数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (1): 39-46.doi: 10.1016/S0252-9602(10)60020-3
付应雄, 李落清
FU Ying-Xiong, LI Luo-Qing
摘要:
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 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=eiαB1(eix}B2(eix) and v takes the form ei(x2πN/b+α)B1(eix)B2(eix), where B1 and B2 are Blaschke products.
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