Solution Manual For Fundamentals Of Signals And Systems Using The Web And Matlab, 3rd Edition

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Chapter1:FundamentalConceptsL1G)’@x(O)=py)+p)af,111®=2(1-)py)-L1-11)0030431©x0=220+2060+2(1-10)oy)171@x0=1p,0-2(1-l)ps)»©H)=Yp-2k+15)k=11.1(ii)Simplesttodefine#andxasfollows,thenusetheMATLABcommandplot(t,x)@)1=[-3-2-2-1-111223];x=[0011221100i(®)1=[-5-4-1145];x=[001100];(©)t=[-7-6-6-3-3033667];x=[00224642200],@1=[3-220223];x=[0042400];©1=[-10011223344556];x=[001100110011007;1.2(a)@==,soAt<w/m=1;sin(n(Atn))=0forallintegers»whenAz=2(®)0=7,a=0.1,50AtS—==025aon?+7?(©)@=/4,a=1,50At<z=06177we+(x/4)+StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.EE

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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3.continued©delt=.11delt=.5[!Eo=)WII-AMABy_5101520“lo5101Time(sec)Time(sec)s20;delt=.9delt=1.51)Wi*|“05-1\1015200510<1520Time(sec)Time(sec)OB)TRx=0.]in\NA20Sg=a2._=:~~./RaNEEoeGLgo\8.T=BN\\1\,[0]2468101214161820Time(sec)(©)1iL.05NSr=2=cLTkes0=.-0.5[+]12345678910Time(sec)|©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storagein2Revasystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.--etnCERCCEAAREAEEenoemereebreRAEI

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1.3(a)(a)t=[-1005];2x=[0011];plot(t,x,"LineWidth',1.5);Psaxis([-1502])Eaxlabel('Time(sec)')ylabel('u(t)')0title('(a)")[]24Time(sec)(b)t=[-1012345];®x=[0012345];.plot(t,x,'LineWidth',1.5);axis([-1505])eg,xlabel('Time(sec)')ylabel('r(t)")title("(b)')Lo2nTime(sec)©t=(-100223];(©)x=[001100];2plot(t,x,'LineWidth',1.5);axis([-1502])4xlabel('Time(sec)')>ylabel('x(t)")title('(c)")0024Time(sec)(d tl=(-100];x1=[0010];10@£2=0:.01:5;x2=10*exp(-3*t2);t=[t1t2];€sx=[x1lx2];plot(t,x,'LineWidth',1.5);axis([-150101)0fn)24xlabel('Time(sec)')Time(sec)ylabel("x(t)'")~title('(d)")(e) ti=[-100];x1=[0031;a(e)t2=0:.01:5;%2=3*exp(~£2).*cos(2+t2);2t=[tlt2];£1x=[x1lx2];0plot(t,x,'LineWidth',1.5);axis([-15-131)B—24xlabel('Time(sec)')Time(sec)ylabel('x(t)')title('(e)')3©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.eteeeSeeeAlerts

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1.3continued(£)tl=[-100];x1=[003];@t2=0:.01:5;0x2=3*exp(t2).*cos(2*t2);t=[tlt2];§-200x=[x1x2];plot(t,x,'LineWidth',1.5);-400axis([-15-450100])024xlabel('Time(sec)')Tiylabel("x(t)")meEe9)title('(£)")(@)t=-1:.01:5;@%=2*sin(3*t-pi/2)-2cos(2*t);plot(t,x,'LineWidth',1.5);goaxis([-15-33])xlabel('Time(sec)').2ylabel("x(t)")024title('(g)")Time(sec)(h)t=-1:.01:5;[0x=sin(5*t)+sin(pi*t);2plot(t,x,'LineWidth',1.5);axis([-15-22])xlabel("Time(sec)')goylabel('x(t)')title('(h)")2024Time(sec)1.4(a)xX)4,~/:X(t)Jo~/Ja3¢-t+StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.SEEeteetLeAii.

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/.4Conbinvedcc)ale)xt)*5al)4ACE14d)a]®t={-2-1-111334];2©x=[0011-1-100];plot(t,x,'LineWidth',1.5);1axis([-24-22])goxlabel("Time(sec)')ylabel("x(t)")-1title('(a)')22[}]24Time(sec)(b)t=[-1011224];(b)x=[00-11100];2plot(t,x,'LineWidth',1.5);1axis([-13-22])xlabel('Time(sec)')goylabel('x(t)')title('(b)")-2El[]123Time(sec)(ec)tl=[01];x1=[00];t2=1:.01:2;x2=2*(£21);©t3=[23);6x3=[22];t4=3:.01:5;4x4=2%t4-4;=t=[tlt2t3ta];jax=[x1x2x3x4];plot(t,x,'LineWidth',1.5);0axis([05-17])xlabel('Time(sec)')[)12345ylabel("x(t)"')Time(sec)title('(c)")I~StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.SEiEE

