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⚪ᫀ1.3⚪1.ᑨᑡ ᔲᕜḄᕜḄ Ḅ ⚣!"ᐳᕜ(1)/(/)=4-3sin(5^r)4-sin(30^r)(2)f(r)=COS(10R)COS(30R)(3)f(t)=cos(lO^r)-cos(20r)(4)/(/)=cos(2r)-V2cos(2r--) (i)A1=nQ3062227r242>?"ᐳᕜ"A/=lx'=«,C15/r5 ⚣/o=H=[=2.5"z1(),(2)f(t)=cos(10^r)cos(30^)=0.5(cos20/rt+COS40R)%_C]_20_1nQ402222O12O1>?,"ᐳᕜ”=n,—=1x----—sCi20%10Q=10z ⚣P)(3)ᵫTUVᑖXḄ⚣᳛Z[\=]^ᳮᦪ>?^ab "ᐳᕜᡠdeᕜ%20Ḅ(4)UVᑖXf⚣᳛Ḅ ⚣f=l/nHz>?"ᐳᕜ7=-1=O5afo2.ᢣ ijkᑡ lmno᳛ mnpX mnqro᳛ ]rps⁚(1)w(z)+5u(t—1)—2u(t—2)(2)u(t)+5u(t—1)—6w(/—2)(3)e~5'u(t)(4)(e-5'+l)n(r) (1)xy⚪2z(a)ᡠ|}ᯠo᳛ P=\im—=limU\'\dt+/36dr+116dr]=16Wr^oo2TJ-TT^OC/,LJ()JIJ2⚪2z(b)⚪2z(a)
1(2)xy⚪2z(b)ᡠ|}ᯠpX E=fl2Jr+1262Jr=1x1+62x1=37JবpX £=lim«)2=7-d=--!^-"=0.1JT—>coJOJ0[00(4)o᳛ }ᯠᨵP=1W3.ᕜ ⚪z3ᡠ| Ḅo᳛ᕜT=7,VᕜḄpX£={42
2(1)(4/+2)%)=65(f)H(2)e'£(4-2t)=0.5e-3W-2)=0.5^(r-2)(3)sin(2/+—)1=4rᙠ®ᑖ¶(-3,6)ᑁ)(3)(3(6_²)¸+4)+25(2/+4)]=1()¹+4)+23"+2)]=2(4)%()35º3/)=3(54(3=1È^^=1)J-00tJ-00'…3t7. ⚪z7lḄ Ḅ▤¼ᦪxy4\/HV,LA>/111>036o23460246⚪7z,()£(“£(')Ḅxyᑖ⚪7z(a)ஹ(t))ஹ(c)ᡠ|/.,(O¿)Ê)(4)ভ(4)4/31--------------2-|-4/3°^--------3.2Q6tt012|46(a)ভ(4)(0ভ(b)⚪7z8.ÃT⚪8zlḄ /(f),dᔜÅÆz(1)y(t)=f(t+3)(2)x(f)=/(2-2)বg(t)=f(2-2t)(4)Ç)=/(—0.5f—1)3
3(5)f(t)(ᏔᑖX)e(6)AQ)(ᑖX)2ஹ03⚪8z9.ᕜ ⚪9zᡠ| Ḅo᳛ᕜT=7,>«)=%+L54
4ᐸpXᓾ,du(%+15ÒÓJ Ḅo᳛P=WWT310.ᵨ Õ ᡈ▤× ⊤|⚪10zlḄ i ÙḄpX(a)J\(t)=2G6(f-3)+2G3(f-3)ÛdÜᡂÝVÞypXE,=4x2+16x2+4x2=48J(b)/2(=2G6-3)+2(-3)ÛdÜᡂVÞy!VÝàyáÈpX=4x6+-x4x2+2xix2x4=34.67J-32(c)-প=63(-3)-20(—3),ÛdÜᡂVÞy!UVÝàyáÈpXR=16x2+-xl6x4=53.33J3311. ᑡ Ḅxy(1)f,(t)=u[COS7lt];ফf(t)=-[u(t+2)-u(t-2)];2(3)f3(t)=sin—u(2-t)J;ভf4(t)^G2(t)sgn(t);মf5=G6(t)Q2(t-2)5(6)f6(t)=u(2-11l)sin(^z)ᔜ Ḅxy⚪11zᡠ|5
5AC)⚪iiz12. ᑡ®ᑖ(1)cosCf3'ª—6)]H(2)f[3(t+2)-3(t-2)]dtJ-co4J-oo(3)(4—2)5«_4)ᵫH(4)-2)5(x-f)dt(a)cosã©)-5(f)]=-1HJ-CO4(b)©[3(t+2)-3(t-2)]dt^u(/+2)-u(r-2)J-00(c)/3(4_/2»«_4)=8/,ஹ98(x-2)x=2(d)f3(t-2)3(x-t)dt=\J—[0xw213. ᑡᔜ Ḅxy(1)()=(+1)“()(2)f(n)=n[w(n)-w(n-5)]2(3)/(«)=(-0.5)3(4)()=2-"“()ᔜxy⚪13zᡠ|6
64æ()44321234n1234n⚪13z14.ÃT⚪14zlḄ /Q),dᔜÅÆzfপ(a)Pª"(f+3)H(b)(c)f(t)=f(2-2t)53(d))=/(-ã-1)H(e)(f)(ᏔᑖX)H(f)f(t)(ᑖX)0:ᔜxy⚪14zᡠ|()224⚪14z7
715. ᑡçᦪḄᔁ®®ᑖ/H(/)*/-(?)প(=""3)/(/)=«(/);2ফì)=È=V"ᕲবP)="(“/(r)=eî,M(f)2ভ/()="(—)/(0=«0-5)2(5)=/(r)=M(r-l)-w(r-2)2ð প/G)=""(£),f(t)=«(/)H2[")*/2(=£e-3rw(r)xw(/-T)JT=Je-3rt/T=-1e^=|(l-e-3,)w(03o3/3*(=""ñ"=Ò-*=6-3Qò"%(বP(/)=%("f(t)=e~'u(t)2f\(f)*%C)=#3*ôõ)=M)*öeZr="(f)*(1"")=((1"ᓅ=9+ø=(-1ù((4)(f)=u(t-1),f(r)=u(t-5)2f\0*/(0=w(f-1)*-5)=(r-6)u(t-6)2ম/i(f)=/M(U/2")="(fT)-"(f2)#k=9Ḅû)=È1)ü-2)/i(0*/2(0=:©"()*-1)3(t-2)]=^(r-1)2u(t-1)-^(^-2)2u(t-2)16.ýþ8
