频谱分析精讲信号处理必学.docx
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频谱分析精讲信号处理必学.docx
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频谱分析精讲信号处理必学
IntroductiontoSpectralAnalysisandMatlab
IRISSummerInternOrientation,2008
Introduction
TheobjectofthislabistoexploretherelationshipbetweenthetimedomainandthefrequencydomainusingMATLAB.Youwillfirstlookatpuresinewavesasafunctionoftimeandtheirrepresentationinthefrequencydomain,andthenexaminesomeearthquakedata.
MATLABisacommonlyusedcommercialpackagedesignedtomanipulateandplotallsortsofdata.TheMATLABintroductionstates:
MATLABisahigh-performancelanguagefortechnicalcomputing.Itintegratescomputation,visualization,andprogramminginaneasy-to-useenvironmentwhereproblemsandsolutionsareexpressedinfamiliarmathematicalnotation.Typicalusesinclude:
∙Mathandcomputation
∙Algorithmdevelopment
∙Modeling,simulation,andprototyping
∙Dataanalysis,exploration,andvisualization
∙Scientificandengineeringgraphics
MATLABisaninteractivesystemwhosebasicdataelementisanarraythatdoesnotrequiredimensioning.Thisallowsyoutosolvemanytechnicalcomputingproblems,especiallythosewithmatrixandvectorformulations,inafractionofthetimeitwouldtaketowriteaprograminascalarnoninteractivelanguagesuchasC.
Today'slabwilluseonlyafewofthefeaturesofferedbyMATLAB,butshouldgiveyouenoughofanintroductiontoallowyoutounderstandthebasicsyntax,input/outputandplotting.
PARTONE-ARTIFICIALDATA
FirstloginandstartupMATLABononeoftheLinuxmachinesbyopeningaterminalwindowandtypingmatlab.
MATLABcommands
TheinitialwindowsthatappearincludeCommand,Workspace,andCommandhistory.Manyoperationscanbeperformedeitherinthecommandwindoworviathedropdownmenus.Inmostcasesthecommandwindowcommandswillbelistedhere.
MATLABisdesignedforeasymatrixmanipulation.Thematricesarelikeexcelspreadsheetsinthatcolumnsandrowscanbemanipulatedasaunitorasindividualelements.
TogetafeelfortheMATLABmatrixsyntax,clickonMATLABhelp,gotogettingstarted,thenMatricesandMagicSquares.
Loadingandlistingdata
FirstcopyallfilesfromtheCDprovidedtoyourhomedirectory.
Inthetextthatfollows,linesbeginningwith“>>“arecommandstobeenteredintothecommandwindow.
>>loadsine_waves(loadsthefilesine_wavesfromthecurrentdirectory)
>>who(liststhevariablesyouhaveloadedorcreated)
>>whos(liststhevariablesyouhaveloadedorcreatedandtheirsizes)
Thefilesine_waveshas512rowsand4columns.Thefirstcolumnisthetimeinseconds.
Columns2-4aretheamplitudesof3differentsinewaves,assampledatthetimeslistedincolumn1.
Toviewtheelementsofamatrixoranyvariable,simplytypeitsname.
>>sine_waves
ordoubleclickonthevariablenameinWorkspace.
Youcanalsoselectasinglecolumn:
>>sine_waves(:
1)
orjustafewelements:
>>sine_waves(1:
3,1)
⇒Whatisthesamplingintervalofthedata(ie.thetimeinsecondsbetweensuccessivesamples)?
⇒Howmanysamplesaretherepersecond?
⇒Themaximumsignalfrequencythatcanbecorrectlyobservedishalfthesamplingfrequency.ThisiscalledtheNyquistfrequency.WhatistheNyquistfrequencyinthiscase?
Plottingdata
Toplotdata,usetheplotcommandandselectthecolumnsyouwanttoplotagainsteachother:
>>plot(sine_waves(:
1),sine_waves(:
2));(plotscolumn1(time)vscolumn2(amplitude))
Youcanmanipulatetheplotusingtheplotmenuitems.Forexample,Edit,axispropertiesletsyouchangethexandyscales,andaddaxeslabelsandatitle.
Oryoucanusethecommands:
>>axis([05-1.51.5]);
>>xlabel('Time(sec)')
>>ylabel('amplitude')
>>title('sinewave')
Toplotasecondlineontheplot:
>>holdon(holdaxesonforlaterplots,holdoffallowsreplottingofnewdata,thedefaultistocleartheploteachtime(holdoff))
>>plot(sine_waves(:
1),sine_waves(:
3),':
');(plotscolumn1vscol3usingadottedline)
Ifyouwanttoaddalineaty=0,whichmakesiteasiertodeterminefrequency,createanewvariablecalledzero_line,andfillitwithzeros:
>>zero_line=zeros(512,1);
>>plot(sine_waves(:
1),zero_line);
Ifyouwanttolookattheplotsindependently:
>>clf(clearsthegraphicswindow)
thenreplotasdesired.
ToopenmultiplewindowssoyoucanlookatplotsseparatelyselectNewFigureintheFileplotmenu
Youcanfindthex,yvalueofanypointonthegraphbyselectingdatacursorintheToolsmenu
⇒Whatarethefrequenciesofthesinewavesincolumn2andcolumn3?
⇒Whataretheirrelativeamplitudes(iewhatistheratiooftheiramplitudes)?
Comparingindividualsinewavestotheirsum
Column4isthesumoftheamplitudesofcolumns2and3.Toplotcolumn4:
>>plot(sine_waves(:
1),sine_waves(:
4),'--');(plotsadashedline)
Notethatwhileyoucantellthattheresultingwavecontainsmorethanonefrequency,itishardertoestimatetherelativeamplitudesofthetwofrequencieswhentheyaresummedtogether.
