# Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty$ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_\pi ^\infty$ , there is a more general definition of the Hilbert transform, which is based on the abstract H 1-BMO(ℝ) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty$ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi ,0}^\infty$ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

, Volume 49 (3) – Oct 15, 2013
27 pages

/lp/springer_journal/characterization-of-the-peak-value-behavior-of-the-hilbert-transform-0XxgumbFrG
Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946013030010
Publisher site
See Article on Publisher Site

### Abstract

The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty$ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_\pi ^\infty$ , there is a more general definition of the Hilbert transform, which is based on the abstract H 1-BMO(ℝ) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty$ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi ,0}^\infty$ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2].

### Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 15, 2013

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