8প/,(/)*tu(t)=[t+e~'-l)w(r)ফ9)*Ò%]=(l-e-fM)-(1-1È-1) È)ð প<(*È(f)=1+e~'-lù(f), >(f)ÿ?[(0*tu(t)=[t+e-'-1'«)2(*)=(1-/)=/“)(2))*]=(1-0-)-(1-03-1),) {/3*!%(]}'=$(*!%)-63]=*=/3-' —[(1-/)/«)_(1+6«|>(-1)]=8(t)+e~'u(t)-e~'8(t]-8(t-l)+e-(,-l)J(/-1)dt=b(f)+/“(/)_U(f)_g_l)_eY-%-1)+"(i)b(f-1)=/8_”:,-1)ᡠ<(-/(0*H%(/)="'"")-"(1>-1)?@*!'(B)]=(1—/—(1+eYT>}«+1)DᐭF ,GH-(1-/1”)-(1-/(+“-1)=e-'u(t)-e-['~'}u(t-i)//)=M)i(—)17.LMNᑡ(),()ḄQB(")*@প<(")=&()="()(2)£()="()B&()=V)-W-1)X N 9
9প$()=/2()=() /Z()*%()=()*"()=(+AG)ফfi(n)=u(n\f(n)=S(n)-S(n-\]2 /()*]()="()*6()-4-1)]=()-("T)18.LM/()=/()Bf(n)=b"u(n),$(")*/2()2 /1()*/2()=`"()*"()=£a/7r=0cd“e*/2fe=/fe*]eg`h=/£]£|c=/?d/()*()=`“()*/()=1jk7=£/=(+1»”/=0»=0Fm ᙠ“20ᡂpBᦑG+-/ஹ,)*()=s"()*/()=(b—a(+1»@a=bca=b=1du(n)*u(n)=(n+l)w(n)19.LM$()=siny)="()B()=b()+ab(+1),/1()*()*/;() 10
10♦()*♦()*(")=sin*}—a8[n-1)]=sin;r*~)+-1)]=sin)*anu(n)-a/l+1w(n)*8(n—1)=sinnᑍ*anu(n)-a/,+1u(n)*8{n-1)=sinn/r*-1)]=sin4*{◖()+(-1)]+(+1)}=sinn7r*3(n)-sinn/rgᵨᑮឋ uEnF=3fe+f+1ea⁚()=&&(”)=5()2.3⚪ 1,%fLegendree⚗ @=1ᙠf-1,1eᑁᦪ ᙠf-1,1eᑁBᨵfiL@¢=£¢=¥’¦£@*ὡ@=©ᓝ+«=ZfhTe1>0£ᓾe*ὡপ®-?e=f|/-¯eᓃ=oB/@*ὡপ=£@/³e=f/+Z/eᓃ=L6@´@f|µ-|fe=f¶Teᓃ=0"প*ὡপ=£f|/+„j=f¹6_/+|º=0ᙠf-1,1eᑁ»¼©ZR@PZffe¢=0fi*jeᙠf-1,1eᑁᦪ11
112. 05/,85f2...,«5f¿ef11Àᦪee,ᙠf0,2ÂeḄᦪᔲÄᜓḄᦪÆ ᙠf0,2ÇeᑁBᨵ0cÉwnµ2È£[cos(mQr)cos(Q()B)df=cÉmw00Tcº=n=0 cosf,cos2fB…,cosmnÀᦪeÊᙠf0,21eḄᦪËÌÄᜓḄÍÀ ᙠᦪÎcosB,cos2,…,cosrnÀᦪeÊÐBÑᙠᦪsinmQrmW0»¼ £Tsinf7?2QrcosnQrdr=0,ÒÓᡠᨵḄmÔn003.⚪3ÕÖ×ØÙÚᑡ"f=Û@ḄᢣᦪᏝÞßᦪÔàáᏝÞßᦪ=-88(t-T)aºe⚪3Õ :$=,ᑁf2012=—»a=—12 1)ᑭᵨêëó :ÍÀôõÀöêÀᦪ,ᡠ<%=0Z,2PA•c,A5+”S21*smn^tdt=+,FéRT0TOT0ᐔ2ᐔ,
13l=2)ᑭᵨ3⚪Ḅñò A,0872fAᑣ/প=Z/(f-Bz)=Zþ+7+")][“"—")+"+(+1)")]>n=-son=-ot>110,A=/«=if-+l4[+Z{[A-£"a]3f-7 -b”+l7 }10=-ooஹ--+AZSQ'0/^—?A+A±1+22ZCOSQ=—COSn=/XT11”oஹ”"n=I7n=l•■-Ac5=F£Lcosn^rdr=:£sin□/10M=11n=\%;=5A,ᡠmz@“,/ஹ=5A+A^ Tsin5.DEᕜGHI/5ḄKLᕜGLM⚪5PᡠQ᪷TUᑡᔜXYZḄ⌕\]^f)ᙠ4'aᕜGḄLM-]T2(1)f(r)dᏔfᦪ,klᨵᏔnoL (2)/5dᏔfᦪ,klᨵnoL (3)/(f)dᏔfᦪ,lᨵᏔnoLpnoL;(4)f(r)dfᦪ,klᨵᏔnoL (5)fপdfᦪ,klᨵnoL(6)fপdfᦪ,lᨵᏔnoLpnoL
14⚪5Prs1)/5dᏔfᦪklᨵᏔnoL2)/5dᏔfᦪklᨵnoL⚪5P(a)3)/5dᏔfᦪlᨵᏔnoLpnoLᡈ⚪5P(d)4)/5dfᦪklᨵᏔnoL5)/«)dfᦪklᨵnoL6.ᕜGHI/5Ḅuv⚣x⚪6PᡠQ\ᐸz{fᦪ⊤Q}14
15⚪6Prs᪷T~=£+£■="('£“)\a,=F.+F.=2τI-τb-j(F-F_)=-2j333/(r)=2cosr-2jsin3r7.DEᕜGMHIপ5⚪7PᡠQ\s(1)5Ḅᦪ~=0.5|is,T=1(is,4=IV,ᑣx◞p~(2)5Ḅᦪ~=1-52T=3.s,A=3V,ᑣx◞p~?(3)55ḄL~(4)(/)LপḄznoL~⚪7Prs/(0=—+*Sa(--)cosnQt27T(1)x◞~Q,='=2xl()64(md/s)ᡈ=1000KHZT\24τ~%τ=—=41()64(4/5)ᡈ8—=2000/^^qr.(2)ᳮ\sx◞~,==2ஹ1()6(sd/s)ᡈ/=USK”Z33~Bg=—=-xlO6^-(raJ/s)ᡈB=—=&KHzf2T3r32215
16¤¥Sa¦lx0.5x3x5a(—)2Aq1(3)SaᓲSa(ᔩ=7T\T?T3xL5xlxS(7¨l\5)'32§4᙮¯1x0.5x3xSa(^y^)So(©:Sa(«)(4)11Tl¬T“2°“±Sxl.SxlxS^3^5)28.\⚪8PᡠQ³L´µᾭ·ḄᏝ¹º»ᣚ½]^⚣xP⚪8Prs/(/)=E\u(t+1)--1)]cos/fTTC)TE[u(t+-)-u(t——)])=ETSCI()222TTITTT(TIcos-r]=Rb(a+—)+5(a——)]rTTITT7VL7l22T=—ErSa(—)*n\8{Q+-)+J(/2--)]22TTcᔆCT2Ercos—ὅ□Á+9+SÂ'>=2CT41-(—)29]7T9.ÅÆUᑡHIḄᏝ¹º»ᣚ(1)ejtsgn(3-2z)(2)(3)e”"(-f+l)dtjri(4)|cos(—),H<1(5)...2-0,|z|>lJ+4rs22-i-n(1)F[sgn(r)]=——=>Ìsgn(3-2]=----e2jQjc16
172-/-(/?-!)=>Ì’sgn(3-2r)]=---------e2j(a-D(2)প]3]Qe-^u(t)]=j^e23Ae-2'u(t)]=e2_22_dtj/2+2(3)ᵨ“(]=Ò(0)+-!-=/'”(7+1)O/02)[R(—2)----1—]jCÔ0-2)e2c,(-/+l)ode"-M'1"(23=—^ߟe°-Ù'=-——ML2-jC,2-jC..2£rcos^^cos(—),kl<>^________2_4cos0c,Ô)ஹcᐔஹ(4)-----—=Sa(£2+—-)+Sa(0-—)|/|>1Û-(0)2]^[1-(—)2]220,117tr=27tE=1(5)Ü~2.ÝÞឋ_>2-627t-e~2^=%"NRC+42*+4210.âᑖäᑭᵨUᑡæXçèéê“5-ë(Q)+(1)ᑭᵨìIfᦪu(f)=g+gsgn(/) (2)ᑭᵨMᾭ·î▲(r->8) (3)ᑭᵨñᑖòᳮu(t)=;8[T)6T;J-co(4)ᑭᵨᓫvᢣᦪfᦪî▲[M5=lime-'"/>0]QTOrs(1)ᶍTTHTCT-j-Q(2)u{t+—)—u[tw(z)—u{t—r)zSa(_,2CTீ07ஹ-øsm(^)TCjsin(/2r)2j./2rrSa(---)e2=---J=---;--------sin2T———[cos—/2-/sin2V22~T~Tsin(12r)2J1-cosQT_sin(/2r)jcosQuCC2-CCjC17
18.h.mrS0a{/-A---r)ஹeF2®=lvim-s-i-n-(-/-2--r-)+---1----lvim-c--o-s--f-2-r-38218CjQ-8jQsin(/2r)11=hm----------+-----=ú>()+----38ᑍQjQjC(3)ᶍ(4)-^~j^=limlim[—1-------,~.-(a+jn),]F[u(t)]=limlimeedTez->aoa-»0°T8T()a+jCa+jC=_Lf=———lim^—(cos/2r-jsinQ)—8jCjC…4=sin/2rcosQ1sin/2r1w+lim------------lim-----------=--------FTVlim---------=--------F71bisC18jCjC-8ᑍQjCii.ü/5ḄᏝ¹º»ᣚ~ý2=g[G2“a-Q,+G2“a+A]⚪iiPᡠQ\ÿ FYjCA1/2_Q)+a0Q⚪111/ஹasinat-G(12)^-———=>2U22at2L7tat /ஹsinat=—Sa(at)cos£4/=----cosQj7t7lt⚪1118
1912. প!■ὡ$2&=R$C&+jX$&+, Ḅ./⚪12$a&ᡠ3+4ᨵ , Ḅ./⚪12$b&ᡠ378$9&/W4fQ)=|iz(+z(-1)J»G(2B)+F,(-2W)2 (□)=ᵬ22)+ὡ*(2I)=2R(20)13.4 /(f)!M(j2),NOPᑡ ḄᏝSTUᣚ:(D/(l-f)(2)(l-)/(l-f)(3)/(2?-5)fপ—M(-□)]=FLMi-jdெFYTW=ᓱdF"dCdC$3&^M$_/`14. abᾭd, ḄᏝSTUᣚeF&0&*Sa?$h&+iᵨᨵᐵOᳮ7/,$?&= 0—2r&cos$2ri2*4x$0+q1&+—A&t22I4J|ETSa2$$a+a&&z&c-/2r$a+%&+$a2Y$!&"/?.—&!4I4JI4J_19
20=Sa?(^2a+Sa2((°L”215.4 / cF(./Q),NOPᑡ ḄᏝSTUᣚ(D(T(2r)(2)(/-2)/(/)(3)(z-2)/(-2f)(4)/^2dt….0ஹ$&J2প/(2r)]=j-j—d(oa122dCফ(r-2)/()o21f,প=]~º.+a)_5(/+»)-º._¼)+5!|)+⚪16(a)ᡠ3r-r,2222-sin(0(7ᧅ)11, (0)=o8Esinp^lln(^pl),%0)=0F'g=^j/2(r-r)(“ᡠ¿=ᨌ☄Â]Ãᑠ]Ã)=(3)ÅÆ=24!+“)"23ÂÇÈÉÊញsin si /⚪16(b)ᡠ3⚪16(a)⚪16(b)17. ▤Í ḄᏝSTUᣚe 6Î+Ï(Q);ÑÒஹÓÒÔᦪḄᏝSTUᣚecos(f^jf)<->][5(/2+R))+<5(X2—Q))]^sin(Qj)<->jTC[3(/2+R))—<5(/2—R))]°7ᓫÙÑÒsin(R)/)”)ᓫÙÓÒcos(f^/)w(/)ḄᏝSTUᣚsin(f^/)w(r)———j7r[3(i2+Q))-b(-"))]*——+^(/2)2[_”221
22if____i___________+Û(0+R)—b(0—4)]>51j(0+%)j(0-q)=Üh0+A)ᵪ0-A)]+ᑍ4Þᳮß7:cos(f^)z)w(0——+Q))+3(C—Q))+(OY)18.7F(2)=----J----ḄᏝSTáUᣚ(a+jA)--,j--------------=——-~-<^^n,w(0+jCdn\_a+jQ\{a+j®?1F(jC)=te-a'u(t)(a+jC)2â!ãä/(f)=eT+(f)*e-"(f)=■KO19.7 /«)=Zç"""+è—3+é,ḄᏝêUᣚTT ᕜìe2T,ᑣ,=í,£î/প—n=2k+l^,tT'=᪵⚣22
23᳛£(D/(3r)ফ.fC)ব/«)*/ফ)(4)/(0+/2(0GH=100"Z-200HzJGfপcF(jC)প,⚣;,-M⚣+Nᜧf=3,=300Z,f2=600”zফ/2T6_LF(M)*F(2),⚣;,-M⚣+Nᜧ0U(”)ḄVW2%£=2f=200WZ,f>2f=400Hzcব/(f)*/(2f)c,F(”)Ḅf=1007/Z,F(]Ḅ2222f"=200Hz,^—F(jQ)*_F"`)Ḅ/=/+/"=100HZ+200HZ=300WZ,°2TT22rf>2f=600Hzsc.1(4)/T+/2«)a/(2)+——F(jC)*F(jC),⚣+Nᜧ0U(2)ḄVW,c2f'=2f=200Hz,/>2f=400Hzcc3.1e⚪gi<6ᑡiᦪḄklkmnᣚpqrᦈtu(1)—(1-e~(x1)(")a\7(2)(r3—2vবe"cos(Q+)T(4)5(-T(5)te~2tu(t)G23
