通信系统习题.docx

通信系统习题.docx

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通信系统习题

Tut2

1．Expressthepulsesshowninthefigurebymeansoftheunitstepfunctionu（t）,andcalculatetheirenergy.

（1）ThepulsecanbeMathematicallydescribedas:

Usingtheunitstepfunctionnotation,thiscanbeexpressedas:

Withoutusingtheseparateintervaldefinitions,but

relyingonthepropertiesoftheunitstepfunction,thiscanbewrittenas:

Thepulseenergyisdefinedas:

Tofindthevalueofthisintegral,wecandivideitinto3separateintegralswithlimits[T,3T],[3T,4T],[4T,6T],respectively.Hence,theenergyisgivenby:

（2）Thepulsecanbeexpressedas:

Intermsoftheunitstepfunction,

thiscanbewrittenas:

Thepulseenergyisequalto:

（3）Thepulsecanbeexpressedas:

Intermsoftheunitstepfunction,this

canbewrittenas:

Andtheenergyisequalto2T.

2.Evaluatethefollowingintegrals

3.Findtheoutputofthetime-invariantlinearsystemhavingtheimpulseresponseh（t）shownintheFigure（a）,inresponsetotheinputsignalx（t）presentedintheFigure（b）.

Inordertocalculatetheconvolutionx（t）*h（t）,weneedfirsttofindthemirroredversionofthesignalx（t）,changethevariabletoλ,i.e.findx（-k）,aspresented

Thatmirroredversionoftheinputsignalisthenshiftedandsignalx（t-λ）isconsidered.

Itisclearlyvisiblethatfort<3thesignalx（t-k）andtheimpulseresponseh（t）donotoverlapforanyλ.

Hencetheintegraloftheirproductisequaltozero.

Thedifferentoverlappingpossibilitiesareillustratedinthefigure,andfinallyfort>7,x（t-λ）andh（t）stopoverlappingforanyλ

Therefore,theoutputsignaly（t）canbeexpressedas:

Afterperformingintegration,weget

Theobtainedoutputsignaly（t）isplottedbelow.Itisworthtonoticethatthesignaly（t）issmootherthantheinputsignalx（t）,andthatthetimelimitspropertyholds.

4.Determinetheauto-correlationfunctionofthefollowingsignal（whereA>0）:

Solution:

Thewaveformis

shownasfollows

Firstlynotethattheautocorrelationofarealfunctionisalwayssymmetric-thusyouonlyneedtodeterminetheshapeoftheautocorrelationforr<0orr>0.Secondly,notethatthetime-shiftedimageofthetriangularpulsedoesnotoverlaptheoriginalfunctionatallifr>Torr>-TThereforetheautocorrelationiszerointhoseregions.Wecanjustworkouttheautocorrelationfunctionfor0

Thisintegralistrivialtoevaluate，whichgivesyou

Theleft-handsideisjustamirror-imageoftheright：

youcansubstitutet=-tintoEquation3toderivethis：

Thusthefullexpressionfortheautocorrelationis

ThisisshowngraphicallyinFigure2.

5.Determinetheauto-correlationfunctionofthesignal

giventheauto-correlationfunctionofaperiodicsignalx（t）isdefinedas:

whereTistheperiodofthesignalx（t）

Solution:

Tut3

1.FindtheFourierseriesexpansionofthetriangularwaveform

Thecyclicallyrepeatedpulsecanbeexpressedas:

withtheboundariesoftheinterval【a，b】being-0.570and0.5T0,respectively.

Thecoefficientsaredeterminedby:

Thepulsex（t）isanevenfunctionoftime,soforany

valuesoftandnwithintheconsideredinterval

Therefore,

whichresultsinallcoefficientsbnbeingequalto0;n

1,2,....

Thecoefficienta0equalsto

Thisresultfollowsdirectlyfromthefactthata0representsthed.c.term,whichisclearlyequalto0intheconsideredcase.

Forothervaluesofn,n=1,2,...,wehave

Thefirstintegralwithinthebracketsequalstozeroforanyvalueofn,asweintegratetherecos（x）functionoveranintegernumberofperiods.

•Thesecondandthethirdtermsareofthesameform,andcanbecomputedusingtheformula

Hence,weget

•Thelatestmeansthat

Exampleplotsofthe

truncatedFourierseries

expansionforthe

triangularwavefor

（a）3terms,

（b）5terms,

（c）50terms

2.FindtheFouriertransformoftherectangularpulse

anddrawitsmagnitudeandphasespectra.

FromthedefinitionofFouriertransform,wehave

Hence,theW（f）isarealfunctionoffrequency.Thisisanexpectedresult,becausethew（t）isaneven

functionoftime.

Thusthemagnitudespectrum|W（f）|isexpressedas:

TofindthephasespectrumΘ（f）,letusfirstconsiderthefactthatforittobeandoddfunction,wemusthave

Therefore,0（f）=0ifW（f）

>0and0（f）=±kn„otherwise;kisanodd

integer.

Inaddition,sincew（t）isarealsignal,0（f）=-0（-f）.

3,FindtheFouriertransformofthesocalled'radiopulses'w1（t）andw2（t）definedas:

•TheFouriertransformW,（f）canbefoundsubstitutingw（t）=n（t）and0=0intothemodulationtheorem.Thissubstitutionyields:

•Forw2（t）,weneedtotake0=-0.5π,whichresultsin:

4.FindtheFouriertransformoftherectangularwavew（t）showninthefigurebelowanddrawthemagnitudespectrum|W（f）|inthefrequencyrange-300Hzto300Hz.

