References
- Da ZX. Modern signal processing. 2nd ed. Beijing: Tsinghua University Press; 2002. p. 362.
- Dembo A, Cover TM. Information theoretic inequalities. IEEE Trans. Inform. Theory. 1991;37:1501–1508.
- Hardy G, Littlewood JE, Polya G. Inequalities. 2nd ed. London: Cambridge University Press; 1952.
- Heinig H, Smith M. Extensions of the Heisenberg-Weyl inequality. Int. J. Math. Math. Sci. 1986;9:185–192.
- Loughlin PJ, Cohen L. The uncertainty principle: global, local, or both? IEEE Trans. Signal Porcess. 2004;52:1218–1227.
- Mustard D. Uncertainty principle invariant under fractional Fourier transform. J. Aust. Math. Soc. Ser. B. 1991;33:180–191.
- Majernik V, Eva M, Shpyrko S. Uncertainty relations expressed by Shannon-like entropies. CEJP. 2003;3:393–420.
- Ozaktas HM, Aytur O. Fractional Fourier domains. Signal Process. 1995;46:119–124.
- Stern A. Sampling of compact signals in offset linear canonical transform domains. Signal Image Video Process. 2007;1:259–367.
- Shinde S, Gadre VM. An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 2001;49:2545–2548.
- Xu GL, Wang XT, Xu XG. Three uncertainty relations for real signals associated with linear canonical transform. IET Signal Process. 2009;3:85–92.
- Aytur O, Ozaktas HM. Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform. Opt. Commun. 1995;120:166–170.
- Cohen L. Time-frequency analysis: theory and applications. Upper Saddle River (NJ): Prentice Hall; 1995.
- Iwo BB. Entropic uncertainty relations in quantum mechanics. Accardi L, Von Waldenfels W, editors. Quantum probability and applications II, Vol. 1136, Lecture notes in mathematics. Berlin: Springer; 1985. p. 90.
- Iwo BB. Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A. 2006;74:052101.
- Iwo BB. Rényi entropy and the uncertainty relations. Adenier G, Fuchs CA, Yu A, editors. Foundations of probability and physics, Khrennikov, AIP Conference Proceedings 889. Melville: American Institute of Physics; 2007. p. 52–62
- Maassen H. A discrete entropic uncertainty relation, quantum probability and applications. Lecture notes in mathematics. Berlin: Springer; 1990. p. 263–266.
- Maassen H, Uffink JBM. Generalized entropic uncertainty relations. Phys. Rev. Lett. 1988;60:1103–1106.
- Rényi A. On measures of information and entropy. Proc. Fouth Berkeley Symp. Math. Stat. Probab. 1960;1: 547–561.
- Stern A. Uncertainty principles in linear canonical transform domains and some of their implications in optics. J. Opt. Soc. Am. A. 2008;25:647–652.
- Wódkiewicz K. Operational approach to phase-space measurements in quantum mechanics. Phys. Rev. Lett. 1984;52:1064–1067.
- Dang P, Deng GT, Qian T. Uncertainty principle involving phase derivative. J. Funct. Anal. 2013;265:2239–2266.
- Ozaktas HM, Zalevsky Z, Kutay MA. The fractional Fourier transform with applications in optics and signal processing. Chichester: Wiley; 2001.
- Kou KI, Xu RH, Zhang YH. Paley-Wiener theorems and uncertainty principles for the windowned linear canonical transform. Math. Methods Appl. Sci. 2012;35:2122–2132.
- Sharma KK, Joshi SD. Uncertainty principles for real signals in linear canonical transform domains. IEEE Trans. Signal Process. 2008;56:2677–2683.
- Dang P, Deng GT, Qian T. A tighter uncertainty principle for linear cannonical transform in terms of phase derivative. IEEE Trans. Signal Process. 2013;61:5153–5164.
- Hitzer E, Mawardi B. Uncertainty principle for the Clifford geometric algebra Cln,0, n = 3(mod4) based on Clifford Fourier transform. In: Qian T, Vai MI, Xu Y, editors. The Springer (SCI) Book Series ‘Applied and Numerical Harmonic Analysis’. Switzerland: 2007. p. 47–56.
- Hitzer E, Mawardi B. Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n = 2(mod4) and n = 3(mod4). Adv. Appl. Clifford Algebras. 2008. doi:10.1007/s00006-008-0098-3. Online First, 25 May.
- Mawardi B, Hitzer E. Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra Cl3,0 Adv. Appl. Clifford Algebras. 2006;16:41–61.
- Mawardi B, Hitzer E. Clifford algebra Cl3,0-valued wavelet transformation, Clifford wavelet uncertainty inequality and Clifford Gabor wavelets. Int. J. Wavelet Multiresolution Inf. Process. 2007;5:997–1019.
- Yang Y, Qian T, Sommen F. Phase derivative of monogenic functions in higher dimensional spaces. Complex Anal. Operator Theory. 2012;6:987–1010.
- Yang Y, Dang P, Qian T. Space-frequency analysis in higher dimensions and applications. Annali di Matematica 2014;1–16. doi:10.1007/s1023-014-0406-6
- Yang Y, Kou KI. Uncertainty principles for hypercomplex signals in the linear canonical transform domains. Signal Process. 2014;95:67–75.
- Clifford WK. Preliminary sketch of bi-quaternions. Proc. London Math. Soc. 1873;4:381–395.
- Clifford WK. Mathematical papers. London: Macmillan; 1882.
- Brackx F, Delanghe R, Sommen F. Clifford analysis. Boston (MA): Pitman; 1982.
- Delanghe R, Sommen F, Soucek V. Clifford algebra and spinor valued functions. Dordrecht: Kluwer; 1992.
- Korn P. Some uncertainty principle for time-frequency transforms for the Cohen class. IEEE Trans. Signal Process. 2005;53:523–527.
- Ozaktas HM, Kutay MA, Zalevsky Z. The fractional Fourier transform with applications in optics and signal processing. New York (NY): Wiley; 2000.