Abstract
We consider the discrete form of the one-dimensional phase retrieval (1-D DPhR) problem from the point of view of input magnitude data. The direct method can provide a solution to the 1-D DPhR problem if certain conditions are satisfied by the input magnitude data, namely the corresponding trigonometric polynomial must be nonnegative. To test positivity of a trigonometric polynomial a novel DFT-based criterion is proposed. We use this DFT criterion for different sets of input magnitude data to evaluate whether the direct method applied to the 1-D DPhR problem leads to a solution in all explored cases.
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Notes
L should be much greater than N; as we shall see in Sect. 5, \(L=2^{13}N\) is a good choice.
References
Hurt, N.E.: Phase Retrieval and Zero Crossings. Kluwer Academic Publishers, Dordrecht (1989)
Salman, M.S., Eleyan, A., Deprem, Z., Çetin, A.E.: Phase retrieval of sparse signals from Fourier Transform magnitude using non-negative matrix factorization. In: IEEE GlobalSIP 2013, Austin, TX, USA, pp. 1113–1116, 3–5 Dec 2013
Quatieri, T.F., Oppenheim, A.V.: Iterative techniques for minimum phase signal reconstruction from phase or magnitudes. IEEE Trans. Acoust. Speech Signal Process. ASSP–29(6), 1187–1193 (1981)
Yegnanarayana, B., Saikia, D.K., Krishnan, T.R.: Significance of group delay functions in signal reconstruction from spectral magnitude and phase. IEEE Trans. Acoust. Speech Signal Process. ASSP–32, 610–623 (1984)
Dumitrescu, B.: Positive Trigonometric Polynomials and Signal Processing Applications. Springer, Dordrecht (2007)
Cadzow, J.A., Sun, Y.: Sequences with positive semidefinite Fourier transforms. IEEE Trans. Acoust. Speech Signal Process. ASSP–34(6), 1502–1510 (1986)
Rusu, C., Astola, J.: About positive trigonometric polynomials and 1-D discrete phase retrieval problem. In: Proceedings of EUSIPCO 2012, Bucharest, pp. 1169–1173, 27–31 Aug 2012
Rusu, C., Astola, J.: About magnitude input data in 1-D discrete phase retrieval problem. In: Proceedings of EUSIPCO 2007, Poznan, pp. 611–615. 3–7 Sept 2007
Siljak, D.D.: Algebraic criteria for positive realness relative to the unit circle. J. Frankl. Inst. 295(6), 469–476 (1973)
Henrici, P.: Applied and Computational Complex Analysis, vol. I: Power Series-Integration-Conformal Mapping-Location of Zeros. Wiley, New-York (1974)
Rusu, C., Astola, J.: A note on argument principle. IEEE Signal Process. Lett. 11(10), 817–820 (2004)
Acknowledgments
The work of first author has been supported by Grant PAV3M PN-II-PT-PCCA-2013-4-1762.
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Rusu, C., Astola, J. On the existence of the solution for one-dimensional discrete phase retrieval problem. SIViP 11, 195–202 (2017). https://doi.org/10.1007/s11760-016-0919-0
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DOI: https://doi.org/10.1007/s11760-016-0919-0