ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 3, pp. 217–238.
Pleiades Publishing, Inc., 2012.
Original Russian Text
H. Boche, U.J. M¨onich, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 3, pp. 23–46.
On the Hilbert Transform
of Bounded Bandlimited Signals
H. Boche and U. J. M¨onich
Lehrstuhl f¨ur Theoretische Informationstechnik, Technische Universit¨at M¨unchen, Germany
Received September 14, 2011
Abstract—In this paper we analyze the Hilbert transform and existence of the analytical
signal for the space B
of bandlimited signals that are bounded on the real axis. Originally,
the theory was developed for signals in L
(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 B
and that the Hilbert transform is not a bounded operator on B
nevertheless possible to deﬁne the Hilbert transform for the space B
. We use a deﬁnition that
is based on the H
–BMO(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 B
is still bandlimited but not necessarily
bounded. With these results we continue the work of [1,2].
In signal processing and communication theory, the Hilbert transform is an important operator
with many applications [1, 3–5]. For example, the calculation of the analytical signal, which was
introduced in , requires the Hilbert transform. In an analytical signal
the imaginary part v is the Hilbert transform of the real part u. Based on the analytical signal it is
possible to deﬁne the instantaneous amplitude and frequency of a signal [1,5]. Although there are
other possibilities to deﬁne the instantaneous amplitude and frequency [5, 6], it was shown in 
that the only deﬁnition that satisﬁes certain physical requirements is the deﬁnition based on the
Hilbert transform and the analytical signal.
Classically, the Hilbert transform of a signal f is deﬁned as the principal value integral:
t − τ
t − τ
t − τ
Of course, the above deﬁnition of the Hilbert transform only makes sense if the integral converges.
Numerous authors have studied the convergence behavior of the integral, and it is known that
Supported in part by the German Research Foundation (DFG), grant nos. BO 1734/13-2 and BO