PROJECT TITLE :
On Two-Dimensional Hilbert Integral Equations, Generalized Minimum-Phase Signals, and Phase Retrieval - 2018
One-dimensional (1-D) causal signals admit Hilbert integral relations between the important and imaginary components of their Fourier spectra. For one-D minimum-part signals, the log-magnitude and phase spectra additionally admit such Hilbert relations. During this Project, we have a tendency to extend these results to a pair of-D signals. We 1st establish the Hilbert integral equations for 2-D initial-quadrant signals. For continuous-domain a pair of-D signals, we have a tendency to gift the partial Hilbert transform relations between the important and imaginary parts of the spectrum. For the discrete-domain counterpart, we show that the partial Hilbert rework will not suffice, that motivates us to introduce the composite Hilbert transform. Considering the problem of part retrieval, we tend to show that, in both 1-D/2-D, and continuous/discrete domains, there exists a generalized class of signals, that subsumes the well-known category of 1-D minimum-section signals with a rational transfer perform, for which Hilbert integral relations exist between the log-magnitude and phase spectra. We tend to introduce them below the nomenclature of generalized minimum-section signals. For this class of signals, part retrieval is doable by using the Hilbert transform while not the necessity of a rational transfer function. Furthermore, we have a tendency to show that a generalized minimum-part signal admits a stable convolutional inverse, that additionally belongs to the same class. Simulation results demonstrating correct reconstruction of 1-D and 2-D generalized minimum-phase signals from their magnitude spectra are presented to support the theoretical developments.
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