Whereas orthogonal wavelets come from one orthogonal basis set, the biorthogonal wavelets project from different basis sets. Each basis set is correspondingly weighted to form filters, either highpass or lowpass, which form the constituents of quadrature mirror filter QMF banks. Consequently, these filters can be used to design wavelets, the differently weighted parameters contributing respective wavelet properties which influence the performance of the transforms in applications, for example multicarrier modulation.
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Hence the regularity of the primal atoms are related to the primal filters. The number of vanishing moments of a wavelet is determined by its dual filter.
It corresponds to the approximating power of the dual multiresolution sequence. This is why it is preferred to synthesize a decomposition filter h with many vanishing moments, and possibly with a small support. On the other hand, this same filter h determines the regularity of j , and hence of y.
According to Lemma 1 , we have. Definition For an arbitrary function , if then is symmetrical at , where is a fixed point. If then is antisymmetrical at. If the Fourier transform of satisfies then it is called that has linear phase. Moreover, if , where are fixed points and is a real-value function, then it is called that has generalized linear phase, where is called phase of. Theorem 2.
Assume that has generalized linear phase and is phase of. Then the inverse Fourier transform of can be written as follows: Taking the complex conjugation on both sides of the above equation and being real-value, we have On the other side, if by taking the Fourier transform on its both sides, we have Let , then is real-value, and has generalized linear phase, where is phase of.
So, if is a real-value function, then and That is, is symmetrical or antisymmetrical at. Theorem 3. Assume that is a two-scale sequence of , and is defined in its discrete Fourier transform. According to Lemma 3 , Theorem 1 , and the length-preserving projection p , it is shown obviously. And the dual scaling function satisfies. And the scaling function and the dual can be generated by. According to Lemma 5 , the corresponding two-scale sequences and can be obtained as follows:.
According to equation 25 ,. In order to show the figures of scaling function and the dual , we choose a logarithmic curve , such as. According to the length-preserving projection in Section 2. Since the birth of bitcoin on January 3, , it has become a hot issue in economy and finance. In this section, according to multiresolution analysis on a logarithmic curve , we can decompose and reconstruct Bitcoin transaction data.
For convenience, the figure of raw data denoted by SP and its graph is given by a solid curve in Figure 5. By observing the graph, the daily data shows a significant logarithmic growth trend. In Figure 5 , the period is from April 30, , to December 1, , and the dotted logarithmic curve indicates its growth trend. The logarithmic trend curve of raw data SP and time can be estimated by the ordinary least squares estimation. The equation of logarithmic trend curve can be obtained as follows:.
In Figure 6 b , the low-frequency a4 captures the approximate data of SP on the logarithmic trend curve in a 3-dimension space, compared to traditional approximate data in Figure 6 a. It provides more information, including logarithmic growth trend curve of Y -axis and wavelet approximate data of Z -axis.
The 3-dimension space is composed of time as X -axis, logarithmic trend curve as Y -axis, and data d 1 as Z -axis. It provides more information, including regression growth trend of Y -axis and volatility of Z -axis.
In Figure 8 c , d 2 is turned to XOY plane. In Figure 9 , the third high-frequency data d3 captures volatility with a period length of 8 to 16 days on logarithmic trend curve. The interpretation is similar to Figure 7. Moreover, the fourth high-frequency data d4 captures volatility with a period length of 16 to 32 days on logarithmic trend curve in Figure These are multiresolution analysis of the data SP on logarithmic trend curve. The blue line indicates low-frequency data and high-frequency data d 1, d 2, d 3, and d 4.
And the red line indicates its logarithmic trend curve. By reconstruction algorithm and length-preserving projection, we reconstruct the low-frequency data a4, high-frequency data d 1, and high-frequency data d 2, d 3, and d 4 to obtain the reconstructed data shown in Figure A 3-dimension space is composed by time as X -axis, logarithmic trend curve as Y -axis, and data as Z -axis.
It provides more information, including logarithmic growth trend of Y -axis, reconstructed data of Z -axis, and error data of Z -axis. The red line indicates its logarithmic growth trend curve. According to the length-preserving projection and Euler discretization method, biorthogonal wavelet filters can be lifted onto logarithmic curves.
Under the length-preserving projection via Euler discretization scheme, biorthogonal wavelet filters on curves can be projected to biorthogonal wavelet filters on real axis. And the construction of biorthogonal wavelet filters on is equivalent to biorthogonal wavelet filters on by length-preserving projection. The properties of biorthogonal wavelet filters on are also discussed in the paper, and some conclusions are similar to biorthogonal wavelet filters on.
Moreover, an example and figures are given. These may have a new inspiration for dealing with some data on manifolds or manifolds learning in signal processing. Bitcoin transaction data were used to support this study. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors.
Read the winning articles. Journal overview. Special Issues. Academic Editor: Basil K. Received 16 Aug
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