TofindtheFouriertransformofthewavew（t）wewillusetheformula

Therefore,weneedfirsttofindtheFouriertransformofthepulseh（t）whichisrepeatedatregularintervalstogeneratethewavew（t）.

Thepulseh（t）isarectangularpulsewithawidthT=10ms

Thepulsesh（t）arerepeatedeveryT0=30ms.

Hence,assumingthattimetisinseconds,thewavew（t）canbeexpressedas:

anditsFouriertransformW（f）isgivenby:

wherethefundamentalfrequencyf0=1/T0=33.333Hz,and/-/（f）istheFouriertransformofasinglepulse

•FromthetableofFouriertransforms

SubstitutingthisintotheformulaforW（f）yields:

BecausethedifferenttermsofthesumintheformulaforW（f）donotoverlap,wecantakethemagnitudeof|W（f）|asequaltothesumofthemagnitudesofdifferentterms.

Theweightsofimpulsesforf=nf0;n=0,±1,±2,canbepresentedintheformofatable

Weightsofspectrallinesforthemagnitudespectrum|W（f）|

•Plotofthemagnitudespectrumofthewavew（t）

Tut5

1.Amixerisusedtomultiplytwosignalsx（i）andy（i）.Plotthemagnitudespectrumoftheoutputofthemixerif

whereA=2mV,fm=2kHz,B=1mVandfc=200kHz.Repeatyourconsiderationsifthemagnitudespectrumofx（i）isasgiveninFig.1,andy（i）isthesameaspreviously.

Solution：

Inthefirstcase,themixer'soutputz（i）isgivenby:

Utilizingthetrigonometricidentity:

weget:

Because

thelatestcanberewrittenintheform

Substitutingthenumericalvaluesfortheconstantsyields:

andthemagnitudespectrumofz（i）isgiveninFig.2.

Figure2:

Magnitudespectrumofwaveformatthemixer'soutput.

Tofindthemagnitudespectruminthesecondcase,weneedtoutilizethefrequencytranslationpropertyofFouriertransform,whichstates

Intheconsideredcase,0=0.Themagnitudespectrumofthez（i）isgiveninFig.3.

Figure3:

Magnitudespectrumofthez（i）.

2.Drawthemagnitudespectrumofthesignalxout（t）attheoutputofanon-lineardevice,havingacharacteristic:

Assumethattheinputsignalconsistsofasumofabasebandsignalandacarrierwavehavingitsfrequencymuchhigherthanthemaximumfrequencyinthebasebandcomponent.Repeatyouranalysis,assumingthatthebasebandsignalsoftheform:

Solution

Inthefirstcase,theconsideredinputsignalxin（i）isgivenbytheequation:

wherem（i）isabasebandsignal.Substitutingthistotheformulafortheoutputvoltageofthenon-lineardeviceyields:

Utilizingthetrigonometricidentity:

weget:

Theplotofthemagnitudespectrumofxoui（i）isgiveninFig.4.

Figure4:

Plotofthemagnitudespectrumattheoutputofanon-lineardeviceiftheinputsignalisasuperpositionofasinusoidalwaveoffrequencyfcandthesignalm（t）havingthe

magnitudespectrum|M（f）|.

Inthesecondpartoftheproblemthexin（t）isoftheform:

Withoutthelossofgenerality,letusassumeherethatthiscorrespondstothesituationwherethesignalm（t）consideredinthepreviouscasehasamagnitudespectrumasshowninFig.5.

Figure5:

Magnitudespectrumofthesignalm（t）composedof

twosinusoids.

Allofthederivationsperformedforthefirstpartoftheproblemareexactlythesameinthispart.However,thesignalm（t）isnowoftheform:

Therefore,tofindallthespectralcomponentsofthesignalat

theoutputofthenon-lineardevice,weneedtoworkoutthe

spectralcomponentsofSubstitutingthepreviousequationfora2m方（t）weget:

Theplotofthemagnitudespectrumofthexout（t）isgiveninFig.6,andthemagnitudesofthespectralcomponentsarelistedinTable1.

Figure6:

Magnitudespectrumofthexout（t）.

Table1:

Magnitudesofspectralcomponentsforxout（t）.

3.Considerapulsew（t）giveninFig.l,andmultiplyitbyasinusoid

Istheresultingwaveformabase-bandoraband-passwaveform?

Iftheresultingwaveformisaband-passone,finditsband-passrepresentationintermsof:

thecomplexenvelopeg（t）,therealandimaginarypartsoftheenvelopex（t）andy（t）,themagnitudeoftheenvelopeR（t）andthephaseΘ（t）.

Solution

Theresultingwaveformisgivenby:

Todetermineifitisabase-bandorband-passwaveform,oneneedstocheckitsmagnitudespectrum.Itcanbecomputedusingtherealsignalfrequencytranslationtheorem[Couch].

Applicationofthattheoremyields:

whereW（f）denotestheFouriertransformofthepulsew（t）derivedinthepreviousexample.AsitisvisibleinFig.3,themagnitudeofWf）decaysrapidlywithincreaseoff,andwecanconsiderwithaverygoodaccuracythat