Phase Retrieval for Wide Band Signals
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(2020) 26:54
Phase Retrieval for Wide Band Signals Philippe Jaming1 · Karim Kellay1 · Rolando Perez III1,2 Received: 5 December 2019 / Revised: 17 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given f ∈ L 2 (R) with Fourier transform in L 2 (R, e2c|x| dx), we find all functions g ∈ L 2 (R) with Fourier transform in L 2 (R, e2c|x| dx), such that | f (x)| = |g(x)| for all x ∈ R. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these constraints force uniqueness of the solution. Keywords Phase retrieval · Hardy spaces Mathematics Subject Classification 30D05 · 30H10 · 42B10 · 94A12
1 Introduction The phase retrieval problem refers to the recovery of the phase of a function f using given data on its magnitude | f | and a priori assumptions on f . These problems are widely studied because of their physical applications in which the quantities involved are identified by their magnitude and phase, where the phase is difficult to measure while the magnitude is easily obtainable. Some physical applications of phase retrieval
Communicated by Gabriel Peyre.
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Philippe Jaming [email protected] Karim Kellay [email protected] Rolando Perez III [email protected]
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Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 33400 Talence, France
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Institute of Mathematics, University of the Philippines Diliman, 1101 Quezon City, Philippines 0123456789().: V,-vol
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problems include works related to astronomy [12], lens design [13], X-ray crystallography [30], inverse scattering [31], and optics [32]. More physical examples were given in the survey articles of Luke et al. [27] Klibanov et al. [25] and the book of Hurt [19]. A more recent overview of the phase retrieval problem was given by the article of Grohs et al. [17], which discussed a more general formulation of the phase retrieval problem using Banach spaces and bounded linear functionals, as well as results related to the uniqueness and stability properties of the problem. Phase retrieval problems have recently been given more interest because of progress in the discrete (finite-dimensional) case, starting with the work of Candès et al. [9] and of Waldspurger et al. [36], which formulated the phase retrieval problem as an optimization problem (though the investigation of optimization tools in phase retrieval algorithms seems to go back at least to the work of Luke et al. [5,8,27]). On the other hand, phase retrieval problems devoted to the continuous (infinite-dimensional) case have been solved in various settings, such as, for one-dimensional band-limited functions [1,2,37], for funct
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