Chapter1Answers1.1ConvertingfrompolartoCartesiancoordinates:1.2convertingfromCartesiantopolarcoordinates:,,,,,1.3.(a)=,=0,because(b),.Therefore,===,=(c)=cos(t).Therefore,===,=(d),.Therefore,==0，because<.(e)=,=1.therefore,==,=.(f)=.Therefore,===,=1.4.(a)Thesignalx[n]isshiftedby3totheright.Theshiftedsignalwillbezeroforn<1,Andn>7.(b)Thesignalx[n]isshiftedby4totheleft.Theshiftedsignalwillbezeroforn<-6.Andn>0.(c)Thesignalx[n]isflippedsignalwillbezeroforn<-1andn>2.(d)Thesignalx[n]isflippedandtheflippedsignalisshiftedby2totheright.ThenewSignalwillbezeroforn<-2andn>4.(e)Thesignalx[n]isflippedandtheflippedandtheflippedsignalisshiftedby2totheleft.Thisnewsignalwillbezeroforn<-6andn>0.1.5.(a)x(1-t)isobtainedbyflippingx(t)andshiftingtheflippedsignalby1totheright.Therefore,x(1-t)willbezerofort>-2.(b)From(a),weknowthatx(1-t)iszerofort>-2.Similarly,x(2-t)iszerofort>-1,Therefore,x(1-t)+x(2-t)willbezerofort>-2.(c)x(3t)isobtainedbylinearlycompressionx(t)byafactorof3.Therefore,x(3t)willbezerofort<1.(d)x(t/3)isobtainedbylinearlycompressionx(t)byafactorof3.Therefore,x(3t)willbezerofort<9.1.6(a)x1(t)isnotperiodicbecauseitiszerofort<0.(b)x2[n]=1foralln.Therefore,itisperiodicwithafundamentalperiodof1.(c)x3[n]isasshownintheFigureS1.6.Therefore,itisperiodicwithafundamentalperiodof4.1.7.(a)=Therefore,iszerofor>3.(b)Sincex1(t)isanoddsignal,iszeroforallvaluesoft.(c)Therefore,iszerowhen<3andwhen.(d)Therefore,iszeroonlywhen.1.8.(a)(b)(c)(d)1.9.(a)isaperiodiccomplexexponential.(b)isacomplexexponentialmultipliedbyadecayingexponential.Therefore,2()txisnotperiodic.（c）isaperiodicsignal.==.isacomplexexponentialwithafundamentalperiodof.(d)isaperiodicsignal.ThefundamentalperiodisgivenbyN=m()=Bychoosingm=3.Weobtainthefundamentalperiodtobe10.(e)isnotperiodic.isacomplexexponentialwith=3/5.Wecannotfindanyintegermsuchthatm()isalsoaninteger.Therefore,isnotperiodic.1.10.x(t)=2cos(10t＋1)-sin(4t-1)PeriodoffirsttermintheRHS=.PeriodoffirsttermintheRHS=.Therefore,theoverallsignalisperiodicwithaperiodwhichtheleastcommonmultipleoftheperiodsofthefirstandsecondterms.Thisisequalto.1.11.x[n]=1+−PeriodoffirsttermintheRHS=1.1-3-14……1……-10……-4……111-1n………5………x3[n]0-1-2-3123X[n]nFigureS1.1210-1210-1t1-2g(t)2-3-3tFigureS1.14x(t)PeriodofsecondtermintheRHS==7（whenm=2）PeriodofsecondtermintheRHS==5(whenm=1)Therefore,theoverallsignalx[n]isperiodicwithaperiodwhichistheleastcommonMultipleoftheperiodsofthethreetermsinnx[n].Thisisequalto35.1.12.Thesignalx[n]isasshowninfigureS1.12.x[n]canbeobtainedbyflippingu[n]andthenShiftingtheflippedsignalby3totheright.Therefore,x[n]=u[-n+3].ThisimpliesthatM=-1andno=-3.1.13y(t)===Therefore1.14Thesignalx(t)anditsderivativeg(t)areshowninFigureS1.14.Therefore)ThisimpliesthatA=3,t=0,A=-3,andt=1.1.15(a)Thesignalx[n],whichistheinputtoS,isthesameasy[n].Therefore,y[n]=x[n-2]+x[n-3]=y[n-2]+y[n-3]=2x[n-2]+4x[n-3]+(2x[n-3]+4x[n-4])=2x[n-2]+5x[n-3]+2x[n-4]Theinput-outputrelationshipforSisy[n]=2x[n-2]+5x[n-3]+2x[n-4](b)Theinput-outputrelationshipdoesnotchangeiftheorderinwhichSandSareconnectedseriesreversed..WecaneasilyprovethisassumingthatSfollowsS.Inthiscase,thesignalx[n],whichistheinputtoSisthesameasy[n].Thereforey[n]=2x[n]+4x[n-1]2=2y[n]+4y[n-1]=2(x[n-2]+x[n-3])+4(x[n-3]+x[n-4])=2x[n-2]+5x[n-3]+2x[n-4]Theinput-outputrelationshipforSisonceagainy[n]=2x[n-2]+5x[n-3]+2x[n-4]1.16(a)Thesystemisnotmemo...
