数字信号处理（姚天任江太辉第三版）课后习题答案

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2.1ᑨᑡᑡᔲᕜᑡ஺Ḅᨬᕜ஺(1)x(n)=Acos(—zz+—)86n(2)x(n)=e(--7r)83+71(3)x(n)=Asin(——nH—)43/(1)0᯿234ᑡḄ5678x(n)=Acos(•+e),;<஺=
஺=>?=@ᨵᳮᦪᡠE8co5ᕜᑡ஺ᨬᕜFGN=g᝕=16(஽L5)஺1271(2)0᯿NᢣᦪᑡḄ5678x(n)=exp[b+0y]n,;——
16%XᳮᦪᡠEY8CDᕜᑡ஺37rJr7T37rTC(3)0᯿234ᑡḄ5678x(n)=Acos(\2+°),^x(n)=Asin(——஻+—)=Acos(-------n----)“43243=Acos(—n--),;<ᗂ=b஺=>c=?ᨵᳮᦪᡠEᕜᑡ஺ᨬᕜFG464co3QNf=8(gL3)2.2ᙠi2.2jx(n)kh(n)ᑖnoឋqrstuḄvᐭkᓫyL᪵{|஺}~ᑡḄx(n)kh(n)ḄoឋᔁE;ᑮtuḄv

1/ᑭᵨoឋᔁ78y(n)=ᑍx(k)h(஻-k)k=-(n-l)+8(n-2)y(n)=-2b(n)+53(nT)=5(n-3)1-a"+l(c)y(n)=k)Q'A5—k)£Q'Iu(n)k=-A:=-oo1-a|VII.2?)Ḅi.ᵪ1.22.3}~oឋoឋᔁ(1)y(n)=u(n)*u(n)(2)y(n)=Anu(n)*u(n)00/:(1)y(n)=k)k=-cooo=Z”(᝕)-k}~(n+1),nNOk=0ᓽy(n)=(n+l)u(n)(2)y(n)=£/“(¦஻(஻5A)A=-oo

2001-2,7+=u(k)u(n-k)=----,n^0k=01—/Lᓽy(n)=1^u(n)2.4iP2.4ᡠ¨ḄᓫyL᪵{|ᑖn©h1(n)kJi?(n)Ḅ«oឋqrstuḄ¬ὶ®¯x(n)=u(n),%(n)=3(n)-3(n-4),h(n)=a"u(n),|a|=u(n)-u(n-4)y(n)
0(n)*h(n)28=y^aku(k)[u(n-k)-u(n-k-4)]A:=-oo=,n23k=n-32.5®¯5oឋqrstuḄᓫyL᪵{|©h(n)=a-஻u(-n),0

32.6»¼½oឋᔁ¾¿Àᣚ᳛ஹÃᔠ᳛kᑖÅ᳛஺¼½(1)ÀᣚÆX(n)*y(n)=^x(k)y(n-k)k=-coÈk=n-t,ᡠEt=n-k,^-8oឋᔁ78sᡂx(n)*y(n)=x(n-1)y[n-(n-1)]/=—008=x(n-=y(n)*x(n)/=—coÀᣚÆ;¼.(2)ÃᔠÆ[x(n)*y(n)]*z(n)=[Z'-—g)]*z(n)k=-r=-oo00Zx(k)Zy(m)z(n-k-m)᝕=T»inxZx(k)[y(n-k)*z(n-k)]k=-

42.7ᑨᑡtuᔲ©oឋtuஹqoឋtuஹÏtuஹ=Ðtu஺E¼½x(n)sinÑ(l)y(n)=2x(n)+3(2)y(n)=(3)y(n)=(4)y(n)=k=-

