Abstract
The generalized conjugate phase retrieval problem aims to reconstruct a complex signal x ∈ ℂn from quadratic measurements x* A1x,…, x* Amx, where A1,…, Am ∈ ℝn×n are real symmetric matrices. The equivalent formulation for generalized conjugate phase retrieval along with the minimal measurement number required for accurate retrieval (up to a global phase factor as well as conjugacy) are derived in this paper. We present a set of nine vectors in ℝ4 and prove that it is conjugate phase retrievable on ℂ4. This result implies the measurement number bound 4n − 6 is not optimal for some n, which confirms a conjecture in the article by Evans and Lai (2019).
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11701098), China Scholarship Council Grant (Grant No. 201908440044), Natural Science Foundation of Guangdong (Grant No. 2016A030313710) and Science and Technology Program of Guangzhou (Grant No. 201607010170). The author thanks Professor Changqing Cheng for hospitality and generosity during the visit in State University of New York at Binghamton. The author also thanks the reviewers for their thorough and helpful comments.
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Dan, W. The minimal measurement number for generalized conjugate phase retrieval. Sci. China Math. 65, 655–664 (2022). https://doi.org/10.1007/s11425-020-1757-6
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DOI: https://doi.org/10.1007/s11425-020-1757-6