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(Top) https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem
1 Introduction https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Introduction
2 Aliasing https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Aliasing
3 Derivation as a special case of Poisson summation https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Derivation_as_a_special_case_of_Poisson_summation
4 Shannon's original proof https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Shannon's_original_proof
4.1 Notes https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Notes
5 Application to multivariable signals and images https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Application_to_multivariable_signals_and_images
6 Critical frequency https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Critical_frequency
7 Sampling of non-baseband signals https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Sampling_of_non-baseband_signals
8 Nonuniform sampling https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Nonuniform_sampling
9 Sampling below the Nyquist rate under additional restrictions https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Sampling_below_the_Nyquist_rate_under_additional_restrictions
10 Historical background https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Historical_background
10.1 Other discoverers https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Other_discoverers
10.2 Why Nyquist? https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Why_Nyquist?
11 See also https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#See_also
12 Notes https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Notes_2
13 References https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#References
14 Further reading https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Further_reading
15 External links https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#External_links
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Shannon–Hartley theoremhttps://en.wikipedia.org/wiki/Shannon–Hartley_theorem
https://en.wikipedia.org/wiki/File:Bandlimited.svg
Fourier transformhttps://en.wikipedia.org/wiki/Fourier_transform
theoremhttps://en.wikipedia.org/wiki/Theorem
signal processinghttps://en.wikipedia.org/wiki/Signal_processing
continuous-time signalshttps://en.wikipedia.org/wiki/Continuous-time_signal
discrete-time signalshttps://en.wikipedia.org/wiki/Discrete-time_signal
sample ratehttps://en.wikipedia.org/wiki/Sample_rate
bandwidthhttps://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
mathematical functionshttps://en.wikipedia.org/wiki/Mathematical_function
Fourier transformhttps://en.wikipedia.org/wiki/Continuous_Fourier_transform
interpolateshttps://en.wikipedia.org/wiki/Interpolates
sample ratehttps://en.wikipedia.org/wiki/Sampling_(signal_processing)
band-limitedhttps://en.wikipedia.org/wiki/Bandlimiting
§ Sampling of non-baseband signalshttps://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Sampling_of_non-baseband_signals
compressed sensinghttps://en.wikipedia.org/wiki/Compressed_sensing
Bochner's theoremhttps://en.wikipedia.org/wiki/Bochner's_theorem
[1]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-1
Nyquist frequencyhttps://en.wikipedia.org/wiki/Nyquist_frequency
aliasinghttps://en.wikipedia.org/wiki/Aliasing
sampling ratehttps://en.wikipedia.org/wiki/Sampling_(signal_processing)
bandwidthhttps://en.wikipedia.org/wiki/Bandwidth_(signal_processing)
Harry Nyquisthttps://en.wikipedia.org/wiki/Harry_Nyquist
Claude Shannonhttps://en.wikipedia.org/wiki/Claude_Shannon
[2]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:2-2
Kinnosuke Ogurahttps://en.wikipedia.org/wiki/Kinnosuke_Ogura
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
E. T. Whittakerhttps://en.wikipedia.org/wiki/E._T._Whittaker
[4]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:1-4
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=1
Samplinghttps://en.wikipedia.org/wiki/Sampling_(signal_processing)
Shannon'shttps://en.wikipedia.org/wiki/Claude_Shannon
[5]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-Shannon49-5
hertzhttps://en.wikipedia.org/wiki/Hertz
bandlimithttps://en.wikipedia.org/wiki/Bandlimiting
aliasinghttps://en.wikipedia.org/wiki/Aliasing
sinusoidalhttps://en.wikipedia.org/wiki/Sine_wave
Nyquist ratehttps://en.wikipedia.org/wiki/Nyquist_rate
Nyquist frequencyhttps://en.wikipedia.org/wiki/Nyquist_frequency
sampling equipmenthttps://en.wikipedia.org/wiki/Analog-to-digital_converter
https://en.wikipedia.org/wiki/File:Sinc_function_(normalized).svg
