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"Sinc" redirects here. For the designation used in the United Kingdom for areas of wildlife interest, see . For the signal processing filter based on this function, see .
and , the cardinal sine function or sinc function, denoted by sinc(x), has two slightly different definitions.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
{\displaystyle \operatorname {sinc} (x)={\frac {\sin(x)}{x}}~.}
and , the normalized sinc function is commonly defined for x ≠ 0 by
The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale.
{\displaystyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}~.}
In either case, the value at x = 0 is defined to be the limiting value sinc(0) = 1.
causes the
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 x.
The normalized sinc function is the
with no scaling. It is used in the concept of
a continuous bandlimited signal from uniformly spaced
of that signal.
The only difference between the two definitions is in the scaling of the
(the ) by a factor of π. In both cases, the value of the function at the
at zero is understood to be the limit value 1. The sinc function is then
everywhere and hence an .
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".
The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue .
The real part of complex sinc Re(sinc z) = Re(sin z/z).
The imaginary part of complex sinc Im(sinc z) = Im(sin z/z).
The absolute value | sinc z | = | sin z/z |.
of the unnormalized sinc are at non-zero integer multiples of π, 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 ξ 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 nth extremum with positive x-coordinate is
{\displaystyle x_{n}\approx (n+{\tfrac {1}{2}})\pi -{\frac {1}{(n+{\frac {1}{2}})\pi }}~,}
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 ξ0 = (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
{\displaystyle {\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)}
and is related to the
Γ(x) through ,
{\displaystyle {\frac {\sin(\pi x)}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}~.}
discovered that
{\displaystyle {\frac {\sin(x)}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right)~.}
of the normalized sinc (to ordinary frequency) is ( f ),
{\displaystyle \int _{-\infty }^{\infty }\operatorname {sinc} (t)\,e^{i2\pi ft}\,dt=\operatorname {rect} (f)~,}
is 1 for argument between -1/2 and 1/2, and zero otherwise. This corresponds to the fact that the
is the ideal (, meaning rectangular frequency response) .
This Fourier integral, including the special case
{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1\,\!}
(cf. ) and not a convergent , as
{\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty ~.}
The normalized sinc function has properties that make it ideal in relationship to
functions:
It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero
The functions xk(t) = sinc(t - k) (k integer) form an
functions in the
L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency fH = 1/2).
Other properties of the two sinc functions include:
The unnormalized sinc is the zeroth-order spherical
of the first kind, j0(x). The normalized sinc is j0(πx).
{\displaystyle \int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} (x)\,\!}
where Si(x) is the .
λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear
{\displaystyle x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.\,\!}
The other is cos(λx)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \,\!\rightarrow \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}(x)\,dx=1~,}
where the normalized sinc is meant.
{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}\,\!}
{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}~.}
The normalized sinc function can be used as a , meaning that the following
{\displaystyle \lim _{a\rightarrow 0}{\frac {\sin \left({\frac {\pi x}{a}}\right)}{\pi x}}=\lim _{a\rightarrow 0}{\frac {1}{a}}{\textrm {sinc}}\left({\frac {x}{a}}\right)=\delta (x)~.}
This is not an ordinary limit, since the left side does not converge. Rather, it means that
{\displaystyle \lim _{a\rightarrow 0}\int _{-\infty }^{\infty }{\frac {1}{a}}{\textrm {sinc}}\left({\frac {x}{a}}\right)\varphi (x)\,dx=\varphi (0)~,}
φ(x) with .
In the above expression, as a → 0, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/πx, regardless of the value of a.
This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0, and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the .
All sums in this section refer to the unnormalized sinc function.
The sum of sinc(n) over integer n from 1 to ∞ equals π - 1/2.
{\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} (n)=\operatorname {sinc} (1)+\operatorname {sinc} (2)+\operatorname {sinc} (3)+\operatorname {sinc} (4)+\cdots ={\frac {\pi -1}{2}}}
The sum of the squares also equals π - 1/2.
{\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} ^{2}(n)=\operatorname {sinc} ^{2}(1)+\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)+\operatorname {sinc} ^{2}(4)+\cdots ={\frac {\pi -1}{2}}}
When the signs of the
alternate and begin with +, the sum equals 1/2.
{\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} (n)=\operatorname {sinc} (1)-\operatorname {sinc} (2)+\operatorname {sinc} (3)-\operatorname {sinc} (4)+\cdots ={\frac {1}{2}}}
The alternating sums of the squares and cubes also equal 1/2.
{\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{2}(n)=\operatorname {sinc} ^{2}(1)-\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)-\operatorname {sinc} ^{2}(4)+\cdots ={\frac {1}{2}}}
{\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{3}(n)=\operatorname {sinc} ^{3}(1)-\operatorname {sinc} ^{3}(2)+\operatorname {sinc} ^{3}(3)-\operatorname {sinc} ^{3}(4)+\cdots ={\frac {1}{2}}}
Unnormalized sinc(x):
{\displaystyle \operatorname {sinc} (x)={\frac {\sin(x)}{x}}=\sum _{n=0}^{\infty }{\frac {\left(-x^{2}\right)^{n}}{(2n+1)!}}}
The product of 1-D sinc functions readily provides a
sinc function for the square, Cartesian, grid (): sincC(x, y) = sinc(x)sinc(y) whose
of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian
(e.g., ) is a function whose
of that lattice. For example, the sinc function for the hexagonal lattice is a function whose
of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor-product. However, the explicit formula for the sinc function for the , ,
and other higher-dimensional lattices can be explicitly derived using the geometric properties of
and their connection to .
