PHASE RETRIEVAL OF TIME-LIMITED SIGNALS

The problem of reconstructing a signal ψ（x） from its magnitude |ψ（x）| is of considerable interest to engineers and physicists.This article concerns the problem of determining a time-limited signal f with period 2π when |f（eix）| is known for x ∈ [-π,π].It is shown that the conditions |g（eix）| = |f（eix）| and |g（ei（x+b））-g（eix）| = |f（ei（x+b））-f（eix）|,b = 2π,together imply that either g = wf or g = vf,where both w and v have period b.Furthermore,if 2bπ is irrational then the functions w and v reduce to some constants c1 and c2,respectively;if 2bπ is rational then w takes the form w=eiαB1（eix）B2（eix） and v takes the form ei（x2πN/b+α）B1（eix）B2（eix）,where B1 and B2 are Blaschke products.