24প|}~T—T1s(s+cr)ᦈt;06>maxf0,-6^](2)L{(/-2+1]9)}=4_21+5ss641s3-4s+6------1—=-------5453SS4ᦈt;0bு0(3)L{e"cos(Q+e)“(f)}=L[e~***s,(cosCtcos0-sinCtsin)T}_(s+a)cos-Osin(s+a)2+ᦈt;08>-a.(4)L{)-ᔆ)}=1a=qs+2s+2ᦈt;0b>-2(S+2)ᦈt;0>a22ᑭᵨklkmnᣚឋ<6ᑡ,-Ḅklkmnᣚপ/b=tsintu(t)(2)/(r)=sin[T-u(t-V)](3)/(0=t[u(t)-u(t-2)](4)/(f)=[t3+te-3'cosফ)]”(£)G(1)0sin——a5+1ᑭᵨ⚣;ᑖឋᨵfsinfg-[a]=J'ஹdss2+\(.1+1)22sᓽFG)=T+1)2(2)/(r)=sinEu(t)+sin-l)w(r-1)247171-s1(1+)UG)52a+/+V22S+7
25(3)f(t)-tu(t)—(t—2)“(f—2)—2u(/—2)y—1/ஹ11—2s2—2,sF(s)=---eccesss52—4(4)0fcos2,aa£-----(/+4)223!᪷¦klkmnᣚ:;⚣§ឋᨵ¨ஹ6(S+3)2-4F(5)=—H-------------54[(5+3>+4]23<6ᑡiᦪḄklkm©nᣚiτ--s.ªߟ2s-I-|.y(1),■(2)¯°(3)(±J-5+1\+esGe e'(1)ᡝ'(S)=——+--+——5+15+15+1᪷¦:µឋ.1g~se~2s“)=LE++¸77T=e''u(t)+eri%-1)+e~('~2)u(t-2)(2)¹U(s)ᳮᡂJdᕜ¾¿À/(s)=a°±1+0—sl-e-2sᑣ/Q)ÁÂaᕜ¾ᓫÄiᦪ03(f)—SQ—1)ஹᕜ¾T=2Ḅᕜ¾iᦪᡠ%f(t)=S(t)-—1)+b(f—2)—3Q—3)+…8=Z(T)%(T)4=025
261-~s1_~s1_~senee(3)0b(s)=(------)2=---------------------------=ᱏমᵫ(s)sssᵫᔁÊËᳮÌ/}=H«Í*HÍᐸÏHQ)=L-'{F(5)}=uQ)—Q—1)1ᡠ%=3Q)-bQ—1)]-(r-l)w(r-l)]=tu(t)—(t—l)u(f—1)—(Z-l)»(f-1)+(f-2)u(f—2)=tu(t)—2(f—l)u(?—1)+(?-2)u(f—2)4ᵨÒᑖᑖÀ'<6ᑡiᦪḄklkm©nᣚ52+4s+24(1W)=ফÓS)=(s+l)(s+2)G+1)(5-+2)33s+925+10(3)U(s)=ভ/G)=5(52+9)S2+4s+13:}1ÍᵫÖU}sÍÏ×==2,✌ᐜᵨÚ◀'ÜÝ==U+4s+2J+-----+-----=(s+1)(5+2)(s+1)(5+2)5ÞᑖÀMßᡂÒᑖᑖÀN(s)_k+kx2Z)(5)(s+1)(5+2)5+1S+2ᐸÏ—à=-1$+25=-1…+2ᗞ=2s=-2S+1s=-212ᑣãÀ0/}SÍ=1—-----------1------------s+1s+2}2ÍãÀMßᡂÒᑖᑖÀ44-4-4—4FG)=---------------------------=------------1--------------------1--------------------1------------(5+1)(5+2)35+1(5+2)3($+2)2S+226
27ᡠ%fb=(4e-'-2t2e-2'-4te-2'-4e-2')u(t)E(s)=3G+9=)+3(3)s(s2+9)s/+9d+9f(t)=(1+sin3t-co2s+10=2(s+2)3(4)s-S2+4s+13-($+2)2+3?(5+2)2+32/(f)=[(2coSt+2sin3t)e~2']u(t)5<6ᑡiᦪklkm©nᣚ/(f)Ḅåæ154+1পU(s)=ফFG)=বF(s)=s+252(5+2)2s~+2s+1G(1)/(()+)=limsF(s)=limsߟ!—=15—>COST85+2(2)ᵫÖU(s)ÁᨵᳮᑖÀéêÁÞᑖÀᑭᵨÚ◀'¹ᐸᑖ0452+1F(5)=5-2+sf+2s245+1ᑣ/(0J=limsF(s)=lim5———-°45—>005—>00s+2s(3)/(0)=limsF(s)=lims—£--------=—+182S~+2S+126<6ᑡiᦪklkm©nᣚëTḄìæ5+s~+2s+1প/ব=ফF(s)=52(52+4)(s+1)(5+3)(5+5):(l)S2($2+4)=0,=îS]=52=°S3=2J,S4=-2jᨵîᙠðñᓃᦑêôᵨìæËᳮ/পõìæ(2)04=1k2=3,.==-5/(s)ḄîᙳᙠsḄøùú☢ᦑüýìæËᳮþ27
28v3+c2+2s+lᨵ/(oo)=limsF(s)-lims—:------------------=0zo…(5+1)(5+3)(5+5)71—6' « Ḅ!ᦪ#1F5 Re(5)>0S(6+l))ᐸᏝ,-.ᣚ123G Ḅᦈ6ᙶ᪗9=0,3" ᙠ=>ᨵ?▤B0,ᙠCDEF☢ᨵ?▤B-1I3" JK#LᑖᑖNCOP(s)=!17+TSᵫN3.5-6 O1—"' •" ḄᏝ,-.ᣚ#3□ =/5 |“+Z0 11+\(C)gi4.3 ⚪ 1.)^_`aᑡḄZ.ᣚdᦈ6e(1)2r, (2)(3)2-,,w(-n)(4)b()(5)3(1)(6)2-"[u(n)-w(n-10)]1:পZT[2-“()]=o,(p"=Z2fZ-"=9”Ctn=-29{2"z"1,Izl^-Bᑖ(1)x()=RN()CN=4(2)x(n)=Ar"cos((yn+(p)u(h),r=0.9,a>=0.5^rad,(p=0.25^rad00n0=0,1,2,3tz3(z-l)=0,B#2Z1,2=0ClBᑖ⚪21(a)ᡠCz=lᜐḄB(2)x(ri)=Arncos(a)n+(p)u(n)=Arn[ei6^nej(p+e~i(^ne~j4—:--r------2---r]2n=0n=021-B2=¡(%--¢CB2z?=re-z=re~^3COSBᑖ⚪21(b)ᡠ(3)£y()=&()Cᑣx(n+1)=y(n)*y(ri)29
