Editing Nyquist–Shannon sampling theorem (section)

==Nonuniform sampling==
The sampling theory of Shannon can be generalized for the case of [[nonuniform sampling]], that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition.<ref>{{cite book | editor-last =Marvasti | editor-first =F. | title =Nonuniform Sampling, Theory and Practice | publisher =Kluwer Academic/Plenum Publishers | date =2000 | location =New York}}</ref> Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction.

The general theory for non-baseband and nonuniform samples was developed in 1967 by [[Henry Landau]].<ref>{{cite journal |first=H. J. |last=Landau |title=Necessary density conditions for sampling and interpolation of certain entire functions |journal=Acta Mathematica |volume=117 |issue=1 |pages=37–52 |year=1967 |doi=10.1007/BF02395039 |doi-access=free }}</ref> He proved that the average sampling rate (uniform or otherwise) must be twice the ''occupied'' bandwidth of the signal, assuming it is ''a priori'' known what portion of the spectrum was occupied.

In the late 1990s, this work was partially extended to cover signals for which the amount of occupied bandwidth is known but the actual occupied portion of the spectrum is unknown.<ref>
For example, {{cite thesis |first=P. |last=Feng |title=Universal minimum-rate sampling and spectrum-blind reconstruction for multiband signals |degree=Ph.D. |institution=University of Illinois at Urbana-Champaign |year=1997 }}</ref> In the 2000s, a complete theory was developed
(see the section [[#Sampling below the Nyquist rate under additional restrictions|Sampling below the Nyquist rate under additional restrictions]] below) using [[compressed sensing]]. In particular, the theory, using signal processing language, is described in a 2009 paper by Mishali and Eldar.<ref>{{cite journal | citeseerx = 10.1.1.154.4255 | title = Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals | first1 = Moshe | last1 = Mishali | first2 = Yonina C. | last2 = Eldar | journal = IEEE Trans. Signal Process. |date=March 2009 | volume = 57 | issue = 3 | pages = 993–1009 | doi = 10.1109/TSP.2009.2012791 | bibcode = 2009ITSP...57..993M | s2cid = 2529543 }}</ref> They show, among other things, that if the frequency locations are unknown, then it is necessary to sample at least at twice the Nyquist criteria; in other words, you must pay at least a factor of 2 for not knowing the location of the [[spectrum]]. Note that minimum sampling requirements do not necessarily guarantee [[Numerical stability|stability]].