This paper investigates the generalized uncertainty principles of fractional Fourier transform (FRFT) for concentrated data in limited supports. The continuous and discrete generalized uncertainty relations, whose bounds are related to FRFT parameters and signal lengths, were derived in theory. These uncertainty principles disclose that the data in FRFT domains may have much higher concentration than that in traditional time-frequency domains, which will enrich the ensemble of generalized uncertainty principles.
Folland, G.B. and Sitaram, A. (1997) The Uncertainty Principle: A Mathematical Survey. The Journal of Fourier Analysis and Applications, 3, 207-238. http://dx.doi.org/10.1007/BF02649110
Loughlin, P.J. and Cohen, L. (2004) The Uncertainty Principle: Global, Local, or Both? IEEE Transactions on Signal Processing, 52, 1218-1227. http://dx.doi.org/10.1109/TSP.2004.826160
Dembo, A., Cover, T.M. and Thomas, J.A. (2001) Information Theoretic Inequalities. IEEE Transactions on Information Theory, 37, 1501-1508. http://dx.doi.org/10.1109/18.104312
Zhang, X.D. (2002) Modern Signal Processing. 2nd Edition, Tsinghua University Press, Beijing, 362.
Tao, R., Qi, L. and Wang, Y. (2004) Theory and Applications of the Fractional Fourier Transform. Tsinghua University Press, Beijing.
Maassen, H. (1988) A Discrete Entropic Uncertainty Relation, Quantum Probability and Applications. V, Springer-Verlag, New York, 263-266.
Birula, I.B. (1985) Entropic Uncertainty Relations in Quantum Mechanics, Quantum Probability and Applications II, In: Accardi, L. and von Waldenfels, W., Eds., Lecture Notes in Mathematics 1136, Springer, Berlin, 90.
Shinde, S. and Vikram, M.G. (2001) An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain. IEEE Transactions on Signal Processing, 49, 2545-2548. http://dx.doi.org/10.1109/78.960402
Mustard, D. (1991) Uncertainty Principle Invariant under Fractional Fourier Transform. Journal of the Australian Mathematical Society (Series B), 33, 180-191. http://dx.doi.org/10.1017/S0334270000006986
Stern, A. (2007) Sampling of Compact Signals in Offset Linear Canonical Transform Domains. Signal, Image and Video Processing, 1, 359-367. http://dx.doi.org/10.1007/s11760-007-0029-0
Aytur, O. and Ozaktas, H.M. (1995) Non-Orthogonal Domains in Phase Space of Quantum Optics and Their Relation to Fractional Fourier Transforms. Optics Communications, 120, 166-170. http://dx.doi.org/10.1016/0030-4018(95)00452-E
Stern, A. (2008) Uncertainty Principles in Linear Canonical Transform Domains and Some of Their Implications in Optics. Journal of the Optical Society of America A, 25, 647-652. http://dx.doi.org/10.1364/JOSAA.25.000647
Sharma, K.K. and Joshi, S.D. (2008) Uncertainty Principle for Real Signals in the Linear Canonical Transform Domains. IEEE Transactions on Signal Processing, 56, 2677-2683. http://dx.doi.org/10.1109/TSP.2008.917384
Zhao, J., Tao, R., Li, Y.L. and Wang, Y. (2009) Uncertainty Principles for Linear Canonical Transform. IEEE Transactions on Signal Processing, 57, 2856-2858. http://dx.doi.org/10.1109/TSP.2009.2020039
Xu, G., Wang, X. and Xu, X. (2009) Three Cases of Uncertainty Principle for Real Signals in Linear Canonical Transform Domain. IET Signal Processing, 3, 85-92. http://dx.doi.org/10.1049/iet-spr:20080019
Xu, G., Wang, X. and Xu, X. (2009) New Inequalities and Uncertainty Relations on Linear Canonical Transform Revisit. EURASIP Journal on Advances in Signal Processing, 2009, Article ID: 563265.
