Nyquist–Shannon sampling theorem: Difference between revisions

Revision as of 04:00, 11 February 2025

Revision as of 06:49, 15 February 2025

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A mathematically ideal way to interpolate the sequence involves the use of [[sinc function]]s. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original ___location of the sample <math>nT,</math> with the amplitude of the sinc function scaled to the sample value, <math>x(nT).</math> Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method uses the [[Dirac comb#Sampling and aliasing|Dirac comb]] and proceeds by [[Convolution|convolving]] one sinc function with a series of [[Dirac delta]] pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as ''interpolation error''.

Practical [[digital-to-analog converter]]s produce neither scaled and delayed [[sinc function]]s, nor ideal [[Dirac pulse]]s. Instead they produce a [[Step function|piecewise-constant]] sequence]] of scaled and delayed [[rectangular function|rectangular pulses]] (the [[zero-order hold]]), usually followed by a [[lowpass filter]] (called an "anti-imaging filter") to remove spurious high-frequency replicas (images) of the original baseband signal.

==Aliasing==