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1.5®@)ti=[-100];X1=(001];15@£2=0:0.01:2;’x2=exp(-t2);1t3=2:.01:4;X3=exp(t3-4);£05t4=[451;xx4=[00];0t=[tlt2t3t4];x=[x1x2x3x4];-05plot(t,x,'LineWidth',1.5);’[}24axis([-15-.51.5])Time(sec)xlabel("Time(sec)')ylabel('x(t)")title(*(a)")(b)tl=-1:.05:pi;xl=cos(t1);1(®)t2=pi:.05:3%pi/2;$x2=-cos(t2);1t3=[3*pi/25);0.5x3=[00];st=[t1t2£3];x0x=[x1x2x3);05plot(t,x,'LineWidth',1.5);-1axis([~15-1.51.5])xlabel("Time(sec)')AS)rlylabel("x(t)"')Tititle('(b)")me(sec)1.6(a)Sincex(1)=w(t-c),replacing£by£+¢givesWr)=x(t+c).(©)()v(r)=e720)=65-2_6p))G@)vO)=(+2-(+2)+1=+3143|Gi)v(t)=sin(2(+7/4))=sin(21+7/2)=cos(21)17(a)un)I-5-¢+-3a.y©/a23¢&sn+StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeiobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.

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1.7continvedrn]Te+33li-5fF<3aoroo7/7A3+£7tc}x(n)2.5to0.3%0.135~~n~-3-x-taa3rs)x{"1py,0.3%n-.5~~Lo./as(e)xn)7672[3Pa”-3xoyesa3|vsli”T=StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.ES

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1.7Condinved27°(£)OWSRS/~7_0.707-¢3aoslel°EY3Pad~0.707.—*~!“FOpron9)JxCn){-1n-¢-3-37EY3-=Ixn)b3.137,0.8!7~~~£-¢«3a+5°o/a13n-1-0.456_-[5~~<).23N-0.835~1.94.7(4)x(n}(--E”-sfo-3-22/a32%5|Vv4T=StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.a

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17()(a)forn=1:5;(a)u(n)=0;2end;forn=6:21;1.5u(n)=1;end;E11n=-5:15;>stem(n,u,'filled")05axis([-51502])Xlabel('n'")0ylabel('uln]')5051015title('(a)")n(b)forn=1:5;(b)X(n)=0;15end;forn=6:21;x(n)=n-6;10end;=n=-5:15;=stem(n,x,'filled")5xlabel('n")ylabel('r([n]")title('(b)")3051015(c)forn=1:5;nx(n)=0;end;1©forn=6:21;x(n)=.5"(n-6);0.8end;n=-5:15;=06stem(n,x,'filled')04xlabel('n')ylabel(*x[n]"')0.2title('(c)')o-5051015(d)forn=1:5;nx(n)=0;end;1(d)forn=6:21;x(n)=(-.5)"(n-6);end;0.5n=-5:15;=stem(n,x,"filled"')=xlabel('n')0|ylabel('x[n]"')title('(d)")-05-5051015n©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,:PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.AeEPCRAAeAABEeersteeteeeeeeee

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1.7(j)continued4PE.10(e)(¢)forn=1:5;x(n)=0;end;E,forn=6:21;®x(n)=2%(n-6);end;on=-5:15;"stem(n,x,'filled")s051015xlabel('n’")nylabel('x[n]")®title('(e)")10.5(£)n=-5:15;Eox=sin(pi*n/4);=stem(n,x,'filled')-0.5xXlabel('n")'ylabel('x[n]')-1title('(£)')-5051015n1[€)(g)n=-5:15;05x=sin(pi*n/2);=0stem(n,x,'filled')xXxlabel('n'")-0.5ylabel('x[n]")1title('(g)")-5051015n0)2(h)(h)n=-5:15;x=(.9.%n).*(sin(pi*n/4)+cos(p1i*n/4));Eostem(n,x,"filled')x®xlabel('n')-1ylabel('x[n]")title('(h)"*2mn5051015(i)u(1)=0;nforn=2:10;"u(n)=1;20end;forn=11:21;15u(n)=0;=end;=1n=-5:15;0.5stem(n,u,"filled")axis([-51502])0xlabel('n')505°1015ylabel('x[n]')ntitle('(i)")+StudyXYJo©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.

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(a)x(n)r4bY3+ss&a-r|On-/x[n)3bY[4bkSd@7/7A3+5&4c)xCn)21an“a-/°oFYE]+xCn)|305.53~22013.02ss.\ENENON4n-=!oa3-|VvT=StudyXY1/©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.

