第三章作业答案-17-25

P122-1293.17-3.253.17x(t)X(jω)1x(t)x(t)=0t02{}ttjedejX−∞∞−=∫ωωπω)(Re21x(t)1x(t)x(t)X(jω){)(Re2)()()(ωjXtxtxtxFe→←−+=}F(3-53)tetxtx−=−+2)()(x(t)=0,t0x(t))(2)(tuetxt−=3.183-38x(t)X(jω)1X(jω)θ(ω)τEx(t)t2X(0)3∫∞∞−ωωdjX)(4∫∞∞−⋅ωωωωωdejXj2sin2)(5∫∞∞−ωωdjX2)(3-386Re{X(jω)}1x(t)x1(t)T1=τ/2t0t0=τ/2E))2()(1τ−=tExtx,x(t)22sin2)(τωωτωωjeEjX−=2)(τωωθ−=[])()()(τ−−=tutuEtx20)2()(ωττωωττωjtjeSaEdtEejX−−==∫2)(τωωθ−=2ττωτωτωωτωωτωωωEeSaEeEjXXjj====−→−→→20200)2(lim2sin2lim)(lim)0(ττEEdtX==∫0)0(3)(2)(txdejXtjπωωω=∫∞∞−0=t)0(2)(xdjXπωω=∫∞∞−)(tx0=tππωωExxdjX=+=+−∞∞−∫2)0()0(2)(4)1()1()(1−−+=tututx)(2)()(11ωωSajXtxFT=→←ωωωsin2)(011)(11=→←=jXelsettxF)()()()(*)()(11ωωωjFjXjXtxtxtfFT=→←=F(3-54))(2)(1ωωSajX=)}(*)({2)()(21txtxdeSajXtjπωωωω=∫∞∞−2=t[]≥≤−===∞∞−∫3431)1(210)(*)(2sin2)(212τπττπτπωωωωωEEtxtxdejXtj5τππωω2222)(2)(EdttxdjX==∫∫∞∞−∞∞−6−−+=−+=→←)2()2(22)()()()](Re[ττωtutuEtxtxtxjXeIFTRe{X(jω)}τE/2t-τ3.19ωωωωcos3sin)(⋅=jHh(t)ωωωωωωωωωωωωωω3sin23sin223sincos3sin)(jjjjeeeejH−−+=+=⋅=ωω111sin201)(TTtTttxF→←==−++==elsetttxtxthT042412214)1()1()(31ωωω)3sin()(1=jH=3035.0)(1ttth[]ωωωωωωωjjejHejHjHjH−+==)()(21cos)()(111[]≥≤=−++=404225.025.0)1()1(21)(11tttththth3.20LTI31)(+=ωωjjHx(t)x(t))()()(43tuetuetytt−−−=)4)(3(14131)(ωωωωωjjjjjY++=+−+=ωωωωjjHjYjX+==41)()()(x(t){})()()(41tuejXFtxt−−==ω3.21LTI23)()(2++=ωωωωjjjjHτEx(t)t3-38123x(t)3381)()(23)()(2ωωωωωωjXjYjjjjH=++=[()()(]23)2ωωωωωjXjjYjj=++dttdxtydttdydttyd)()(2)(3)(22=++2112223)()(2+−+=++=ωωωωωωjjjjjjHh(t){})()2()()(21tueejHFthtt−−−−==ω33.18(1)ωωτωτω2sin2)(2jEejX−=()jeeejjEjjjEejHjXjjjjj212111)2)(1(2)()()(222τωτωωττωωωωωωωωωω−−−−⋅−+−+=++⋅==Y{}[])()()()()()()(2)(21τωττ−−−−==−−−−−−−tueetueeEjYFtytttt)]()([)(τ−−=tutuEtx)]()([)(τδδ−−=ttEdttdx)1()(ωτjFTeEdttdx−−→←)1()2111(23)()1()(2ωτωτωωωωωjjeEjjjjeEj−−−+−+=++−=Y)(][)()()()(2)(2τττ−−−−=−−−−−−tueeEtueeEtytttt3.22LTI)()()(6)(5)(22txdttdxtydttdydttyd+=++1h(t)2)()(tutetxt−=3)()(2tuetxt−=4kttxkkπ100cos)21()(50∑==1)(1)()(ωδjXttxFT=→←=213265)(1)()()(2+−+=+++==ωωωωωωωωjjjjjjXjYjHFh(t){})()2()()(231tueejHFttt−−−−==ωh2)()(tutetxt−=)()(1tuetxt−=11)(1+=→←−ωωjjXeFTt21)1(1)()(+=→←=−ωωωωjdjdXjtejXFTt35.02115.0)3)(2)(1(1)()()(+++−+=+++==ωωωωωωωωωjjjjjjjHjXjY)()2(21)}({)(321tueeejYFtyttt−−−−+−==ω3)()(2tuetx−=21)(2+=→←−ωωjjXeFTt32)2(122)3()2(1)()()(22+−+−+=+++==ωωωωωωωωωjjjjjjjHjXjY)()22()(322tueteetyttt−−−−−=4∑∑=−+=+==50100100150)()21()100cos()21()(ktkjtkjkkkeetktxπππtkjketxπ100)(=0≠kπππππkjtkjkekjkjekjHty100100)1002110032()100()(+−+==0=k1)(0=tx61)(0=ty∑≠−=++−++=5051001)1002110032()21(61)(kkkjkekjkjtyπππ3.23x(t)h(t)12)()(tuetxt−=)()(3tuetht−=)()(tutetxt−=)()(3tuetht−=3)1()1()(),1()1()(−−+=−−+=tututhtututx1ωωjjXtuetxFt+=→←=−11)()()(ωωjjHtuethFt+=→←=−31)()()(3+−+=++=→←∗ωωωωωωjjjjjHjXthtxF311121)3)(1(1)()()()()()(21)()(3tueethtxtt−−−=∗22)1(1)11{)()(ωωωjdjdjtutetxFt+=+→←=−]3111)1(2[41)3()1(1)()(22+++−+=++→←∗ωωωωωjjjjjthtxF)()2(41)(*)()(3tueetethtxtyttt−−−+−==3ωωωπδωωωωsin2)(1)()()1()1()(=+−=→←−−+=−jeejXtututxjjFωωωsin2)()1()1()(=→←−−+=jHtututhF()ωωωωω224sin2)()()()(SajHjXthtxF==→←∗−=202)21()(1τττtttEtx(P1003-7))4(2)(21ωττωSaEjX=4=τ2=E)()(4)4(2)(221ωωωττωjYSaSaEjX===)()(1tytx=−==202)21(2)(*)()(tttthtxty3.24x(t))(ωjXp(t)0ω0)(ωjkkeatpk∑∞−∞==1)()()(tptxty⋅=2)(ωjX3-39p(t)y(t))2cos()(ttp=ttpcos)(=ttp2cos)(=)()(∑∞−∞=−=kkttpπδ)2()(∑∞−∞=−=kkttpπδ)4()(∑∞−∞=−=kkttpπδ1))(()(0ωωωkjXetxFTtjko−→←∑∑∑∞−∞=∞−∞=∞−∞=−→←==kkFTktjkkktjkkkjXaetxaeatxtptx))(()()()()(000ωωωω3-39wX(jw)-1112)(21)2cos()(22tjtjeettp−+==))]21(())21(([21)(2)()()(22++−→←+=−ωωjXjXeetxtptxFTtjtj)(21cos)(jtjteettp−+==))]1(())1(([21)(2)()()(++−→←+=−ωωjXjXeetxtptxFTjtjt)(212cos)(22tjtjeettp−+==))]2(())2(([21)(2)()()(22++−→←+=−ωωjXjXeetxtptxFTtjtj∑∑∞−∞=∞−∞==−=kktjkekttp21)()(ππδ∑∑∞−∞=∞−∞=−→←=kFTkktjkjXetxtptx))2((1)(1)()(2ωππ∑∑∞−∞=∞−∞==−=kjktkekttpππδ21)2()(πωππ1))((21)(21)()(=−→←=∑∑∞−∞=∞−∞=kFTkjktkjXetxtptx∑∑∞−∞=∞−∞==−=ktkjkekttp241)4()(ππδ∑∑∞−∞=∞−∞=−→←=kFTktkjkjXetxtptx))2((41)(41)()(2ωππ3.25LTI[])()(3tueetxtt−−+=y(t)[])(22)(4tueetytt−−−=1)(ωjH2h(t)31)3)(1()2(23111)(ωωωωωωjjjjjjX+++=+++=)4)(1(64212)(ωωωωωjjjjj++=+−+=YωωωωωωωωjjjjjjXjYjH+++=+++==423223)4)(2()3(3)()()(2h(t){})(2323)()(421tueejHFttt+==−−−ωh386)(39)4)(2()3(3)(2+++=+++=ωωωωωωωjjjjjjjH)(9)(3)(8)(6)(22txdttdxtydttdydttyd+=++