The minimal measurement number for generalized conjugate phase retrieval
Given a complex signal \(x\in C^n\), written as a row vector, this paper deals with the problem, to recover \(x\) from \(m\) measurements of the form \(x^*A_kx\; k=1,\ldots,m\). In particular, the author investigates lower bounds \(m^ast\) for the number of these measurements. Here, the \(A_k\) are real symmetric matrices, \(x^ast=\overline{x}^T\) and recovery means that \(x\) may be reconstructed up to a global phase factor \(e^{i\theta}\) and complex conjugation. In this sense, the question is to recover equivalence classes of signals, where \(x\equiv y\) iff \(y=e^{i\theta}x\) and/or \(y=\overline{x}\). The paper generalizes a similar problem formulated in [\textit{L. Evans} and \textit{C.-K. Lai}, Linear Algebra Appl. 587, 45--69 (2020; Zbl 1437.42048)], where \(A_k=a_k\cdot a_k^T\), with real \(m\)-dimensional column vectors \(a_k\). A formulation of a corresponding continuous problem may be found in [\textit{C.-K. Lai} et al., J. Fourier Anal. Appl. 27, No. 6, Paper No. 89, 23 p. (2021; Zbl 1479.94089)]. The author provides estimates for \(m^ast\) and gives an example for \(m^ast=9\) and \(n=4\). The main tool for obtaining the results is an equivalent formulation of the phase retrieval problem given in theorem 3.1. The formulation relates the recovery problem to special properties of certain bilinear forms of the \(A_k\).
scientific article; zbMATH DE number 7490485
scientific article