Regularity between Wavelet Transform of Output Signal and Input Signal or Transmitting System. On Daubechies' Orthonormal Wavelets.
Akira Sone, Shizuo Yamamoto, Arata MASUDA, Akira Nakaoka, Ryuichi Ashino · TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series C · 1995
The singularities of the output signals from a physical system depend upon both the singularities of input signals and the regularizing property of the physical system. Wavelet analysis is adequate for analysis of these singularities. In this paper, wavelet analysis, using Daubechies' compactly supported orthonormal wavelet basis, is applied to signals having impulse singularities in certain derivatives. To analyze singularities of this kind, the wavelet function must have regularity and vanishing moments. Daubechies' wavelet functions are parameterized by N, which is related to the regularities of wavelets and to their vanishing moments. For several signals having an impulse singularity in certain derivatives, the minimum integer N for analysis of singularities of this kind is determined by numerical calculations, and for several output signals passing once or twice through, a transmitting system expressed by a second-order differential equation, the minimum integer N for analysis of singularities of this kind is also determined by numerical calculations. From these examples, we conclude that choosing appropriate N, the parameter of Daubechies' wavelet function, her compactly supported orthonormal wavelet basis, is applicable for detection of impulse singularities in certain derivatives of input signals if both the regularizing property of the physical system and impulse singularities in certain derivatives of their output signals are known.