30zxz =[yz f,xz =zT[yz r¦#Y(z)=tz4-lz\z-l)-4_i14_i7¦¨OᑮX(z)=~~-]2=4[---]2z(z-1)ZZ-l2ᑍkB#2Z|=0.z=1.B#2z=eJ4,k=0,1,2,3t2kᙠz=lᜐḄBCᦈ6e#202¸®ᦈ6e«¬¸®»¼Ḅ½aᑡ1 ¾ᦈ6e#lzl<0.5¿Cᵫᦈ6e¯O½aᑡ#D¶aᑡ.3_2X(z)=-1+l-2z-'12À⊤4-1¯Ox(n)=-[3•(g)"+2•2"]u(-n-1)2 ¾ᦈ6e#0.53132X(z)=-+——p=X|(z)+X?(z)1i11-2z1-2Zᵫᦈ6e¯OX1(z)«¬Ḅ½aᑡ#¶aᑡCÃX2(z)«¬Ḅ½aᑡ#D¶aᑡCÀ⊤4-1¯Ox(n)=3•(g)”5)-2•2"u(-n-1)(3)¾ᦈ6e#Izl>2¿Cᵫᦈ6e¯O½aᑡ#¶aᑡ32rXQ)=j—571—yÀ⊤4-1¯Ox()=3(g)"+2-2"W(n)4.x()="%()C0aI-azt-2(2)ZT[nx(n)]=-z—X(z)=_f\z\>adz(l-az)(3)ZT[a-"“(—)]=Na-"z-"=£a"z"-——f1Zl2«¬Ḅ½aᑡx()12X(z)ᨵ³?B2z,=0.5,%2=2,ᡠ_ᑭᵨLᑖᑖNËÌJK#:31
32A+4X(z)=bU7ᐸ4=------Z___(I-2Z-')L=-I=23TÍ14=-----------t——,-i(1-2/)(Ît/)2%ᡠ_X(z)=X,(z)+X(z)=-4^+^^-112l-2z-'+.l-il-2zi1-il”zI-P(1)ᦈ6e0.52«¬Ḅ½aᑡx(),ᵫᦈ6e¯OX[(z)ஹX2Q)«¬Ḅ½aᑡÑ#¶aᑡCÀ⊤4-1¯Ox()=(g)"()2"()6.ᑖÄᵨÒ◀ÔஹLᑖᑖNÔ)_`X(z)ḄÕ.ᣚ2Lt1পX(z)=-f—,kl>-117-221—z4l-2z~'1(2)X()=—^ߟ,kl<-Z11Ù21--7Z412(1)LᑖᑖNÔ2X(z)ᨵ³?B2z=0.5,Z2=-.5,ᡠ_ᑭᵨLᑖᑖNËÌJK1#232
331-14+3X(z)=—f-42•(1-*')£=A6-(l+^-Z-|)l._5=-|=0(14r)(14r)2oᡠ_156+6X(z)=ᵫᦈ6eIZl>,¯O½aᑡ#¶aᑡCÀ⊤4-1¯O2_ =Ú3"+3-3"]" o2o2Ò◀Ô41211J——ZH---Z12161-3.1-SLL)12'16)33
34(2)LᑖᑖNÔ2X(z)ᨵ³?B2Z1=0.5,Z2=-0.5,ᡠ_ᑭᵨLᑖᑖNËÌJK#:\-2zA]+-2Xফx.1_1l--z~21+2Z424l-2z-'--(1-^--13Z)L4x~~1-------r=052z~')221-211-K._5=-d+2")"-]1x)(1+2/)0-2ᡠ_35xfÙ2+1——z1-z42ᵫᦈ6eIZ1ÝÞ¯O½aᑡ#D¶aᑡCÀ⊤3-2¯O2x()1)Ò◀Ô-4z^+3223—1ß4+1~~rz~2/-2z-1+l4/-2z-1+8zl-8z1-4Z2______-8z+4z2-8z+32z34Z2-32Z34Z2-16Z432Z3+16Z4x(n)n>0X(-1)x(-2)x(-3)x(-4)x(-5)CCC08-432-16128…7.áâãឋåaᑡx(n)Ḅæᐵ!ᦪᵨ`N⊤:00%(⊈)=Zx()x(n+m)n=-oo34
35éᵨx()ḄZ.ᣚX(z)êᏝ,-.ᣚXd)ᑖÄ⊤æᐵ!ᦪḄZ.ᣚR,\C)êᏝ,-.ᣚR"")8p:r(m)=^x(n)x(n+m)xvn=-oo0000ooooR«(z)=XZx()x(+í)=Zx()£x("+í)z-Mm=-/1=—oon=-m=oo£?'=+2,ᑣ0000R"(z)=£x()£x(/)z-'"'+"”=-00l'=-00=£x()z"í=x(z-')x(z)“=□0wf=-00ooᡈὅ(m)=x(n)x(n+m)=x(m)*x(-m)“=-00qফ=X(/)X(z)RW)=R“(z)“X(T)X(e*¦#x()µåaᑡCX(e5)=X*(eC),¦¨¨,(e"")=IX(-3)|28.áX(e"")êY(ejM)ᑖĵx()êy()ḄᏝ,-.ᣚCé)`ᑡaᑡḄᏝ,-.ᣚ:(1)ó(0)(2)/()(3)x(-n)(4)x(n)*y(n)(5)x(n)-y(n)(6)nx(n)(7)x(2n)(8)x2(n)x(/t/2),n=Ꮤᦪ(9)x(n)=<90,n=ᦪ:00(1)FT[x(n-n)]=^x(n-n)e~ja3f,0Qrt=-oo£/?'=0C='+oCᑣ35
