Consistency (statistics)

Name: Jodi Junker
My age: 32
Country: Netherlands
Home town: Oudkarspel
Post code: 1724 Xg
Street: Waterlelie 22

my page - www.hostgator1centcoupon.info

In mathematics, physics and engineering, 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 by

$\mathrm {sinc} (x)={\frac {\sin(x)}{x}}.\,\!$

In digital signal processing and information theory, the normalized sinc function is commonly defined by

The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale.

$\mathrm {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}.\,\!$

In either case, the value at $x=0$ is defined to be the limiting value: $\mathrm {sinc} (0)=1$.

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 π). 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 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 analytic everywhere.

The term "sinc" Template:IPAc-en 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 local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.

The zero crossings of the unnormalized sinc are at non-zero 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 n-th extremum with positive x-coordinate is

$x_{n}\approx (n+{\tfrac {1}{2}})\pi -{\frac {1}{(n+{\frac {1}{2}})\pi }}$

where odd n lead to a local minimum and even n to a local maximum. Besides the extrema at x_n, the curve has an absolute maximum at ξ₀ = (0,1) and because of its symmetry to the y-axis extrema with x-coordinates −x_n.

The normalized sinc function has a simple representation as the infinite product

${\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)\,\!$

and is related to the gamma function $\Gamma (x)$ by Euler's reflection formula:

${\frac {\sin(\pi x)}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}.\,\!$

Euler discovered that

${\frac {\sin(x)}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right).$

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f),

$\int _{-\infty }^{\infty }\mathrm {sinc} (t)\,e^{-i2\pi ft}\,dt=\mathrm {rect} (f),\,\!$

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

$\int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\mathrm {rect} (0)=1\,\!$

is an improper integral and not a convergent Lebesgue integral, as

$\int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty .$

The normalized sinc function has properties that make it ideal in relationship to 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 x_k(t) = sinc(t−k) (k integer) form an orthonormal basis for bandlimited functions in the function space L²(R), with highest angular frequency ωH = π (that is, highest cycle frequency ƒ_H = 1/2).

Other properties of the two sinc functions include:

• The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, $\scriptstyle j_{0}(x)$. The normalized sinc is $\scriptstyle j_{0}(\pi x)$

• $\int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\mathrm {Si} (x)\,\!$

where Si(x) is the sine integral.

• λ sinc(λ x) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation

$x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.\,\!$

The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.

• $\int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \,\!\rightarrow \int _{-\infty }^{\infty }\mathrm {sinc} ^{2}(x)\,dx=1.$

where the normalized sinc is meant.

• $\int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}\,\!$

• $\int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}\,\!$

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:

$\lim _{a\rightarrow 0}{\frac {1}{a}}{\textrm {sinc}}(x/a)=\delta (x).$

This is not an ordinary limit, since the left side does not converge. Rather, it means that

$\lim _{a\rightarrow 0}\int _{-\infty }^{\infty }{\frac {1}{a}}{\textrm {sinc}}(x/a)\varphi (x)\,dx=\varphi (0),$

for any smooth function $\scriptstyle \varphi (x)$ with compact support.

In the above expression, as a  approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approaches zero for any nonzero value of x. 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 Gibbs phenomenon.

Summation

Summation of sinc(x) over x from 1 to $\infty$ equals ${\tfrac {\pi -1}{2}}$.

$\sum _{x=1}^{\infty }\mathrm {sinc} (x)=\mathrm {sinc} (1)+\mathrm {sinc} (2)+\mathrm {sinc} (3)+\mathrm {sinc} (4)+\cdots ={\frac {\pi -1}{2}}$

When the signs of the addends alternate and begin with +, the sum equals ${\tfrac {1}{2}}$.

$\sum _{x=1}^{\infty }(-1)^{x+1}\,\mathrm {sinc} (x)=\mathrm {sinc} (1)-\mathrm {sinc} (2)+\mathrm {sinc} (3)-\mathrm {sinc} (4)+\cdots ={\frac {1}{2}}$

Multidimensions

The tensor-product of 1-D sinc functions readily provides a multivariate sinc function for the square, Cartesian, grid (Lattice): ${\rm {sinc}}_{\rm {C}}(x,y)={\rm {sinc}}(x){\rm {sinc}}(y)$ whose Fourier transform is the indicator function of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian lattice (e.g., hexagonal lattice) is a function whose Fourier transform is the indicator function of the Brillouin zone of that lattice. For example, the sinc function for the hexagonal lattice is a function whose Fourier transform is the indicator function 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 hexagonal, body centered cubic, face centered cubic and other higher dimensional lattices can be explicitly derived using the geometric properties of Brillouin zones and their connection to zonotopes.

For example, a hexagonal lattice can be generated by the (integer) Linear span of the vectors $u_{1}=\left[{\begin{array}{c}1/2\\{\sqrt {3}}/2\end{array}}\right]$ and $u_{2}=\left[{\begin{array}{c}1/2\\-{\sqrt {3}}/2\end{array}}\right]$. Denoting $\xi _{1}=2/3u_{1},\xi _{2}=2/3u_{2},\xi _{3}=-2/3(u_{1}+u_{2})$ and $\mathbf {x} =\left[{\begin{array}{c}x\\y\end{array}}\right]$, one can derive the sinc function for this hexagonal lattice as:

${\rm {sinc}}_{\rm {H}}(\mathbf {x} )=1/3\left(\cos(\pi \xi _{1}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{2}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{3}\cdot \mathbf {x} )+\cos(\pi \xi _{2}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{3}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{1}\cdot \mathbf {x} )+\cos(\pi \xi _{3}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{1}\cdot \mathbf {x} ){\rm {sinc}}(\xi _{2}\cdot \mathbf {x} )\right)$.

This construction can be used to design Lanczos window for general multidimensional lattices.

See also

• Borwein integral
• Dirichlet integral
• Anti-aliasing filter
• Sinc filter
• Lanczos resampling
• Whittaker–Shannon interpolation formula
• Winkel tripel projection (cartography)

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.