On Two-Dimensional Hilbert Integral Equations, Generalized Minimum-Phase Signals, and Phase Retrieval

Shenoy, Basty Ajay and Mulleti, Satish and Seelamantula, Chandra Sekhar (2018) On Two-Dimensional Hilbert Integral Equations, Generalized Minimum-Phase Signals, and Phase Retrieval. In: IEEE TRANSACTIONS ON SIGNAL PROCESSING, 66 (14). pp. 3906-3917.

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Abstract

One-dimensional (1-D) causal signals admit Hilbert integral relations between the real and imaginary parts of their Fourier spectra. For 1-D minimum-phase signals, the log-magnitude and phase spectra also admit such Hilbert relations. In this paper, we extend these results to 2-D signals. We first establish the Hilbert integral equations for 2-D first-quadrant signals. For continuous-domain 2-D signals, we present the partial Hilbert transform relations between the real and imaginary parts of the spectrum. For the discrete-domain counterpart, we show that the partial Hilbert transform does not suffice, which motivates us to introduce the composite Hilbert transform. Considering the problem of phase retrieval, we show that, in both 1-D/2-D, and continuous/discrete domains, there exists a generalized class of signals, which subsumes the well-known class of I-D minimum-phase signals with a rational transfer function, for which Hilbert integral relations exist between the log-magnitude and phase spectra. We introduce them under the nomenclature of generalized minimum-phase signals. For this class of signals, phase retrieval is possible by using the Hilbert transform without the requirement of a rational transfer function. Furthermore, we show that a generalized minimum-phase signal admits a stable convolutional inverse, which also belongs to the same class. Simulation results demonstrating accurate reconstruction of 1-D and 2-D generalized minimum-phase signals from their magnitude spectra are presented to support the theoretical developments.

Item Type:	Journal Article
Publication:	IEEE TRANSACTIONS ON SIGNAL PROCESSING
Publisher:	IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC, 445 HOES LANE, PISCATAWAY, NJ 08855-4141 USA
Additional Information:	Copyright of this article belong to IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC, 445 HOES LANE, PISCATAWAY, NJ 08855-4141 USA
Department/Centre:	Division of Electrical Sciences > Electrical Engineering
Date Deposited:	16 Jul 2018 14:57
Last Modified:	16 Jul 2018 14:57
URI:	http://eprints.iisc.ac.in/id/eprint/60168

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