Signal reconstruction from two close fractional Fourier power spectra

T. Alieva, M.J. Bastiaans, L. Stankovic

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53 Citations (Scopus)
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Abstract

Based on the definition of the instantaneous frequency (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 permits 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 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 applications 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 languageEnglish
Pages (from-to)112-123
Number of pages12
JournalIEEE Transactions on Signal Processing
Volume51
Issue number1
DOIs
Publication statusPublished - 2003

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Signal reconstruction
Power spectrum
Derivatives
Signal analysis
Optics
Signal to noise ratio
Fourier transforms

Cite this

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title = "Signal reconstruction from two close fractional Fourier power spectra",
abstract = "Based on the definition of the instantaneous frequency (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 permits 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 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 applications 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.",
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Signal reconstruction from two close fractional Fourier power spectra. / Alieva, T.; Bastiaans, M.J.; Stankovic, L.

In: IEEE Transactions on Signal Processing, Vol. 51, No. 1, 2003, p. 112-123.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Signal reconstruction from two close fractional Fourier power spectra

AU - Alieva, T.

AU - Bastiaans, M.J.

AU - Stankovic, L.

PY - 2003

Y1 - 2003

N2 - Based on the definition of the instantaneous frequency (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 permits 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 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 applications 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.

AB - Based on the definition of the instantaneous frequency (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 permits 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 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 applications 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.

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JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

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