Editing Digital signal processing (section)

== Signal sampling ==
{{Main|Sampling (signal processing)}}

To digitally analyze and manipulate an analog signal, it must be digitized with an [[analog-to-digital converter]] (ADC).<ref>{{cite journal |last=Walden |first=R. H. |date=1999 |title=Analog-to-digital converter survey and analysis |journal=IEEE Journal on Selected Areas in Communications |volume=17 |issue=4 |pages=539–550 |doi=10.1109/49.761034}}</ref> Sampling is usually carried out in two stages, [[discretization]] and [[Quantization (signal processing)|quantization]]. Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude.  Quantization means each amplitude measurement is approximated by a value from a finite set.  Rounding [[real numbers]] to integers is an example.

The [[Nyquist–Shannon sampling theorem]] states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal. In practice, the sampling frequency is often significantly higher than this.<ref>{{cite journal |last1=Candes |first1=E. J. |last2=Wakin |first2=M. B. |date=2008 |title=An Introduction To Compressive Sampling |journal=IEEE Signal Processing Magazine |volume=25 |issue=2 |pages=21–30 |doi=10.1109/MSP.2007.914731|bibcode=2008ISPM...25...21C |s2cid=1704522 |url=https://resolver.caltech.edu/CaltechAUTHORS:CANieeespm08 }}</ref> It is common to use an [[anti-aliasing filter]] to limit the signal bandwidth to comply with the sampling theorem, however careful selection of this filter is required because the reconstructed signal will be the filtered signal plus residual [[aliasing]] from imperfect [[stop band]] rejection instead of the original (unfiltered) signal.

Theoretical DSP analyses and derivations are typically performed on [[discrete-time signal]] models with no amplitude inaccuracies ([[quantization error]]), created by the abstract process of [[Sampling (signal processing)|sampling]]. Numerical methods require a quantized signal, such as those produced by an ADC.  The processed result might be a frequency spectrum or a set of statistics.  But often it is another quantized signal that is converted back to analog form by a [[digital-to-analog converter]] (DAC).