5=|x(n)I|sin[—(n-k)+—]|36"i=-oo=T[x(n-t)]ᡠEØtuqrstu஺Òx(n)=M<8y(n)=ZM=8,ᡠEØtuYÏtu஺2=-00=y(n)ÛLÜGÝᙠkÞßḄvᐭx(n),YLÜGàᩭḄvᐭᦑØtu=Ðtu஺(4)Òyi(n)=*y2(n)=2ᐗ(ᦇ)ᵫGk=nok=noy(n)=T[axi(n)+bx(n)]=^[axi(/:)+bx2(^)]2k=no=aEx,^)+bE%2(k)=ayi(n)+by(n)2k=nok=noᦑØtuoឋtu஺=y(n-k)==k=nom=no+tWT[x(n-t)]=Zx(m-ì)k="oᡠEØturstu஺Òx(n)=M,ᑣlimy(n)=lim(n-n(,)M=oo,ᡠEØtuYÏtu஺í×᧕ïn2n0஺ᑣØtu=Ðtuðñò஺ᑣØ=Ðtuq=Ðtu஺

6(5)Òy](n)=x](n)g(n),y(n)=x(n)g(n),ᵫG22y(n)=T[aXj(n)+bx(n)]=(axj(n)+bx(n))g(n)22=ax,(n)g(n)+b(n)=ay)(n)+by(n)22ᦑtuoឋtu஺=y(n-k)=x(n-k)×T[x(n-k)]=x(n-k)g(n)Wy(n-k)ᡠEturstu஺Ò|x(n)|WM<8,ᑣᨵ!y(n)|=|x(n)g(n)|=M|g(n)IᡠEóg(n)ᨵ▲õØtuÏtu஺=y(n)ÛLÜGÝᙠkÞßḄvᐭx(n),YLÜGöᩭḄvᐭᦑØtu=Ðtu஺2.8÷øᑡᔜoឋqrstuḄ=ÐឋkÏឋ(1)h(n)=2wu(-n)(4)h(n)=(—)wu(n)2(2)h(n)=-aHu(-n-l)(5)h(n)=—u(n)n(3)h(n)=0(6)h(n)=2஻R஻u(n)00/(1)=©ᙠn<0õh(n)=2஻NO,ᦑØtuY=Ðtu஺0000=©S=2lh(n)|=£2"ᦑØtuÏtu஺«=-<»”=0(2)=©ᙠn<0õh(n)WO,ᦑØtuY=Ðtu஺00-]00=©S=£|h(n)|=Z|a"|=£a'",ᦑØtuÛᨵᙠIa|>1õýÏtu஺n=-oo"=-oo/i=oo(3)=©ᙠn<0õh(n)WO,ᦑØtuY=Ðtu஺oo00=©S=Zlh(n)l=Zl^(n+n)|=l

7(6)ᙠnீ0h(n)=O,ᦑ஺00N-1S=Z|h(n)|=Z12"|=2"-1<8,ᦑ஺n=-co”=02.9+,y(n)-2cos/7y(n-l)+y(n-2)=0,3y(0)=0,y(l)=l,456য8“஺:sin°5=⚪?@ABᑖDEḄᱯHDEa2-2cos[3•a+1=0ᵫᱯHDE4JᱯH᪷a!=cosp+jsin/3-ej/i,a2=cos/?-jsin/?=@ABᑖDEḄOPy(n)=cj"+c"=ce*"+ce-▬22t2TᐭVWᩩYJy(0)=C|+c=02y(l)=c]eipn+ce~ipn=12ᵫ]^_Jᑮ11=8_1C'~ejfi"-e-jfin~2sin/3',-sin.bC1cC2TᐭOPd_ᨬfJᑮy(n)=C|e""+c,e-W=hi(ej+e3)=ᒹm122sin/7sin/72.10+,y(n)+2ay(n-l)+0(n-2)=0,3y(0)=0,y(1)=3,y(2)=6,y(3)=36,4y(n)P✌ᐜqVWᩩY4rDEsJᦪacb[y(2)+2qyপ+y(0)=6+6Q=0j[y(3)+2ay(2)+by(1)=36+12஻+3ᓃ=0{4ra=-l,b=-8|}DEy(n)-2y(n-l)-iy(n-2)=0ᵫᱯHDE஺2-2a-8=04JᱯH᪷a=4,a=-212

