Wavelets in real time digital audio processing: Analysis and sample implementations Master’s thesis Presented at the Department of Computer Science IV Prof. Dr. W. Effelsberg Advisor Dipl.-Math Claudia Schremmer University of Mannheim in May 2000 by Florian Bömers from Bremen in collaboration with I3 Srl., Rome, Italy Contents Contents ABBREVIATIONS .............................................................................................III INDEX OF FIGURES ......................................................................................... V INDEX OF TABLES AND EQUATIONS........................................................... VI 1 INTRODUCTION.........................................................................................1 1.1 Motivation.....................................................................................................................................1 1.2 Limits of this thesis.......................................................................................................................1 1.3 Structure of the Thesis .................................................................................................................2 2 BASIC CONCEPTS ....................................................................................3 2.1 Fundamentals of Digital Audio ...................................................................................................3 2.2 AD-DA Conversion.......................................................................................................................6 2.3 Analysis-Resynthesis ..................................................................................................................10 3 THE SHORT TIME FOURIER TRANSFORM ...........................................11 3.1 Overview .....................................................................................................................................11 3.2 Discrete Analysis and Resynthesis ............................................................................................13 3.3 Frequency Bands ........................................................................................................................13 3.4 Windowing ..................................................................................................................................14 3.5 Time/Frequency Uncertainty ....................................................................................................17 3.6 Spectral Representation.............................................................................................................18 3.7 STFT for processing Musical Signals .......................................................................................21 4 THE WAVELET TRANSFORM.................................................................24 4.1 Introduction ................................................................................................................................24 4.2 Constant Q Filter Bank Analysis ..............................................................................................25 4.3 Filter Bank Wavelet Transform................................................................................................26 4.4 Wavelet Functions ......................................................................................................................34 4.5 Connection of Filter Banks and Wavelet Functions ................................................................38 4.6 Properties of the Wavelet Transform .......................................................................................41 4.7 Wavelet Applications .................................................................................................................42 4.8 The WT for processing real-time Musical Signals...................................................................42 5 CHOOSING A WAVELET FOR PROCESSING MUSICAL SIGNALS .....44 5.1 Requirements..............................................................................................................................44 5.2 Common Wavelets and their Properties...................................................................................46 5.3 Decision .......................................................................................................................................50 6 COMPUTER-BASED ALGORITHM OF THE WAVELET TRANSFORM .51 6.1 Algorithm ....................................................................................................................................51 6.2 Implementation...........................................................................................................................54 6.3 Problems and Solutions..............................................................................................................55 - I - Wavelets in real time digital audio processing 7 APPLICATIONS OF WAVELETS IN REAL TIME DIGITAL AUDIO ........61 7.1 Coding Style................................................................................................................................ 61 7.2 The Audio Framework .............................................................................................................. 61 7.3 Implemented Extensions ........................................................................................................... 67 7.4 Implemented Filters................................................................................................................... 69 7.5 The GUI ...................................................................................................................................... 74 7.6 Noise Reduction.......................................................................................................................... 78 7.7 Equalizing................................................................................................................................... 83 8 CONCLUSION ..........................................................................................86 BIBLIOGRAPHY ............................................................................................. VII APPENDIX A - CLASS DESCRIPTION......................................................... A-1 A.1 Wavelet classes ....................................................................................................................A-1 A.2 Framework Core Classes....................................................................................................A-1 A.3 Framework Extensions .......................................................................................................A-5 A.4 Filters ...................................................................................................................................A-6 A.5 Windows GUI ......................................................................................................................A-8 APPENDIX B – CLASS INHERITANCE TREES ...........................................B-1 B.1 Core Interfaces and Classes ............................................................................................... B-2 B.2 Platform-dependent Interfaces .......................................................................................... B-3 B.3 High-level Classes................................................................................................................ B-3 B.4 Windows Implementation .................................................................................................. B-3 APPENDIX C – AUDIO FRAMEWORK CHAINS ..........................................C-1 C.1 Audio Chain of the GUI......................................................................................................C-1 C.2 An example Audio Chain....................................................................................................C-1 APPENDIX D – THE CD-ROM.......................................................................D-1 D.1 Directory “Bibliography”...................................................................................................D-1 D.2 Directory “Program” ..........................................................................................................D-1 D.3 Directory “Readers” ...........................................................................................................D-1 D.4 Directory “Source”..............................................................................................................D-2 D.5 Directory “Thesis” ..............................................................................................................D-2 D.6 Directory “Unsorted Info”..................................................................................................D-2 D.7 Audio part............................................................................................................................D-2 - II - Abbreviations Abbreviations 3D Three Dimensional AD Analog to Digital ADC Analog to Digital Converter AIFF Audio Interchange File Format CD Compact Disc CD-ROM Compact Disc Read Only Memory Codec Coder/Decoder CPU Central Processing Unit CWT Continuous Wavelet Transform DA Digital to Analog DFT Discrete Fourier Transform DWT Discrete Wavelet Transform etc. and so on (lat.: et cetera) f. and following pages FFT Fast Fourier Transform Fig. Figure FIR Finite Impulse Response (filter) FT Fourier Transform FWT Fast Wavelet Transform GUI Graphical User Interface Hi-fi