The fractional Fourier transform Fθ(w)\documentclass[12pt]{minimal}
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\begin{document}$$F_{\theta }(w)$$\end{document} with an angle θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} of a function f(t) is a generalization of the standard Fourier transform and reduces to it when θ=π/2.\documentclass[12pt]{minimal}
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\begin{document}$$\theta =\pi /2. $$\end{document} It has many applications in signal processing and optics because of its close relations with a number of time-frequency representations. It is known that the Wigner distribution of the fractional Fourier transform Fθ(w)\documentclass[12pt]{minimal}
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\begin{document}$$F_{\theta }(w)$$\end{document} may be obtained from the Wigner distribution of f by a two-dimensional rotation with the angle θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} in the t-w\documentclass[12pt]{minimal}
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\begin{document}$$t-w$$\end{document} plane The fractional Fourier transform has been extended to higher dimensions by taking the tensor product of one-dimensional transforms; hence, resulting in a transform in several but separable variables. It has been shown that the Wigner distribution of the two-dimensional fractional Fourier transform Fθ,ϕ(v,w)\documentclass[12pt]{minimal}
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\begin{document}$$F_{\theta ,\phi }(v,w)$$\end{document} may be obtained from the Wigner distribution of f(x, y) by a simple four-dimensional rotation with the angle θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} in the x-y\documentclass[12pt]{minimal}
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\begin{document}$$x-y$$\end{document} plane and the angle ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} in the v-w\documentclass[12pt]{minimal}
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\begin{document}$$v-w$$\end{document} plane. The aim of this paper is two-fold: (1) To introduce a new definition of the two-dimensional fractional Fourier transform that is not a tensor product of two copies of one-dimensional transforms. The new transform, which is more general than the one that exists in the literature, uses a relatively new family of Hermite functions, known as Hermite functions of two complex variables. (2) To give an explicit matrix representation of a four-dimensional rotation that verifies that the Wigner distribution of the new two-dimensional fractional Fourier transform Fθ,ϕ(v,w)\documentclass[12pt]{minimal}
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\begin{document}$$F_{\theta ,\phi }(v,w)$$\end{document} may be obtained from the Wigner distribution of f(x, y) by a four-dimensional rotation. The matrix representation is more general than the one for the tensor product case and it corresponds to a four-dimensional rotation with two planes of rotations, one with the angle (θ+ϕ)/2\documentclass[12pt]{minimal}
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\begin{document}$$(\theta +\phi )/2$$\end{document} and the other with the angle (θ-ϕ)/2\documentclass[12pt]{minimal}
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\begin{document}$$(\theta -\phi )/2$$\end{document}.