For example, a
can be generated by the (integer)
of the vectors
{\displaystyle u_{1}=\left[{\begin{array}{c}{\frac {1}{2}}\\{\frac {\sqrt {3}}{2}}\end{array}}\right]\quad {\text{and}}\quad u_{2}=\left[{\begin{array}{c}{\frac {1}{2}}\\-{\frac {\sqrt {3}}{2}}\end{array}}\right].}
{\displaystyle \xi _{1}={\tfrac {2}{3}}u_{1},\quad \xi _{2}={\tfrac {2}{3}}u_{2},\quad \xi _{3}=-{\tfrac {2}{3}}(u_{1}+u_{2}),\quad \mathbf {x} =\left[{\begin{array}{c}x\\y\end{array}}\right],}
one can derive the sinc function for this hexagonal lattice as:
{\displaystyle {\begin{aligned}\operatorname {sinc} _{\rm {H}}(\mathbf {x} )={\tfrac {1}{3}}{\big (}&\cos(\pi \xi _{1}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{2}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{3}\cdot \mathbf {x} )+{}\\&\cos(\pi \xi _{2}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{3}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{1}\cdot \mathbf {x} )+{}\\&\cos(\pi \xi _{3}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{1}\cdot \mathbf {x} )\operatorname {sinc} (\xi _{2}\cdot \mathbf {x} ){\big )}\end{aligned}}}
This construction can be used to design
for general multidimensional lattices.
(cartography)
; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), , , Cambridge University Press,  ,  
Poynton, Charles A. (2003). Digital video and HDTV. Morgan Kaufmann Publishers. p. 147.  .
Woodward, P. M.; Davies, I. L. (March 1952).
(PDF). Proceedings of the IEE - Part III: Radio and Communication Engineering. 99 (58): 37–44. :.
Woodward, Phillip M. (1953). Probability and information theory, with applications to radar. London: Pergamon Press. p. 29.  .  .
Euler, Leonhard (1735). .
Robert B ;
(December 2008). "Surprising Sinc Sums and Integrals". American Mathematical Monthly. 115 (10): 888–901.
Baillie, Robert (2008). "Fun with Fourier series". : [].
Ye, W.; Entezari, A. (June 2012). . IEEE Transactions on Image Processing. 21 (6): . :.  .当前位置： &
sinc函数的英文
英文翻译sinc functions:&&&&function:&&&&森克:&&&&sinc滤波器:&&&&sinc函数; 辛克函数:&&&&sinc函数:&&&&sinc filter:&&&&[数学] function◇函数计算机 function ...:&&&&primary function:&&&&fortran function:&&&&helper function
例句与用法And denied the way to get a interpolation - operator by theory and experiment , which just simply cuts out and samples the sinc function . 2另外，从理论和实验上否定了“简单地通过对sinc函数进行截取、采样来获得插值算子”的方法。 3 . adaptive lms method of time delay estdriation and time delay estanation method of single vallable adaptive based on sinc function are studied3 )研究自适应lms时延估计方法和利用sinc函数构成的单变量自适应时延估计方法，并对其性能进行了分析。 A new osculatory rational interpolation kernel function is established , which is different from the classical linear interpolation kernel functions . generally , it is a more accurate approximation for the ideal interpolation function than other linear polynomial interpolants functions . simulation results are also presented to demonstrate the superior performance of this new interpolation kernel function本文构造了一个全新的图像插值核函数?自适应切触有理插值核函数，同现有的线性插值核函数相比，其空域特性和频域特性均最接近合肥工业大学博士论文理想插值核函数sinc函数。 &&
相邻词汇热门词汇
sinc函数的英文翻译，sinc函数英文怎么说，怎么用英语翻译sinc函数，sinc函数的英文意思，发音，例句，用法和解释由查查在线词典提供，版权所有违者必究。
&&&&&&&&&&&&&&&&
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All rights reserved抽样信号_百度百科
抽样信号也被称为抽样函数或Sa(t)函数，是指sint与t之比构成的函数。其定义为：
抽样信号是指正弦函数和自变量之比构成的函数，其表达式为
。抽样函数是一个，在t的正、负两方向振幅都逐渐衰减，当t =
时，函数值等于零。
Sa(t)函数还具有一下性质：
与Sa(t)函数类似的是sinc(t)函数，它的表达式为：
有些书中将两种符号通用，即Sa(t)也可用sinc(t)表示。
企业信用信息如何计算sinc函数
我心属畅26
斜度定义说斜度是指直线或平面对另一直线或平面倾斜的程度,一般以直角三角形的两直角边的比值来表示；正切函数是直角三角形中,对边与邻边的比值.也是直角三角形的两直角边的比值,所以常常用直角三角形的正切函数表示斜度.如果知道一个直角三角形的三边的长度,计算斜边的斜度就是计算两直角边的比值,即直角三角形的正切函数（tanθ=y/x）值.你学没有学过正弦定理和余弦定理!如果学过就应该知道,这个定理肯定能解决这个问题.正弦定理：a/sinA=b/sinB=c/sinC=2R（2R就是三角形的外接圆的直径）通过这个定理你想计算哪个角（就是你说的边的斜度）,就把这个角对着的边除以斜边的长度,得出个数值,然后查正弦表,就可以找到这个角的角度了.说的更明白点：把正弦定理的公式变一下就得：sinA=a/2R,a和2R是已知的,也就得到了这个角的正弦值,查正弦表就可以查出这个正弦值对应的角度是多少度!也就得到了结果.斜度是夹角的正切函数.既然知道三边的长度,就简单了,没必要再推算角度、三角函数了.设：两个直角边分别为 a、b斜边对a边的斜度为 —— b/a斜边对b边的斜度为 —— a/b
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