Lesson03SamplingV5.docx
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Lesson03SamplingV5
SamplingTheorem
Lesson02[IEEE]03[UF]
What’sitallabout!
∙Shannon’s(Nyquist)SamplingTheorem.
∙Signalreconstruction(interpolation).
∙Practicalinterpolation.
∙Samplingmodalities(critical,over,under).
IntroductiontotheSamplingTheorem
ClaudeShannon(1916-2001)
OneofthemostimportantscientificadvancementsofthefirsthalfofthetwentiethcenturyisattributabletoClaudeShannonofBellLaboratories.ManyofShannon'sinventionsremainwithustoday,othersareforgotten.Oneofhismoreamusingcreationswasablackboxthat,whenactivatedwithaswitch,wouldextendagreenhandoutwardsandturntheswitchoff.OfgreatervalueishiscelebratedandenduringSamplingTheorem(seefactoid)
Shannonworkedforthetelephonecompanyand,assuch,wasinterestedinmaximizingthenumberofbillablesubscribersthatcouldsimultaneouslyuseacoppertelephoneline,thetechnologyoftheday.Shannon’sinventionwastosampletheindividualsubscriber’sconversations,interlacethesamplestogether,placethemallonacommontelephonewire,finallyreconstructingtheoriginalmessageatthereceiverafterde-interlacingthesamples.Todaywerefertothisprocessastime-divisionmultiplexing(TDM).Shannonestablishedtherulesthatgovernthesamplingandsignalreconstructionprocedure.Withoutareconstructionrule,however,Shannon’slaborswouldhaveheldnovaluetothetelephonecompany.
TheSamplingTheoremiscoretotheunderstandingandpracticeofDSP.ThetheorembothenablesandconstrainstheperformanceofthetypicalDSPsystemsuggestedinFigure1,asystemconsistingofanADC,DAC,digitalorDSPprocessor,plusanalogsignalconditioningfilters(i.e.,anti-aliasingandreconstructionfilter).TheSamplingTheoremalsomotivatestheneedforthesesignalconditioningfilters.
Factoid:
SomeattributethesamplingtheoremtoClaudeShannon,andotherstoHarryNyquist.Nyquistsuggestedthesamplingtheoremin1928,whichwasmathematicallyprovenbyShannonin1949(MSThesis).Somebooksusetheterm"NyquistSamplingTheorem",andothersuse"ShannonSamplingTheorem"torefertotheunderlyingtheoryofsampling.
Figure1:
TypicalDSPsystemconsistingofaninputsignalconditioningstage(anti-aliasingfilter),ADC,DSPmicroprocessor,sampleandhold(S&H)DAC,andoutputsignalconditioningfilter.
Analog
Filter
Antialiasing
Filter
Figure1:
DSPsystemconsistingofaninputsignalconditioner(anti-aliasingfilter),ADC,DSPmicroprocessor,DAC,andoutputsignalconditioner(reconstructionfilter).(redraw)
Shannon’sSamplingTheorem(theenablingtechnology)
TheSamplingTheoremstatesthatifaband-limitedsignalx(t),whosehighestfrequencyisboundedfromabovebysomefmax,isperiodicallysampledatsomeratefs,where
fs>2·fmax;Ts=1/fs1.
thentheoriginalsignalx(t)canbereconstructedfromthesamplevaluesx(t=kTs)=x[k].Theparametersandterminologyofthisprocessarereportedintheinsert.
Asapointofterminology,thefollowingnamingconventionsareoftenused:
2·fmax=NyquistsamplerateinSa/s.
fs/2=NyquistfrequencyinHz.
