Extension of the Hilbert transform
- 1 March 2012
- conference paper
- conference paper
- Published by Institute of Electrical and Electronics Engineers (IEEE)
- p. 3697-3700
- https://doi.org/10.1109/icassp.2012.6288719
Abstract
The Hilbert transform is an important operator in signal processing, e.g., the definition of the “analytical signal” uses the Hilbert transform. In this paper we analyze the Hilbert transform for bounded bandlimited signals in B ∞ π. Although the common integral representation of the Hilbert transform may diverge for certain signals in B ∞ π, it is possible to define the Hilbert transform meaningfully for bounded signals. We employ a definition that is based on the H 1 -BMO(ℝ) duality. The problem of this abstract definition is that there exists no constructive procedure to calculate the Hilbert transform. However, for the subspace of bounded bandlimited signals, we can give an explicit formula for the calculation of the Hilbert transform. Further, we show that the Hilbert transform of a bounded bandlimited signal is still bandlimited but not necessarily bounded.Keywords
This publication has 8 references indexed in Scilit:
- Hilbert TransformsPublished by Cambridge University Press (CUP) ,2009
- The Bedrosian identity for functionsJournal of Mathematical Analysis and Applications, 2008
- Theory of Analytic Modulation SystemsBell System Technical Journal, 1978
- Analytic signals and product theorems for Hilbert transformsIEEE Transactions on Circuits and Systems, 1974
- Characterizations of bounded mean oscillationBulletin of the American Mathematical Society, 1971
- Toward a unified theory of modulation part I: Phase-envelope relationshipsProceedings of the IEEE, 1966
- Some properties of functions of exponential typeBulletin of the American Mathematical Society, 1938
- Some theorems on Fourier transforms and conjugate trigonometric integralsTransactions of the American Mathematical Society, 1936