Back
 JSIP  Vol.4 No.3 , August 2013
Generalized Parseval’s Theorem on Fractional Fourier Transform for Discrete Signals and Filtering of LFM Signals
Abstract: This paper investigates the generalized Parseval’s theorem of fractional Fourier transform (FRFT) for concentrated data. Also, in the framework of multiple FRFT domains, Parseval’s theorem reduces to an inequality with lower and upper bounds associated with FRFT parameters, named as generalized Parseval’s theorem by us. These results theoretically provide potential valuable applications in filtering, and examples of filtering for LFM signals in FRFT domains are demonstrated to support the derived conclusions.
Cite this paper: X. Wang, G. Xu, Y. Ma, L. Zhou and L. Wang, "Generalized Parseval’s Theorem on Fractional Fourier Transform for Discrete Signals and Filtering of LFM Signals," Journal of Signal and Information Processing, Vol. 4 No. 3, 2013, pp. 274-281. doi: 10.4236/jsip.2013.43035.
References

[1]   P. J. Loughlin and L. Cohen, “The Uncertainty Principle: Global, Local, or Both?” IEEE Transactions on Signal Processing, Vol. 52, No. 5, 2004, pp. 1218-1227. doi:10.1109/TSP.2004.826160

[2]   A. Dembo, T. M. Cover and J. A. Thomas, “Information Theoretic Inequalities,” IEEE Transactions on Information Theory, Vol. 37, No. 6, 2001, pp. 1501-1508. doi:10.1109/18.104312

[3]   G. B. Folland and A. Sitaram, “The Uncertainty Principle: A Mathematical Survey,” The Journal of Fourier Analysis and Applications, Vol. 3, No. 3, 1997, pp. 207-238. doi:10.1007/BF02649110

[4]   K. K. Selig, “Uncertainty Principles Revisited,” Technische Universitat Munchen, Munchen, 2001. http://www-lit.ma.tum.de/veroeff/quel/010.47001.pdf

[5]   X. D. Zhang, “Modern Signal Processing,” 2nd Edition, Tsinghua University Press, Beijing, 2002, p. 362.

[6]   R. Tao, L. Qi and Y. Wang, “Theory and Application of the Fractional Fourier Transform,” Tsinghua University Press, Beijing, 2004.

[7]   H. Maassen, “A Discrete Entropic Uncertainty Relation,” Quantum Probability and Applications V, Springer-Verlag, New York, 1988, pp. 263-266.

[8]   I. B. Birula, “Entropic Uncertainty Relations in Quantum Mechanics,” In: L. Accardi and W. von Waldenfels, Eds., Quantum Probability and Applications II, Lecture Notes in Mathematics 1136, Springer, Berlin, 1985, p. 90.

[9]   S. Shinde and M. G. Vikram, “An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain,” IEEE Transactions on Signal Processing, Vol. 49, No. 11, 2001, pp. 2545-2548. doi:10.1109/78.960402

[10]   D. Mustard, “Uncertainty Principle Invariant under Fractional Fourier Transform,” Journal of the Australian Mathematical Society Series B, Vol. 33, 1991, pp. 180-191. doi:10.1017/S0334270000006986

[11]   A. Stern, “Sampling of Compact Signals in Offset Linear Canonical Transform Domains,” Signal, Image and Video Processing, Vol. 1, No. 4, 2007, pp. 359-367.

[12]   O. Aytur and H. M. Ozaktas, “Non-Orthogonal Domains in Phase Space of Quantum Optics and Their Relation to Fractional Fourier Transform,” Optics Communications, Vol. 120, 1995, pp. 166-170. doi:10.1016/0030-4018(95)00452-E

[13]   A. Stern, “Uncertainty Principles in Linear Canonical Transform Domains and Some of Their Implications in Optics,” Journal of the Optical Society of America A, Vol. 25, No. 3, 2008, pp. 647-652. doi:10.1364/JOSAA.25.000647

[14]   K. K. Sharma and S. D. Joshi, “Uncertainty Principle for Real Signals in the Linear Canonical Transform Domains,” IEEE Transactions on Signal Processing, Vol. 56, No. 7, 2008, pp. 2677-2683. doi:10.1109/TSP.2008.917384

[15]   J. Zhao, R. Tao, Y. L. Li and Y. Wang, “Uncertainty Principles for Linear Canonical Transform,” IEEE Transaction on Signal Process, Vol. 57, No. 7, 2009, pp. 2856-2858. doi:10.1109/TSP.2009.2020039

[16]   G. L. Xu, X. T. Wang and X. G. Xu, “Three Cases of Uncertainty Principle for Real Signals in Linear Canonical Transform Domain,” IET Signal Processing, Vol. 3, No. 1, 2009, pp. 85-92. doi:10.1049/iet-spr:20080019

