Phase retrieval of finite Blaschke projection

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 reconstructP(f)= n-ary sumation k=0 infinity⟨f,B{a0,a1, horizontal ellipsis ,ak}⟩B{a0,a1, horizontal ellipsis ,ak}by the intensity measurements{|⟨f,Bk1⟩|,|⟨f,Bk2⟩|,|⟨f,Bk3⟩|:k >= 1}, whereflies in Hardy spaceScript capital H2(D)such thatf(a(0))=0,B{a0,a1, horizontal ellipsis ,ak}andBkiare all the finite MBPs. We establish the condition onBkisuch thatP(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.