A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix
Abstract
:1. Introduction
2. Primary Definitions and Theorems
Algorithm 1: FFT-based fast method for linear convolution. |
Input: the signal, the signal, Output: the linear convolution of f and h, 1. Set 2. Appending an array of zeros to the end of h to obtain so that the length of is n. 3. Appending an array of zeros to the end of f to obtain so that the length of is n 4. |
3. A Class of Algorithms for Continuous Wavelet Transform
3.1. Time-Domain Algorithm for CWT
Algorithm 2: Time-domain fast algorithm for CWT. |
Input: the signal, mother wavelet, scale, a T, where is the support of . Output:wavelet coefficients, 1. Let 2. Let , where the overline represents the operation of complex conjugate; 3. Let ; 4. Let 5. ; 6. . |
3.2. Frequency-Domain Algorithm for CWT
Algorithm 3: Frequency-domain fast algorithm for CWT (In real application, the case is enough for the calculation of CWT). |
Input: the signal, scale, a , the Fourier transform of the Morlet wavelet T, where is the support of . Output: wavelet coefficients, 1. Let 2. Define , where the overline represents the operation of complex conjugate; 3. Let ; 4. If M is even, Let 5. else 6. end if 7. ; 8. switch(m) case 0 ; case 1 ; case 2 . |
4. Numerical Experiments
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Chi, C.T.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Math. Phys. Eng. Sci. 1998, 454, 903–995. [Google Scholar] [CrossRef]
- Daubechies, I.; Heil, C. Ten Lectures on Wavelets. Comput. Phys. 1992, 6, 1671. [Google Scholar] [CrossRef]
- Gilles, J. Empirical Wavelet Transform. IEEE Trans. Signal Process. 2013, 61, 3999–4010. [Google Scholar] [CrossRef]
- Amezquita-Sanchez, J.P.; Adeli, H. A new music-empirical wavelet transform methodology for time frequency analysis of noisy nonlinear and non-stationary signals. Digit. Signal Process. 2015, 45, 55–68. [Google Scholar] [CrossRef]
- Daubechies, I.; Lu, J.; Wu, H.T. Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal. 2011, 30, 243–261. [Google Scholar] [CrossRef]
- Garcia-Perez, A.; Amezquita-Sanche, J.P.; Dominguez-Gonzale, A.; Sedaghat, R.; Osornio-Rio, R.; Romero-Troncos, R.J. Fused empirical mode decomposition and wavelets for locating combined damage in a truss-type structure through vibration analysis. J. Zhejiang Univ. Sci. A 2013, 14, 615–630. [Google Scholar] [CrossRef]
- Boashash, B.; Khan, N.A.; Ben-Jabeur, T. Time frequency features for pattern recognition using high-resolution TFDs: A tutorial review. Digit. Signal Process. 2015, 40, 1–30. [Google Scholar]
- Glowacz, A. DC Motor Fault Analysis with the Use of Acoustic Signals, Coiflet Wavelet Transform, and K-Nearest Neighbor Classifier. Arch. Acoust. 2015, 40, 321–327. [Google Scholar] [CrossRef]
- Chen, J.; Rostami, J.; Tse, P.W.; Wan, X. The Design of a Novel Mother Wavelet that is Tailor-made for Continuous Wavelet Transform in Extracting Defect-Related Features from Reflected Guided Wave Signals. Measurement 2017, 110, 176–191. [Google Scholar] [CrossRef]
- Chamanzar, A.; Shabany, M.; Malekmohammadi, A.; Mohammadinejad, S. Efficient Hardware Implementation of Real-Time Low-Power Movement Intention Detector System Using FFT and Adaptive Wavelet Transform. IEEE Trans. Biomed. Circuits Syst. 2017, 11, 585–596. [Google Scholar] [CrossRef] [PubMed]
- Lee, W.K.; Ratnam, M.M.; Ahmad, Z.A. Detection of chipping in ceramic cutting inserts from workpiece profile during turning using fast Fourier transform (FFT) and continuous wavelet transform (CWT). Precis. Eng. 2017, 47, 406–423. [Google Scholar] [CrossRef]
- Ni, S.H.; Li, J.L.; Yang, Y.Z.; Yang, Z.T. An improved approach to evaluating pile length using complex continuous wavelet transform analysis. Insight Non-Destr. Test. Cond. Monit. 2017, 59, 318–324. [Google Scholar] [CrossRef]
- Rafiei, M.H.; Adeli, H. A novel unsupervised deep learning model for global and local health condition assessment of structures. Eng. Struct. 2018, 156, 598–607. [Google Scholar] [CrossRef]
