Phase retrieval of time-limited signals

The problem of reconstructing a signal ϕ(x) from its magnitude |ϕ(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 , where both w and v have period b. Furthermore, if is irrational then the functions w and v reduce to some constants c1 and c2, respectively; if is rational then w takes the form and v takes the form , where B1 and B2 are Blaschke products.