Nyquist–Shannon sampling theorem: Difference between revisions

Revision as of 03:48, 18 February 2025

Revision as of 15:20, 18 March 2025

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As depicted, copies of <math>X(f)</math> are shifted by multiples of the sampling rate <math>f_s = 1/T</math> and combined by addition. For a band-limited function <math>(X(f) = 0, \text{ for all } |f| \ge B)</math> and sufficiently large <math>f_s,</math> it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous <math>X(f).</math> Any frequency component above <math>f_s/2</math> is indistinguishable from a lower-frequency component, called an ''alias'', associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), <math>x(t)</math> is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a [[lowpass filter]], and in this application it is called an [[anti-aliasing filter]].

[[File:ReconstructFilter.pngsvg|thumb|right|upright=1.8|Spectrum, <math>X_s(f)</math>, of a properly sampled bandlimited signal (blue) and the adjacent DTFT images (green) that do not overlap. A ''brick-wall'' low-pass filter, <math>H(f)</math>, removes the images, leaves the original spectrum, <math>X(f)</math>, and recovers the original signal from its samples.]]

[[File:Nyquist sampling.gif|upright=1.8|thumb|right|The figure on the left shows a function (in gray/black) being sampled and reconstructed (in gold) at steadily increasing sample-densities, while the figure on the right shows the frequency spectrum of the gray/black function, which does not change. The highest frequency in the spectrum is half the width of the entire spectrum. The width of the steadily-increasing pink shading is equal to the sample-rate. When it encompasses the entire frequency spectrum it is twice as large as the highest frequency, and that is when the reconstructed waveform matches the sampled one.]]