8@ABᑖDEJOPy(n)=c]a]"+c2a2"=c14"+c2(-2")TᐭVWᩩYJy(n)=Cja)+ca=4a=3222ᵫ]8_Jᑮ11'222bcτcc2TᐭOPd_ᨬfJᑮy(n)=c,«,"+ca"=1[4"-(-2)"]222.11ᵨᱯH᪷c⌴4PᑡBᑖDEy(n)-y(n-l)-y(n-2)=0,3y(0)=1,yপ=1PᵫᱯHDEa2—஺-1=04JᱯH᪷1+V5_1-5/5a.=------*a122=C,(ᓃ"+஺2(ᓃ"OPy(n)=c]a]"+c2a2''222TᐭVWᩩYJc,+c—12q()+஺2(-)=1J-V5qi+4rC]=_2ᩈ-C2:2ᨬfJᑮOPy(n)=c,(^^)n-C2(ᓃ)"2>/5-2V51/+"+I/1-

9+l1
F)2.12iḄ᪾᝞P2.12ᡠ4Ḅᓫ¡¢᪵¤¥h(n)cᓫ¡▤§¤¥

10BPq{,y(n)=x(n)+஻y(nT)4ᓫ¡¢᪵¤¥ªx(n)=S(n),|ᨵh(n)=5(n)+/Jh(n-l)q¬Jᑮ᜛’u(n)▤§¤¥y(n)=h(n)*u(n)=±᜛"6(k)u(n-k)k=0=£(n)I-2.13´µᑡx(n)ḄᏝ·¸¹ᣚX(e>),4ᑡᔜµᑡḄᏝ·¸¹ᣚP(1)F[ax](n)+bx(n)]=aX,(eiw)+bX(e¿)22(2)F[x(n-k)]=e->*X(e>)(3)F[eM"x(n)]=X[e"f°)]ভF[x(-n)]=X(ef)মF[x*(n)]=X*(ef)যF[x*(-n)]=X*(e>)(7)

11(8)jIm[x(n)]=L[X(e”')-X*(e-)]2X(e")*X(e"')2ᐔ/ஹdx(ejw)(10)j—~-dw2.14Ḅឋᵫ!ᑡ#ᑖ%&'(y(n)--y(n-l)=x(n)+—x(n-1)22(1))*Ḅᓫ,-᪵/0h(n)(2)ᵨ(1)2ᑮḄ4)5ᐭ7x(n)=e8"9Ḅ/0(3))Ḅ⚣᳛/07F7T(4))<5ᐭx(n)=cos(>n+>)Ḅ/024P(1)ªX(n)=6(n),Jᑮh(n)-h(n-l)/2=8(n)+8(n-l)/2ᵫ|ḄÆឋÈɹᦑᵫ]_Jrh(n)=h(n-l)/2+6(n)+8(n-l)/2,n0⌴ÊËrh(-l)=0h(0)=h(-l)/2+8(0)=1h(l)=h(0)/2+l/2=lh(2)=h(l)/2=l/2h(3)=1h(2)=(l)2h(4)=1h(2)=(1)3

12h(n)=6(n)+(-)n-'u(n-l)ᡈh(n)=(1)n[u(n)-u(n-l)]Í{bBᑖDEᵨᓫ¡ÎÏËÐ⊤ᡂ(1-D)h(n)=(l+D)8()nᵫ¬Jᑮh(n)=[(l+^D)/(l-1D)]8()=[1+D+-1D2+(1)2D3+...+(1/-'D3+...]8n=6(n)+6(n-l)+-8(n-2)+-5(n-3)+...+(Ö×$(n-l)+•••22=8(n)+(1)nu(n-l)2)bX(஻)=ejwnTᐭy(n)=x(n)*y(஻)Jᑮ1+-Dy(”)=J"*[—b(n)]1--D21+-1D=—2—Jwn1e1——D2i/1V

