TY - JOUR

T1 - Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

AU - Boche, H.

AU - Mönich, U. J.

N1 - Funding Information:
1The material in this paper was presented in part at the 2012 European Signal Processing (EUSIPCO-2012). 2 Supported in part by the German Research Foundation (DFG), grant no. BO 1734/13-2. 3 Supported by the German Research Foundation (DFG), grant no. MO 2572/1-1.

PY - 2013

Y1 - 2013

N2 - 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 Boπ 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 Boπ, 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 Boπ 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 Bπ,0∞ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2].

AB - 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 Boπ 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 Boπ, 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 Boπ 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 Bπ,0∞ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2].

UR - http://www.scopus.com/inward/record.url?scp=84888327659&partnerID=8YFLogxK

U2 - 10.1134/S0032946013030010

DO - 10.1134/S0032946013030010

M3 - Article

AN - SCOPUS:84888327659

SN - 0032-9460

VL - 49

SP - 197

EP - 223

JO - Problems of Information Transmission

JF - Problems of Information Transmission

IS - 3

ER -