仙农采样定律就是对一个有限带宽的信号x(t)采样,只有采样频率大于2倍信号最高频率fmax,样本x(t=kTs)=x[k].才能够恢复出原始信号x(t)。
等于也不行,等于的情况下,如果对正弦波采样,两倍的采样率,刚好采在t=0处,采样出来都是0,所以不行。
ItshouldbestronglynotedthatthesamplingratemustbegreaterthantheNyquistsamplerate(2fmax)andnotequalto2fmax.Theimportanceofthestatementcanbeillustratedbyconsideringasimplecosinewavex(t)=cos(πt/fs).Ifsampledataratefsbeginningandt=0,theresultingtime-seriesisx[k]={1,-1,1,…,(-1)k,…}asillustratedinFigure2.Itistemptingtoassumethatasignalwithacosingenvelopecouldhaveproducedthesamplevalues.IsthisthenacounterexampleofShannon’ssamplingtheorem?
Considersaslightmodificationofthepreviousobservationbyusingx(t)=sin(πt/fs).Samplingx(t)atthesameratefs,wouldproduceatime-seriesx[k]={0,0,0,…}whichwouldbeinterpreted(incorrectly)asx(t)=0.ItshouldbeunderstoodexactlywhatShannonstated.Shannon’sSamplingTheoremsimplystatesthatifyousampleataratefs>2·fma,,itcanbeassuredthattheoriginalsignalcanbetheoreticallyreconstructedfromitssamplevalues.Itmakesnoclaimsbeyondthis.Whatremainstoestablishedisameansofreconstructingtheoriginalsignalformitssamplevalues.Thisprocessisalsocalledinterpolation.
Figure2:
Sampleratehypothetical.(left)cosinesampledatfs,and(right)sinesampledatfs.
Example:
SamplingRate
Thehumanearcanpotentiallydetectsoundsacrossa20Hzto20kHzfrequencyrange.Basedonfirstprinciples,itwouldfollowthattheminimumsampleratemustbegreaterthan40 kHz.Astandard44.1 kHzmultimediaADCwouldobviouslysatisfythisrequirement.Itshouldbeappreciated;however,thata20 kHzsignalsampledat44.1 kHzproducesonly2.205samplesperperiod.AccordingtoShannon,thisissufficienttoguaranteesignalrecovery.人耳朵能听到声音信号频率范围是20Hz----20kHz,根据采样定律,采样率应该大于40kHz,标准采样率定为44.1kHz,就可以原样恢复人能听到的声音信号。
Endofexample---------------------------------
SignalReconstruction信号恢复
Shannonassumedthatananalogsignalx(t)canbereplacedwithasetofperiodicallysamplevaluesx[k]thatformatimeseriesx[k].Thereconstructionofx(t)fromitssamplevaluesisachievedbyaprocessinvolvingtheuseofaninterpolatingfilter插值滤波器.Interpolationcanbedefinedintermsoftheinteractionofadiscrete-timetime-seriesandaananaloginterpolatingfilterhavingagivenimpulseresponse.Shannon’sinterpolatingfilterhasanimpulseresponseisgivenby:
仙农插值滤波器的冲激响应
.2
Formally,theoutputofaShannoninterpolatortoanarbitraryinputtime-seriesismathematicallydefinedbyaprocesscalledlinearconvolution卷积asmotivatedbelow:
数字样本信号x[k]和仙农插值滤波器冲激响应h(t)卷积,就可得到原始模拟信号。
3
Theinterpolationprocess,describedinEquation3,isgraphicallyinterpretedinFigure3.Theinputsignalx(t)issampledatarateconsistentwithShannon’sSamplingTheoremtoproduceatime-seriesx[k].Thetime-seriesisthenpassedthroughShannon’sinterpolationfilterthatperformstheconvolutionoperationdescribedbyEquation3.Theoutcomeisthereconstructedsignalx(t)whichresidesincontinuous-time.