13\+-e-jw^)=-2—H2(4)ᵫ(3){,-?@௃BiCr7—_1-J—.rr22argHe=arctan1+—e-arctan1=-2arctan—y(஻)=W(ejcosஹ+arg["(eJW)]ᦑ:7171-COS-2arctan242.15ÜiÆឋÈɹᵫᑡBᑖDEÝÞy(n)-ay(n-1)=x(n)-bx(n-1)ßàáᡂᐰOḄbã(b^a),ᡠäᐰOᢣᐸ⚣᳛¤¥Ḅé⚣᳛ᗂëᐵḄíᦪḄ஺Pªx(n)=B(n),ᑣh(n)=ah(n-l)=^(n)-b8(n-l)ᡈh(n)=ah(n-l)+®(n)-®(n-l),n^0ᵫ|ÆឋḄÈɹᦑï]_⌴ÊËJrh(-l)=0h(0)=lh(l)=ah(0)-b^(0)=a-b2h(2)=ah(l)=a'-ab32h(3)=ah(2)=a-a'b

14h(n)=ah(n-l)=a"-an%,n0h(n)=a"u(n)-anXbu(n-1)ᡈḄ⚣᳛ᱯឋ1133)=2=_81ஹ(11)—a**Tbu(n_lie-®NLane-m.bEjanTe-m1e+=l-ae-iw°l-ae-iw1-be-W=l-ae-jwðñḄᱯឋòDl-be-»0|2|H@3)|=óô(l-beWa-b*)(l-ae-j»)(l-aei«)=1--13^-3+q2|=l-ae-*w-aeiw+a2õ-43Ti3+ipIbl2^-^——---=l-ae-j°-ae_jw+lai211ö⌱¢2=øᡈ6=/,ᑣᨵ|H(e/")|2=|b|2,ᓽñû¤¥ü|⚣᳛¤¥ëᐵḄíᦪᦑᐰO஺2.16(1)ឋḄᓫ,DE/07h(n)=a"u(n),ᐸGa7Hᦪ?J0),Y(e>),TabY(e>)=H(e>)X(e>)

15P(l)y(n)=f஻(k)x(஻iA)8=£a?u(k)0%(n-k)1Z=-00\-apA07Q)6T\-ap-x—a/T'BnNO46")ᑣ)y(n)=(1᜛(2)X(e'")=£/772=w=0@A.H(e"")=£aZT°1n=0\-ae-j(0/a"___B6")஺~#Yd)?a-pP1a—᜛)a-஺l-ae~j6)a-pe-,o>1aBᵫ&B)a-(31-ae~j(0\-pe1=X(e>)H(e>

16X(eJwn)=Z00X*(eZ)=(Zx*(N)ejwn)*Xx*(")eJwnn=-oon=-oo1X“)X*(e)dw271ᑍ001x(m)e-jwn][£X*(n)e/u]dw271n=—ooITஹ1tZX(Z)ZX*(஻)eJW(n-m)dwm=-co2஻ᐸ^717C—27C,♦•஻mejw(n-m)dwjyv[n—nt)jw(n-in)--------=0,•…nwmnm130cX(e)Xjw)dwZx(n)x*(H)2bn=—ooDYd:QOZX(n)x*(/?)n=-co£X(஻)[ijw)e-jwndw]n--oo2n-n1XX(")[X(ej'')eg஻dw]1-Kn=-oo2711X*(e>H)Zx(஻)e-jwndw2711-nn=—oo1ᑍX(ejw)X*(ejw)dw2n-712.18i◤⌕lm▲opq

