Inequalities in Mellin-Fourier signal analysis. (English) Zbl 1046.94001
Debnath, Lokenath, Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhäuser (ISBN 0-8176-4104-1/hbk). Applied and Numerical Harmonic Analysis, 289-319 (2001).
A modified Mellin transform is the so-called scale transform of \(x\in L^2(\mathbb{R}_+)\) defined by
\[
\widehat x(s)= \int^\infty_0 x(f) f^{2\pi si- 1/2} df\qquad (s\in\mathbb{R}).
\]
This transform is a natural complement to the Fourier transform for wideband analytic signals. In this paper, limitations for the simultaneous localization of \(x\) and \(\widehat x\) are investigated. Several inequalities are discussed, based on various measures of spread (Heisenberg-type inequalities for variance-like measures, Hirschman-type inequalities for entropy). The same issue of maximally concentrating a signal in both scale and frequency domains is also addressed via spread measures, which are applied directly to joint scale-frequency distributions. Finally, a new uncertainty relation for the wavelet transform is presented.
For the entire collection see [Zbl 0996.00017].
For the entire collection see [Zbl 0996.00017].
Reviewer: Manfred Tasche (Rostock)
MSC:
94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
94A17 | Measures of information, entropy |
42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
44A15 | Special integral transforms (Legendre, Hilbert, etc.) |