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Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

Chen Lung-Hui

Journal of Applied Analysis

2021

Vol. 27, nr 2

259--268

In this paper, we discuss how to partially determine the Fourier transform F(z) = 1 ∫ −1 f(t)eizt dt, z ∈ ℂ, given the data |F(z)| or arg F(z) for z ∈ ℝ. Initially, we assume [−1, 1] to be the convex hull of the suport of the signal f . We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F(z). Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness.We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform.We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.