[17]   G. L. Xu, X. T. Wang and X. G. Xu, “Uncertainty Inequalities for Linear Canonical Transform,” IET Signal Processing, Vol. 3, No. 5, 2009, pp. 392-402. doi:10.1049/iet-spr.2008.0102

[18]   G. L. Xu, X. T. Wang and X. G. Xu, “Generalized Entropic Uncertainty Principle on Fractional Fourier Transform,” Signal Processing, Vol. 89, No. 12, 2009, pp. 2692-2697. doi:10.1016/j.sigpro.2009.05.014

[19]   G. L. Xu, X. T. Wang and X. G. Xu, “New Inequalities and Uncertainty Relations on Linear Canonical Transform Revisit,” EURASIP Journal on Advances in Signal Processing, Vol. 2009, 2009, Article ID: 563265. doi:10.1155/2009/563265

[20]   G. L. Xu, X. T. Wang and X. G. Xu, “On Uncertainty Principle for the Linear Canonical Transform of Complex Signals,” IEEE Transactions on Signal Processing, Vol. 58, No. 9, 2010, pp. 4916-4918. doi:10.1109/TSP.2010.2050201

[21]   G. L. Xu, X. T. Wang and X. G. Xu, “The Logarithmic, Heisenberg’s and Short-Time Uncertainty Principles Associated with Fractional Fourier Transform,” Signal Process, Vol. 89, No. 3, 2009, pp. 339-343. doi:10.1016/j.sigpro.2008.09.002

[22]   S. C. Pei and J. J. Ding, “Eigenfunctions of Fourier and Fractional Fourire Transforms with Complex Offsets and Parameters,” IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 54, No. 7, 2007, pp. 1599-1611.

[23]   S. C. Pei, M. H. Yeh and T. L. Luo, “Fractional Fourier Series Expansion for Finite Signals and Dual Extension to Discrete-Time Fractional Fourier Transform,” IEEE Transactions on Circuits and System II: Analog and Digital Signal Processing, Vol. 47, No. 10, 1999, pp. 2883-2888.

[24]   R. Somaraju and L. W. Hanlen, “Uncertainty Principles for Signal Concentrations,” Proceeding of 7th Australian Communications Theory Workshop, Perth, 1-3 February 2006, pp. 38-42. doi:10.1109/AUSCTW.2006.1625252

[25]   D. L. Donoho and P. B. Stark, “Uncertainty Principles and Signal Recovery,” SIAM Journal on Applied Mathematics, Vol. 49, No. 3, 1989, pp. 906-930. doi:10.1137/0149053

[26]   D. L. Donoho and X. Huo, “Uncertainty Principles and Ideal Atomic Decomposition,” IEEE Transactions on Information Theory, Vol. 47, No. 7, 2001, pp. 2845-2862. doi:10.1109/18.959265

[27]   M. Elad and A. M. Bruckstein, “A Generalized Uncertainty Principle and Sparse Representation in Pairs of Bases,” IEEE Transactions on Information Theory, Vol. 48, No. 9, 2002, pp. 2558-2567. doi:10.1109/TIT.2002.801410

[28]   A. Averbuch, R. R. Coifman, D. L. Donoho, et al., “Fast and Accurate Polar Fourier Transform,” Applied and Computational Harmonic Analysis, Vol. 21, No. 2, 2006, pp. 145-167. doi:10.1016/j.acha.2005.11.003

[29]   S. C. Pei and J. J. Ding, “Eigenfunctions of the Offset Fourier, Fractional Fourier, and Linear Canonical Transforms,” Journal of the Optical Society of America A, Vol. 20, 2003, pp. 522-532. doi:10.1364/JOSAA.20.000522

[30]   L. Qi, R. Tao, S. Zhou and Y. Wang, “Detection and Parameter Estimation of Multicomponent LFM Signal Based on the Fractional Fourier Transform,” Science in China Series F: Information Sciences, Vol. 47, No. 2, 2004, pp. 184-198.

[31]   S, C. Pei and J-J. Ding, “Closed-Form Discrete Fractional and Affine Fourier Transforms,” IEEE Transactions on Signal Processing, Vol. 48, 2000, pp. 1338-1356. doi:10.1109/78.839981

[32]   A. Kutay, H. M. Ozaktas, O. Ankan and L. Onural, “Optimal Filtering in Fractional Fourier Domains,” IEEE Transactions on Signal Processing, Vol. 45, No. 5, 1997, pp. 1129-1143. doi:10.1109/78.575688

 
 
Top