1、NyquistShannonsamplingtheoremX ( )BBFig. 1: Magnitude of the Fourier transform of a bandlimitedfunctionIntheeldofdigitalsignalprocessing,thesamplingthe-orem is a fundamental bridge between continuous-timesignals (often called “analog signals”) and discrete-timesignals (often called “digital signals”
2、). It establishes asucientconditionbetweenasignalsbandwidthandthesample rate that permits a discrete sequence of samplesto capture all the information from the continuous-timesignal.Strictly speaking, the theorem only applies to a class ofmathematicalfunctionshavingaFouriertransformthatiszerooutside
3、ofaniteregionoffrequencies(seeFig1).Intuitivelyweexpectthatwhenonereducesacontinuousfunction to a discrete sequence and interpolates back toa continuous function, the delity of the result dependson the density (or sample rate) of the original samples.The sampling theorem introduces the concept of a
4、sam-ple rate that is sucient for perfect delity for the classof functions that are bandlimited to a given bandwidth,such that no actual information is lost in the samplingprocess. It expresses the sucient sample rate in termsof the bandwidth for the class of functions. The theo-rem also leads to a f
5、ormula for perfectly reconstructingtheoriginalcontinuous-timefunctionfromthesamples.Perfect reconstruction may still be possible when thesample-ratecriterionisnotsatised,providedothercon-straintsonthesignalareknown. (SeeSamplingofnon-basebandsignalsbelow,andCompressedsensing.)The name NyquistShannon
6、 sampling theorem honorsHarry Nyquist and Claude Shannon. The theoremwas also discovered independently by E. T. Whit-taker, by Vladimir Kotelnikov, and by others. Soit is also known by the names NyquistShannonKotelnikov, WhittakerShannonKotelnikov, WhittakerNyquistKotelnikovShannon, and cardinal the
7、orem ofinterpolation.1 IntroductionSampling is the process of converting a signal (for ex-ample, a function of continuous time and/or space) intoa numeric sequence (a function of discrete time and/orspace). Shannonsversionofthetheoremstates:1If a function x(t) contains no frequencies higher than Bhe
8、rtz,itiscompletelydeterminedbygivingitsordinatesataseriesofpointsspaced1/(2B)secondsapart.Asucientsample-rateistherefore2B samples/second,or anything larger. Equivalently, for a given sample ratefnull, perfect reconstruction is guaranteed possible for abandlimitB fnull/2.When the bandlimit is too hi
9、gh (or there is no ban-dlimit),thereconstructionexhibitsimperfectionsknownasaliasing. Modernstatementsofthetheoremaresome-times careful to explicitly state that x(t) must containno sinusoidal component at exactly frequency B, or thatB must be strictly less than the sample rate. Thetwo thresholds, 2B
10、 and fnull/2 are respectively called theNyquist rate andNyquist frequency. Andrespectively,theyareattributesofx(t)andofthesamplingequipment.The condition described by these inequalities is calledthe Nyquist criterion, or sometimes the Raabe condi-tion. Thetheoremisalsoapplicabletofunctionsofotherdom
11、ains,suchas space, inthecaseofadigitizedimage.Theonlychange,inthecaseofotherdomains,istheunitsofmeasureappliedtot,fnull,andB.1.00.80.60.40.2-0.2-6 -4 -2 2 4 6sin(x)xxFig. 2: The normalized sinc function: sin(x) / (x) . showingthe central peak at x= 0, and zero-crossings at the other integervalues of
12、 x.ThesymbolT =1/fnulliscustomarilyusedtorepresenttheinterval between samples and is called the sample pe-riodorsampling interval . Andthesamplesoffunction12 2 ALIASINGx(t)arecommonlydenotedbyxn=x(nT)(alternatively“xn“ in older signal processing literature), for all integervalues of n. The mathemati
