The Balian-Low Theorem for the Windowed Quaternionic Fourier Transform
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In this paper we present the Balian-Low theorem for the twosided windowed quaternionic Fourier transform (WQFT), a theorem which expresses the fact that time-frequency concentrations are incompatible with non-redundancy whenever Gabor systems form orthonormal bases or frames. Since uncertainty principles are closely connected with representations of the kernel of the Fourier transform under consideration, we construct a suitable representation for the kernel of our two-sided WQFT which in turn provides suitable Gabor systems. We proceed by deriving several important properties of the WQFT, such as shift and modulation operators, a reconstruction formula, orthogonality relations and a Heisenberg uncertainty principle for the WQFT. Finally, we establish the Balian-Low theorem for Gabor orthonormal bases associated with discrete versions of the kernels of the WQFT and of the right-sided WQFT.
KeywordsGabor system Balian-Low theorem Quaternionic Fourier transform Windowed quaternionic Fourier transform
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- 2.W. Czaja, A. M. Powell, Recent developments in the Balian-Low theorem, Harmonic Analysis and Applications. Birkhäuser, Boston (2006), 79–100.Google Scholar
- 3.I. Daubechies, Ten lectures on wavelets. CBMS-NSF, Regional conference series in applied mathematics, SIAM Publication, Philadelphia, 1992.Google Scholar
- 4.K. Gröchenig, Foundations of time-frequency Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.Google Scholar
- 5.T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Germany, 1999.Google Scholar
- 6.T. A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio, Texas, 1993, 1830–1841.Google Scholar
- 8.Bülow T., Felsberg M., Sommer G.: Non-commutative hypercomplex Fourier transforms of multidimensional signals. In: Sommer, G. (eds) Geometric Computing with Clifford Algebras., pp. 187–207. Springer, Heidelberg (2001)Google Scholar
- 12.P. Bas, N. Le Bihan, J. M. Chassery, Color image watermarking using quaternion Fourier transform. In: Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP, Hong-Kong, (2003), 521–524.Google Scholar