信号与系统第二版课后习题解奥本海姆(6-7-9)答．SignalandSystemChap66.1Consideracontinuous-timeLTIsystemwithfrequencyresponseandreal?S)H(j??ej(|(Hj))?|Himpulseresponseh(t).Supposethatweapplyaninputtothis??))?cos(?t(xt00system.Theresultingoutputcanbeshowntobeoftheform)?t?Ax(ty(t)0WhereAisanonnegativerealnumberrepresentinganamplitude-scalingfactorandisatimet0delay.(a)ExpressAintermsof.?|j)|H((b)Expressintermsof?t)j(SH00Solution:(a)For)ty()?(Ax?tt02SignalandSystemSo?jt???ej)?AXY(j()0?)jY(?tj??AeH(j?)?0?)jX(So?|j)?|H(A(b)for??tj?)?(SH0?SH)(jSo?t?0?6.3ConsiderthefollowingfrequencyresponseforacausalandstableLTIsystem:?j?1??H(j)?j1?(a)Showthat,and?A)||H(j?determinethevaluesofA.(b)Determinewhichofthefollowingstatementsistrueabout,thegroupdelayofthe??)(system.(Note,whereis?????)Hj())/djS(d()??(SHexpressedinaformthatdoesnotcontainanydiscontinuities.)1.???0?for)?0(2.???0?(0)?for3???0for?()0?3SignalandSystemSolution:2??1(a)for?1?)||H(j?2??1SoA=1???)??j??(?)?(1?j1)?H(j(b)for???)(???2)?arctg(arctg)arctg(?)2dj?H(????()?2???d1So???0?)?0for(6.5Consideracontinuous-timeidealbandpassfilterwhosefrequencyresponseis???3|?|?1,?cc??(j)H?elsewhere,0?(a)Ifh(t)istheimpulseresponseofthisfilter,determineafunctiong(t)suchthat?tsinc)t)h(t?g(?t(b)Asisincreased,dosethe?cimpulseresponseofthefiltergetmoreconcentratedorlessconcentratedabouttheorigin?Solution(a)Method1.Let4SignalandSystem1???)j?G(X(h(jt)(?x)t)tg()?H(j?)?2Theyareshowninthefigures,where???1,?tsinc?c{)x(t?j?X(?)????0,tcSowecanget?????????)][((?2?j)?2)?2G)g(t?2cos(2t)?(cccUsingtheinverseFTMethod2.definition,itisobtained1??3?????tjjtcc??}ed?d{)(ht?e?2??3?cc11????}t2t{sin}sint{sin?3?t?}{2coscccc??ttmoreconcentrated.(b)5SignalandSystem6SignalandSystemChap77.1Areal-valuedsignalx(t)isknowtobeuniquelydeterminedbyitssampleswhenthesamplingfrequencyis.Forwhat??10,000?svaluesofisguaranteed??)X(jtobezero?Solution:Accordingtothesamplingtheoremww?2Ms?110000Thatis?5000??w?wMs22Soif,?5000w?w?0jw)?X(M7.2Acontinuous-timesignalx(t)isobtainedattheoutputofanideallowpassfilterwithcutofffrequency.If??1,000?cimpulse-trainsamplingisperformedonx(t),whichofthefollowingsamplingperiodswouldguaranteethatx(t)canberecoveredfromitssampledversionusinganappropriate7SignalandSystemlowpassfilter?(a)3?10?0.5?T(b)3?102?T?(c)4?10T?Solution:?1000??QwwcMFromthesamplingtheorem,,thatis?2000?2ww??Ms??22?310?T??s?20002wMtheconditions(a)and(c)are?satisfiedwiththesamplingtheorem,(b)isnotsatisfied.7.3Thefrequencywhich,underthesamplingtheorem,mustbeexceededbythesamplingfrequencyiscalledtheNyquistrate.DeterminetheNyquistratecorrespondingtoeachofthefollowingsignals:(a)??)sin(4,000tcos(2,000)t?1??t)(x?t)sin(4,000(b)?)(xt?t8SignalandSystem?tsin(4,000)(c)2)x(t)?(?tSolution:(a)??)sin(4000cos(2000tt)?1x(t)??Q???40002000),4000??wmax(0,?MtheNyquistrateis?8000?w?2w?Ms?sin(4000t)(b)?t)x(?t?4000Q?wMtheNyquistrateis?80002w?w??Ms2?)sin(4000t(c)???)x(t???t??2?1)sin(4000t???Q))cos(8000t(1???t)x(??22??t2t???8000w??MtheNyquistrateis?16000w?w?2?Ms7.4Letx(t)beasignalwith.DeterminetheNyquistrate?0Nyquistrateforeachofthefollowingsignals:(a)1)?x)?(t(xtdx)t((b)dt(c)2)(xt(d)?ttx()cos0Solution:9SignalandSystem(a)welet1)t?)?x((t)?x(ty1So??j??j????)j)?(1(j)?e?eXX(j((Yj))?X1SotheNyquistrateofsignal(a)is.?0dx(t)(b)welet?y(t)2dtSo???)()?jjXY(j2SotheNyquistrateofsignal(b)is.?0(c)welet2)tx((t)?y31So???)j)*X(j)?(X(jY3?2SotheNyqu...