Whittaker–Shannon interpolation formula: Difference between revisions

Given a sequence of real numbers, ''x''[''n''], the continuous function''':'''

:<math>x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot, {\rm sinc}\left(\frac{t - nT}{T}\right)\,</math>

(where "sinc" denotes the [[normalized sinc function]]) has a [[Fourier transform]], ''X''(''f''), whose non-zero values are confined to the region |''f':''' |f| &nbsp;≤ 1/2T.&nbsp;1/(2''T''). When the parameter ''T'' has units of seconds, the '''bandlimit''', 1/2T(2''T''), has units of cycles/sec ([[hertz]]). When the ''x''[''n''] sequence represents time samples, at interval ''T'', of a continuous function, the quantity ''f''_''s'' = 1/''T'' is known as the [[sample rate]], and ''f''_''s''/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]].'')

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]]''':'''

:<math> x(t) = \left( \sum_{n=-\infty}^{\infty} T\cdot \underbrace{x(nT)}_{x[n]}\cdot \delta \left( t - nT \right) \right) *\circledast

\left( \frac{1}{T}{\rm sinc}\left(\frac{t}{T}\right) \right). </math>

This is equivalent to filtering the impulse train with an ideal (''brick-wall'') [[low-pass filter]] with gain of 1 (or 0&nbsp;dB) in the passband. If the sample rate is sufficiently high, this means that the baseband image (the original signal before sampling) is passed unchanged and the other images are removed by the brick-wall filter.

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&nbsp;<&le;&nbsp;''p''&nbsp;<&nbsp;∞, that is

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.

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^p 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.

{{Use dmy dates|date=SeptemberMay 20102014}}

*http://www.stanford.edu/class/ee104/shannonpaper.pdf-->

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