13、cally ideal way to interpo-late the sequence involves the use of sinc functions, likethoseshowninFig2. Eachsampleinthesequenceisre-placedbyasincfunction,centeredonthetimeaxisattheoriginallocationofthesample,nT,withtheamplitudeofthesincfunctionscaledtothesamplevalue,xn. Subse-quently,thesincfunctions
14、aresummedintoacontinuousfunction. Amathematicallyequivalentmethodistocon-volveonesincfunctionwithaseriesofDiracdeltapulses,weightedbythesamplevalues. Neithermethodisnumer-ically practical. Instead, some type of approximation ofthe sinc functions, nite in length, is used. The imper-fections attributa
15、ble to the approximation are known asinterpolation error.Practical digital-to-analog converters produce neitherscaledanddelayedsincfunctions,noridealDiracpulses.Instead they produce a piecewise-constant sequence ofscaled and delayed rectangular pulses (the zero-orderhold),usuallyfollowedbyan“anti-im
16、aginglter”tocleanupspurioushigh-frequencycontent.2 AliasingMainarticle: AliasingWhenx(t)isafunctionwithaFouriertransform,X(f):Fig. 3: The samples of two sine waves can be identical, when atleast one of them is at a frequency above half the sample rate.X(f) def= 11x(t) e i2 ft dt;the Poisson summatio
17、n formula indicates that the sam-ples,x(nT),ofx(t)aresucienttocreateaperiodicsum-mationofX(f). Theresultis:which is a periodic function and its equivalent represen-tationasaFourierseries,whosecoecientsareTx(nT).This function is also known as the discrete-time Fouriertransform(DTFT)ofthesequenceTx(nT
18、),forintegersn.As depicted in Figure 4, copies of X(f) are shifted bymultiples of fs and combined by addition. For a band-limitedfunction(X(f)=0forall|f|B),andsucientlyFig. 4: X(f) (top blue) and XA(f) (bottom blue) are continuousFouriertransformsoftwodierentfunctions,x(t)andxA(t)(notshown). When th
19、e functions are sampled at rate fs, the images(green) are added to the original transforms (blue) when one ex-amines the discrete-time Fourier transforms (DTFT) of the se-quences. In this hypothetical example, the DTFTs are identical,which means the sampled sequences are identical, even thoughthe or
20、iginal continuous pre-sampled functions are not. If thesewere audio signals, x(t) and xA(t) might not sound the same.But their samples (taken at ratefs) are identical and would leadto identical reproduced sounds; thus xA(t) is an alias of x(t) atthis sample rate.large fs, it is possible for the copi
21、es to remain distinctfrom each other. But if the Nyquist criterion is not sat-ised, adjacent copies overlap, and it is not possible ingeneraltodiscernanunambiguous X(f). Anyfrequencycomponentabove fs/2isindistinguishablefromalower-frequencycomponent,calledanalias,associatedwithoneof the copies. In s
22、uch cases, the customary interpola-tiontechniquesproducethealias,ratherthantheoriginalcomponent. Whenthesample-rateispre-determinedbyother considerations (such as an industry standard), x(t)isusuallylteredtoreduceitshighfrequenciestoaccept-ablelevelsbeforeitissampled. Thetypeoflterrequiredis a lowpa
23、ss lter, and in this application it is called ananti-aliasinglter.Fig. 5: Spectrum, Xnull(f), of a properly sampled bandlimited sig-nal (blue) and the adjacent DTFT images (green) that do notoverlap. A brick-wall low-pass lter, H(f), removes the images,leaves the original spectrum, X(f), and recover
24、s the original sig-nal from its samples.33 Derivation as a special case ofPoisson summationFromFigure5,itisapparentthatwhenthereisnooverlapof the copies (aka “images”) of X(f), the k = 0 term ofXs(f)canberecoveredbytheproduct:X(f) = H(f) Xs(f); where:H(f) def=1 jfj fs B:Atthispoint,thesamplingtheore
25、misproved,sinceX(f)uniquelydeterminesx(t).All that remains is to derive the formula for reconstruc-tion. H(f)neednotbepreciselydenedintheregionB,fsBbecauseXnull(f)iszerointhatregion. However,theworst case is when B = fs/2, the Nyquist frequency. Afunctionthatissucientforthatandalllessseverecasesis:H
