Sampling (signal processing): Difference between revisions

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For functions that vary with time, let ''S''(''t'') be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every ''T'' seconds, which is called the '''sampling interval''' or '''sampling period'''.<ref>{{cite book | title = Communications Standard Dictionary | author = Martin H. Weik | publisher = Springer | year = 1996 | isbn = 0412083914 | url = https://books.google.com/books?id=jxXDQgAACAAJ&q=Communications+Standard+Dictionary }}</ref>&nbsp; Then the sampled function is given by the sequence:

: ''S''(''nT''), &nbsp; for integer values of ''n''.

{{anchor|Sampling rate}}The '''sampling frequency''' or '''sampling rate''', ''f''_s, is the average number of samples divided by the interval length overobtained in whichone occursecond, thus {{nowrap|1=''f''_s = 1/''T''}}, with the unit ''samplesamples per second'', sometimes referred to as [[hertz]], for example e.g. 48&nbsp;kHz is 48,000 ''samples per second''.

Reconstructing a continuous function from samples is done by interpolation algorithms. The [[Whittaker–Shannon interpolation formula]] is mathematically equivalent to an ideal [[low-pass filter]] whose input is a sequence of [[Dirac delta functions]] that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant (''T''), the sequence of delta functions is called a [[Dirac comb]]. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with ''s''(''t''). That mathematical abstraction is sometimes referred to as ''impulse sampling''.<ref>{{cite book |title=Signals and Systems |author=Rao, R. |isbn=9788120338593 |url=https://books.google.com/books?id=4z3BrI717sMC |publisher=Prentice-Hall Of India Pvt. Limited|year=2008 }}</ref>