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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 Sinc function ： ウィキペディア英語版 Sinc function In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by , has two slightly different definitions. In mathematics, the historical unnormalized sinc function is defined for $x\; \backslash ne\; 0$ by :$\backslash operatorname(x)\; =\; \backslash frac~.$ In digital signal processing and information theory, the normalized sinc function is commonly defined for $x\; \backslash ne\; 0$ by :$\backslash operatorname(x)\; =\; \backslash frac~.$ In either case, the value at = 0 is defined to be the limiting value = 1. The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of ). As a further useful property, all of the zeros of the normalized sinc function are integer values of . The normalized sinc function is the Fourier transform of the rectangular function with no scaling. This function is fundamental in the concept of reconstructing the original continuous bandlimited signal from uniformly spaced samples of that signal. The only difference between the two definitions is in the scaling of the independent variable (the x-axis) by a factor of ''π''. In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function. The term "sinc" is a contraction of the function's full Latin name, the ''sinus cardinalis'' (cardinal sine).〔 It was introduced by Phillip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication", in which he said the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own", and his 1953 book "Probability and Information Theory, with Applications to Radar". == Properties == The zero crossings of the unnormalized sinc are at non-zero multiples of , while zero crossings of the normalized sinc occur at non-zero integers. The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(''ξ'')/''ξ'' = cos(''ξ'') for all points where the derivative of sin(''x'')/''x'' is zero and thus a local extremum is reached. A good approximation of the ''x''-coordinate of the ''n''-th extremum with positive ''x''-coordinate is :$$ x_n \approx (n+\tfrac12)\pi - \frac1 ~, where odd ''n'' lead to a local minimum and even ''n'' to a local maximum. Besides the extrema at ''xn'', the curve has an absolute maximum at  = (0,1) and because of its symmetry to the ''y''-axis extrema with ''x''-coordinates −''xn''. The normalized sinc function has a simple representation as the infinite product :$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash left(1\; -\; \backslash frac\backslash right)$ and is related to the gamma function through Euler's reflection formula, :$\backslash frac\; =\; \backslash frac~.$ Euler discovered that :$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash cos\backslash left(\backslash frac\backslash right)~.$ The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect( ), :$\backslash int\_^\backslash infty\; \backslash operatorname(t)\; \backslash ,\; e^\backslash ,dt\; =\; \backslash operatorname(f)~,$ where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the sinc filter is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter. This Fourier integral, including the special case :$\backslash int\_^\backslash infty\; \backslash frac\; \backslash ,\; dx\; =\; \backslash operatorname(0)\; =\; 1\backslash ,\backslash !$ is an improper integral and not a convergent Lebesgue integral, as :$\backslash int\_^\backslash infty\; \backslash left|\backslash frac\; \backslash right|\backslash ,\; dx\; =\; +\backslash infty\; ~.$ The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimited functions: * It is an interpolating function, i.e., sinc(0) = 1, and sinc(''k'') = 0 for nonzero integer ''k''. * The functions ''xk''(''t'') = sinc(''t'' − ''k'') (''k'' integer) form an orthonormal basis for bandlimited functions in the function space ''L''2(R), with highest angular frequency ''ω''H = ''π'' (that is, highest cycle frequency ''ƒ''H = 1/2). Other properties of the two sinc functions include: * The unnormalized sinc is the zeroth order spherical Bessel function of the first kind, . The normalized sinc is . * $\backslash int\_0^x\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash operatorname(x)\; \backslash ,\backslash !$ :where Si(''x'') is the sine integral. * (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation ::$x\; \backslash frac\; +\; 2\; \backslash frac\; +\; \backslash lambda^2\; x\; y\; =\; 0.\backslash ,\backslash !$ :The other is cos(λ ''x'')/''x'', which is not bounded at ''x'' = 0, unlike its sinc function counterpart. * $\backslash int\_^\backslash infty\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash pi\; \backslash ,\backslash !\; \backslash rightarrow\; \backslash int\_^\backslash infty\; \backslash operatorname^2(x)\backslash ,dx\; =\; 1~,$ :where the normalized sinc is meant. * $\backslash int\_^\backslash infty\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash frac\; \backslash ,\backslash !$ * $\backslash int\_^\backslash infty\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash frac\; ~.$ 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「Sinc function」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. 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