sinc functionhttps://en.wikipedia.org/wiki/Sinc_function
[6]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-6
Discrete-time Fourier transform#Relation to Fourier Transformhttps://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Relation_to_Fourier_Transform
sinc functionshttps://en.wikipedia.org/wiki/Sinc_function
Dirac combhttps://en.wikipedia.org/wiki/Dirac_comb#Sampling_and_aliasing
convolvinghttps://en.wikipedia.org/wiki/Convolution
Dirac deltahttps://en.wikipedia.org/wiki/Dirac_delta
digital-to-analog convertershttps://en.wikipedia.org/wiki/Digital-to-analog_converter
sinc functionshttps://en.wikipedia.org/wiki/Sinc_function
Dirac pulseshttps://en.wikipedia.org/wiki/Dirac_pulse
piecewise-constant sequencehttps://en.wikipedia.org/wiki/Step_function
rectangular pulseshttps://en.wikipedia.org/wiki/Rectangular_function
zero-order holdhttps://en.wikipedia.org/wiki/Zero-order_hold
lowpass filterhttps://en.wikipedia.org/wiki/Lowpass_filter
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=2
Aliasinghttps://en.wikipedia.org/wiki/Aliasing
https://en.wikipedia.org/wiki/File:CPT-sound-nyquist-thereom-1.5percycle.svg
Fourier transformhttps://en.wikipedia.org/wiki/Fourier_transform
periodic summationhttps://en.wikipedia.org/wiki/Periodic_summation
Discrete-time Fourier transform#Relation to Fourier Transformhttps://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Relation_to_Fourier_Transform
https://en.wikipedia.org/wiki/File:AliasedSpectrum.png
Fourier serieshttps://en.wikipedia.org/wiki/Fourier_series
discrete-time Fourier transformhttps://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
lowpass filterhttps://en.wikipedia.org/wiki/Lowpass_filter
anti-aliasing filterhttps://en.wikipedia.org/wiki/Anti-aliasing_filter
https://en.wikipedia.org/wiki/File:ReconstructFilter.svg
https://en.wikipedia.org/wiki/File:Nyquist_sampling.gif
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=3
Eq.1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#math_Eq.1
rectangular functionhttps://en.wikipedia.org/wiki/Rectangular_function
Eq.1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#math_Eq.1
[A]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-7
Whittaker–Shannon interpolation formulahttps://en.wikipedia.org/wiki/Whittaker–Shannon_interpolation_formula
Undersamplinghttps://en.wikipedia.org/wiki/Undersampling
low-pass filterhttps://en.wikipedia.org/wiki/Low-pass_filter
Dirac combhttps://en.wikipedia.org/wiki/Dirac_comb
digital-to-analog convertershttps://en.wikipedia.org/wiki/Digital-to-analog_converter
zero-order holdhttps://en.wikipedia.org/wiki/Zero-order_hold
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[a]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-8
sinc functionshttps://en.wikipedia.org/wiki/Sinc_function
Whittaker–Shannon interpolation formulahttps://en.wikipedia.org/wiki/Whittaker–Shannon_interpolation_formula
recthttps://en.wikipedia.org/wiki/Rectangular_function
[7]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-9
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↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-8
Eq.2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#math_Eq.2
Eq.1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#math_Eq.1
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=6
Multidimensional samplinghttps://en.wikipedia.org/wiki/Multidimensional_sampling
https://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks_small.jpg
Moiré patternhttps://en.wikipedia.org/wiki/Moiré_pattern
https://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks.jpg
pixelshttps://en.wikipedia.org/wiki/Pixel
vector-valued functionshttps://en.wikipedia.org/wiki/Vector-valued_function
image sensorhttps://en.wikipedia.org/wiki/Image_sensor
moiré patternhttps://en.wikipedia.org/wiki/Moiré_pattern
optical low-pass filterhttps://en.wikipedia.org/wiki/Optical_low-pass_filter
low-pass filterhttps://en.wikipedia.org/wiki/Low-pass_filter
downsampleshttps://en.wikipedia.org/wiki/Downsampling
moiré patternhttps://en.wikipedia.org/wiki/Moiré_pattern
modulation transfer functionhttps://en.wikipedia.org/wiki/Optical_transfer_function
spatial anti-aliasinghttps://en.wikipedia.org/wiki/Spatial_anti-aliasing
graphics processing unithttps://en.wikipedia.org/wiki/Graphics_processing_unit
smartphonehttps://en.wikipedia.org/wiki/Smartphone
digital signal processinghttps://en.wikipedia.org/wiki/Digital_signal_processing
https://en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svg
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edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=8
[5]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-Shannon49-5
