Lesson03SamplingV5.docx

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