Application of the continuous method for solving operator equations to the approximate solution of the amplitude-phase problem

Abstract

Background. 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 restored. At the second stage, the Fourier transform of the reconstructed signal is calculated and the phase of the signal spectrum is approximately calculated. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gato (or Freshe) derivatives of nonlinear operators. Results and conclusions. To restore the original signal, spline-collocation schemes with splines of zero and first orders are proposed. Computing schemes are implemented by a continuous method for solving nonlinear operator equations.

Keywords

• amplitude-phase problem
• ill-posed problems
• continuous method for solving operator equations
• numerical methods