177Tªᳮ£¤¥rsᘤḄ⚣᳛(«¬/)ᢚᔠᡂᦪz°⚣᳛±²(«¬)³´ᦪzrsᘤh(n)ḄTTTIT⚣᳛(«¬)⌕ᜧᦑ᦮:Ḅ⚣᳛ᵫᦪzrsᘤh(n)Ḅ⚣᳛(«¬)ᩭ·T஺¸88ᐸᣚ©ᡂ¹▭⚣᳛ᓽ¸£==10000Hzmᐭ2A=?,¾-ᑮZ=625HzTTTTᳮ£¤¥rsᘤḄ⚣᳛À(«¬/)ᣚ©ᡂ¹▭⚣᳛Á-ᑮ¡ᓽᵫᐵ=2²f,-ᑮc,110000…„f.=——=-----=500HzMM2T22.19Åᑡ;ᑡḄZBᣚ4ᦈÈ°(1)8(n—m)ফ(Ë“஻(”)(3)a஻u(-nT)(4)(1)"[H(n)-M(n-10)](5)cos(a)n)u(n)[)ªF(1)X(z)=£6(/-m)zn=znm1im>0tx(n)LÒ;ᑡᦈÈ°±0<|zlW8,ÕÖ×᩽×±0(m▤)Úim<0t'x(n)L⌮Ò;ᑡᦈÈ°±0WIzIW8,Ö×±0(m▤)Õ᩽×Úim=0,X(z)=l,ᦈÈ°±OWIzIW8,ÝÕÖ×ÞÕ᩽×ফXফ=X(£fu(n)z”=f(')=—]—n=-oo\)஻=0\J1„-1X(n)Làá;ᑡ³ḄZBᣚḄᦈÈ°Lâã±RஹḄᙊḄ᜜çè°éêx(n+1)_1Rஹ=lim“TOOx(஻)2T(n)íLÒ;ᑡîᨵIzI=8,ᦑᦈÈ°±Iz|ð8஺Ö×±0,᩽×2

18X(n)íLÒ;ᑡîᨵIzI=8,ᦑᦈÈ°±ñ<|zI<0°oÖ×±0,᩽×±ò஺(3)228—00x(z)=ZDz"=Z("Z1)"n=oon=-\—008,=£(Jz)"=♦(7)"=À[=TTn--1n-\\—aZ1-QZX(n)Lôá;ᑡ³ḄZBᣚḄᦈÈ°Lâãõö+ḄᙊḄᑁçè°éê—஻11Rஹ+øimjLlim"("+ùLidn—>oox(-(n+1))…sax(஻)íL⌮Ò;ᑡîᨵlzl=0,ᦑᦈÈ°±OMzIMalÖ×±0,᩽×±ú(4)X(z)=£-j[u(n)-u(n-10)]znn="ooûüø-(2z)7°-SI2JZ-1—(2z)TX(n)Lᨵ▲ÿᑡḄZᣚᨵ⚗ᦑᦈ0Mo0(5)X(z)=Ecos0o஻)i/5)z=S----------------z/I=-00n=ooZ001001஻=oZ«=oF111__(----d----H-----d----)=21-eJW°zl-e~JW°z1--Icosw01,2z"cosw+z-20x(஻)./0ᑡḄZᣚḄᦈ.123-ḄᙊḄ᜜789:

19X(71+l)?COS[W(H+1)]0X(n)COS(ᔩಘ=1x(n)J.KLᑡMNᨵτZ1=00,ᦑᦈ14ZK8,(JCOSW,QjwAo/""o᩽eeo2.20XYᑡᑡḄZᣚᦈ᩽ᑖ[\(1)x(n)=alHl,0

2011—X(n)./0ᑡḄZᣚḄᦈ.12RḄᙊḄ᜜789:Xx(n+1)R_=limXX->00%(஻)I_τeii/e+x(n)J./0ᑡMNᨵ14=°0,ᦑᦈeீW8஺0,᩽e஺00X(z)=£Ar"COS(G0஻+*)"(஻)[-"71=-0000---------------------zߟ"H=0nApjv>oo0-j᜛=2,——£+———£(re…஺z2஻=02஻=oAe"]Ae-"]21-reMJ21-re~i(0°z~x-j(co-(O)./(6?-69)\-1.-i(pAren+rez)z+e~21-rz-1(eM+e-*஺)+r2z-2cos(p-rz~1cos(&)-(p)()=A1-2rzcosco+rznQߟX(n)./0ᑡḄZᣚḄᦈ.12R3,ḄᙊḄ᜜789:A„cos[%(஻+1)+0]..x(n+1)r+IRxlim—:-------=lim----------------------------஻-8x(n)A„cos(69+°)r0%()J.KLᑡMNᨵ|z|=8,ᦑᦈIH

