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Phase retrieval of finite Blaschke projection
Author(s) -
Li Youfa,
Zhou Chunxu
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6603
Subject(s) - phase retrieval , unimodular matrix , mathematics , scalar (mathematics) , fourier transform , hardy space , phase (matter) , projection (relational algebra) , generalization , intensity (physics) , phase space , mathematical analysis , algorithm , pure mathematics , geometry , optics , physics , quantum mechanics
Phase retrieval by Fourier measurements is a classical application in coherent diffraction imaging, and the modified Blaschke products (MBPs) are the generalization of linear Fourier atoms. Motivated by this, we investigate the phase retrieval modeled as to reconstruct P ( f ) = ∑ k = 0 ∞ ⟨ f , B { a 0 , a 1 , … , a k } ⟩ B { a 0 , a 1 , … , a k }by the intensity measurements { | ⟨ f , Bk 1⟩ | , | ⟨ f , Bk 2⟩ | , | ⟨ f , Bk 3⟩ | : k ≥ 1 } , where f lies in Hardy spaceℋ 2 ( D ) such that f ( a 0 )=0,B { a 0 , a 1 , … , a k }andBk iare all the finite MBPs. We establish the condition onBk isuch that P ( f ) can be determined, up to a unimodular scalar, by the above measurements. A byproduct of our result is that the instantaneous frequency of the target can be exactly reconstructed by the above intensity measurements. Moreover, a recursive algorithm for the phase retrieval is established. Numerical simulations are conducted to verify our result.