26、(f) = rect( ffs)=1 jfj fs2 ;whererect()istherectangularfunction. Therefore:X(f) = rect(ffs)Xs(f)= rect(Tf) 1n= 1T x(nT) e i2 nTf (fromEq.1 ,above).= 1n= 1x(nT) T rect(Tf) e i2 nTf| z Ffsinc(t nTT )g:2The inverse transform of both sides produces theWhittakerShannoninterpolationformula:x(t) =1n= 1x(nT
27、) sinc(t nTT);which shows how the samples, x(nT), can be combinedtoreconstructx(t).FromFigure5,itisclearthatlarger-than-necessaryvalues of fs (smaller values of T), called oversam-pling, have no eect on the outcome of the recon-structionandhavethebenetofleavingroomforatransitionbandinwhichH(f)isfree
28、totakeinterme-diatevalues. Undersampling,whichcausesaliasing,isnotingeneralareversibleoperation.Theoretically, the interpolation formula can beimplemented as a low pass lter, whose im-pulse response is sinc(t/T) and whose input is1n= 1x(nT) (t nT);whichisaDiraccombfunction modulated by the signal sa
29、mples. Practi-cal digital-to-analog converters (DAC) implementan approximation like the zero-order hold. In thatcase, oversampling can reduce the approximationerror.4 Shannons original proofPoisson shows that the Fourier series in Eq.1 producesthe periodic summation of X(f), regardless of fs and B.S
30、hannon,however,onlyderivestheseriescoecientsforthecasefs=2B.VirtuallyquotingShannonsoriginalpa-per:Let X(!) bethespectrumof x(t): Thensince X(!) is assumed to be zero outside thebandj !2 j2B,considerthefamilyof sinusoids (depicted in Fig. 8) generated by dierentvaluesofinthisformula:x(t) = cos(2 Bt+
31、 )cos( ) = cos(2 Bt) sin(2 Bt)tan( ); /2 /2:Withfs=2BorequivalentlyT =1/(2B),thesamplesaregivenby:x(nT) = cos( n) sin( n)| z 0tan( ) = ( 1)nregardlessofthevalueof. Thatsortofambiguityisthereasonforthe strict inequalityofthesamplingtheoremscondition.7 Sampling of non-baseband sig-nalsAsdiscussedbySha
32、nnon:1A similar result is true if theband does not start at zero fre-quency but at some higher value,andcanbeprovedbyalineartrans-lation (corresponding physically tosingle-sideband modulation) of thezero-frequency case. In this casethe elementary pulse is obtainedfrom sin(x)/x by single-side-bandmod
33、ulation.Thatis,asucientno-lossconditionforsamplingsignalsthatdonothavebasebandcomponentsexiststhatinvolvesthe width of the non-zero frequency interval as opposedtoitshighestfrequencycomponent. SeeSampling(signalprocessing)formoredetailsandexamples.AbandpassconditionisthatX(f)=0,forallnonnegativef ou
34、tsidetheopenbandoffrequencies:(N2 fs;N + 12 fs);for some nonnegative integer N. This formulation in-cludesthenormalbasebandconditionasthecaseN=0.The corresponding interpolation function is the impulseresponse of an ideal brick-wall bandpass lter (as op-posed to the ideal brick-wall lowpass lter used
35、 above)withcutosattheupperandloweredgesofthespeciedband, which is the dierence between a pair of lowpassimpulseresponses:(N+1) sinc(N + 1)tT)N sinc(NtT):6 10 HISTORICAL BACKGROUNDOther generalizations, for example to signals occupyingmultiplenon-contiguousbands,arepossibleaswell. Eventhemostgenerali
36、zedformofthesamplingtheoremdoesnot have a provably true converse. That is, one cannotconcludethatinformationisnecessarilylostjustbecausetheconditionsofthesamplingtheoremarenotsatised;fromanengineeringperspective,however,itisgenerallysafetoassumethatifthesamplingtheoremisnotsatisedtheninformationwill
37、mostlikelybelost.8 Nonuniform samplingThe sampling theory of Shannon can be generalizedfor the case of nonuniform sampling, that is, samplesnot taken equally spaced in time. The Shannon sam-plingtheoryfornon-uniformsamplingstatesthataband-limitedsignalcanbeperfectlyreconstructedfromitssam-ples if th
