A least square method based on data fitting is proposed to construct a new lifting wavelet, together with the nonlinear idea and redundant algorithm, the adaptive redundant lifting transform based on fitting is firstly stated in this paper. By variable combination selections of basis function, sample number and dimension of basis function, a total of nine wavelets with different characteristics are constructed, which are respectively adopted to perform redundant lifting wavelet transforms on low-frequency approximate signals at each layer. Then the normalized lP norms of the new node-signal obtained through decomposition are calculated to adaptively determine the optimal wavelet for the decomposed approximate signal. Next, the original signal is taken for subsection power spectrum analysis to choose the node-signal for single branch reconstruction and demodulation. Experiment signals and engineering signals are respectively used to verify the above method and the results show that bearing faults can be diagnosed more effectively by the method presented here than by both spectrum analysis and demodulation analysis. Meanwhile, compared with the symmetrical wavelets constructed with Lagrange interpolation algorithm, the asymmetrical wavelets constructed based on data fitting are more suitable in feature extraction of fault signal of roller bearings.
References
[1]
Sweldens, W. The lifting scheme: A custom-design construction of biorthogonal wavelet. Appl. Comput. Harmon. Anal. 1996, 3, 186–200.
[2]
Sweldens, W. The lifting scheme: A construction of second generation wavelet constructions. SIAM J. Math. Anal. 1997, 29, 511–546.
[3]
Daubechies, I.; Sweldens, W. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl. 1998, 4, 247–269.
[4]
Shim, M.; Laine, A.F. Overcomplete lifted wavelet representations for multiscale feature analysis. Proceedings of the 1998 International Conference on Image Processing (ICIP), Chicago, IL, USA, 4–7 October 1998; pp. 242–246.
[5]
Yang, X.Y.; Shi, Y.; Chen, L.H.; Quan, Z.F. The lifting scheme for wavelet bi-frames-theory, structure, and algorithm. IEEE Trans. Signal Process. 2010, 19, 612–624.
[6]
Jacek, S.; Tomasz, Z.; Roman, R.R. Image denoising using scale-adaptive lifting schemes. Proceedings of the 2000 International Conference on Image Processing (ICIP), Vancouver, BC, Canada, 10–13 September 2000; pp. 288–291.
[7]
Li, Z.; He, Z.J.; Zi, Y.Y.; Wang, Y.X. Customized wavelet denoising using intra- and inter-scale dependency for bearing fault detection. J. Sound Vib. 2008, 313, 342–359.
[8]
Bao, W.; Zhou, R.; Yang, J.G.; Yu, D.R.; Li, N. Anti-aliasing lifting scheme for mechanical vibration fault feature extraction. Mech. Syst. Signal Process. 2009, 23, 1458–1473.
[9]
Lee, C.M.; Ko, C.N. Short-term load forecasting using lifting scheme and ARIMA models. Expert Syst. Appl. 2011, 38, 5902–5911.
[10]
Amiri, M.; Rabiee, H.R. A novel rotation/scale invariant template matching algorithm using weighted adaptive lifting scheme transform. Pattern Recog. 2010, 43, 2485–2496.
[11]
Quellec, G.; Lamard, M.; Cazuguel, G.; Cochener, B.; Roux, C. Adaptive nonseparable wavelet transform via lifting and its application to content-based image retrieval. IEEE Trans. Image Process. 2010, 19, 25–35.
Sweldens, W.; Schr？der, P. Building Your Own Wavelets at Home. Proceedings of the CAM Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 96), New Orleans, LA, USA, 4–9 August 1996; pp. 72–130.
[14]
Xue, Y.; Geng, M.Y. Chapter Six: Function approximation. Numerical Analysis; Beijing University of Technology Press: Beijing, China, 2003; pp. 205–232.
[15]
Holschneider, M.; Kronland-Martinet, R.; Morlet, J.; Tchamitchian, P. A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform. Proceedings of the International Conference on Wavelets, Time-Frequency Methods and Phase Space, Marseille, France, 14–18 December 1987; p. 286.
Claypoole, R.L.; Baraniuk, R.G. Flexible Wavelet Transforms Using Lifting, Available online: http://scholarship.rice.edu/handle/1911/19805 (accessed on 23 March 2012).
[18]
Gao, L.X.; Yang, Z.J.; Cai, L.G.; Wang, H.Q.; Chen, P. Roller bearing fault diagnosis based on nonlinear redundant lifting wavelet packet analysis. Sensors 2011, 11, 260–277.