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ooEEEE12aAeASoptember1.8(¢)(a)(8)n=-5:6;2x=[000001-1-1-10001;1stem(n,x,'filled’)=axis([-56-22])=0xlabel('n')ylabel('x[n]"')-1title('(a)")2-505n(b)n=-5:6;(b)x=[000010123000];4stem(n,x,'filled")3axis([-56-14])xlabel('n")=2ylabel("x[n]")®4title('(b)")01s05n(c)n=-5:6;©x=[000020100000];3stem(n,x,'filled")axis([-56-13])2xlabel('n'):ylabel('x[n]'")E1title('(c)")[1]-1-5[1]5n(d)forn=1:4;(d)x(n)=0;150end;x(5)=exp(-.8);100forn=6:12;=Xx(n)=exp(.8*(n-6))+1;xend;50n=-5:6;stem(n,x,'filled')0Xlabel('n')-5[]510ylabel('x[n]")n|title("(d)")12©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.eerete055mmemertresrte

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272L_2_419@)[=""=2GH=-—"—=25L="2C2ignalisperiodicwii@5-2,0yryrS572s5»sothesignalisperiodicwithfundamentalperiodequalto57;=10.Jo(——)a)2t=0:.01:30;Ailx=cos(pi*t)+cos(4*pi*t/5);plot(t,x,"LineWidth',1.5)=xlabel("Time(sec)')x0ylabel("x(t)")title('(a)")2!051015202530Time(sec)2%22_2Lh_15b)[==5=L=2lo2ignalisperiodicwii(LFpyL555,2083»50thesignalisperiodicwithfandamentalperiodequalto.ST=2'fea®)t=0:.01:10;2x=cos(2*pi*(t-4))+sin(S*pi*t);plot(t,x,'LineWidth',1.5)=xlabel('Time(sec)')x0|ylabel("x(t)*)title('(b)")-20246810Time(sec)272z_7%015.qoe©)[==5h="2022Aato2,4iignali©ipy1,70SGABx=foranypositiveintegers¢andr,sothesignalisnotperiodic.(©)2t=0:.01:20;|X=C0s(2*pi*t)+sin(10%t);|plot(t,x,'LineWidth',1.5)=olxlabel('Time(sec)')xylabel('x(t)")\title('(c)")2J[]5101520Time(sec)(@)10==foranypositiveintegersand7,sothesignalisnotperiodic.n=1:31;1@x=sin(10*n);n=0:30;stem(n,x,"filled")=xlabel('n')=0.ylabel(*x[n]")title('(d)")»[4]51015202530n+StudyXY/3©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.teee7555Ateres.

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1.9continued107_27qN.APTI©3==;Withq=>5andr=3,sothesignalisperiodicwithfundamentalperiodequaltor=3.n=1:11;13(©x=sin(10*pi*(n~-1)/3);n=0:10;stem(n,x,'filled')xlabel('n')goylabel('x[n]")title('(e)")-10246810n.(0)Isthereapositiveintegerrsuchthatx[»+7]=x[n]?Sincex[n+r]=cos(w(n+7)?)=cos?+n+r)thesignalisperiodicif7(2nr+72)=2mwheremisaninteger,or(2nr+72)=2m.Ifr=2,m=2(n+1),whichisanintegerforanyintegervalueofm,sor=2isthefundamentalperiod.Togeneralize,consider22theperiodicityofof=)forsomeintegeri.Thisisaperiodicsignalifror=m.Thesignalisperiodicifiseven,inwhichcasethefundamentalperiodis7=i,andthesignalisperiodicififiisodd,inwhichcasethefundamentalperiodisr=2i.2f-f)n=1:11;1¥=cos(pi*n."2);n=0:10;stem(n,x,"filled")Zoxlabel('n')ylabel('x[n]")title('(£)")10246810n1.10(a)Whenx[n]=é[n],yn)=47(Ont—11+-+-+51n-N+1)1n=012.,N-1Sim={w",1,2,...,0,otherwise14+StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.OU

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11.10(b)Whenx[n]=u[n],yn]=(utrsn=1J+--+uln—N+1)RForn<0,yn]=0.1For0<n<A-,Hr=FornxN,y[n}=1.1(c)Whenx[n]=rn],y[n]=ryy+r{n=1]+:+r[n—N+1})Forn<0,y[n]=0.For0<n<N-1,1Arl=—(ntn-1+n=2+-+2+1+0)PerformingtheadditionnAnl+p2++2+1+00+1+2++n2+n+nn+n+nto+n+n+n=nm)showsthat2(n+7—1+n—2+---+2+1+0)=n(n+1)andthus1(n@+)\_nn+])n]=—|——=|=——=for0<n<N-1.Mn]ol))oNfor0<n<N-1.1Finally,forn>N,yn]=lren—len-24tn-N+2en-N+1).Viaanadditionprocesssimilartotheonegivenabove,wehavethatstnl=(Hh)=2VltornN.1.11BytheresultinProblem1.10,){r]Seiforn>NandthusN-1..Nnl=n-—o=r[n-(N-1)/2]forn>N,whichshowsthattheN-pointMAfilterdelaystherampr{n]by(V-1)/2timeunits.L12n=1:69;x=5*sin(pi*n/10+pi/4);forn=6:69;y(n)=(1/6)*sum(x(n-5:n));end;n=6:69;plot(n,y(n),n,y(n),'o’,'LineWidth',1.5)gridxlabel('n")ylabel('y[n]")15I~StudyXY©2007PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThismaterialisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermissions),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.a—

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