Xu, G., Wang, X. and Xu, X. (2009) Generalized Entropic Uncertainty Principle on Fractional Fourier Transform. Signal Processing, 89, 2692-2697. http://dx.doi.org/10.1016/j.sigpro.2009.05.014
Xu, G., Wang, X. and Xu, X. (2009) Uncertainty Inequalities for Linear Canonical Transform. IET Signal Processing, 3, 392-402. http://dx.doi.org/10.1049/iet-spr.2008.0102
Xu, G., Wang, X. and Xu, X. (2009) The Logarithmic, Heisenberg’s and Short-Time Uncertainty Principles Associated with Fractional Fourier Transform. Signal Process, 89, 339-343. http://dx.doi.org/10.1016/j.sigpro.2008.09.002
Xu, G., Wang, X. and Xu, X. (2010) On Uncertainty Principle for the Linear Canonical Transform of Complex Signals. IEEE Transactions on Signal Processing, 58, 4916-4918. http://dx.doi.org/10.1109/TSP.2010.2050201
Xu, G., Wang, X. and Xu, X. (2010) Novel Uncertainty Relations in Fractional Fourier Transform Domain for Real Signals. Chinese Physics B, 19, Article ID: 014203. http://dx.doi.org/10.1088/1674-1056/19/1/014203
Somaraju, R. and Hanlen, L.W. (2006) Uncertainty Principles for Signal Concentrations. Proceedings of 7th Australian Communications Theory Workshop, Perth, 1-3 February 2006, 38-42. http://dx.doi.org/10.1109/ausctw.2006.1625252
Donoho, D.L. and Stark, P.B. (1989) Uncertainty Principles and Signal Recovery. SIAM Journal on Applied Mathematics, 49, 906-931. http://dx.doi.org/10.1137/0149053
Donoho, D.L. and Huo, X. (2001) Uncertainty Principles and Ideal Atomic Decomposition. IEEE Transactions on Information Theory, 47, 2845-2862. http://dx.doi.org/10.1109/18.959265
Elad, M. and Bruckstein, A.M. (2002) A Generalized Uncertainty Principle and Sparse Representation in Pairs of Bases. IEEE Transactions on Information Theory, 48, 2558-2567. http://dx.doi.org/10.1109/TIT.2002.801410
Averbuch, A., Coifman, R.R., Donoho, D.L., Eladd, M. and Israeli, M. (2006) Fast and Accurate Polar Fourier Transform. Applied and Computational Harmonic Analysis, 21, 145-167. http://dx.doi.org/10.1016/j.acha.2005.11.003
Pei, S.C. and Ding, J.J. (2007) Eigenfunctions of Fourier and Fractional Fourire Transforms with Complex Offsets and Parameters. IEEE Transactions on Circuits and Systems I: Regular Papers, 54, 1599-1611. http://dx.doi.org/10.1109/TCSI.2007.900182
Pei, S.C., Yeh, M.H. and Luo, T.L. (1999) Fractional Fourier Series Expansion for Finite Signals and Dual Extension to Discrete-Time Fractional Fourier Transform. IEEE Transactions on Signal Processing, 47, 2883-2888. http://dx.doi.org/10.1109/78.790671
Pei, S.C. and Ding, J.J. (2003) Eigenfunctions of the Offset Fourier, Fractional Fourier, and Linear Canonical Transforms. Journal of the Optical Society of America A, 20, 522-532. http://dx.doi.org/10.1364/JOSAA.20.000522
Qi, L., Tao, R., Zhou, S. and Wang, Y. (2004) Detection and Parameter Estimation of Multicomponent LFM Signal Based on the Fractional Fourier Transform. Science in China Series F: Information Sciences, 47, 184-198. http://dx.doi.org/10.1360/02yf0456
Pei, S.C. and Ding, J-J. (2000) Closed-Form Discrete Fractional and Affine Fourier Transforms. IEEE Transactions on Signal Processing, 48, 1338-1356. http://dx.doi.org/10.1109/78.839981
Xia, X.G. (1996) On Bandlimited Signals with Fractional Fourier Transform. IEEE Signal Processing Letters, 3, 72-74. http://dx.doi.org/10.1109/97.481159
Tao, R., Li, Y.L. and Wang, Y. (2010) Short-Time Fractional Fourier Transform and Its Applications. IEEE Transactions on Signal Processing, 58, 2568-2580. http://dx.doi.org/10.1109/TSP.2009.2028095
Tao, R., Deng, B., Zhang, W.Q. and Wang, Y. (2008) Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain. IEEE Transactions on Signal Processing, 56, 158-171. http://dx.doi.org/10.1109/TSP.2007.901666
Wang, X., Xu, G., Ma, Y., Zhou, L. and Wang, L. (2013) Generalized Parseval’s Theorem on Fractional Fourier Transform for Discrete Signals and Filtering of LFM Signals. Journal of Signal and Information Processing, 4, 274-281. http://dx.doi.org/10.4236/jsip.2013.43035