Abstract:

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(e^ix)| is known for x ∈[-π, π]. It is shown that the conditions |g(e^ix)|=|f(e^ix)| and |g(e^i(x+b))-g(e^ix)|=|f(e^i(x+b))-f(e^ix)|, b≠2π, together imply that either g=wf or g=vf, where both w and v have period b. Furthermore, if b/2π is irrational then the functions w and v reduce to some constants c₁ and c₂, respectively;  if b/2π is rational then  w takes the form w=e^iαB₁(e^ix}B₂(e^ix) and v takes the form e^{i(x2πN/b+α)}B₁(e^ix)B₂(e^ix), where B₁ and B₂ are Blaschke products.

Key words: phase retrieval, time-limited signal, Blaschke product, Hardy space on the unit circle