Real phase retrieval from unordered partial frame coefficients
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Real phase retrieval from unordered partial frame coefficients Fusheng Lv1 · Wenchang Sun1
Received: 13 April 2017 / Accepted: 21 September 2017 © Springer Science+Business Media, LLC 2017
Abstract We study the signal recovery from unordered partial phaseless frame coefficients. To this end, we introduce the concepts of m-erasure (almost) phase retrievable frames. We show that with an m-erasure (almost) phase retrievable frame, it is possible to reconstruct (almost) all n-dimensional real signals up to a sign from their arbitrary N − m unordered phaseless frame coefficients, where N stands for the element number of the frame. We give necessary and sufficient conditions for a frame to be m-erasure (almost) phase retrievable. Moreover, we give an explicit construction of such frames based on prime numbers. Keywords Phase retrieval · Frames · Erasure recovery Mathematics Subject Classification (2010) 42C15 · 46C05
1 Introduction The problem of signal reconstruction is important in engineering and sciences. When we recover signals from measurements, some information might be lost due to
Communicated by: Gitta Kutyniok This work was partially supported by the National Natural Science Foundation of China (11371200, 11525104 and 11531013). Wenchang Sun
[email protected] Fusheng Lv [email protected] 1
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
F. Lv, W. Sun
various reasons. For instance, in speech processing [4, 26], X-ray crystallography [14, 25] and quantum communication [16, 17], we need to recover signals without the phase information since the measurement apparatus can not record it. In literatures, this is referred to the problem of phase retrieval. On the other hand, erasures occur often in digital transmission since the network is not stable. And many works [7, 11, 12, 19–21, 23, 24] have been done to recover signals from erased frame coefficients. In 2006, Balan, Casazza and Edidin [1] studied the phase retrieval problem for general frames. Since then, it has received considerable attention and various related problems have been studied. In particular, there are many works on the minimal number of measurements [1, 3, 6, 8, 13, 15, 16, 22, 27, 28], recovery algorithms [8–10] and conditions for the invertibility and robustness for the reconstruction [2, 3, 18]. And in [5], Bodmann, Casazza, Edidin and Balan showed that phase retrieval is also possible if some frame coefficients are lost. Let f be a signal in an n-dimension Hilbert space H. The phase retrieval problem N is to recover f from phaseless measurements {|f, ϕj |}N j =1 , where {ϕj }j =1 is a frame in H. Note that for any unimodular ω in T = {ω : |ω| = 1}, {|ωf, ϕj |}N j =1 = N N N {|f, ϕj |}j =1 . Hence the map A : H/T → R+ , A(f ) = {|f, ϕj |}j =1 is well defined. We say that {ϕj }N j =1 is phase retrievable for H if A is injective. It was shown in [1] that N = 2n − 1 is the minimal number for {ϕj }N j =1 to be phase retrievable for n n R . For the case of H = C , the question becomes co
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