single-sideband modulationhttps://en.wikipedia.org/wiki/Single-sideband_modulation
signalshttps://en.wikipedia.org/wiki/Signal_(information_theory)
basebandhttps://en.wikipedia.org/wiki/Baseband
samplinghttps://en.wikipedia.org/wiki/Sampling_(signal_processing)
FM radiohttps://en.wikipedia.org/wiki/FM_broadcasting
MHzhttps://en.wikipedia.org/wiki/Megahertz
bandpass filterhttps://en.wikipedia.org/wiki/Bandpass_filter
brick-wallhttps://en.wikipedia.org/wiki/Brick-wall_filter
lowpass filterhttps://en.wikipedia.org/wiki/Lowpass_filter
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=9
nonuniform samplinghttps://en.wikipedia.org/wiki/Nonuniform_sampling
[8]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-10
Henry Landauhttps://en.wikipedia.org/wiki/Henry_Landau
[9]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-11
[10]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-12
Sampling below the Nyquist rate under additional restrictionshttps://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Sampling_below_the_Nyquist_rate_under_additional_restrictions
compressed sensinghttps://en.wikipedia.org/wiki/Compressed_sensing
[11]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-13
spectrumhttps://en.wikipedia.org/wiki/Spectrum
stabilityhttps://en.wikipedia.org/wiki/Numerical_stability
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=10
Undersamplinghttps://en.wikipedia.org/wiki/Undersampling
sufficient conditionhttps://en.wikipedia.org/wiki/Necessary_and_sufficient_condition
Whittaker–Shannon interpolation formulahttps://en.wikipedia.org/wiki/Whittaker–Shannon_interpolation_formula
necessary conditionhttps://en.wikipedia.org/wiki/Necessary_and_sufficient_condition
compressed sensinghttps://en.wikipedia.org/wiki/Compressed_sensing
passband techniquehttps://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#Sampling_of_non-baseband_signals
linear optimization programhttps://en.wikipedia.org/wiki/Linear_programming
lossy compressionhttps://en.wikipedia.org/wiki/Lossy_compression
[12]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-14
quantizationhttps://en.wikipedia.org/wiki/Quantization_(signal_processing)
random signalhttps://en.wikipedia.org/wiki/Random_signal
quantizationhttps://en.wikipedia.org/wiki/Quantization_(signal_processing)
[13]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-15
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=11
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
[2]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:2-2
Kinnosuke Ogurahttps://en.wikipedia.org/wiki/Kinnosuke_Ogura
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
[4]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:1-4
E. T. Whittakerhttps://en.wikipedia.org/wiki/E._T._Whittaker
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
[3]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:0-3
Harry Nyquisthttps://en.wikipedia.org/wiki/Harry_Nyquist
[14]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-16
[15]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-17
[16]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:02-18
[16]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:02-18
[16]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:02-18
[16]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:02-18
[16]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:02-18
[5]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-Shannon49-5
Karl Küpfmüllerhttps://en.wikipedia.org/wiki/Karl_Küpfmüller
[17]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-19
sine integralhttps://en.wikipedia.org/wiki/Sine_integral
Claude E. Shannonhttps://en.wikipedia.org/wiki/Claude_Shannon
[18]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:3-20
[19]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:4-21
[5]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-Shannon49-5
[20]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-22
A Mathematical Theory of Communicationhttps://en.wikipedia.org/wiki/A_Mathematical_Theory_of_Communication
[18]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:3-20
[19]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-:4-21
[B]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-23
communication theoryhttps://en.wikipedia.org/wiki/Communication_theory
[21]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-24
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=12
[22]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-25
[23]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-26
Herbert Raabehttps://en.wikipedia.org/wiki/Herbert_Raabe?action=edit&redlink=1
[24]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-EM-27
