4.7 Summary
This chapter begins with a description of the discretization of the scale and transla-
tion para meters. The MRA and orthogonal wavelet transform are then introduced in
Sect. 4.2. Af ter that, we describe in Sect. 4.3 the dual-scale equation and its
associated wavelet filter pair. The Mallat algorithm for implementing the DWT is
then discussed in Sect. 4.4, followed by the introduction of some commonly used
wavelets in Sect. 4.5. Some typical applications of the DWT are shown in Sect. 4.6.
4.8 References
Abbasion S, Rafsanjani A, Farshidianfar A, Irani N (2007) Rolling element bearings multi fault
classification based on the wavelet denoising and support vector machine. Mech Syst Signal
Process 21:2933 2945
Addison N (2002) The illustrated wavelet transform handbook. Taylor & Francis, New York
Burt P, Adelson E (1983) The Laplacian pyramid as a compact image code. IEEE Trans Commun
31:482 540
Cohen A, Daubechies I, Feauveau, JC (1992) Biorthogonal bases of compactly supported wave
lets. Commun Pure Appl Math 45:485 560
Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia
Donoho DL (1995) De noising by soft thresholding. IEEE Trans Inform Theory, 41(3): 613 627
Donoho DL; Johnstone IM (1995) Adapting to unknown smoothness via wavelet shrinkage. J Am
Stat Assoc 90(432):1200 1244
Fu S, Muralikrishnan, Raja J (2003) Engineering surface analysis with different wavelet bases.
ASME J Manuf Sci Eng 125(6):844 852
Gao R, Yan R (2006) Non stationary signal processing for bearing health monitoring. Int J Manuf
Res 1(1):18 40
Haar A (1910) Zur theorie der orthgonalen funktionensysteme. Math Annalen 69:331 371
Kim JS, Lee JH, Kim JH, Baek J, Kim SS (2010) Fault detection of cycle based signals using
wavelet transform in FAB processes. Int J Precision Eng Manuf 11(2):237 246
Lou X, Loparo KA (2004) Bearing fault diagnosis based on wavelet transform and fuzzy inference.
Mech Syst Signal Process 18:1077 1095
Mallat SG (1989a) A theory of multiresolution signal decomposition: the wavelet representation.
IEEE Trans Pattern Anal Machine Intell 11(7) 674 693
Mallat SG (1989b) Multiresolution approximations and wavelet orthonormal bases of L
2
(R). Trans
Am Math Soc 315:69 87
Mallat SG (1998) A wavelet tour of signal processing. Academic, San Diego, CA
Nikolaou NG, Antoniadis IA (2002) Rolling element bearing fault diagnosis using wavelet
packets. NDT&E Int 35:197 205
Rafiee J, Rafiee MA, Tse PW (2010) Application of mother wavelet functions for automatic gear
and bearing fault diagnosis. Expert Syst Appl 37:4568 4579
Shakher C, Ishtiaque SM, Singh SK, Zaidi HN (2004) Application of wavelet transform in
characterization of fabric texture. J Text Inst 95(1 6):107 120
Sugumaran V, Ramachandran KI (2009) Wavelet selection using decision tree for fault diagnosis
of roller bearings. Int J Appl Eng Res 4(2):201 225
Witkin A (1983) Scale space filtering. In: Proceedings of international joint conference on artificial
intelligence, Karlsruhe, Germany, pp 1019 1023
Zhou SY, Sun BC, Shi JJ (2006) An SPC monitoring system for cycle based waveform signals
using haar transform. IEEE Trans Automat Sci Eng 3(1):60 72
68 4 Discrete Wavelet Transform