38、e average sampling rate satises the Nyquistcondition.3 Therefore, although uniformly spaced sam-ples may result in easier reconstruction algorithms, it isnotanecessaryconditionforperfectreconstruction.The general theory for non-baseband and nonuniformsampleswasdevelopedin1967byLandau.4 Heprovedthat,
39、 to paraphrase roughly, the average sampling rate(uniformorotherwise)mustbetwicetheoccupied band-width of the signal, assuming it is a priori known whatportionofthespectrumwasoccupied. Inthelate1990s,thisworkwaspartiallyextendedtocoversignalsofwhenthe amount of occupied bandwidth was known, but thea
40、ctualoccupiedportionofthespectrumwasunknown.5In the 2000s, a complete theory was developed (see thesection Beyond Nyquist below) using compressed sens-ing. In particular, the theory, using signal processinglanguage, is described in this 2009 paper.6 They show,amongotherthings,thatifthefrequencylocat
41、ionsareun-known,thenitisnecessarytosampleatleastattwicetheNyquist criteria; in other words, you must pay at least afactorof2fornotknowingthelocationofthespectrum.Notethatminimumsamplingrequirementsdonotneces-sarilyguaranteestability.9 Sampling below the Nyquist rateunder additional restrictionsMaina
42、rticle: UndersamplingThe NyquistShannon sampling theorem provides asucient condition for the sampling and reconstructionof a band-limited signal. When reconstruction is donevia the WhittakerShannon interpolation formula, theNyquist criterion is also a necessary condition to avoidaliasing,inthesenset
43、hatifsamplesaretakenataslowerrate than twice the band limit, then there are some sig-nals that will not be correctly reconstructed. However,iffurtherrestrictionsareimposedonthesignal,thentheNyquistcriterionmaynolongerbeanecessarycondition.A non-trivial example of exploiting extra assumptionsaboutthe
44、signalisgivenbytherecenteldofcompressedsensing, which allows for full reconstruction with a sub-Nyquist sampling rate. Specically, this applies to sig-nals that are sparse (or compressible) in some domain.As an example, compressed sensing deals with signalsthat may have a low over-all bandwidth (say
45、, the eec-tive bandwidth EB), but the frequency locations are un-known,ratherthanalltogetherinasingleband,sothatthepassbandtechniquedoesntapply. Inotherwords,thefre-quency spectrum is sparse. Traditionally, the necessarysamplingrateisthus2B.Usingcompressedsensingtech-niques,thesignalcouldbeperfectly
46、reconstructedifitissampledatarateslightlylowerthan2EB.Thedownsideofthisapproachisthatreconstructionisnolongergivenbyaformula,butinsteadbythesolutiontoaconvexop-timizationprogramwhichrequireswell-studiedbutnon-linearmethods.10 Historical backgroundThesamplingtheoremwasimpliedbytheworkofHarryNyquist i
47、n 1928 (“Certain topics in telegraph transmis-sion theory”), in which he showed that up to 2B inde-pendentpulsesamplescouldbesentthroughasystemofbandwidth B;buthedidnotexplicitlyconsidertheprob-lemofsamplingandreconstructionofcontinuoussignals.About the same time, Karl Kpfmller showed a simi-lar res
48、ult,7 and discussed the sinc-function impulse re-sponse of a band-limiting lter, via its integral, the stepresponse Integralsinus; thisbandlimitingandreconstruc-tion lter that is so central to the sampling theorem issometimesreferredtoasa Kpfmller lter (butseldomsoinEnglish).Thesamplingtheorem,essen
49、tiallyadualofNyquistsre-sult, was proved by Claude E. Shannon in 1949 (“Com-municationinthepresenceofnoise”). V.A.Kotelnikovpublished similar results in 1933 (“On the transmissioncapacity of the ether and of cables in electrical com-munications”, translation from the Russian), as did themathematicianE.T.Whittakerin1915(“ExpansionsoftheInterpolation-Theory”,“TheoriederKardinalfunktio-nen”), J. M. Whittaker in 1935 (“Interpolatory functiontheory”), and Gabor in 1946 (“Theor
