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1、Tut21．Expressthepulsesshowninthefigurebymeansoftheunitstepfunctionu(t),andcalculatetheirenergy.(1)ThepulsecanbeMathematicallydescribedas:Usingtheunitstepfunctionnotation,thiscanbeexpressedas:Withoutusingtheseparateintervaldefinitions,butrelyingonthepropertiesoftheunitstepfunction,thiscanbewritt
2、enas:Thepulseenergyisdefinedas:Tofindthevalueofthisintegral,wecandivideitinto3separateintegralswithlimits[T,3T],[3T,4T],[4T,6T],respectively.Hence,theenergyisgivenby:(2)Thepulsecanbeexpressedas:Intermsoftheunitstepfunction,thiscanbewrittenas:Thepulseenergyisequalto:(3)Thepulsecanbeexpressedas:I
3、ntermsoftheunitstepfunction,thiscanbewrittenas:Andtheenergyisequalto2T.2.Evaluatethefollowingintegrals3.Findtheoutputofthetime-invariantlinearsystemhavingtheimpulseresponseh(t)shownintheFigure(a),inresponsetotheinputsignalx(t)presentedintheFigure(b).Inordertocalculatetheconvolutionx(t)*h(t),wen
4、eedfirsttofindthemirroredversionofthesignalx(t),changethevariabletoλ,i.e.findx(-k),aspresentedThatmirroredversionoftheinputsignalisthenshiftedandsignalx(t-λ)isconsidered.Itisclearlyvisiblethatfort<3thesignalx(t-k)andtheimpulseresponseh(t)donotoverlapforanyλ.Hencetheintegraloftheirproductisequal
5、tozero.Thedifferentoverlappingpossibilitiesareillustratedinthefigure,andfinallyfort>7,x(t-λ)andh(t)stopoverlappingforanyλTherefore,theoutputsignaly(t)canbeexpressedas:Afterperformingintegration,wegetTheobtainedoutputsignaly(t)isplottedbelow.Itisworthtonoticethatthesignaly(t)issmootherthantheinp
6、utsignalx(t),andthatthetimelimitspropertyholds.4.Determinetheauto-correlationfunctionofthefollowingsignal(whereA>0):Solution:ThewaveformisshownasfollowsFirstlynotethattheautocorrelationofarealfunctionisalwayssymmetric-thusyouonlyneedtodeterminetheshapeoftheautocorrelationforr<0orr>0.Secondly,no
7、tethatthetime-shiftedimageofthetriangularpulsedoesnotoverlaptheoriginalfunctionatallifr>Torr>-TThereforetheautocorrelationiszerointhoseregions.Wecanjustworkouttheautocorrelationfunctionfor0