36FT[x(n-n)]=£x(')e-w"+")=e-"M°X(eW)0n=&a>(2)FT[/()]=£x"()e-m=[fx()e7s]*=X*(eR)/}=-0On=-0000(3)FT[x(-)]=ZN)*/5z?=-co£'xCöIj00FT[x(-)]=Ex(')eW=X(e*)M=-CC(4)FHx()*y()]=X(/3)Y(3)00÷øx()*y()=^x(m)y(n-m)l=TK>FT[x(n)*y(n)]=ᑍ[^x(m)y(n-m)^~jM,=-oo"]=ùootu=n-m,öij88FT[x(n)*y()]=ó[(í)ûüýþÿ "k=-(x>m=-co=£y(k)eTdYx(m)e-jM"k=—cc>,=-oo=X(ejro)Y(-3)00(5)FT[x(n)•y(n)]=^x(n)y(n)e~jiann=-ooV%(»)[—rY(ei37(6)X(e%)=o ᑮn=-m'E♦)]=jdX(e”")dco8(7)FT[x(2)]=Zx(2)e-n=-oon'=2n,ᑣFT[x(2n)]=Zx(')e-3"'/2'3Ꮤᦪ=J70()+(-l)"x()]eTn=-“=-00ᑭᵨ(5)⚪CDEx()=y()ᑣFT[x2(n)]=—X(e>)*X(e>)=—X(e”)X(e"3-"))d(y'22L(9)FT[/()]=Jx(-)e-jm"=-□0”3ᏔᦪE'=n/2,—oo<389.TUX(ew"L0,g<1691< X(eJm)ḄᏝ\]^_ᣚx(n)oab4)=dᔆ/3%0=^^2J-v)mi10.gឋij_klḄ⚣᳛opH(ef3)T”(/3)IejeM,Dᓫtuᑡopw()xuᑡyz{x()=Acos(g+§)Ḅopy(n)=x()*/?()=AIH(eJa>0)Icos[3912.⚪¾12ᡠÀḄuᑡÁ)ḄᏝ\]_ᣚᵨX(e"")⊤ÀjÃÄ Xi))fÅᡂÇᑡÈÉ(1)X(e°)7(2)£(3)X(e")b(4)Ë¡ÌÍᏝ\]_ᣚxÎRe[X(/3)]ḄiÐuᑡÑ(")7(5)IX(e"0)Fd7(6)ÒRdCO7ab(1)X(e”)=Zx(")=6=Ó3(2)£X(ey,aW=x(0)-2^=4^oo7(3)X(e")=be*=Z(T)"M)=2n=-407(5)£lX(e>)l2J®=2^^lx(72)l2=28L“n=-3ã6ädXd)..4—:---=FrTr[f-jnx(n)]do),dX(ej(a)c7'rI-------Idco-27inx(n)I2=316ådcon=-313.y ÇuᑡḄᏝ\]_ᣚ(1)%()æç(Ó3)(2)x(n)=7S(+l)+b()+gb(-l)2(3)x(n)=anu(n),0M=-O0é.111(2)X(eja,)=ஹ2(*R=/°+1+*ᑓ=1+—(&+eT©)=l+coso2n=-222(3)X(e>)=£8”()”{=£(17-=•~~—3«=-oo«=o1-aeoc3(4)XK3)=Z[“X+3)—(-4)ïᑘ=Ye~jmn=-oon=-3£3"{—133+Xe-w=2*"'"'+2e""'n=0n=-3n=0=1/33_e-j4J/l-L40
41,7.7.7.,7Ó/,/j-G)fsin(—g)e2(e2—e2)_&iii«—i-j-(oj-a)-j-a).1ஹe2(e2-e2)sinz(-^)ᡈὅ4(n)=u(n+3)-w(n-4)=/?(n+3)7X(ej<0)=£/(+3ñ4n=-co61_c"জFn/?(n)]=£^=-——7n=o1-ecoi_0j70X4(i)=d'(+3)"R=^7"3M=-OC1-e.7.7.7.1.7.7.ஹ“oj-co-j-co-j-coJ-(D-j-a>Sin(-G)e2(e2-e2)e2(e22)ஹ2-----------------e1------------------=--------.1.1.1i,i.1i-J—Cl)J—CO-J—(I)-1—(0J—(I)-1—(0.z1\e2(e2—e2)e2(e2-e2)s”14.(1)x()xᏔuᑡ(2)x()xuᑡᑖôᑖ᪆ö ᙠøùᎷÇᐸx()ḄᏝ\]_ᣚḄឋ§abE8x(e"")=Zx()""Mn=-(1)X()xᏔ¦ᦪX(T)=£x()e-"Mn=-42X(e"")==Zx()[cos69+jsinco]n=—con=—ooᵫ¢x()Ꮤ¦ᦪx()sinG¦ᦪÿ coy^x(n)sin69=0n=-oo00X(e")=y^x(n)cosa)n=-xX(e”“)ᦪ ḄᏔᦪ!"#$x()Ꮤᦪ& '(ḄᏝ*+,ᣚX(eJ°)0Ꮤᦪ(2)x()ᦪ $☢4567 ᵫ9x(n):ᑡ X(-3)ᐹᨵᐳA'BឋD ᓽX(e 0)=X*(i)8COX(eJa>)=y^x(n)e~j03n=os&+jsinco}n=-oo/i=-coᵫ9X()ᦪ X()COSGᦪ N 8ZX()COSG=0n=-X(ejM)=j^x(n)sin6y“=—COX(e"")RSᦪ Ḅᦪ15.Wx(n)=&() XYx()ḄᐳZ'B:ᑡ[()\ᐳ]^'B:ᑡx(ri),`ᑖbᵨod⊤fgh%()=:য()+l4(n)) 4(n)=g(p()n⁐(n”))%()\%()Ḅrs⚪15gdᡠfᓝ4()wx()■142TMTTT° -2-10123-0.5
43⚪15gd16.Wx()=a"u(n),0)=1+COS0KY:ᑡ()ᐸᏝ*+,ᣚH(T)gh“R(eW)=l+coso=l+e""+e-w=FT0()]=£,()]‘n=-oon=-I2,heW=\1 n=0n=1.20,n<01,n=0h(n)-<1,n=1e2h() /?>00,ᐸ“eHd)=£/2()e-W'=l+e-j0)=2e-Ja,/2cos-n=-oo243
4418.:ᑡ/?(“):ᑡ (0)=1,ᐸᏝ*+,ᣚḄSH[জ)--sinY:ᑡh(n)ᐸᏝ*+,ᣚ"(e"")ghH=-sinco=-—(eja-e-j<0)2j¡0,()]=/"/(i)=—g(en=-11-5,n=-l4()=<0,=01,—,n=1I20,H<01,n=0h(n)=vh(n),n=01,n=12ho(H),>00,ᐸH93)=£%()e-"M=1+e~°=2e-j(0/2cos-?l=-cc219.W£¤Ḅᓫ¦:ᑡ§(5)= 5),0<«<1,¨ᐭ:ᑡx(n)=5(ª)+23(n-2)«ᡂ.ᑡᔜ⚪(1)Y7£¤¨7:ᑡy(n)=x(n)*h(n)(2)ᑖbY7x()ஹh(n)\y(n)ḄᏝ*+,ᣚg:(1)y(n)=h(n)*x(n)=anu(n)*[8(h)+28(n—2)]=anu(n)+2an~2u(n—2)X(eW)=fU()+2S(”—2)]e-w=l+2e72e(2)“=~cc1\-ae-ja,n=-4520.4³~´=2cos(2af),µKfOOHz #¸᪵⚣᳛£=400"z'x´¼½¸᪵ ¾ᑮ¸᪵ÀÁ&´\&ÂÃᦣÀÁx() X«ᡂ☢ᔜ⚪h(1)ᑏ7&(f)ḄᏝ*+,ᣚ⊤ÆX/jQ)(2)ᑏ7Ç´\x()Ḅ⊤Æ(3)ᑖbY7È«)ḄᏝ*+,ᣚ\x(")ḄᏝ*+,ᣚgh(1)X_(jQ)=£x“(t)e-jn,dt=£2cosine-fiVdt=£(/,'+ËÌ'kcdt$µᢣᦪᦪḄᏝ*+,ᣚÎÏᙠ ÑᐭÒᦪbᦪ ᐸᏝ*+,ᣚÔ⊤fᡂhX“(/Q)=2Õ3(Q—Q0)+5(Q+QO)]800ফ£“«)=ZzQWQ—")=£2cos(QnrW-/zT)0n=-oon=-oox(n)=2cos(QnT),noo<465.3é⚪êg1.