information cost functions. In Proceedings of the IEEE International Symposium on Time-Frequency and Time-Scale Analysis. IEEE Press, 13--16.

[56]

Carl Taswell and Kevin C. McGill. 1994. Algorithm 735: Wavelet transform algorithms for finite-duration discrete-time signals. ACM Transactions on Mathematical Software 20, 3, 398--412.

Digital Library

[57]

P. P. Vaidyanathan. 1993. Multirate Systems and Filter Banks. Prentice-Hall, Englewood Cliffs, NJ.

Digital Library

[58]

Martin Vetterli, Jelena Kovačević, and Vivek K. Goyal. 2014. Foundations of Signal Processing. Cambridge University Press, New York, NY. http://www.fourierandwavelets.org/.

[59]

Mladen Victor Wickerhauser. 1994. Adapted Wavelet Analysis from Theory to Software. Wellesley-Cambridge Press, Wellesley, MA.

Digital Library

Cited By

Kampik MRoj JDróżdż Ł(2024)Estimation of the Resultant Expanded Uncertainty of the Output Quantities of the Measurement Chain Using the Discrete Wavelet Transform AlgorithmApplied Sciences10.3390/app1409369114:9(3691)Online publication date: 26-Apr-2024
https://doi.org/10.3390/app14093691
Yang GLuan BSun JNiu JLin HWang L(2024)Sparrow search mechanism-based effective feature mining algorithm for the broken wire signal detection of prestressed concrete cylinder pipeMechanical Systems and Signal Processing10.1016/j.ymssp.2024.111270212(111270)Online publication date: Apr-2024
https://doi.org/10.1016/j.ymssp.2024.111270
Köhldorfer LBalazs PCasazza PHeineken SHollomey CMorillas PShamsabadi M(2024)A Survey of Fusion Frames in Hilbert SpacesSampling, Approximation, and Signal Analysis10.1007/978-3-031-41130-4_11(245-328)Online publication date: 5-Jan-2024
https://doi.org/10.1007/978-3-031-41130-4_11
Show More Cited By

Index Terms

Discrete Wavelet Transforms in the Large Time-Frequency Analysis Toolbox for MATLAB/GNU Octave
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Computation of transforms

Recommendations

On the Dual-Tree Complex Wavelet Packet and M -Band Transforms

The two-band discrete wavelet transform (DWT) provides an octave-band analysis in the frequency domain, but this might not be ldquooptimalrdquo for a given signal. The discrete wavelet packet transform (DWPT) provides a dictionary of bases over which ...
On the shiftability of dual-tree complex wavelet transforms

The dual-tree complex wavelet transform (DT-CWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-CWT which, among other things, offers a direct ...
Construction of Hilbert transform pairs of wavelet bases and Gabor-like transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 42, Issue 4

July 2016

185 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2956571

Editor:
Michael A. Heroux
Sandia National Laboratories, USA

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 14 June 2016

Accepted: 01 October 2015

Received: 01 May 2015

Published in TOMS Volume 42, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Ministry of Education, Youth and Sport of the Czech Republic (MŠMT) National Sustainability Program
Austrian Science Fund (FWF) START-project FLAME (Frames and Linear Operators for Acoustical Modeling and Parameter Estimation; Y 551-N13)

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

11
Total Citations
View Citations
341
Total Downloads

Downloads (Last 12 months)9
Downloads (Last 6 weeks)1

Reflects downloads up to 07 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Kampik MRoj JDróżdż Ł(2024)Estimation of the Resultant Expanded Uncertainty of the Output Quantities of the Measurement Chain Using the Discrete Wavelet Transform AlgorithmApplied Sciences10.3390/app1409369114:9(3691)Online publication date: 26-Apr-2024
https://doi.org/10.3390/app14093691
Yang GLuan BSun JNiu JLin HWang L(2024)Sparrow search mechanism-based effective feature mining algorithm for the broken wire signal detection of prestressed concrete cylinder pipeMechanical Systems and Signal Processing10.1016/j.ymssp.2024.111270212(111270)Online publication date: Apr-2024
https://doi.org/10.1016/j.ymssp.2024.111270
Köhldorfer LBalazs PCasazza PHeineken SHollomey CMorillas PShamsabadi M(2024)A Survey of Fusion Frames in Hilbert SpacesSampling, Approximation, and Signal Analysis10.1007/978-3-031-41130-4_11(245-328)Online publication date: 5-Jan-2024
https://doi.org/10.1007/978-3-031-41130-4_11
Balazs PCasey S(2023)Sampling and Frame Expansions for UWB Signals2023 International Conference on Sampling Theory and Applications (SampTA)10.1109/SampTA59647.2023.10301377(1-5)Online publication date: 10-Jul-2023
https://doi.org/10.1109/SampTA59647.2023.10301377
Gao SSun ZZhu JFan YLu C(2022)Using wavelet analysis to characterize the transition from bubbling to turbulent fluidizationPowder Technology10.1016/j.powtec.2022.117269401(117269)Online publication date: Mar-2022
https://doi.org/10.1016/j.powtec.2022.117269
Escola JSouza UBrito L(2022)Discrete Wavelet Transform in digital audio signal processingComputers and Electrical Engineering10.1016/j.compeleceng.2022.108439104:PAOnline publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1016/j.compeleceng.2022.108439
Tian Z(2020)A Method to Predict Random Time-Delay of Networked Control SystemIETE Journal of Research10.1080/03772063.2020.176890768:5(3503-3513)Online publication date: 27-May-2020
https://doi.org/10.1080/03772063.2020.1768907
Goyal KSharma D(2020)A MATLAB Suite for Second Generation Wavelets on an Interval and the Corresponding Adaptive GridActa Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications10.1007/s10440-019-00299-5169:1(279-321)Online publication date: 1-Oct-2020
https://dl.acm.org/doi/10.1007/s10440-019-00299-5
Zhang MLiu SChen SChen YXu GQian D(2018)Focus Energy Determination of Mining Microseisms Using Residual Seismic Wave Attenuation in Deep Coal MiningShock and Vibration10.1155/2018/38543292018:1Online publication date: 21-Feb-2018
https://doi.org/10.1155/2018/3854329
Sun BCui LLi WKang XZhang M(2018)A Space-Scale Estimation Method based on continuous wavelet transform for coastal wetland ecosystem services in Liaoning Province, ChinaOcean & Coastal Management10.1016/j.ocecoaman.2018.02.019157(138-146)Online publication date: May-2018
https://doi.org/10.1016/j.ocecoaman.2018.02.019
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents