信号与系统奥本海姆英文版课后答案chapter9

CHAPTER9ANSWERS91（A）THEGIVENINTEGRALMAYBEWRITTENAS50TJEDIF5,THENTHEINTEGRALDOESCONVERGEBTHEGIVENINTEGRALMAYBEWRITTENAST05TJEIF5,THENTHEFUNCTIONGROWSTOWARDSASTDECREASESTOWARDSANDTHEGIVENINTEGRALDOESNOT5TECONVERGEBUTIF5,THENTHEFUNCTIONGROWSTOWARDSASTDECREASESTOWARDSANDTHEGIVENINTEGRAL5TEDOESNOTCONVERGEIF5,THEREFORE,THEGIVENINTERNALCONVERGESWHEN5,THENTHEFUNCTIONGROWSTOWARDSASTDECREASETOWARDSANDTHEGIVENINTEGRAL5TEDOESNOTCONVERGEBUTIF5RESBBYUSINGEG93,WECANEASILYSHOWTHATGTAUTHASTHELAPLACETRANSFORM5T0GS05STAETHEROCISSPECIFIEDASMAX5,RESINCEWEAREGIVENTHATTHEROCISESRES3,WEKNOWTHATRE3THEREARENOCONSTRAINTSONTHEIMAGINARYPARTOF94WEKNOWFORMTABLE92THAT,RES111SIN2LTXEUTXWEALSOKNOWFORMTABLE91THATXTXSSTHEROCOFXSISSUCHTHATIFWASINTHEROCOF,THENWILLBEINTHEROCOFXSPUTTINGTHE0S1XS0TWOABOVEEQUATIONSTOGETHER,WEHAVEXTTLXS1,FROMPROPERTY5INSECTION92WEKNEWTHATXTCANNOTBEARIGHTSIDESIGNALDYESSINCETHESIGNALISABSOLUTELYINTEGRABLE,THEROCMUSTINCLUDE,THEAXISFURTHERMORE,XSJHASAPOLEATS2THEREFORE,ONEVALIDROCFORTHESIGNALCOULDBEII20BECAUSEBOTHARERIGHTHANDSIGNALS916TAKINGTHELAPLACETRANSFORMOFBOTHSIDESOFTHEGIVENDIFFERENTIALEQUATIONS,WEOBTAIN123SXYTHEREFORE,223SSXHATAKINGTHELAPLACETRANSFORMOFBOTHSIDESOFTHEGIVENEQUATION,WEHAVEGSSHSHSSUBSTITUTINGFORHSFROMABOVE,112223SSGTHEREFORE,GSHAS2POLESBWEKNOWTHATHS22STHEREFORE,HSHASPOLESATIFTHESYSTEMHASTOBESTABLE,THENANDJ,31,231JTHEREALPARTOFTHEPOLESHASTOBELESSTHANZEROFORTHISTOBETRUE,WEREQUIRETHATIE,02/0917THEOVERALLSYSTEMSHOWINFIGURE917MAYBETREATEDASTWOFEEDBACKSYSTEMOFTHEFORMSHOWNINFIGURE931CONNECTEDINPARALLELBYCARRYINGOUTANANALYSISSIMILARTOTHATDESCRIBEDINSECTION981,WEFINDTHESYSTEMFUNCTIONOFTHEUPPERFEEDBACKSYSTEMTOBE82/41SSHSIMILARLY,THESYSTEMFUNCTIONOFTHELOWERFEEDBACKSYSTEMIS/2THESYSTEMFUNCTIONOFTHEOVERALLSYSTEMISNOW1602321SSSINCEHSYS/XS,WEMAYWRITEXYTAKINGTHEINVERSELAPLACETRANSFORM,WEOBTAINDTXTYDTTY3602918AFROMPROBLEM320,WEKNOWTHATDIFFERENTIALEQUATIONRELATINGTHEINPUTANDOUTPUTOFTHERLCCIRCUITIS2DYTTXTAKINGTHELAPLACETRANSFORMOFTHISWHILENOTHINGTHATTHESYSTEMISCAUSALANDSTABLE,WEOBTAIN21YSXSTHEREFORE,2,H12EBWENOTETHATHSHASTWOPOLESATANDITHASNOZEROSINTHEFINITESPLANE3SJSJFROMSECTION94WEKNOWTHATTHEMAGNITUDEOFTHEFOURIERTRANSFORMMAYBEEXPRESSEDAS113LENGTHOFVCRMTJLENGTHOFVCRMTJ22WESEETHATTHERIGHTHANDSIDEOFTHEABOVEEXPRESSIONINCREASESWITHINCREASING|UNTIL|REACHESTHENITSTARTSDECREASINGAS|INCREASINGEVENFURTHERITFINALLYREACHES0FOR|12THEREFOREISAPPROXIMATELYLOWPASS|HJCBYREPEATINGTHEANALYSISCARRIEDOUTINPROBLEM320ANDPARTAOFTHISPROBLEMWITHR,31WECANSHOWTHAT21,YSX05ESDWEHAVE33VECTLNFROM05JECTLNFROJ2WESEETHATWHEN|ISINHEVICINITY00005,THERIGHTHANDSIDEOFTHEABOVEEQUATIONTAKESONEXTREMELYLARGEVALUEONEITHERSIDEOFTHISVALUEOF|THEVALUEOF|HJ|ROLLSOFFRAPIDLYTHEREFORE,HSMAYBECONSIDEREDTOBEAPPROXIMATELYBANDPASS919ATHEUNILATERALLAPLACETRANSFORMISXS201TSTEUDTSBTHEUNILATERALLAPLACETRANSFORMIS23011TSTXSTEUDTST6SCTHEUNILATERALLAPLACETRANSFORMIS240TTSTXSEUEDS12920INPROBLEM329,WEKNOWTHATTHEINPUTOFTHERLCIRCUITARERELATEDBYTXYDTAPPLYINGTHEUNILATERALLAPLACETRANSFORMTOTHISEQUATION,WEHAVE0SSAFORTHEZEROSTATERESPONSE,SETALSOWEHAVELUX2TE1THEREFORE,YSS1SCOMPUTINGTHEPARTIALFRACTIONEXPANSIONOFTHERIGHTHANDSIDEOFTHEABOVEEQUATIONANDTHENTAKINGITSINVERSEUNILATERALLAPLACETRANSFORM,WEHAVE2TUETTYBFORTHEZEROSTATERESPONSE,ASSUMETHATXT0SINCEWEAREGIVENTHAT,01Y101SSYTAKINGTHEINVERSEUNILATERALLAPLACETRANSFORM,WEHAVETYEUFIGURES921CTHETOTALRESPONSEISTHESUMOFTHEZEROSTATEANDZEROINPUTRESPONSETHISIS2TTYEU921THEPOLEZEROPLOTSFORALLTHESUBPARTSARESHOWNINFIGURES921ATHELAPLACETRANSFORMOFXTISXS230TSTD300/|/|SSTEE2156BUSINGANAPPROACHSIMILARTOTHATSHOWINPARTA,WEHAVE4,LTEUS4ALSO,51,5TJJAND5,LTTJEUESSJFROMTHISWEOBTAIN,555215SIN2LTTTJTJTUSWHERETHEREFORE,E2325IMRA2RIMEIMRFIMRGIMRH2244IMRDRBIMCRIM24531570SIN,549LTTTSEUTESCTHELAPLACETRANSFORMOFISX023TSTXED300/|/|SSTE2156THEREGIONOFCONVERGENCEROCIS2DUSINGANAPPROACHALONGTHELINESOFPARTA,WEOBTAINS92112,LTTEUESUSINGANAPPROACHALONGTHELINESOFPARTC,WEOBTAINS921221,2TFROMTHESEWEOBTAIN,224TLTTTSEUE2ESUSINGTHEDIFFERENTIATIONINTHESDOMAINPROPERTY,WEOBTAIN22284TLTDSEUSINGTHEDIFFERENTIATIONINTHESDOMAINPROPERTYONEQS9211,WEGET221,LTTEUESSUSINGTHEDIFFERENTIATIONINTHESDOMAINPROPERTYONEQS9212,WEGET22,LTTDTHEREFORE,2224,2TLTTTSEUEESFFROMTHEPREVIOUSPART,WEHAVE2221,LTTTSGNOTETHATTHEGIVENSIGNALMAYBEWRITTENASNOTETHAT1XTUT,0LTUTESUSINGTHETIMESHIFTINGPROPERTY,WEGET1,LTTTHEREFORE,1XALL,SLTEUTNOTETHATINTHISCASE,SINCETHESIGNALISFINITEDURATION,THEROCISTHEENTIRESPLANEHCONSIDERTHESIGNALNOTETHATTHESIGNALMAYBE11XTUTXTEXPRESSEDASWEHAVEFROMTHEPREVIOUSPART2,ALLSLTEUTUSINGTHEDIFFERENTIATIONINSDOMAINPROPERTY,WEHAVE,ALL121SSSDEXTTUSINGTHETIMESCALINGPROPERTY,WEOBTAIN,ALL121SLTEXTSTHEN,USINGTHESHIFTPROPERTY,WEHAVE,ALL12STTHEREFORE,ALLS21211SSSSLTEEXTTITHELAPLACETRANSFORMOFISXTU/0XJNOTETHATTHEREFORE,THELAPLACETRANSFORMISTHESAMEASTHERESULTOFTHE3TUPREVIOUSPART922AFROMTABLE92,WEHAVE1SINXTTBFROMTABLE92WEKNOWTHAT2CO3,09LTSTUEUSINGTHETIMESCALINGPROPERTY,WEOBTAINS,TSTHEREFORE,THEINVERSELAPLACETRANSFORMOFISXCO3XTTUCFROMTABLE92WEKNOWTHAT21S,9LTTSEEUSINGTHETIMESCALINGPROPERTY,WEOBTAIN2CO3,1TUTSSTHEREFORE,THEINVERSELAPLACETRANSFORMOFISXTXETDUSINGPARTIALFRACTIONEXPANSIONON,WEOBTAINS2143XSFROMTHEGIVENROC,WEKNOWTHATMUSTBEATWOSIDEDSIGNALTHEREFOREXTTTXEUEUSINGPARTIALFRACTIONEXPANSIONON,WEOBTAINXS213SFROMTHEGIVENROC,WEKNOWTHATMUSTBEATWOSIDEDSIGNAL,THEREFORE,XT3TXTEUFWEMAYREWRITEASXS212/3S22/1USINGTABLE92,WEOBTAIN/COS/SIN3/TTXTETUETUGWEMAYREWRITEASX231SXFROMTABLE92,WEKNOWTHAT2,0LTTUESUSINGTHESHIFTINGPROPERTY,WEOBTAIN1,TESUSINGTHEDIFFERENTIATIONPROPERTY,2,1LTTTTDSUEUETHEREFORE,3TTXT923THEFOURPOLEZEROPLOTSSHOWNMAYHAVETHEFOLLOWINGPOSSIBLEROCSPLOTAOROR2ES2ES2ESPLOTBORPLOTCORPLOTDENTIRESPLANEALSO,SUPPOSETHATTHESIGNALHASALAPLACETRANSFORMWITHROCXTXSR1WEKNOWFROMTABLE91THAT33LTTEXSTHEROCOFTHISNEWLAPLACETRANSFORMISSHIFTEDBY3TOTHELEFTIFISABSOLUTELY1RR3TXEINTEGRABLE,THENMUSTINCLUDETHEAXISJWFORPLOTA,THISISPOSSIBLEONLYIFWAS2ESFORPLOTB,THISISPOSSIBLEONLYIFWASFORPLOTC,THISISPOSSIBLEONLYIFWASFORPLOTD,ISTHEENTIRESPLANER2WEKNOWFROMTABLE92THAT1,LTTEUESALSO,FROMTABLE91WEOBTAIN2,1TXXRESIFISABSOLUTELYINTEGRABLE,THENMUSTINCLUDETHEAXISTE2JWFORPLOTA,THISISPOSSIBLEONLYIFWASFORPLOTB,THISISPOSSIBLEONLYIFWASESFORPLOTC,THISISPOSSIBLEONLYIFWASRFORPLOTD,ISTHEENTIRESPLANER3IFFOR,THENTHESIGNALISALEFTSIDEDSIGNALORAFINITEDURATIONSIGNAL0XT1TFORPLOTA,THISISPOSSIBLEONLYIFWAS2ESFORPLOTB,THISISPOSSIBLEONLYIFWASFORPLOTC,THISISPOSSIBLEONLYIFWASRFORPLOTD,ISTHEENTIRESPLANER4IFFOR,THENTHESIGNALISARIGHTSIDEDSIGNALORAFINITEDURATIONSIGNAL0XT1TFORPLOTA,THISISPOSSIBLEONLYIFWASR2ESFORPLOTB,THISISPOSSIBLEONLYIFWASFORPLOTC,THISISPOSSIBLEONLYIFWASFORPLOTD,ISTHEENTIRESPLANER924ATHEPOLEZERODIAGRAMWITHTHEAPPROPRIATEMARKINGSISSHOWNFIGURES924BBYINSPECTINGTHEPOLEZERODIAGRAMOFPARTA,ITISCLEARTHATTHEPOLEZERODIAGRAMSHOWNINFIGURES924WILLALSORESULTINTHESAMETHISWOULDCORRESPONDTOTHELAPLACETRANSFORMXJW,12S1ESCXJJDWITHTHEPOLEZERODIAGRAMSHOWNBELOWINFIGURES924WOULDHAVETHEPROPERTYTHAT2SHERE,JWJ21/XSE21/FFROMTHERESULTOFPARTB,ITISCLEARTHATMAYBEOBTAINEDBYREFLECTINGTHEPOLESANDZEROS1INTHERIGHTHALFOFTHESPLANETOTHELEFTHALFOFTHESPLANETHEREFORE,1/SXFROMPARTD,ITISCLEARTHATMAYBEOBTAINEDBYREFLECTINGTHEPOLESZEROSINTHERIGHTHALF2XSOFTHESPLANETOTHELEFTHALFANDSIMULTANEOUSLYCHANGINGTHEMTOZEROSPOLESTHEREFORE,21/S925THEPLOTSAREASSHOWNINFIGURES925926FROMTABLE92WEHAVE211,2LTTXEUXSESAND311,3TUSINGTHETIMESHIFTINGTIMESCALINGPROPERTIESFROMTABLE91,WEOBTAIN2211,SLTSEXTXAND3322,SSTTHEREFORE,USINGTHECONVOLUTIONPROPERTYWEOBTAIN2312SLTEYTXTY927FROMCLUES1AND2,WEKNOWTHATISOFTHEFORMXAXSABFURTHERMORE,WEAREGIVENTHATONEOFTHEPOLESOFISSINCEISREAL,THEXS1JXTPOLESOFMUSTOCCURINCONJUGATERECIPROCALPAIRSTHEREFORE,ANDA1BJAND1HSJFROMCLUE5,WEKNOWTHATTHEREFORE,WEMAYDEDUCETHATAND08X6A26SLETDENOTETHEROCOFFROMTHEPOLELOCATIONSWEKNOWTHATTHEREARETWORPOSSIBLECHOICESOFMAYEITHERBEORWEWILLNOWUSER1ESSCLUE4TOPICKONENOTETHAT22LTTYEXYXTHEROCOFISSHIFTEDBY2TOTHERIGHTSINCEITISGIVENTHATISNOTABSOLUTELYYSYTINTEGRABLE,THEROCOFSHOULDNOTINCLUDETHETHISISPOSSIBLEONLYOFJWAXISISR1928ATHEPOSSIBLEROCSAREI2ESIIIIIIV1SBIUNSTABLEANDANTICAUSALIIUNSTABLEANDNONCAUSAL（III）STABLEANDNONCAUSALIVUNSTABLEANDCAUSAL929AUSINGTABLE92,WEOBTAIN1,XSESAND,2HBSINCE,WEMAYUSETHECONVOLUTIONPROPERTYTOOBTAINYTXHT1YSXSTHEROCOFISECPERFORMINGPARTIALFRACTIONEXPANSIONON,WEOBTAINYS12STAKINGTHEINVERSELAPLACETRANSFORM,WEGETTTYEUDEXPLICITCONVOLUTIONOFANDGIVESUSXHTHTD20EFORT0T2U930FORTHEINPUT,THELAPLACETRANSFORMISXT1XSETHECORRESPONDINGOUTPUTHASTHELAPLACETRANSFORMTTYTEU22,0YSSTHEREFORE,1,HEXSNOW,THEOUTPUTHASTHELAPLACETRANSFORM3TTYTU1236,0YSSTHEREFORE,THELAPLACETRANSFORMOFTHECORRESPONDINGINPUTWILLBE16,03YSXESHTAKINGTHEINVERSELAPLACETRANSFORMOFTHEPARTIALFRACTIONEXPANSIONOFWEOBTAIN1,XS3124TXTUE931ATAKINGTHELAPLACETRANSFORMOFBOTHSIDESOFTHEGIVENDIFFERENTIALEQUATIONANDSIMPLIFYING,WEOBTAIN21YSXTHEPOLEZEROPLOTFORISASSHOWNINFIGURES931HBTHEPARTIALFRACTIONEXPANSIONOFISS1/32IIFTHESYSTEMISSTABLE,THEROCFORHASTOBETHEREFOREHS12ESTTHTEUIIIFTHESYSTEMISCAUSAL,THEROCFORHASTOBETHEREFORE213TTIIIIFTHESYSTEMISNEITHERSTABLENORCAUSAL,THEROCFORHASTOBES1ESTHEREFORE,213TTHTEU932IFPRODUCES,THENALSO,BYTAKINGTHELAPLACETRANSFORMOFBOTHSIDESOF2TXE/6TY2/6HTHEGIVENDIFFERENTIALEQUATIONWEGET4SBSINCE,WEMAYDEDUCETHATTHEREFORE1/H124SS933SINCE,TTTXEUE11XEWEAREALSOGIVENTHAT2SHSINCETHEPOLESOFAREAT,ANDSINCEISCAUSAL,WEMAYCONCLUDETHATTHEROCOFJHTHSISNOW1E21YSXSTHEROCOFWILLBETHEINTERSECTIONOFTHEROCSOFANDTHISISXSH1ESWEMAYOBTAINTHEFOLLOWINGPARTIALFRACTIONEXPANSIONFORY2/56/SWEMAYREWRITETHISAS2/14YS012REIMFIGURES931NOTHINGTHATTHEROCOFISANDUSINGTABLE92,WEOBTAINYS1ES24COIN55TTTYEUU934WEKNOWTHAT11,0LTXTXSTHEREFORE,HASAPOLEATNOW,THELAPLACETRANSFORMOFTHEOUTPUTOFTHESYSTEMWITHS1YT1XTASTHEINPUTIS11YHSINCEINCLUE2,ISGIVENTOBEABSOLUTELYINTEGRABLE,MUSTHAVEAZEROATWHICHCANCELSHS0SOUTTHEPOLEOFAT1XS0WEALSOKNOWTHAT22,0LTXTUSETHEREFORE,HASTWOPOLESATNOW,THELAPLACETRANSFORMOFTHEOUTPUTOFTHESYSTEMS2YTWITHASTHEINPUTIS2T2YHXSSINCEINCLUE3,ISGIVENTOBENOTABSOLUTELYINTEGRABLE,DOESNOTHAVETWOZEROSATHSTHEREFORE,WECONCLUDETHATHASEXACTLYONEZEROAT0S0FROMCLUE4WEKNOWTHATTHESIGNAL2DHTPTISFINITEDURATIONTAKINGTHELAPLACETRANSFORMOFBOTHSIDESOFTHEABOVEEQUATION,WEGET2PSHSTHEREFORE,2SINCEISOFFINITEDURATION,WEKNOWTHATWILLHAVENOPOLESINTHEFINITESPLANEPTPSTHEREFORE,ISOFTHEFORMS,12NIIAZHWHERE,REPRESENTTHEZEROSOFHERE,ISSOMECONSTANTIZ,SAFROMCLUE5WEKNOWTHATTHEDENOMINATORPOLYNOMIALOFHASTOHAVEADEGREEWHICHISHSEXACTLYONEGREATERTHANTHEDEGREEOFTHENUMERATORPOLYNOMIALTHEREFORE,12ASSINCEWEALREADYKNOWTHATHASAZEROAT,WEMAYREWRITETHISAS0S2ASFROMCLUE1WEKNOWTHATISFROMTHIS,WEMAYEASILYSHOWTHATTHEREFORE,H012SSINCETHEPOLESOFAREATANDSINCEISCAUSALANDSTABLE,THEROCOFIS1JHTHS1E935AWEMAYREDRAWTHEGIVENBLOCKDIAGRAMASSHOWNINFIGURES935FROMTHEFIGURE,ITISCLEARTHAT1FSYTHEREFORE,SIMILARLY,THEREFORE,/FTDYT/ETDFT21/ETDYTFROMTHEBLOCKDIAGRAMITISCLEARTHAT21166YEFTTHEREFORE2116YSSYNOW,LETUSDETERMINETHERELATIONSHIPBETWEENANDTHISMAYBEDONEBYCONCENTRATINGONTHELOWER1YTXHALFOFTHEABOVEFIGUREWEREDRAWTHISINFIGURES935FROMEXAMPLE930,ITISCLEARTHATANDMUSTBERELATEDBYTHEFOLLOWINGDIFFERENTIALEQUATION1T21DYTXTHEREFORE,12XSYUSINGTHISINCONJUNCTIONWITHEQS9351,WEGET6TAKINGTHEINVERSELAPLACETRANSFORM,WEOBTAIN226DYTDXTT（B）THETWOPOLESOFTHESYSTEMAREAT1SINCETHESYSTEMISCAUSALTHEROCMUSTBETOTHERIGHTOFS1THEREFORE,THEROCMUSTINCLUDETHEAXISHENCE,THESYSTEMJISSTABLE936AWEKNOWTHATANDARERELATEDBY1YS246S1YTAKINGTHEINVERSELAPLACETRANSFORM,WEGET12DYTTYTBSINCE1S/,FFXT1/S241/S263YT（E）XT1/S2YT1121/S2FFIGURES936SYSTEMSYSTEM1/S1/S1/S11/S156YTA1/S21/S561/S1/S101BC1/S21/S41/SYTXTFIGURES937XT1/S226181/SYTFIGURES936CSINCEFS21/,DYTFEETDFROMPARTA,46YETHEEXTENDEDBLOCKDIAGRAMISASSHOWNINFIGURES936FTHEBLOCKDIAGRAMISASSHOWNINFIGURES936GTHEBLOCKDIAGRAMISASSHOWNINFIGURES936THETHREESUBSYSTEMSMAYBECONNECTEDINPARALLELASSHOWINTHEFIGUREABOVETOOBTAINTHEOVERALLSYSTEM937THEBLOCKDIAGRAMSARESHOWNINFIGURES937938AWEMAYREWRITEASHS331122JJSCLEARLYMAYBETREATEDASTHECASCADECOMBINATIONOFFOURFIRSTORDERSUBSYSTEMSHSCONSIDERONEOFTHERESUBSYSTEMSWITHTHESYSTEMFUNCTION1S312JTHEBLOCKDIAGRAMFORTHISISASSHOWINFIGURES938CLEARLY,ITCONTAINSMULTIPLICATIONSWITHCOEFFICIENTSTHATARENOTREALBWEMAYWRITEASSH12221HSSTHEBLOCKDIAGRAMFORMAYBECONSTRUCTEDASACASCADEOFTHEBLOCKDIAGRAMSOFANDASSHOWINFIGURES9381S2CWEMAYREWRITEASSH1321432SHSSTHEBLOCKDIAGRAMFORMAYBECONSTRUCTEDASAPARALLELCOMBINATIONOFTHEBLOCKDIAGRAMSOFANDASSHOWINFIGURES9383S4939AFOR,THEUNILATERALANDBILATERALLAPLACETRANSFORMSAREIDENTICAL1TX2,1SESXXBHERE,USINGTABLE92ANDTIMESHIFTINGPROPERTYWEGET3,22ETHEUNILATERALLAPLACETRANSFORMIS1/S231JYT1/SXTYT1/S1/S11/S2XT1/S1YTXT1/S213101/S1/S1/S1/S01111FIGURES9383,132SESXCWEHAVEGS121SSTAKINGTHEINVERSEUNILATERALTRANSFORM,WEOBTAIN132TUETTRCLEARLY,FORTG0940TAKINGTHEINVERSEUNILATERALTRANSFORMOFBOTHSIDESOFTHEGIVENDIFFERENTIALEQUATION,WEGET61060223SXYSYYSS9401AFORTHEZEROSTATERESPONSE,ASSUMETHATALLTHEINITIALCONDITIONSAREZEROFURTHERMORE,FORMTHEGIVENXTWEMAYDETERMINE,4SX4SEFORMEQS9401,WEGET1623YTHEREFORE,423SSTAKINGTHEINVERSEUNILATERALLAPLACETRANSFORMOFTHEPARTIALFRACTIONEXPANSIONOFTHEABOVEEQUATION,WEGET161324TUETTUETTYBFORTHEZEROINPUTRESPONSE,WEASSUMETHATXS0ASSUMINGTHATTHEINITIALCONDITIONSAREASGIVEN,WEOBTAINFROMS9401523SSTAKINGTHEINVERSEUNILATERALLAPLACETRANSFORMOFTHEABOVEEQUATION,WEGETTUEYCTHETOTALRESPONSEISTHESUMOFTHEZEROSTATEANDAEROINPUTRESPONSES2161734TUETTTT941LETUSFIRSTFINDTHELAPLACETRANSFORMOFTHESIGNALWEHAVEXYDTXSYSXASINCEFORANEVENSIGNAL,WECANCONCLUDETHATLTXTLTXTTHEREFORE,XSXTXTSBSINCEFORANODDSIGNAL,WECANCONCLUDETHATTHERE,LTXTTBFIRSTOFALLNOTETHATFORASIGNALTOBEEVEN,ITMUSTBEEITHERTWOSIDEDORFINITEDURATIONTHEREFORE,IFHASPOLES,THEROCMUSTBEASTRIPINTHESPLANECFORMPLOTA,WEGET1SAXTHEREFORE,XTHEREFORE,XTISNOTEVENINFACTITISODDFORPLOTB,WENOTETHATTHEROCCANNOTBECHOSENTOCORRESPONDTOATWOSIDEDFUNCTIONXTTHEREFORE,THISSIGNALISNOTEVENFORMPLOTC,WEGET12SASJXTHEREFORE,12SXASXTHEREFORE,XTISEVENPROVIDEDTHEROCISCHOSENTOBE10ETHISSTATEMENTISFALSECONSIDERTHESIGNALTHENTX00DTETSXSSBOTHINTEGRALSONTHERIGHTHANDSIDECONVERGEFORANYVALUEOFS0943WEAREGIVENTHATHTISCAUSALANDSTABLETHEREFORE,ALLPOLESAREINTHELEFTHALFOFTHESPLANEANOTETHATSHGDTHGLNOW,GSHASTHESAMEPOLESASHSANDHENCETHEROCFORGSREMAINSTHESAMETHEREFORE,GTISALSOGUARANTEEDTOBECAUSALANDSTABLEBNOTETHATSRTRLNOTETHATRSDOSENOTHAVEAPOLEATS0ONLYIFHSHASAZEROATS0THEREFORE,WECANNOTGUARANTEETHATRTISALWAYSCAUSALANDUNSTABLE944ANOTETHAT,ALENTTSNTCTHEREFORE,STSNS10INORDERTODETERMINETHEROC,LETUSFIRSTFINDTHEPOLESOFXSCLEARLY,THEPOLESOCCURWHENTHISIMPLIESTHATTHEPOLESSATISFYTHEFOLLOWINGEQUATION,21,021KEJSTKTAKINGTHELOGARITHMOFBOTHSIDEOFTHEABOVEEQUATIONANDSIMPLIFYING,WEGET,TJKTHEREFORE,THEPOLESALLLIEONAVERTICALLINEPARALLELTOTHEJWAXISPASSINGTHOUGHS1SINCETHESIGNALISRIGHTSIDED,THEROCISRES1CTHEPOLEZEROPOLESISASSHOWNINFIGURES944CTHEMAGNITUDEOFTHEFOURIERTRANSFORMISGIVENBYTHEPRODUCTOFTHERECIPROCALOFTHEJW0FIGURES9442/4/1XXXXXREIMLENGTHSOFTHEVECTORSFROMTHEPOLESTOTHEPOINTJWTHEPHASEOFISGIVENBYTHENEGATIVEOFJWTHESUMOFTHEANGLESOFTHESEVECTORSCLEARLYFROMTHEPOLEZEROPLOTABOVEITISCLEARTHATBOTHTHEMAGNITUDEANDPHASEHAVETOVARYPERIODICALLYWITHAPERIODOF2/945ATAKINGTHELAPLACETRANSFORMOFTHESIGNALXT,WEGET1213/2/SSTHEROCALSO,NOTETHATSINCEXTISALEFTSIDEDSIGNAL,THEROC1ESFORXSIS2NOW,21YSSXHWEKNOWTHATTHEROCOFYSHASTOBETHEINTERSECTIONOFTHEROCSOFXSANDHSTHISLEADSUSTOCONCLUDETHATTHEROCOFHSISEBTHEPARTIALFRACTIONEXPANSIONOFHSIS21STHEREFORE,TTHEUCISANEIGENFUNCTIONOFTHELTISYSTEMTHEREFORE,3TE320TTYH946SINCEYTISREAL,THETHIRDINPUTMUSTBEOFTHEFORMSINCEXTISOFTHEFORMSANDTHEOUTPUTIS,WEMAYCONCLUDETHAT0STT4418336COSINTTTEE1843HJLETUSTRYTHEN6THEU51SWEMAYEASILYSHOWTHATTHEREFORE,HSASGIVENABOVEISCONSISTENTWITHTHEGIVEN843HJINFORMATION947ATAKINGTHELAPLACETRANSFORMOFYT,WEOBTAIN12SY2ESTHEREFORE,2SHXTHEPOLEZERODIAGRAMFORXSISASSHOWNINFIGURES947NOW,THEROCOFHSIS1ESWEKNOWTHATROCOFYSISATLESTTHEINTERSECTIONOFTHEROCSOFXSANDHSNOTETHATTHEROCCANBELARGERIFSOMEPOLESARECANCELEDOUTBYZEROSATTHESAMELOCATIONINTHISCASE,WECANCHOOSETHEROCOFXSTOBEEITHER20,THEINTEGRALINTHEABOVEEQUATIONIS31TTEEFORT2951SINCEHTISREAL,I