Phase retrieval of time-limited signals

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 i x )| is known for x ∈[−π,π]. It is shown that the conditions | g(e i x )| = | f(e i x )| and | g(e i( x+ b) )− g(e i x )|= f(e i( x+ b) )− f(e i x )|, b≠ 2π, together imply that either g = wf or g = v f ¯ , where both w and v have period b. Furthermore, if b 2 π is irrational then the functions w and v reduce to some constants c 1 and c 2, respectively; if b 2 π is rational then w takes the form w = e i a B 1 ( e i x ) B 2 ( e i x ) ¯ and v takes the form e i ( x 2 π N / b + a ) B 1 ( e i x ) B 2 ( e i x ) ¯ , where B 1 and B 2 are Blaschke products.