The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA
Applied and Computational Mathematics
Volume 7, Issue 3, June 2018, Pages: 161-166
Received: Aug. 9, 2018; Published: Aug. 13, 2018
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Authors
Yuanyuan Zhang, College of Sciences, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
Zhaofeng Li, College of Sciences, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China
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Abstract
In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.
Keywords
Frame Multiresolution Analysis, Polyphase Decomposition, Minimum-Energy Frames
To cite this article
Yuanyuan Zhang, Zhaofeng Li, The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA, Applied and Computational Mathematics. Vol. 7, No. 3, 2018, pp. 161-166. doi: 10.11648/j.acm.20180703.22
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