J. M. Whittakerhttps://en.wikipedia.org/wiki/J._M._Whittaker
Borelhttps://en.wikipedia.org/wiki/Émile_Borel
[Meijering 1]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-28
Kotel'nikovhttps://en.wikipedia.org/wiki/Vladimir_Kotelnikov
Raabehttps://en.wikipedia.org/wiki/Herbert_Raabe?action=edit&redlink=1
[Meijering 2]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-29
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-28
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-29
Vladimir Kotelnikovhttps://en.wikipedia.org/wiki/Vladimir_Kotelnikov
[25]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-30
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=13
Harry Nyquisthttps://en.wikipedia.org/wiki/Harry_Nyquist
Bell Labshttps://en.wikipedia.org/wiki/Bell_Labs
[26]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-31
[27]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-32
[28]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-33
[29]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-34
Blackmanhttps://en.wikipedia.org/wiki/R._B._Blackman
Tukeyhttps://en.wikipedia.org/wiki/J._W._Tukey
[30]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-35
[24]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-EM-27
Nyquist ratehttps://en.wikipedia.org/wiki/Nyquist_rate
Harold S. Blackhttps://en.wikipedia.org/wiki/Harold_Stephen_Black
[31]https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_note-36
Oxford English Dictionaryhttps://en.wikipedia.org/wiki/Oxford_English_Dictionary
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=14
44,100 Hzhttps://en.wikipedia.org/wiki/44,100_Hz
Balian–Low theoremhttps://en.wikipedia.org/wiki/Balian–Low_theorem
Cheung–Marks theoremhttps://en.wikipedia.org/wiki/Cheung–Marks_theorem
Shannon–Hartley theoremhttps://en.wikipedia.org/wiki/Shannon–Hartley_theorem
Nyquist ISI criterionhttps://en.wikipedia.org/wiki/Nyquist_ISI_criterion
Reconstruction from zero crossingshttps://en.wikipedia.org/wiki/Reconstruction_from_zero_crossings
Zero-order holdhttps://en.wikipedia.org/wiki/Zero-order_hold
Dirac combhttps://en.wikipedia.org/wiki/Dirac_comb#Sampling_and_aliasing
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=15
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-7
transform tableshttps://en.wikipedia.org/wiki/Table_of_Fourier_transforms
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-23
Shannon 1949https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#refShannon49
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=16
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-1
arXivhttps://en.wikipedia.org/wiki/ArXiv_(identifier)
1506.03300https://arxiv.org/abs/1506.03300
physics.med-phhttps://arxiv.org/archive/physics.med-ph
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:2_2-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:2_2-1
Kinnosuke Ogurahttps://en.wikipedia.org/wiki/Kinnosuke_Ogura
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-1
3https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-2
4https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-3
5https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-4
6https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:0_3-5
"Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers"https://sweet.ua.pt/pjf/PDF/Ferreira2011h.pdf
cite journalhttps://en.wikipedia.org/wiki/Template:Cite_journal
linkhttps://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:1_4-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:1_4-1
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-Shannon49_5-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-Shannon49_5-1
3https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-Shannon49_5-2
4https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-Shannon49_5-3
Shannon, Claude E.https://en.wikipedia.org/wiki/Claude_Shannon
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/jrproc.1949.232969https://doi.org/10.1109%2Fjrproc.1949.232969
S2CIDhttps://en.wikipedia.org/wiki/S2CID_(identifier)
52873253https://api.semanticscholar.org/CorpusID:52873253
Reprint as classic paper in: Proc. IEEE, Vol. 86, No. 2, (Feb 1998)http://www.stanford.edu/class/ee104/shannonpaper.pdf
Archivedhttps://web.archive.org/web/20100208112344/http://www.stanford.edu/class/ee104/shannonpaper.pdf
Wayback Machinehttps://en.wikipedia.org/wiki/Wayback_Machine
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-6
Orthogonal Transforms for Digital Signal Processinghttps://books.google.com/books?id=F-nvCAAAQBAJ
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1007/978-3-642-45450-9https://doi.org/10.1007%2F978-3-642-45450-9
