The problem of reconstructing a signalφ(x) from its magnitude |φ(x)] isof considerable interest to engineers and physicists. This article concerns the problem of determining a time-limited signal f with period ...The problem of reconstructing a signalφ(x) from its magnitude |φ(x)] isof considerable interest to engineers and physicists. This article concerns the problem of determining a time-limited signal f with period 2π when |f(eix)l is known for x∈[-π,π]. It is shown that the conditions |g(eix)| = |f(eix)| and |g(ci(x+b)) -g(eix)| =f(ei(x+b)) - f(eix)|, b ≠ 27π, together imply that either g = wf or g = v f, where both w and v have period b. Furthermore, if b/2π is irrational then the functions w and v b is rational then w takes the form reduce to some constants c1 and c2, respectively; ifb/2π is rational then w takes the form w=elexB1(e1x)B2(elx)and v takes the form ei(x2πN/b+a)B1(elx)B2(elx),where B1 and B2 are Blaschke products.展开更多
基金Supported by Foundation of Hubei Educational Committee (Q20091004)NSFC (10771053)+1 种基金the National Research Foundation for the Doctoral Program of Higher Education of China (SRFDP) (20060512001)Natural Science 373 Foundation of Hubei Province (2007ABA139)
文摘The problem of reconstructing a signalφ(x) from its magnitude |φ(x)] isof considerable interest to engineers and physicists. This article concerns the problem of determining a time-limited signal f with period 2π when |f(eix)l is known for x∈[-π,π]. It is shown that the conditions |g(eix)| = |f(eix)| and |g(ci(x+b)) -g(eix)| =f(ei(x+b)) - f(eix)|, b ≠ 27π, together imply that either g = wf or g = v f, where both w and v have period b. Furthermore, if b/2π is irrational then the functions w and v b is rational then w takes the form reduce to some constants c1 and c2, respectively; ifb/2π is rational then w takes the form w=elexB1(e1x)B2(elx)and v takes the form ei(x2πN/b+a)B1(elx)B2(elx),where B1 and B2 are Blaschke products.