Parseval Relationship of Samples in the Fractional Fourier Transform Domain

This paper investigates the Parseval relationship of samples associated with the fractional Fourier transform. Firstly, the Parseval relationship for uniform samples of band-limited signal is obtained. Then, the relationship is extended to a general set of nonuniform samples of band-limited signal associated with the fractional Fourier transform. Finally, the two dimensional case is investigated in detail, it is also shown that the derived results can be regarded as the generalization of the classical ones in the Fourier domain to the fractional Fourier transform domain.

[1]  Gozde Bozdagi Akar,et al.  Digital computation of the fractional Fourier transform , 1996, IEEE Trans. Signal Process..

[2]  A. Zayed.  On the relationship between the Fourier and fractional Fourier transforms , 1996, IEEE Signal Processing Letters.

[3]  A. Luthra,et al.  Extension of Parseval's relation to nonuniform sampling , 1988, IEEE Trans. Acoust. Speech Signal Process..

[4]  Juliano B. Lima,et al.  The fractional Fourier transform over finite fields , 2012, Signal Process..

[5]  Ran Tao,et al.  Spectral Analysis and Reconstruction for Periodic Nonuniformly Sampled Signals in Fractional Fourier Domain , 2007, IEEE Transactions on Signal Processing.

[6]  A.I. Zayed,et al.  A convolution and product theorem for the fractional Fourier transform , 1998, IEEE Signal Processing Letters.

[7]  Ran Tao,et al.  The Poisson sum formulae associated with the fractional Fourier transform , 2009, Signal Process..

[8]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

[9]  Lutfiye Durak-Ata,et al.  The discrete fractional Fourier transform based on the DFT matrix , 2011, Signal Process..

[10]  Ahmed I. Zayed.  Hilbert transform associated with the fractional Fourier transform , 1998, IEEE Signal Processing Letters.

[11]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[12]  Qiwen Ran,et al.  Fractionalisation of an odd time odd frequency DFT matrix based on the eigenvectors of a novel nearly tridiagonal commuting matrix , 2011 .

[13]  Deyun Wei,et al.  Generalized Sampling Expansion for Bandlimited Signals Associated With the Fractional Fourier Transform , 2010, IEEE Signal Processing Letters.

[14]  Soo-Chang Pei,et al.  Two dimensional discrete fractional Fourier transform , 1998, Signal Process..

[15]  Ran Tao,et al.  Short-Time Fractional Fourier Transform and Its Applications , 2010, IEEE Transactions on Signal Processing.

[16]  Xiang-Gen Xia,et al.  On bandlimited signals with fractional Fourier transform , 1996, IEEE Signal Processing Letters.

[17]  K. Deergha Rao,et al.  Mitigation of UWB Signals Spectral Leakage to the GPS L1 Band via Filtering After Clipping in the UWB Transmitter , 2009 .

[18]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 2000, IEEE Trans. Signal Process..

[19]  Di Xiao,et al.  The parameters estimation and the feature extraction of underwater transient signal , 2011, 2011 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC).

[20]  Tomaso Erseghe,et al.  Unified fractional Fourier transform and sampling theorem , 1999, IEEE Trans. Signal Process..

[21]  Vikram M. Gadre,et al.  An uncertainty principle for real signals in the fractional Fourier transform domain , 2001, IEEE Trans. Signal Process..

[22]  SerbesAhmet,et al.  The discrete fractional Fourier transform based on the DFT matrix , 2011 .

[23]  Chien-Cheng Tseng,et al.  Discrete fractional Fourier transform based on orthogonal projections , 1999, IEEE Trans. Signal Process..

[24]  Zhou Min.  Digital Computation of Fractional Fourier Transform , 2002 .

[25]  Juan J. Trujillo,et al.  Fractional Fourier transform in the framework of fractional calculus operators , 2010 .

[26]  Rajiv Saxena,et al.  Fractional Fourier transform: A novel tool for signal processing , 2013 .

[27]  L. B. Almeida.  Product and Convolution Theorems for the Fractional Fourier Transform , 1997, IEEE Signal Processing Letters.

[28]  Ran Tao,et al.  Research progress of the fractional Fourier transform in signal processing , 2006, Science in China Series F.

[29]  Yuan Feng,et al.  Modeling and characteristic analysis of underwater acoustic signal of the accelerating propeller , 2011, Science China Information Sciences.

[30]  Santanu Manna,et al.  The fractional Fourier transform and its applications , 2012 .

[31]  Roshen Jacob,et al.  Applications of Fractional Fourier Transform in Sonar Signal Processing , 2009 .

[32]  Soo-Chang Pei,et al.  Closed-form discrete fractional and affine Fourier transforms , 2000, IEEE Trans. Signal Process..

[33]  Stephen A. Dyer,et al.  Digital signal processing , 2018, 8th International Multitopic Conference, 2004. Proceedings of INMIC 2004..

[34]  Pablo Irarrazaval,et al.  The fractional Fourier transform and quadratic field magnetic resonance imaging , 2011, Comput. Math. Appl..

[35]  Farrokh Marvasti,et al.  Parseval relationship of nonuniform samples of one- and two-dimensional signals , 1990, IEEE Trans. Acoust. Speech Signal Process..

[36]  A. Lohmann,et al.  RELATIONSHIPS BETWEEN THE RADON-WIGNER AND FRACTIONAL FOURIER TRANSFORMS , 1994 .