TY - GEN

T1 - Solution of the peak value problem for Gabor's theory of communication

AU - Boche, Holger

AU - Monich, Ullrich J.

PY - 2012

Y1 - 2012

N2 - Since, for certain bounded signals, the common integral definition of the Hilbert transform may diverge, it was long thought that the Hilbert transform does not exist for general bounded signals. However, using a definition that is based on the H1-BMO(R) duality, it is possible to define the Hilbert transform meaningfully for the space of bounded signals. Unfortunately, this abstract definition gives no constructive procedure for the calculation of the Hilbert transform. However, if the signals are additionally bandlimited, i.e., if we consider signals in Bπ∞, it was recently shown that an explicit formula for the calculation of the Hilbert transform does exist. Based on this result, we analyze the asymptotic growth behavior of the Hilbert transform of signals in Bπ∞ and solve the peak value problem of the Hilbert transform. It is shown that the order of growth of Hilbert transform of signals in Bπ∞ is at most logarithmic.

AB - Since, for certain bounded signals, the common integral definition of the Hilbert transform may diverge, it was long thought that the Hilbert transform does not exist for general bounded signals. However, using a definition that is based on the H1-BMO(R) duality, it is possible to define the Hilbert transform meaningfully for the space of bounded signals. Unfortunately, this abstract definition gives no constructive procedure for the calculation of the Hilbert transform. However, if the signals are additionally bandlimited, i.e., if we consider signals in Bπ∞, it was recently shown that an explicit formula for the calculation of the Hilbert transform does exist. Based on this result, we analyze the asymptotic growth behavior of the Hilbert transform of signals in Bπ∞ and solve the peak value problem of the Hilbert transform. It is shown that the order of growth of Hilbert transform of signals in Bπ∞ is at most logarithmic.

UR - http://www.scopus.com/inward/record.url?scp=84873128594&partnerID=8YFLogxK

U2 - 10.1109/ISSSE.2012.6374340

DO - 10.1109/ISSSE.2012.6374340

M3 - Conference contribution

AN - SCOPUS:84873128594

SN - 9781467344555

T3 - Conference Proceedings of the International Symposium on Signals, Systems and Electronics

BT - 2012 International Symposium on Signals, Systems, and Electronics, ISSSE 2012

T2 - 2012 International Symposium on Signals, Systems, and Electronics, ISSSE 2012

Y2 - 3 October 2012 through 5 October 2012

ER -