ISBNhttps://en.wikipedia.org/wiki/ISBN_(identifier)
978-3-540-06556-2https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-06556-2
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-9
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-10
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-11
"Necessary density conditions for sampling and interpolation of certain entire functions"https://doi.org/10.1007%2FBF02395039
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1007/BF02395039https://doi.org/10.1007%2FBF02395039
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-12
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-13
arXivhttps://en.wikipedia.org/wiki/ArXiv_(identifier)
0709.1563https://arxiv.org/abs/0709.1563
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
2009ITSP...57..993Mhttps://ui.adsabs.harvard.edu/abs/2009ITSP...57..993M
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/TSP.2009.2012791https://doi.org/10.1109%2FTSP.2009.2012791
S2CIDhttps://en.wikipedia.org/wiki/S2CID_(identifier)
2529543https://api.semanticscholar.org/CorpusID:2529543
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-14
arXivhttps://en.wikipedia.org/wiki/ArXiv_(identifier)
1405.5329https://arxiv.org/abs/1405.5329
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
2016ITIT...62..401Khttps://ui.adsabs.harvard.edu/abs/2016ITIT...62..401K
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/tit.2015.2485271https://doi.org/10.1109%2Ftit.2015.2485271
S2CIDhttps://en.wikipedia.org/wiki/S2CID_(identifier)
47085927https://api.semanticscholar.org/CorpusID:47085927
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-15
arXivhttps://en.wikipedia.org/wiki/ArXiv_(identifier)
1801.06718https://arxiv.org/abs/1801.06718
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
2018ISPM...35c..16Khttps://ui.adsabs.harvard.edu/abs/2018ISPM...35c..16K
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/MSP.2017.2774249https://doi.org/10.1109%2FMSP.2017.2774249
S2CIDhttps://en.wikipedia.org/wiki/S2CID_(identifier)
13693437https://api.semanticscholar.org/CorpusID:13693437
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-16
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-17
Nyquist, Harryhttps://en.wikipedia.org/wiki/Harry_Nyquist
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
1928TAIEE..47..617Nhttps://ui.adsabs.harvard.edu/abs/1928TAIEE..47..617N
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/t-aiee.1928.5055024https://doi.org/10.1109%2Ft-aiee.1928.5055024
Reprint as classic paperhttps://web.archive.org/web/20130926031230/http://www.ieee.org/publications_standards/publications/proceedings/nyquist.pdf
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:02_18-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:02_18-1
3https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:02_18-2
4https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:02_18-3
5https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:02_18-4
"Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers"https://sweet.ua.pt/pjf/PDF/Ferreira2011h.pdf
cite journalhttps://en.wikipedia.org/wiki/Template:Cite_journal
linkhttps://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-19
(English translation 2005)http://ict.open.ac.uk/classics/2.pdf
Archivedhttps://web.archive.org/web/20190521021624/http://ict.open.ac.uk/classics/2.pdf
Wayback Machinehttps://en.wikipedia.org/wiki/Wayback_Machine
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:3_20-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:3_20-1
Shannon, Claude E.https://en.wikipedia.org/wiki/Claude_Shannon
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1002/j.1538-7305.1948.tb01338.xhttps://doi.org/10.1002%2Fj.1538-7305.1948.tb01338.x
hdlhttps://en.wikipedia.org/wiki/Hdl_(identifier)
11858/00-001M-0000-002C-4317-Bhttps://hdl.handle.net/11858%2F00-001M-0000-002C-4317-B
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:4_21-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-:4_21-1
Shannon, Claude E.https://en.wikipedia.org/wiki/Claude_Shannon
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1002/j.1538-7305.1948.tb00917.xhttps://doi.org/10.1002%2Fj.1538-7305.1948.tb00917.x
hdlhttps://en.wikipedia.org/wiki/Hdl_(identifier)
11858/00-001M-0000-002C-4314-2https://hdl.handle.net/11858%2F00-001M-0000-002C-4314-2
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-22
"Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers"https://sweet.ua.pt/pjf/PDF/Ferreira2011h.pdf
cite journalhttps://en.wikipedia.org/wiki/Template:Cite_journal
linkhttps://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-24
