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International symposium on image and signal processing and analysis/ISPA ...

Fast classical and quantum fractional Haar wavelet transforms

Valery Labunets, Ekaterina V. Labunets-Rundblad, Jaakko T. Astola

Published Nov 13, 2002• 1 citation
Publisher

Abstract

The fractional Fourier transform (FRFT) is one-parametric generalization of the classical Fourier transform. FRFT was introduced in the 1980s and found a lot of applications in signal processing. The time and spectral domains are both the special cases of the functional Fourier domain. They correspond to the 0th and 1st fractional Fourier domains, respectively. We introduce the classical and quantum fractional Haar-wavelet transforms and develop corresponding fast algorithms.

DOI
10.1109/ispa.2001.938692

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The fractional Fourier transform (FRFT) is one-parametric generalization of the classical Fourier transform. FRFT was introduced in the 1980s and found a lot of applications in signal processing. The time and spectral domains are both the special cases of the functional Fourier domain. They correspond to the 0th and 1st fractional Fourier domains, respectively. We introduce the classical and quantum fractional Haar-wavelet transforms and develop corresponding fast algorithms.
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Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
It is well known that the fractional Fourier transform (FrFT) is a non-trivial generalization of the classical Fourier transform. This transform has been extensively utilized in signal analysis and image processing. The uncertainty principle is one of the fundamental results related to the FrFT. In the present paper, we first propose the different method in proving a variant of Heisenberg uncertainty principles associated with the FrFT. This method combines the Heisenberg-type uncertainty principle pertaining to the FrFT and the additive property of the FrFT. We then provide an alternative proof of the logarithmic inequality for the FRFT and find that this result differs slightly that in the literature. Finally, we explicitly expand the variant of Heisenberg uncertainty principle to a more general case.
The fractional Fourier transform (FrFT), a fundamental operation in physics that corresponds to a rotation of phase space by any angle, is also an indispensable tool employed in digital signal processing for noise reduction. Processing of optical signals in their time-frequency degree of freedom bypasses the digitization step and presents an opportunity to enhance many protocols in quantum and classical communication, sensing, and computing. In this Letter, we present the experimental realization of the fractional Fourier transform in the time-frequency domain using an atomic quantum-optical memory system with processing capabilities. Our scheme performs the operation by imposing programmable interleaved spectral and temporal phases. We have verified the FrFT by analyses of chroncyclic Wigner functions measured via a shot-noise limited homodyne detector. Our results hold prospects for achieving temporal-mode sorting, processing, and superresolved parameter estimation.
We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 x 2 unitary matrices) in the two-dimensional case [rather than simply U(1) x U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2 = S1 x S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFT's corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite-Gaussian beams and the Laguerre-Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.