Editing Nyquist–Shannon sampling theorem (section)

==Sampling below the Nyquist rate under additional restrictions==
{{Main|Undersampling}}
The Nyquist–Shannon sampling theorem provides a [[necessary and sufficient condition|sufficient condition]] for the sampling and reconstruction of a band-limited signal. When reconstruction is done via the [[Whittaker–Shannon interpolation formula]], the Nyquist criterion is also a necessary condition to avoid aliasing, in the sense that if samples are taken at a slower rate than twice the band limit, then there are some signals that will not be correctly reconstructed. However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a [[necessary and sufficient condition|necessary condition]].

A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of [[compressed sensing]], which allows for full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to signals that are sparse (or compressible) in some domain. As an example, compressed sensing deals with signals that may have a low overall bandwidth (say, the ''effective'' bandwidth <math>EB</math>) but the frequency locations are unknown, rather than all together in a single band, so that the [[#Sampling of non-baseband signals|passband technique]] does not apply. In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus <math>2B.</math> Using compressed sensing techniques, the signal could be perfectly reconstructed if it is sampled at a rate slightly lower than <math>2EB.</math> With this approach, reconstruction is no longer given by a formula, but instead by the solution to a [[Linear programming|linear optimization program]].

Another example where sub-Nyquist sampling is optimal arises under the additional constraint that the samples are quantized in an optimal manner, as in a combined system of sampling and optimal [[lossy compression]].<ref>{{cite journal|last1=Kipnis|first1=Alon|last2=Goldsmith|first2=Andrea J.|last3=Eldar|first3=Yonina C.|last4=Weissman|first4=Tsachy|title=Distortion rate function of sub-Nyquist sampled Gaussian sources|journal=IEEE Transactions on Information Theory|date=January 2016|volume=62|pages=401–429|doi=10.1109/tit.2015.2485271|arxiv=1405.5329|s2cid=47085927 }}</ref> This setting is relevant in cases where the joint effect of sampling and [[Quantization (signal processing)|quantization]] is to be considered, and can provide a lower bound for the minimal reconstruction error that can be attained in sampling and quantizing a [[random signal]]. For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal [[Quantization (signal processing)|quantization]].<ref>{{cite journal |last1=Kipnis |first1=Alon |last2=Eldar |first2=Yonina |last3=Goldsmith |first3=Andrea |title=Analog-to-Digital Compression: A New Paradigm for Converting Signals to Bits |journal=IEEE Signal Processing Magazine |date=26 April 2018 |volume=35 |issue=3 |pages=16–39 |doi=10.1109/MSP.2017.2774249 |arxiv=1801.06718 |bibcode=2018ISPM...35c..16K |s2cid=13693437 }}</ref>