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| | Hello from Switzerland. I'm glad to came across you. My first name is Nicolas. <br>I live in a small town called Ecce Homo in south Switzerland.<br>I was also born in Ecce Homo 27 years ago. Married in November year 2004. I'm working at the university.<br><br>Feel free to surf to my website - [http://www.atinargentina.com.ar/node/253645 cctv security systems perth] |
| The '''Whittaker–Shannon interpolation formula''' or '''sinc interpolation''' is a method to construct a [[continuous-time]] [[bandlimited]] function from a sequence of real numbers. The formula dates back to the works of [[E. Borel]] in 1898, and [[E. T. Whittaker]] in 1915, and was cited from works of [[J. M. Whittaker]] in 1935, and in the formulation of the [[Nyquist–Shannon sampling theorem]] by [[Claude Shannon]] in 1949. It is also commonly called '''Shannon's interpolation formula''' and '''Whittaker's interpolation formula'''. E. T. Whittaker, who published it in 1915, called it the '''Cardinal series'''.
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| ==Definition==
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| [[Image:bandlimited.svg|thumb|right|240px|Fourier transform of a bandlimited function.]]
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| Given a sequence of real numbers, x[n], the continuous function''':'''
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| :<math>x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot {\rm sinc}\left(\frac{t - nT}{T}\right)\,</math>
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| has a [[Fourier transform]], X(f), whose non-zero values are confined to the region''':''' |f| ≤ 1/2T. When parameter T has units of seconds, the '''bandlimit''', 1/2T, has units of cycles/sec ([[hertz]]). When the x[n] sequence represents time samples, at interval T, of a continuous function, the quantity ''f''<sub>''s''</sub> = 1/T is known as the [[sample rate]], and ''f''<sub>''s''</sub>/2 is the corresponding [[Nyquist frequency]]. When the sampled function has a bandlimit, B, less than the Nyquist frequency, x(t) is a '''perfect reconstruction''' of the original function. (''See [[Sampling theorem]].'') Otherwise, the frequency components above the Nyquist frequency ''fold'' into the sub-Nyquist region of X(f), resulting in distortion. (''See [[Aliasing]].'')
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| ==Equivalent formulation: convolution/lowpass filter==
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| The interpolation formula is derived in the [[Nyquist–Shannon sampling theorem]] article, which points out that it can also be expressed as the [[convolution]] of an [[Dirac comb|infinite impulse train]] with a [[sinc function]]''':'''
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| :<math> x(t) = \left( \sum_{n=-\infty}^{\infty} x[n]\cdot \delta \left( t - nT \right) \right) *
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| {\rm sinc}\left(\frac{t}{T}\right). </math>
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| This is equivalent to filtering the impulse train with an ideal (''brick-wall'') [[low-pass filter]].
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| ==Convergence==
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| The interpolation formula always converges [[absolute convergence|absolutely]] and [[uniform convergence|locally uniformly]] as long as
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| :<math>\sum_{n\in\Z,\,n\ne 0}\left|\frac{x[n]}n\right|<\infty.</math>
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| By the [[Hölder inequality]] this is satisfied if the sequence <math>\scriptstyle (x[n])_{n\in\Z}</math> belongs to any of the <math>\scriptstyle\ell^p(\Z,\mathbb C)</math> [[Lp space|spaces]] with 1 < ''p'' < ∞, that is
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| :<math>\sum_{n\in\Z}\left|x[n]\right|^p<\infty.</math>
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| This condition is sufficient, but not necessary. For example, the sum will generally converge if the sample sequence comes from sampling almost any [[stationary process]], in which case the sample sequence is not square summable, and is not in any <math>\scriptstyle\ell^p(\Z,\mathbb C)</math> space.
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| ==Stationary random processes==
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| If ''x''[''n''] is an infinite sequence of samples of a sample function of a wide-sense [[stationary process]], then it is not a member of any <math>\scriptstyle\ell^p</math> or [[Lp space|L<sup>p</sup> space]], with probability 1; that is, the infinite sum of samples raised to a power ''p'' does not have a finite expected value. Nevertheless, the interpolation formula converges with probability 1. Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.
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| Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have an [[autocorrelation function]] and hence a [[spectral density]] according to the [[Wiener–Khinchin theorem]]. A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.
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| ==See also==
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| * [[Aliasing]], [[Anti-aliasing filter]], [[Spatial anti-aliasing]]
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| * [[Fourier transform]]
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| * [[Rectangular function]]
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| * [[Sampling (signal processing)]]
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| * [[Signal (electronics)]]
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| * [[Sinc function]], [[Sinc filter]]
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| {{Use dmy dates|date=September 2010}}
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| ==References==
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| <references />
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| *http://www.stanford.edu/class/ee104/shannonpaper.pdf
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| {{DEFAULTSORT:Whittaker-Shannon Interpolation Formula}}
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| [[Category:Digital signal processing]]
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| [[Category:Signal processing]]
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| [[Category:Fourier analysis]]
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Hello from Switzerland. I'm glad to came across you. My first name is Nicolas.
I live in a small town called Ecce Homo in south Switzerland.
I was also born in Ecce Homo 27 years ago. Married in November year 2004. I'm working at the university.
Feel free to surf to my website - cctv security systems perth