Application of the Continuous Method for Solving Operator Equations to the Approximate Solution to the Amplitude–Phase Problem

Abstract—

The article is devoted to approximate methods for solving the phase problem for one-dimensional and two-dimensional signals. The cases of continuous and discrete signals are considered. The solution of the phase problem consists of two stages. At the first stage, the original signal is reconstructed from the known amplitude of the spectrum. At the second stage, the Fourier transform of the reconstructed signal is calculated and the phase of the signal spectrum is calculated approximately. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations using the theory of stability of solutions to systems of ordinary differential equation. The method is stable under perturbations of the mathematical model parameters and, when solving nonlinear operator equations, does not require the reversibility of the Gateaux (or Frechet) derivatives of nonlinear operators. To restore the original signal, spline-collocation schemes with splines of the zeroth and first orders are proposed. Computing schemes are implemented by a continuous method for solving nonlinear operator equations.