21rcos(4—9)“u73no----------------᩽rereCOS^9001ooZ7—RX(z)X——u(n)Z~n(4)“=—oon!n=0n1-11-21-31-n=1+Z+-j-fZ+3yZ+..+rZ4"...1XRX(n)./0ᑡḄZᣚḄᦈ.12xḄᙊḄ᜜789:x(n+1)R-=limlim---=0Ayn—>oo)28/l+lX(n)J.KLᑡMNᨵZ=co,ᦑᦈ0

221(/_*%+(/(¤)e-jOo—஺)2j1-(ejw°+e~jWQ)z~x+z~2sin°+sin(%,°)zl-2coswz-1+7'ox(n)./0ᑡ,ḄZᣚᦈ.12R0ḄᙊḄ᜜7¥8,9:x(n+l)sin[%(஻+1)+0]=lim-lim——-——N----r——-=1…x…sin(wJ0M+x(஻)J.KLᑡ,ᜧᦑᦈ1¨¨8.0,ᔢ0),᩽sin஺cos%+jsinvvcosw-jsinw.0002.21ᵨ¬®¯XYᑡZᓄḄ⌮ᣚ(1)X(Z)=ߟ5—,|z|¨-14z-'22'1(2)X(Z)=-]2_,||>Z1,3--i,12(3)X(Z)=1~,aZ,|Z|>|a-'|Z~'-a´(1)µᵨᗩ·ᦪ¯஺ᵫᦈ¹º»X|(n).¼0ᑡ஺½KlimX1(z)=lᨵ▲¿ᡠNX|(n).⌮KLᑡ஺ᵨÁ◀¯ÃX1(z)ÄÅᡂÆÇ·ᦪᓽX](z)=È=2z-4z2+8z3-16z4+2U5+...+(-1Ê2"z"+…=£(-1Ë2"z"=S-(-2)3"n=ln=lᨬÍÎᑮ

23X](n)=-2(-2)*n=T,-2,-3......ᡈX](n)=-(-5)"஻(,஻-1)Ñ2Òµᵨ7ᑖᑖÓÄů஺ÃX2ÑzÒÄÅ▰7ᑖᑖÓ1I1--iX2(Zஹ)--~22—~1,24rrl+|z-l+^z-2(1+^-')+(1+^-')4824=41A21,1-il+-zl+-z24ᐸÖ1--zA-2=4A|,]1+-Z-141--Z-14-2=,311+-Z-I2ᵫᦈMº»X2ÑnÒÓ/0ᑡ஺½KlimX2ÑzÒ=l,ᡠNXzÑnÒJ.KLᑡ஺ᵨÁ◀¯ᑖ×ÃXT94+ÄÅᡂØ·ᦪᓽ1+?~77^1+j4-4[1---z"H—---Z3+…+(-)"Z~"+...124821+Ë81!?(-j,"n=0F£711TT+*2-q/+...+(¡)▼+..•]71+4Z001n=£—3(—w)z"n=0"ᵫÜ{ÓÎᑮ஽Ñ஻Ò=UÞßÒ

24(3)µᵨᶇᦪ»ᳮ¯஺âãäᑖḄåäæᦪ3(஻)z=-----i-------=--------n—z-az-açn>0ᵫè»ḄᦈMéåäæᦪᙠâãëᑁíᨵt᩽Kîax(n)=Re5[x(z)zrt"\33aᓃ=(a2-X)an-

