ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 3, pp. 197–223.
Pleiades Publishing, Inc., 2013.
Original Russian Text
H. Boche, U.J. M¨onich, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 3, pp. 3–31.
Characterization of the Peak Value Behavior
of the Hilbert Transform of Bounded
and U. J. M¨onich
Lehrstuhl f¨ur Theoretische Informationstechnik, Technische Universit¨at M¨unchen, Germany
Research Laboratory of Electronics, Massachusetts Institute of Technology, USA
Received January 15, 2013
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 B
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 B
, there is a more general
deﬁnition of the Hilbert transform, which is based on the abstract H
-BMO(R) duality. It was
recently shown in  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 B
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 B
of bounded bandlimited signals that vanish at inﬁnity. By studying the properties of the Hilbert
transform, we continue the work .
The peak value is a basic characteristic of signals. In many applications it is crucial to control
the peak value. For example, in wireless communication systems high peak-to-average power ratios
(PAPRs) are problematic because high peak values can overload power ampliﬁers, which in turn
leads to undesirable out-of-band radiation [3–5]. In this paper we analyze the asymptotic growth
behavior of the Hilbert transform for the space of bounded bandlimited signals, and thereby solve
the peak value problem of the Hilbert transform for this space.
The Hilbert transform is an important operation in numerous ﬁelds, in particular in communica-
tion theory and signal processing. For example, the “analytic signal” , which was used by Dennis
Gabor in his “Theory of Communication” , is based on the Hilbert transform. Further concepts
and theories in which the Hilbert transform is an integral part are the instantaneous amplitude,
phase, and frequency of a signal [6,8–13] and the theory of modulation [6, 14–16].
In an analytic signal ψ(t)=u(t)+iv(t), the imaginary part v is the Hilbert transform of the
real part u, i.e., v = Hu. Based on the analytic signal it is possible to deﬁne the instantaneous
The material in this paper was presented in part at the 2012 European Signal Processing Conference
Supported in part by the German Research Foundation (DFG), grant no. BO 1734/13-2.
Supported by the German Research Foundation (DFG), grant no. MO 2572/1-1.