Phase Retrieval Based on Multiple Fractional Fourier Transform Magnitudes

Phase retrieval is a problem of great importance in various fields in engineering. Inspired by the implementation of Fractional Fourier Transform (FRFT) and its applications in optics, we consider FRFT phase retrieval problem and give according method. Our method can be concluded as using matrix completion method with multiple magnitudes of FRFT to reconstruct the signal. We emphasize that the choice of the rotation angles of FRFT will influence the uniqueness of phase retrieval and illustrate this with a counterexample. Then through numerical tests our method is proved effective in both noised and noise-free situations. And we show that if the rotation angles of FRFTs are not well selected, the numerical test could be invalid because of loss of uniqueness. Furthermore, 4-FRFT method and 5-FRFT method are compared, and 5-FRFT method produces more accurate reconstruction when processing noised signals.