ëì#í:ᑡḄNïDFT,ᙠ,ᣚñò04WN-1ᑁ :ᑡôõ(1)x()=5(ª)(2)x(n)=R(/i),0ö1(ej—N"-+eN)e-J—Nn=0kN)„=021ᱏ1ÿ-=_>eN+-yeAn=0=0/—-J—(/n+/r)Af1-eNi-Ne-----9---------------9------2(,"-*)-1-eN1-eNN,k=m,k=N-mT047N-\1N-[•▢Xয=$%_Nn=02j"=ol-e-Jea°N1-eN(5)N-lXয=£W"4=0,1,…,N-ln=07…N(N-l)%=0()X(k)=2^n=------------=o2kw0()x(A)=0+W(+2W^k++…+(N-l)wyT"▅X(%)=0+W/f++3Wj+CCC+(N-2)+/"+(N-1)N-lN-\X(,--X(R)=ZW"(N-1)=ZW"1-(N-1)=-N/I=1"=0ᡠ )-NX(Zr)=4*0ᓽN(N-l)k=0X(k)=<-Nk=1,2…N-l2.678ᑡXO;,;]N@e",k=m2N(1)X(k)=<@e•~jd,k-N-m20,ᐸBk47
48c?je",k=mN.(2)X(k)=<~Je,k=N-m0,ᐸBNᐸC)DEᦪ)0<?<@,NJᣚLMNO2PQ(1)x()=/OFT[X(A)]iN-\=Ux(W_ᵯW+Me-%"“=0,1,…%—]N[221j(^^mn+O')-j(&/w»+^)=—eN4-eN2\_=cos(-^mn+0)(2)x()="-^jej0W-")"|j(&m/:+^)-j(—ZWJ+^)=—eN—eN,“=0,1,…N—124J=sin(•Z[+0)N3.NON=10Ḅ_`ᨵ▲Ncᑡ1,049*X|(w)(a)0.50246810()I"0.5•0(-0.5-y()81046⚪3Pl4.DFTḄᳮ)ᓽᎷX(k)=DFT[x(n)]DFT[X(n)]=Nx(N-k)Qxয=>>=;aDFT[X()]=£X()WP”n=0N-\NT=0m=QNT/V-l49
50N,m=N—kn=00,mN-kf0(())“᪷n=0fi=0n=nIN,/=01…)r—1,'=01…)N—1,Ur-\N-l2—+w)»H(k)=£Ex(('+IN))NerN1=0'=01N-li-kii,¡lrr-l=»3)e[¢=X4)Z1=0n'=0r/=050
51rf8=ᦪr—1-Jߟ¤rEe/=00,K#ᦪrᡠ rX(-),4=ᦪr”য=\052=0N-\=Zx()ᔾk71=0@=ᦪ()OWkWrN—TN-i!L„r=Zx()w,sn=0=X&r8.x()N®Ḅᨵ▲Ncᑡ)X(k)=DFT[x(n)],Wx()Ḅ¯_®°M⊡²E`´®)µᑮ@`·®Ḅᨵ▲Ncᑡy5)/ஹx(-),=ir,i=0,l,…,N-ly()=jr0,ᐸBnº53rN-\N-ljN-\y-)=£y()W½=£x(-)C=Ex(i)Wk=O,lrrN-ln=0i=0ri=0ᡠ yয=x(য)NRÀয9.X(,=/7Á5)],Â()N7Ḅᨵ▲Ncᑡ)(1)x()=—x(N—l—))ᑣX(0)=0N(2)NᏔᦪ()x()=x(N—1—))ᑣX(Å)=0Q(1)N,N-lN-12N-1X(0)==>()=Z%()-£X(N-1-n)"=°"=o«=o2%L22•=Xx()_£x(/n)=o=0m=0Q(2)NN-lN_N-\22nX(7)=ZX(")%2=^x(n)(-l)"=2^x(2r)-^x(2r+l)2=0n=0r=0r=0%%92=£X(N-l-2r)-+1)r=0r=0%%22ÆÇÈÉ->=£x(2k+1)-£x(2r+1)=0%=0r=Q10._`ᨵ▲Ncᑡx()qy()Ḅ´LMx(n)=0,<0ᡈ8ீy(n)=0,n<0ᡈ2054ºÏᙠÑÒ®Æf(n)=x(n)*y(n))ÓÔPQÕ/1(n)=x(n)*y(n))Öf(n)=IDFT[F(k)]=x(n)©y(n)/5)NO27,/()NO20,×ὅḄᐵ/()=£Ù(+20[)ߟ/?20()ffl=—00ÚᨵᙠÆᕜÜÝÞcᑡCßàáḄ®Æ)âãäf(n)=Ù()ᡠ f(n)=f()=x(n)*y(n),755N-lX|(A)=Zx()W'=0,1,…N_]M=02N-1û2=,=£{=;+22Z=0,lü-2N—l=0m-2n,ᑣM-2NM-lm*2;=ZX=3;WJm=0L,u_ikin=E?@ABᳮᵫEF⚪13EHᡠIḄJ!"#N=5Ḅ%&'ᑡKL,ឋᔁ./L=82Ḅ34ᔁ.ḄIOHPQRᵫS34ᔁ.Ḅ2ᦪL=8U,ឋᔁ.Ḅ"V2AM=9Y12Z[ᙠ]^#0_34ᔁ.ᨵabcdeᙠ]^#1f34ᔁ.Ḅabcdgᜫᡠ]^#1i7Ḅ34ᔁ.Ḅ<=,ឋᔁ.Ḅ<>?*য-5-3-101234567854
56⚪13EH14.no'ᑡx()=r“()0<«<1,ux()ḄZvᣚXQ)ᙠᓫ^ᙊyz{◞}᪵N2}᪵<#X(A)=x(z)Lwk=OL-,N-lᨵ▲"'ᑡ/OET[X(k)]EFᵫSX(e*")=X(z)L/J2#ᕜḄᕜᦪᡠX(/))N=X(z)l^defX(k)z=e'N=N#ᕜ,X(k)Q+ᕜ'ᑡ()ḄDFSᦪᑣ1N-\i1N-l*ᱛQᐭᦻR=X(z)lJNkE=!>()▀"Z=e=W^«=-<»"="001N-loo1N-\=Z'hNnic=-/n=-oo/V&=oᵫS1N-1m=n+IN1,0,ᐸᡠ8x(n)=+IN)/=-ooᵫ⚪OoX(k)=X(k)R(k)Nᡠ᪷ᨵᐵX(k)=XN()Ḅᕜ'ᑡḄDFSᦪḄᐵᨵ(n)=IDFT[X(k)]=x(n)R(n)N8=£X(+IN)RN()/=-0000=2"+%,(+£¤()1=-ᵫS04WN-l,ᡠ55
