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

In this paper, we discuss how to partially determine the Fourier transform F ⁢ (z) = ∫ - 1 1 f ⁢ (t) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ (z) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ (z) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}}. Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support 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) {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. [ABSTRACT FROM AUTHOR]