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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<
r<
Tandwehavealltheinformationrequiredtosolvetheproblem.Whentheimageisshiftedtotherightbyt=tunits,theoverlappingregiongoesfromttoT.Therefore,theautocorrelationis
Thisintegralistrivialtoevaluate,whichgivesyou
Theleft-handsideisjustamirror-imageoftheright:
youcansubstitutet=-tintoEquation3toderivethis:
Thusthefullexpressionfortheautocorrelationis
ThisisshowngraphicallyinFigure2.
5.Determinetheauto-correlationfunctionofthesignal
giventheauto-correlationfunctionofaperiodicsignalx(t)isdefinedas:
whereTistheperiodofthesignalx(t)
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:
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).
Theresultingwaveformisgivenby:
Todetermineifitisabase-bandorband-passwaveform,oneneedstocheckitsmagnitudespectrum.Itcanbecomputedusingtherealsignalfrequencytranslationtheorem[Couch].
Applicationofthattheoremyields:
whereW(f)denotestheFouriertransformofthepulsew(t)derivedinthepreviousexample.AsitisvisibleinFig.3,themagnitudeofWf)decaysrapidlywithincreaseoff,andwecanconsiderwithaverygoodaccuracythat