…
…
ShannonFilters
y[k-2]
x[k-2]
y[k-1]
x[k-1]
y[k]
x[k]
x(t)
ShannonFilters
…
Figure3:
Shannon’sinterpolationprocessshowinghowsamplevaluesareconverted(interpolated)intoacontinuous-timesignalusinganidealsincfilter.表明了样本怎样通过仙农插值恢复到连续信号的
…
Shannon’sinterpolatingfilter,definedbyEquation2,canalsobeinterpretedinthefrequencydomainassuggestedinFigure4.ThefrequencydomainenvelopeofShannon’sinterpolatingfilteristhatofanideallowpass(boxcar)filterhavingapassbanddefinedoverf[0,fs/2) Hz(i.e.,fromDCtotheNyquistfrequency).仙农插值滤波器的幅频特性
|H(f)|
Figure4:
Shannon’sinterpolatingfilterinterpretedinthecontinuous-time(left)andcontinuous-frequencydomain(right).
ShannonInterpolation
Assume,forpurposesofconceptdevelopmentthatananalogsinusoidalsignalx(t)=sin(2πf0t)issampledabovetheNyquistsamplerateatafrequencyfs,wherefs>2f0.Thesampleprocessproducesadiscrete-timesignalx[k].Theoretically,theperfectreconstructionofananalogsignalfromthatsignal’ssamplevaluesrequiresthatShannon’sinterpolatingfilterbeemployedasmotivatedinFigure5.仙农理论插值过程
Figure 5:
Shannon’stheoreticalinterpolatorprocess.
x(t)
Originalanalogsignal
Idealsampler
Discrete-timesignal
x[k]
Shannoninterpolatingfilter
Interpolated(filtered)signal
x(t)
ApproximateInterpolatingFilters近似的插值滤波器
WhileShannon’sinterpolatingfilteriselegant,itpresentsamajorimplementationobstacle.Theinterpolatingfilterh(t)=sinc(t /Ts)isseentobenon-causalbyvirtueofthefactthatthefilter’sresponseisactiveforalltime,-t.Asaresult,Shannon’sinterpolationfilterisnotphysicallyrealizablesincethefilter’simpulseresponseexistsinpre-history(i.e.,t0).ThishascausedDSPtechnologiststosearchforalternativeinterpolatingfiltersthatbehaveinamannersimilartoaShannoninterpolator,butarealsophysicallyrealizable.SuchfilterswouldbeusedtoreplacetheShannonfiltershowninFigure5.仙农插值滤波器的冲激响应在时间轴上-t都有信号,物理上是不能实现的。
就需要找一个近似的滤波器代替仙农滤波器
Inpractice,theidealsignalinterpolationprocessdescribedinFigure5wouldbeperformedusingthetechnologyasshowninFigure 6.Itisrequiredthatthatthesampledsignalbefirstconvertedtoadigitaldatastreamusingananalogtodigitalconverter(ADC).Thedigitizedsamplesareconvertedbackintoanalogformusingadigitaltoanalogconverter(DAC)whichtranslatesdigitalwordsintoanalogsignallevelsthataremaintainedforafullsamplingperiod.Thisprocessiscalledzero-orderhold(ZOH).TheanalogZOHsignalisthenpresentedtoacausalanalogfilterthatapproximatesthebehaviorofaShannoninterpolator.Theoutputisaninterpolatedanalogsignalx0(t)thatapproximatestheoriginalinputx(t).Interpolationerrorsareduetoquantizationeffectsintroducedbythedigitaldevices(ADCandDAC)anddifferencesbetweentheShannonandapproximatinginterpolationfilter.
x(t)
ADC
X[k]
DAC(S&H)
x[k]
Analogsignal–Zeroorderhold.
可以是1阶保持或n阶保持
Approximate
Shannon
Interpolation
Filter
x0(t)~x(t)近似相等
Figure6:
Practicalinterpolationprocess.
Figure6:
Practical(digital)interpolationprocess.
Ultimately,theobjectiveoftheinterpolatingfilteristoproduceanoutputwhichisingoodagreementwiththeanalogsignalformknowledgeonlyofthesignal’ssamplevalues.Traditionally,thishasbeenaccomplished,tovaryingdegreesof
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