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 Probl. Peredachi Inf., 2012, Volume 48, Issue 3, Pages 23–46 (Mi ppi2083)

Information Theory

On the Hilbert transform of bounded bandlimited signals

H. Boche, U. Mönich

Technische Universität München, Germany

Abstract: In this paper we analyze the Hilbert transform and existence of the analytical signal for the space $\mathcal B_\pi^\infty$ of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in $L^2(\mathbb R)$ and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in $\mathcal B_\pi^\infty$ and that the Hilbert transform is not a bounded operator on $\mathcal B_\pi^\infty$, it is nevertheless possible to define the Hilbert transform for the space $\mathcal B_\pi^\infty$. We use a definition that is based on the $\mathcal H^1$$\mathrm{BMO}(\mathbb R)$ duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in $\mathcal B_\pi^\infty$ is still bandlimited but not necessarily bounded. With these results we continue the work of [1,2].

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English version:
Problems of Information Transmission, 2012, 48:3, 217–238

Bibliographic databases:

UDC: 621.391.1+517

Citation: H. Boche, U. Mönich, “On the Hilbert transform of bounded bandlimited signals”, Probl. Peredachi Inf., 48:3 (2012), 23–46; Problems Inform. Transmission, 48:3 (2012), 217–238

Citation in format AMSBIB
\Bibitem{BocMon12} \by H.~Boche, U.~M\"onich \paper On the Hilbert transform of bounded bandlimited signals \jour Probl. Peredachi Inf. \yr 2012 \vol 48 \issue 3 \pages 23--46 \mathnet{http://mi.mathnet.ru/ppi2083} \transl \jour Problems Inform. Transmission \yr 2012 \vol 48 \issue 3 \pages 217--238 \crossref{https://doi.org/10.1134/S0032946012030027} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000310208200002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84870678837} 

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• http://mi.mathnet.ru/eng/ppi/v48/i3/p23

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. H. Boche, U. J. Mönich, “Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals”, Problems Inform. Transmission, 49:3 (2013), 197–223
2. Wunder G., Fischer R.F.H., Boche H., Litsyn S., No J.-S., “The Papr Problem in Ofdm Transmission”, IEEE Signal Process. Mag., 30:6 (2013), 130–144
3. Boche H., Moenich U.J., “The Structure of Bandlimited Bmo-Functions and Applications”, J. Funct. Anal., 264:12 (2013), 2637–2675
4. Boche H., Moenich U.J., “Characterization of the Range of the Hilbert Transform for Bounded Bandlimited Signals and Applications”, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), International Conference on Acoustics Speech and Signal Processing ICASSP, IEEE, 2013, 5388–5391
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