### Abstract

Based on the definition of the instantaneous fre
quency (signal phase derivative) as a local moment of the Wigner
distribution, we derive the relationship between the instantaneous
frequency and the derivative of the squared modulus of the
fractional Fourier transform (fractional Fourier transform power
spectrum) with respect to the angle parameter. We show that the
angular derivative of the fractional power spectrum can be found
from the knowledge of two close fractional power spectra. It per
mits us to find the instantaneous frequency and to solve the phase
retrieval problem up to a constant phase term, if only two close
fractional power spectra are known. The proposed technique is
noniterative and noninterferometric. The efficiency of the method
is demonstrated on several examples including monocomponent,
multicomponent, and noisy signals. It is shown that the proposed
method works well for signal-to-noise ratios (SNRs) higher than
about 3 dB. The appropriate angular difference of the fractional
power spectra used for phase retrieval depends on the complexity
of the signal and can usually reach several degrees. Other applica
tions of the angular derivative of the fractional power spectra for
signal analysis are discussed briefly. The proposed technique can
be applied for phase retrieval in optics, where only the fractional
power spectra associated with intensity distributions can be easily
measured.

Original language | English |
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Title of host publication | Selected Papers on Phase-Space Optics |

Editors | M.E. Testorf, J. Ojeda-Castañeda, A.W. Lohmann |

Place of Publication | Bellingham |

Publisher | SPIE |

Pages | 566-577 |

Number of pages | 12 |

ISBN (Print) | 0-8194-6353-1 |

Publication status | Published - 2006 |

### Publication series

Name | SPIE Milestone series |
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Volume | MS 181 |

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## Cite this

Alieva, T., Bastiaans, M. J., & Stankovic, L. (2006). Signal reconstruction from two close fractional Fourier power spectra. In M. E. Testorf, J. Ojeda-Castañeda, & A. W. Lohmann (Eds.),

*Selected Papers on Phase-Space Optics*(pp. 566-577). (SPIE Milestone series; Vol. MS 181). SPIE.