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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fractional Fourier transform ： ウィキペディア英語版 Fractional Fourier transform In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon,〔E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", ''Proc. Nat. Acad. Sci. USA'' 23, (1937) 158–164. (online )〕 by solving for the Green's function for phase-space rotations, and also by Namias,〔V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," ''J. Inst. Appl. Math.'' 25, 241–265 (1980).〕 generalizing work of Wiener〔N. Wiener, "Hermitian Polynomials and Fourier Analysis", ''J. Mathematics and Physics'' 8 (1929) 70-73.〕 on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups.〔Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," ''IEEE Trans. Sig. Processing'' 42 (11), 3084–3091 (1994).〕 Since then, there has been a surge of interest in extending Shannon's sampling theorem〔Ran Tao, Bing Deng, Wei-Qiang Zhang and Yue Wang, "Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain," ''IEEE Transactions on Signal Processing'', 56 (1), 158–171 (2008).〕〔A. Bhandari and P. Marziliano, "Sampling and reconstruction of sparse signals in fractional Fourier domain," ''IEEE Signal Processing Letters'', 17 (3), 221–224 (2010).〕 for signals which are band-limited in the Fractional Fourier domain. A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber〔D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," ''SIAM Review'' 33, 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)〕 as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT. ==Introduction== The continuous Fourier transform $\backslash mathcal$ of a function is a unitary operator of ''L''2 that maps the function ƒ to its frequential version ƒ̂: :$\backslash hat(\backslash xi)\; =\; \backslash int\_^\; f(x)\backslash \; e^\backslash ,\backslash mathrmx$,   for every real number $\backslash xi$. And ƒ is determined by ƒ̂ via the inverse transform $\backslash mathcal^$ :$f(x)\; =\; \backslash int\_^\; \backslash hat(\backslash xi)\backslash \; e^\backslash ,\backslash mathrm\backslash xi,$   for every real number ''x''. Let us study its ''n''-th iterated $\backslash mathcal^$ defined by $\backslash mathcal^()\; =\; \backslash mathcal\; =\; f$. Their sequence is finite since $\backslash mathcal$ is a 4-periodic automorphism: for every function ƒ, $\backslash mathcal^4\; ()\; =\; f$. More precisely, let us introduce the parity operator $\backslash mathcal$ that inverts time, $\backslash mathcal()\backslash colon\; t\; \backslash mapsto\; f(-t)$. Then the following properties hold: :$\backslash mathcal^0\; =\; \backslash mathrm,\; \backslash qquad\; \backslash mathcal^1\; =\; \backslash mathcal,\; \backslash qquad\; \backslash mathcal^2\; =\; \backslash mathcal,\; \backslash qquad\; \backslash mathcal^4\; =\; \backslash mathrm$ :$\backslash mathcal^3\; =\; \backslash mathcal^\; =\; \backslash mathcal\; \backslash circ\; \backslash mathcal\; =\; \backslash mathcal\; \backslash circ\; \backslash mathcal.$ The FrFT provides a family of linear transforms that further extends this definition to handle non-integer powers of the FT. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fractional Fourier transform」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.