FrequencyDomain
Totransformtothefrequencydomain,calculatetheFouriertransformforthesinewavesincolumns2-4ofsine_waves.YoumustgivetheFouriertransformthesamplingfrequency(inthiscase25Hz).
>>transformed2=fourier(sine_waves(:
2),25);
>>transformed3=fourier(sine_waves(:
3),25);
>>transformed4=fourier(sine_waves(:
4),25);
Foreachsinewave,anewmatrixiscreatedwithfrequency(Hz)incolumn1,amplitudeincolumn2andphaseincolumn3.Lookingatthenumbersintransformed2:
⇒Whatisthesamplingintervalinthetransformeddata(inHz)
⇒Whatisthemaximumfrequency?
⇒DoesthisagreewithyourdeterminationoftheNyquistfrequency?
⇒Beforeplottingthespectra,considerwhatyoumightexpectforthefrequencyresponseofeachsinewave.
Nowplottheamplitudespectrumforthesinewavethatwasincolumn2ofsine_waves:
>>plot(transformed2(:
1),transformed2(:
2));
⇒Atwhatfrequencyisthereamaximum?
Nowplottheamplitudespectrumforthesinewavethatwasincolumn3ofsine_waves.
⇒Whatisthepeakfrequencyofthissinewave?
⇒Whatistherelativeamplitudeofthepeaksforthe2waves?
⇒Howdoesthisratiocomparetoyourmeasuredratioofthesinewaveamplitudes?
Nowplottheamplitudespectrumforthesinewavethatwasincolumn4ofsine_waves,usingadifferentlinesymbol.
⇒Howdothespectralamplitudesofthecombinedsinewavescomparetothespectralamplitudesoftheindividualsinewaves?
Thisshowsthatthecombinationofsinewavesisalinearprocessandanarbitrarilyshapedshapedwavecanbecreatedbytheadditionofasufficientnumberofsinewaves.
Constructionofawaveplot
Nowloadthefilemulti_sine.
Thefilemulti_sineincludes10differentsinewaves(incolumns2-11)whichhavebeenphaseshiftedsothatthereisonetimewhenallthesinewavesareatamaximum.Column1istimeasinsine_waves.Youcanplotthemindividuallyinthetimedomaintoseewhattheylooklike.
Toaddthemalltogetheryoucandoitthelongway:
>>bigwave=multi_sine(:
2)+multi_sine(:
3)+multi_sine(:
4)+multi_sine(:
5)
+multi_sine(:
6)+multi_sine(:
7)+multi_sine(:
8)+multi_sine(:
9)+multi_sine(:
10)
+multi_sine(:
11);
oryoucanuseMATLAB'ssumutility(whichsumscolumns)alongwithitstransposeutility(whichswapsrowsandcolumns):
>>bigwave=sum(multi_sine(:
2:
11)')';
bigwaveisnowthesumofall10sinewaves.
Plotbigwaveinonewindowandinanotherwindowplotthe10sinewavesusedtocreateit.Notehowthesumofcontinuoussinewavesresultsinawavepacketoffiniteduration.
Thisexampleshowshowmanysineandcosinefunctionsareneededtocreateapulse-likewaveform.Tocreateasinglepulse,aninfiniteseriesofsineandcosinefunctionshavetobeaddedtogether.Asinglepulsecanbefoundincolumn13ofmulti_sine.
⇒tryplottingit.
NowcalculateandplottheFourieramplitudespectrumofbigwave(thesamplingintervalisthesameasbefore).
⇒Whatarethefrequenciesofthesinewavesthatmakeupthewavepulseinbigwave?
⇒Whatisthefrequencyresponseofthespikeincolumn13ofmulti_sine?
Why?
PARTTWO-REALDATA
NowyoucanexamineearthquakedatathatwerecollectedduringanearthquakehazardassessmentstudyoftheWellington,NewZealandregion.Thefilequake_dataincludes10secondsofSwaverecordingfrom3differentsitesforthesameearthquake.Column2isdatafromarocksite,column3istherecordingfromasedimentarybasinsiteandthecolumn4seismographwaslocatedonanoldpeatbog(nowhousingdevelopment).Asbefore,timeisincolumn1.
Plotthe3seismogramsandcomparethesignals.Notethemuchloweramplitudeoftherocksiteandthenearlysinusoidalcharacterofthebasinsites.
Timedomain
⇒Whatisthesamplingrate?
⇒WhatistheNyquistfrequency?
Frequencydomain
Transformthetimeseriestothefrequencydomainasbefore(don'tforgettoincludethenewsamplingrate).
⇒Whatisthemaximumfrequencynow?
Firstlookatthefrequencyresponseoftherocksite(column2).
⇒Whatistherangeoffrequenciesinthegroundmotion?
Nowlookatthefrequencyresponseofthegroundmotionaftertheseismicwaveshavetraveledthroughthesoftsedimentsbelowthesitesincolumns3and4.Youmaywanttochangetheaxessothatyoucanfocusonthelowfrequencies.
⇒Notethatthereareclearlydefinedpeaksinfrequencyforcolumns3and4butnotfortherocksite(column2).
⇒Whatisthefrequencyatwhichthereisamaximumforcolumns3and4?
Thesedatawerecollectedinsedimentarybasinswhichcanshakelikeabowlofjello.Abasincanresonateatparticularfrequenciesjustlikeasimpleharmonicoscillator.Theresonancecontinueslongaftertheseismicenergyhasdissipatedatthenearby
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- 关 键 词:
- 频谱 分析 信号 处理