25>0çn=0åäæᦪᙠâãëᑁᨵ{t᩽z=z=0,Kîa3=Res[X(z)zn-1,-]+Res[X(z)zn-',0]33a-iஹð],஻ÞZ=(1,஺Z)Zi-------ñ“-7ò-a"2=0=(\—a~)a—a=—a~

26=0çn<0KX3(Z*Iᙠâãë᜜᩽X3(z)z1ᙠZ=8ᜐᨵl-n22▤᩽ᡠNᨵô(஻)=0,n<0ᨬÍ´Î(Q2-1)L,஻>0x(n)=<=030,/?<0ᦑ.(♦)=(஺——l)a"'u(஻—1)—Qõ(஻)2.22XYᑡZᣚḄ⌮ᣚ1পX(z)=,K|z|<2(l-z-')(l-2z-1)z-5(2)X(z)=,0.5<|z|<2(l-0.5z-')(l-0.5z)(3)X(z)=eÈö÷(1—el-)(4)X(z)=Z(2"j),|a|

27(4)µᵨ7ᑖᑖÓ¯X4(z)1-4z-1+▲l-2z-14=-----p|=-1,A,þ=21l-2z-'1=11-z᪷üᦈlḄ?@
ABᑡ.ᐸD0.5F@
ᑡGᓽn<0IGx(n)=0KH>0I,./0ᦪᨵ1
᩽P0.5ᙠRSᑁ,ᦑ%5(஻)=Res[x(z)z"T,0.5](z-5)z”-6(1)",n>0(l-0.5z)z=0.5|z|<2>F@
⌮ᑡGᓽn20IGx(n)=0;n<0IG./0ᦪᙠRS᜜ᨵ1
᩽P2,aᑖbc⚗eḄ▤gᑖhc⚗eḄ▤i2—(n+1)=l-n>2,ᦑX5(஻)=-Res[X5(z)z"T,2](z-5*=-2"+,஻<0z-0.5n-2ᨬᑮ.6,),«>0m("-2,,=+l,n<0

28p஻)=-6qᡈrᵨᶇᦪ*ᳮ,G./0ᦪ᪷6ᦈ89lzl>c”Ḅᑡ?@
ᑡ஺ᓽn<0IGᙠxv஻w=0IGᙠ஻20I;./0ᦪᙠ/ᑖRSᑁᨵ1
2▤᩽Pz=c",xP(஻)=Rcs[x,(z)z"T,c-T]=%"z"=c~Tnz"~1\,-=nc~Tn,n>0Il=eTᨬᑮnc-T",n>0x--t0,n<0ᡈX/(஻)=nc"T"u(n)(7)ᵫᦈ89<=G>Ḅ?@
ABᑡ஺xv஻w|}~ᑖᑖeᑖGᓽX()_z(2z-a-2)_2-(஺+᝖-11(z-a)(z-b)(l-az"')(l-Z?z-1)A+ᵫ1-az-1ᐸDv-"7ব==1t=a2Ta+b)z\A=(1-^-')X(Z)|7t=bsl-az~'SA>F@J,ᦈ8ᩩ|ZI>a⊤>F@
Bᑡlim-J=1ᨵ▲,1-az/->o1-azᡠt@F@
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29ᵫxᑮ%(஻)=anu(n)>F1-bz”Gᦈ8ᩩIZIீb⊤>F@
§Bᑡ1—aI=0¨ᨵ▲Gᡠ]]t1@>F@
⌮ᑡ©2(஻)஺ᵨ◀,1@ª7ᡂZḄ«¢ᦪGᓽ—l—^^-h-'z-h^z2——ᶪ"-…=-"᝕"=-yb"z'nᵫxᑮx(n)-bnu(-u-l)2=ᨬᑮx(n)=a"u(n)-b"u(-u-1)722.23®X(Z)="+/,0<|z|ீoo,Ḅ⌮¯ᣚ2/±ᡂ²¢ᦪ_2nooi/=l+z+j…=>—z"2!n\µ஻¶0101=E3@1i+y——zn£(-஻)¶£j஻l!ᵫtº»eqX(z)n+i*-"+EGf=l+£◤zᨬx(n)=^(n)+---,-oo<«