Some Historic Remarks On Sampling Theoremhttps://sites.bu.edu/mark/files/2018/02/196.pdf
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-25
Jerri, Abdulhttps://en.wikipedia.org/wiki/Abdul_Jerri
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
1977IEEEP..65.1565Jhttps://ui.adsabs.harvard.edu/abs/1977IEEEP..65.1565J
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/proc.1977.10771https://doi.org/10.1109%2Fproc.1977.10771
S2CIDhttps://en.wikipedia.org/wiki/S2CID_(identifier)
37036141https://api.semanticscholar.org/CorpusID:37036141
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/proc.1979.11307https://doi.org/10.1109%2Fproc.1979.11307
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-26
"The Origins of the Sampling Theorem"http://www.hit.bme.hu/people/papay/edu/Conv/pdf/origins.pdf
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/35.755459https://doi.org/10.1109%2F35.755459
1https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-EM_27-0
2https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-EM_27-1
"A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing"http://bigwww.epfl.ch/publications/meijering0201.pdf
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
2002IEEEP..90..319Mhttps://ui.adsabs.harvard.edu/abs/2002IEEEP..90..319M
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1109/5.993400https://doi.org/10.1109%2F5.993400
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-30
(English translation, PDF)http://ict.open.ac.uk/classics/1.pdf
Archivedhttps://web.archive.org/web/20210301042517/http://ict.open.ac.uk/classics/1.pdf
Wayback Machinehttps://en.wikipedia.org/wiki/Wayback_Machine
Modern Sampling Theory: Mathematics and Applicationshttp://www.ieeta.pt/~pjf/MSTMA/
ISBNhttps://en.wikipedia.org/wiki/ISBN_(identifier)
0-8176-4023-1https://en.wikipedia.org/wiki/Special:BookSources/0-8176-4023-1
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-31
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-32
Theory of Linear Physical Systemshttps://books.google.com/books?id=jtI-AAAAIAAJ
ISBNhttps://en.wikipedia.org/wiki/ISBN_(identifier)
978-0-471-33070-7https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-33070-7
cite bookhttps://en.wikipedia.org/wiki/Template:Cite_book
helphttps://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-33
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-34
Bibcodehttps://en.wikipedia.org/wiki/Bibcode_(identifier)
1954PhT.....7k..17Ghttps://ui.adsabs.harvard.edu/abs/1954PhT.....7k..17G
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1063/1.3061438https://doi.org/10.1063%2F1.3061438
hdlhttps://en.wikipedia.org/wiki/Hdl_(identifier)
2027/mdp.39015002049487https://hdl.handle.net/2027%2Fmdp.39015002049487
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-35
Blackman, R. B.https://en.wikipedia.org/wiki/R._B._Blackman
Tukey, J. W.https://en.wikipedia.org/wiki/J._W._Tukey
The Bell System Technical Journalhttps://en.wikipedia.org/wiki/The_Bell_System_Technical_Journal
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1002/j.1538-7305.1958.tb03874.xhttps://doi.org/10.1002%2Fj.1538-7305.1958.tb03874.x
MRhttps://en.wikipedia.org/wiki/MR_(identifier)
0102897https://mathscinet.ams.org/mathscinet-getitem?mr=0102897
↑https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem#cite_ref-36
edithttps://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&action=edit§ion=17
"Five short stories about the cardinal series"https://www.ams.org/bull/1985-12-01/S0273-0979-1985-15293-0/
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1090/S0273-0979-1985-15293-0https://doi.org/10.1090%2FS0273-0979-1985-15293-0
Küpfmüller, Karlhttps://en.wikipedia.org/wiki/Karl_Küpfmüller
"Utjämningsförlopp inom Telegraf- och Telefontekniken"https://runeberg.org/tektid/1931e/0157.html
Teknisk Tidskrifthttps://en.wikipedia.org/wiki/Teknisk_Tidskrift
178–182https://runeberg.org/tektid/1931e/0182.html
Introduction to Shannon Sampling and Interpolation Theoryhttp://marksmannet.com/RobertMarks/REPRINTS/1999_IntroductionToShannonSamplingAndInterpolationTheory.pdf
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1007/978-1-4613-9708-3https://doi.org/10.1007%2F978-1-4613-9708-3
ISBNhttps://en.wikipedia.org/wiki/ISBN_(identifier)
978-1-4613-9708-3https://en.wikipedia.org/wiki/Special:BookSources/978-1-4613-9708-3
Archivedhttps://web.archive.org/web/20110714042657/http://marksmannet.com/RobertMarks/REPRINTS/1999_IntroductionToShannonSamplingAndInterpolationTheory.pdf
Advanced Topics in Shannon Sampling and Interpolation Theoryhttp://marksmannet.com/RobertMarks/REPRINTS/1993_AdvancedTopicsOnShannon.pdf
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