Nonlinear Signal Decomposition with Bilinear Hilbert Transform: A Framework for Analytical Decision-Making Applications
DOI:
https://doi.org/10.22105/apmtm433Keywords:
Bilinear Hilbert transform, Bedrosian identity, Nonlinear Fourier atomsAbstract
This note emphasizes a point of view on the Bilinear Hilbert Transform (BHT) in the Bedrosian identity, which is originally based on the conventional Hilbert Transform in the theory of analytic signals. We show that the generalized sinc functions from Möbius play a crucial role. We have demonstrated some properties of BHT: Riesz representation theorem, boundedness, and Bedrosian identity for BHT. The generalized sinc function is a special solution of the Bedrosian identity. After that, we consider some of the nonlinear sinc functions, which are also the solutions of the Bedrosian identity. Lastly, the nonlinear sinc function system is orthonormal. In addition, we propose a framework for analytical decision-making applications that leverages the properties of BHT. incorporating this framework into existing real-world systems further enhances the adaptability and responsiveness of decision-making models, positioning the BHT as a critical tool for optimizing processes in dynamic, data-rich environments.
References
B. Boashash(1992). Estimating and interpreting the instantaneous frequency of a signal.I. Fundamentals. Proc. IEEE ,80, 417-430.https://doi.org/10.1109/5.135378
M. Lacey and C. Thiele(1997).Lp estimates for the bilinear Hilbert transform. Proc. Natl. Acad. Sci. USA, 94,33-35.https://doi.org/10.1073/pnas.94.1.33
M. Lacey and C.M. Thiele(1997). Lp estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. Math., 146, 693-724.https://doi.org/10.2307/2952458
A. Buc˘kovska and S.Pilipovic´(2002). An extension of bilinear Hilbert transform to distributions. Integral Transforms Spec.Funct., 13, 1-15.https://doi.org/10.1080/10652460212891
M. Lacey and C. Thiele(1999). On Calderon¡¯s conjecture.Ann. Math., 149, 475-496.https://doi.org/10.2307/120971
C. Muscalu, T. Tao, and C. Thiele(2002). Multi-linear operators given by singular multipliers. J. Amer. Math. Soc., 15, 469-496. https://doi.org/10.48550/arXiv.math/9910039
L. Cohen(1995). Time-frequency analysis. Control Engineering Practice 5(2),292–294.https://doi.org/10.1016/S0967-0661(97)90028-9
N. E. Huang at al(1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. London. A, 454, 903-995.https://doi.org/10.1098/rspa.1998.0193
I. Daubechies(1992). Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics,61.https://doi.org/10.1137/1.9781611970104
B. Picinbono(1997).On Instantaneous Amplitude and Phase of Signals. IEEE Transaction on Signal Processing, 45(3), 552-560.https://doi.org/10.1109/78.558469
T. Qian, Q.H. Chen and L.Q. Li(2005). Analytic unit quadrature signals with nonlinear phase. Physica D: Nonlinear Phenomena, 203, 80-87.https://doi.org/10.1016/j.physd.2005.03.005
T. Qian(2005). Characterization of boundary values of functions in Hardy spaces with applications in signal analysis. Journal of Integral Equations and Applications,17(2),159-198.https://doi.org/10.1216/jiea/1181075323
L. Grafakos(2008). Classical Fourier analysis.New York: Springer.https://doi.org/10.1007/978-0-387-09432-8
E. M. Stein and T. S. Murphy(1993). Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Vol. 3. Princeton University Press.https://doi.org/10.1515/9781400883929-015
Moises Venouziou and Haizhang Zhang(2001). Characterizing the Hilbert transform by the Bedrosian theorem. Journal of
Mathematical Analysis and Applications,49(80),2844-2852.https://doi.org/10.1016/j.jmaa.2007.05.067
Lihua Yang and Haizhang Zhang(2008). The Bedrosian identity for functions. Journal of Mathematical Analysis and Applications, 345 (2), 975-984.https://doi.org/10.1016/j.jmaa.2008.05.005
Metrics