- Perez-Ramirez, C.A.; Amezquita-Sanchez, J.P.; Adeli, H.; Valtierra-Rodriguez, M.; Camarena-Martinez, D.; Romero-Troncoso, R.J. New methodology for modal parameters identification of smart civil structures using ambient vibrations and synchrosqueezed wavelet transform. Eng. Appl. Artif. Intell. 2016, 48, 1–12. [Google Scholar] [CrossRef]
- Leigh, G.M. Fast FIR Algorithms for the Continuous Wavelet Transform From Constrained Least Squares. IEEE Trans. Signal Process. 2012, 61, 28–37. [Google Scholar] [CrossRef]
- Unser, M. Fast Gabor-like windowed Fourier and continuous wavelet transforms. IEEE Signal Process. Lett. 1994, 1, 76–79. [Google Scholar] [CrossRef]
- Yi, H.; Chen, Z.; Cao, Y. High Precision Computation of Morlet Wavelet Transform for Multi-period Analysis of Climate Data. J. Inf. Comput. Sci. 2014, 11, 6369–6385. [Google Scholar] [CrossRef]
- Torrence, C.; Compo, G.P. A Practical Guide to Wavelet Analysis. Bull. Am. Meteorol. Soc. 1998, 79, 61–78. [Google Scholar] [CrossRef]
- Burrus, C.; Parks, T.W. DFT/FFT and Convolution Algorithms: Theory and Implementation; John Wiley & Sons Inc.: New York, NY, USA, 1991. [Google Scholar]
- Jin, X.Q. Preconditioning Techniques for Toeplitz Systems; Higher Education Press: Beijing, China, 2010. [Google Scholar]
- Nagy, J.G.; Plemmons, R.J.; Torgersen, T.C. Iterative image restoration using approximate inverse preconditioning. IEEE Trans. Image Process. 1996, 5, 1151–1162. [Google Scholar] [CrossRef] [PubMed]
- Proakis, J.G.; Manolakis, D.G. Digital Signal Processing: Principles, Algorithms, And Applications; Pearson Education: Delhi, India, 2007; Volume 23, pp. 392–394. [Google Scholar]
- Mallat, S.G. A Wavelet Tour of Signal Processing: The Sparse Way; Academic Press: New York, NY, USA, 2009. [Google Scholar]
- Yi, H.; Shu, H. The improvement of the Morlet wavelet for multi-period analysis of climate data. Comptes Rendus Geosci. 2012, 344, 483–497. [Google Scholar] [CrossRef]
- Jones, D.L.; Baraniuk, R.G. Efficient approximation of continuous wavelet transforms. Electron. Lett. 1991, 27, 748–750. [Google Scholar] [CrossRef]
- Zhao, Y.Y.; Yuan, X.; Wei, Y.H. Realization of Continuous Wavelet Transform of Sequences by MATLAB. J. Sichuan Univ. 2006, 43, 325–329. [Google Scholar]
Algorithm | N | CPU | Entropy | ||||
---|---|---|---|---|---|---|---|
cwt | 1024 | 0.2468 | 8.7125 | − | − | ||
2048 | 0.2560 | 9.3815 | − | − | |||
Zhao’s | 1024 | 0.0495 | 8.7068 | − | − | − | |
2048 | 0.0562 | 9.3762 | − | − | − | ||
Direct | 1024 | 0.4724 | 8.7097 | − | − | − | |
2048 | 0.8697 | 9.3784 | − | − | − | − | |
Overlap-Add | 1024 | 0.0291 | 8.7083 | − | − | − | |
2048 | 0.0359 | 9.3777 | − | − | − | − | |
Algorithm 2 | 1024 | 0.0204 | 8.7083 | − | − | − | |
2048 | 0.0291 | 9.3777 | − | − | − | − | |
Algorithm 3 | 1024 | 0.0155 | 8.7083 | − | − | − | |
2048 | 0.0223 | 9.3777 | − | − | − | − |
CWT | Zhao’s Method | Algorithm 2 | Algorithm 3 | ||
---|---|---|---|---|---|
1 | 5 | 9.5190 | 9.4150 | 9.4144 | 9.4534 |
1 | 6 | 9.5051 | 9.4044 | 9.4042 | 9.4332 |
1 | 7 | 9.4869 | 9.3897 | 9.3893 | 9.4117 |
1 | 8 | 9.4677 | 9.3713 | 9.3709 | 9.3892 |
1.5 | 5 | 9.5012 | 9.4030 | 9.4026 | 9.4304 |
1.5 | 6 | 9.4797 | 9.3832 | 9.3831 | 9.4037 |
1.5 | 7 | 9.4545 | 9.3601 | 9.3598 | 9.3762 |
1.5 | 8 | 9.4323 | 9.3377 | 9.3374 | 9.3513 |
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Yi, H.; Xin, S.-Y.; Yin, J.-F. A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix. Algorithms 2018, 11, 24. https://doi.org/10.3390/a11030024
Yi H, Xin S-Y, Yin J-F. A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix. Algorithms. 2018; 11(3):24. https://doi.org/10.3390/a11030024
Chicago/Turabian StyleYi, Hua, Shi-You Xin, and Jun-Feng Yin. 2018. "A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix" Algorithms 11, no. 3: 24. https://doi.org/10.3390/a11030024
APA StyleYi, H., Xin, S. -Y., & Yin, J. -F. (2018). A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix. Algorithms, 11(3), 24. https://doi.org/10.3390/a11030024