571,n+IN>Q——>/>0u(n+/TV)=<0,/<0Z[ᡭ()=r£ª£-ᩆ(")==&()/=oI-a15.no®'ᑡx()=%()+#,.(«)«ᐸ9x,()/±()J²'ᑡ'ᑡx()ḄZvᣚX(z)ᙠᓫ^ᙊḄµ¶·(co27T)Ḅ<#¸X()Ḅ¹ᦣ»¼½vᣚX(k)¶Ḅ<@ABᳮᵫEF¾N#ᏔᦪX(k)Ḅf¶Jᢣ=,TN—1ᡠuÃḄX(k)Ḅ<ᵫSX(A)=X(z)lᐸ9½Å'=JR,=0,1…,N—1,ᡠÊuÃSX(z)ᙠᓫ^ᙊµz=%,¶·z{◞2yḄË᪵<Z[X(ÌḄf¶Ḅ<ᐰ#¸16.ᵨÎᜐᳮu²ᦪ'ᑡ+Ðᑖ᪆⌕ÐᑖÓ᳛/450HzÖ×ᨬÙ⚣᳛#IkHZ,ÛÜݵᔜßᦪF(1)ᨬàáâ_{/min(2)ᨬᜧ}᪵{◞“ᯠ(3)ᨬY}᪵2ᦪN,Rn(4)ᙠ⚣èéVêvḄëìµ⚣᳛ᑖÓ᳛íÙîḄN<EF(1)noï=50Wz,T=—=—=0.02snmin(4)⚣èéVêvñOᕡḼ}᪵{◞TêvÃôõáâ_{ᜧî#0.04s²c⚣᳛ᑖÓ᳛íÙî(Fv#øᩭḄ1/2)N="ú=80min0.5zs17.ᵨÎᜐᳮu²ᦪ'ᑡ+Ðᑖ᪆⌕ÐᑖÓ᳛/W5HzÖ×ᨬÙ⚣᳛#X=1.25kHzfÛÜݵᔜßᦪF(1)ᨬàáâ_{Tpmin56
58(2)ᨬᜧ}᪵{◞7”(3)ᨬY}᪵2ᦪ"⊈(4)ᙠ⚣èéVêvḄëìµ⚣᳛ᑖÓ᳛íÙîḄN<Eপ=0.2.VF5Z[à=0.2sn(2)Z#⌕122ÿ1_1=0.4x10%ᡠ*axߟ2x1250ব=5F5(4)ᙠ⚣Ḅ !⚣᳛ᑖ$᳛%&'(ᓽ*=2.5[2x1250=10002.518../ᑭᵨA(/i)56N=50ḄFIR;<ᘤ>'?@5ḄᦪBCᑡEF;<ᜐᳮ⌕JKᵨLMNᶇPQRDFTᩭVWᡠXLMNᶇPYZ>[ᐭCᑡEFᑖ?(]⚪_`?56M=100bK᪵d)efgh?ijLMVbdᯠmnoᔜ?qh(n)ḄLds⚪tL=128)uvᔁxyᑮ[{Cᑡ|⊤~??no[{ᨬm.()t{8b`?t{ḄBbK᪵dyᑮ;<[{)(1)JV(2)J8(3)t{ḄBbK᪵6ḄK᪵d6uvᔁxḄ[{Cᑡ,“()ḄCᑡ᪗6012,…,127ᐜ/l()qᔜ?[ᐭḄ¡ឋᔁx£,()ὃ⇋¦,“()dᑮ48d(ᐳ49bd)©ª«6;<[{49dᑮ99d(ᐳ51bd)6©Ḅ;<[{Cᑡy()Ḅ'?ᓽB=51ᡠ6◀¯☢49b©dt{51b©Ḅd±yᑮ²³´µ¶·dḄy()fijLM100-51=49bdᓽV=49«F☢º»>128dḄuvᔁxு%()½¾¿ÀZ©ḄᡃÂÃ⍝57
59y,“()=£%,(+128Å){28()r=—ooÆ6y|5)56N+M-l=50+10(M=149ᡠ"=20Çij127ÈÉ”()=))"(")Ìᯠ49dᑮ99dÍὅÏfÐᡠᡠt{Ḅ51d649ᑮ99dḄÑ()oÒ½ᡠÓ¾ᡠy¾ÔV=49,8=51⌱tஹ“()49-99d«6;<[{19.¿QᵨnoØḄÙ6Úᙳ`ÑÜᦪÝP◤⌕5s,`ÑÜᦪ|P◤⌕ᵨᩭnoß1024dDFT,àáno◤⌕¶âã²ᵨFFTnoᕖå᯿ç᪵noᵨFFTEFèÙᔁx>éEFᜐᳮãêoëVWVãᜐᳮḄéᨬ&⚣᳛ÌN=1024=2ᙢãáno/TḄÜᦪíoÑᦪ6N2=10242ÑÜᦪ|PnoÑᦪ6N(N-1)=1024x1023=1047552Ñánoᡠᵨnoã²T6DT=5x10{*10242+1047552xlO-6=6.290432sDᵨFFTno1024dOFTᡠ◤noã²6T=5xI.x^logzN+NlogzNx10-6F=5X10-6X^^X10+1024X10X10-62=35.84/sèÙᔁxã⌕no'ÑNd"7(ὃ⇋ᑮ"(î=/7 ¦()ïðnoñòᐭᑁò)'ÑNd/FF7ôNÑ⚣᳛ÜᦪÝPᡠno1024dèÙᔁxḄnoã²õ6T,=27+1024ÑÜᦪÝnoã²=71680M+5X1024M=76800M1074ᡠ`ö÷ᜐᳮḄK᪵dᦪ(ᓽK᪵⚣᳛)f',=13333.3Ñ/öᵫK᪵s'76800xlO-6ᳮÃëVãᜐᳮḄéᨬ&⚣᳛658
60f133333f61᪷%/T-FFTḄtuv⌕9\Ḅ()./()ḄN1£xy,{|}~ᓫḄ≈=\ᑮx(")Ḅ2N1yG:/()./()ᙳ:23ᑡᡠJ᪷%ḄᐳᵐOPឋSᵨAN1"9\X1(Z).2(KᐹF'y()=X](")+"2()Y(k)=DFT[y(n)],=0,1…,N—1ᑣXg)=DFT[x()]=%,(A)=Lয+L*(N-A)]{jXk)=DFT[jx(n)]=Y0*)=L^-Y\N-k)]2[Y2N1DFT[X(n)]=X(KSᵫX1().X2)\ᑮX()=Xτ(A)+X2)k=0,1,…,N-1X(k+N)=X{k)—W5NXfk)᪵}AN1/FFTCᡂ2N1DFT.ᯠ⌕=XOḄ,ᵫL)9X|(A),2).X(4Ḅ=(2).(1)¡@%1(n)=x(2n),=*2()=x(2+1),n--1X^k)DFT[x^n)],k=0,LஹN—lX(k)=DFT[x(n)],k=0,l--,N-l22ᑣ¢£¤ᐵ¥¦X(A)=X|(k)+W2)X2(k),=0,1,…,N—l,X(A+N)=X|(Z)—WUNX2(A)ᵫe¦SE§Xa=#®+XgN)]X2(¨)=Jx(k)—X(A+N)]W^ᵫJeᑖ᪆S\ᑮ=}ªFজᵫX(Z)§X1(K.X2)60
62X"*&)+X(…)]X(k)=[X(k)-X(k+N)]W^2ঝᵫX](X.X2(±᪀ᡂN1⚣n3ᑡL/)Y(k)=XSk)+jX(k)=Y(k)+Y(k)2epopᐸ¶·,(k)=X|§)=º»)N1IFFT\ᑮy(n)=IFFT[Y(k)]=Re[y()]+jIm-n=0,1,…,N-1ᵫDFTḄᐳÅOPឋ`Re[y()]=Æ[>()+y*(«)]=DFT[Y(k)]^x,()epjlm[y(n)]=^-[y(n)-y*(n)]=DFT[Y(k')]=jx(n)l>p2ঞᵫx(n).x(n)ᔠᡂx(n)12Xi(T)=Ꮤᦪx(rT)-<,0
