3 April 1997 Hermitian hat wavelet design for singularity detection in the Paraguay river-level data analyses
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Abstract
The direct differentiation of a noisy signal ds/dt is known to be inaccurate. Differentiation can be improved by employing the Dirac (delta) -function introduced into a convolution product denoted by (direct product) and then integrated by parts: ds/dt equals ds/dt (direct product) (delta) equals - s (direct product) d(delta) /dt. The Schwartz Gaussian representation of the delta function is then explicitly used in the differentiation. It turns out that such a convolution approach to the first and the second derivatives produces a pair of mother wavelets the combination of which is the complex generalization of the Mexican hat called a Hermitian hat wavelet. It is shown that the Hermitian filter is a single oscillation wavelet having much lower frequency bandwidth than the Mortlet or Gabor wavelet. As a result of Nyquist theorem, a fewer number of grid points would be needed for the discrete convolution operation. Therefore, the singularity characteristic will not be overly smeared and the noise can be smoothed away. The phase plot of the Hermitian wavelet transform in terms of the time scale and frequency domains reveal a bifurcation discontinuity of a noisy cusp singularity at the precise location of the singularity as well as the scale nature of the underlying dynamics. This phase plot is defined as (theta) (t/a) equals tan-1 [(ds/dt)/(-d2s/dt2] equals tan-1 [((d(delta) (t/a)dt) (direct product) s)/((d2(delta) (t/a)/dt2) (direct product) s)] applied to a real world data of the Paraguay river levels.
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Harold H. Szu, Harold H. Szu, Charles C. Hsu, Charles C. Hsu, Leonardo Deane Sa, Leonardo Deane Sa, Weigang Li, Weigang Li, } "Hermitian hat wavelet design for singularity detection in the Paraguay river-level data analyses", Proc. SPIE 3078, Wavelet Applications IV, (3 April 1997); doi: 10.1117/12.271774; https://doi.org/10.1117/12.271774
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