302.24¾¿*X(z)=z*?ᔲÁ⊤Â
ᑡZ¯ᣚGÃÄᳮᵫÅÆG¨G᝞X(z)ÆÁ⊤Â
ᑡZ¯ᣚGᑣX(z)ÉÊᙠᦈ89ᑁ¾᪆0ᦪ஺Ì?GÍaa,ᙠx(z)=u(x,y)+jv(x,y)=z*=x—jy,Ðᯠᨵº=1W—=-1,ᓽX(z)ÅÓÔÕÖ@×dxdyضÙÚGxX(z)Å?᪆0ᦪGᦑX(z)ÅÆÁ⊤Â
ᑡZ¯ᣚ஺2.25᝞X(z)?x(n)Z¯ᣚGÛ(1)z-"'X(z)?x(n-m)ḄZ¯ᣚ(2)X(a"z)?a"x(n)ḄZ¯ᣚ(3)@z"D?nx(n)ḄZ¯ᣚn=-oon=-a>=Ý"Þ(஻)2-"=//(2)n=-ooaoao(2)£ßà(஻)"=£x(n)(a-'z)-n=X(a-'z)n=-a>n=-oo“aga(3)^/ix(n)z-n=-z—=-z—X(z)n=-ooUNn=-oo2.26Û(1)£x*(஻)z-"=[£=X"(z*)M=-oon=-oo86ফZx(f)z-"=Zx(஻XzT)f=X(zT)n=—oO஻=çoo(3)£Re[x(n)]z-Z,=èKé(஻)+©*(஻)]2@"=K[£x(n)""+fx*(஻)z-"]=K[X(z)+X*(z*)](4n=-cc>n=-ooFF”=-aon=-a>F)

31flm"=t#ফn=-cc“=-oon=-oon=-oo2.27XQ)=—^,ºI>11-Zy(z)=——r,lz\>a\-az1AA%ব=x(z»Q)=--------K--------=—4+—=êব+w,()rz(l-z-')(l-az-')1-z-'1-az-'ᐸDA------------1]=ë@}11-az'1\-a■1.1-«4=-~->,=«=-~K-i-zl-<2\-aᵫFx(n)y(n)ì?ᑡGᦑw(n)í?ᑡGᑡGîW(z)Ḅᦈ89¨lzl>l஺ï᪵GêবḄᦈ89¨lzl>l,îᔩবḄᦈ89¨lzl>a஺ïòᕡḼêবÉবì>FᑡGx<ᵨ◀,ᑖêQ)W2(z)ᡂzḄõ¢ᦪGᓽ]1àW](z)=-------(1+z"+z-2++)=-------Vz-”1—Cl\-Cln=_gW(z)=——(1+az-1+a2z~2+…+ö2-"+…)=—ߟV^n^~w21@஺\—aQn=ᵫºeᑮ1—a÷(“)=----it(n),a>(n)--------a"u(n')2I-a~1-aᨬᑮl-an+1Ct)(n)--ᓺ஻)+஺2(஻)=----------M(H)1-a2.29(1)¨úû?ḄGᡠtᦈ89¨lal

322ᑮ|"("3)|=/,ᓽ56Ḅ89ᱯឋ<ᦪᡠ?@56A•CᐰE56஺2.30(1)᪷I᩽2ᑮx(n)ḄZLᣚX(Z)=-7-^------(z--)(z-2)(z-3)0ᏝTULᣚᦈᡠ?ᓫᙊᙠᦈᑁ0Xᦈ0Z3(z-2)(z-3)3uvwᦪᨵ2C᩽|=2|2=3ᙠvᑖ}~᜜0ᑖ⚗Ḅ▤ᑖ⚗Ḅ▤3-஻ு2(0n<0),ᦑx(஻)=—Res[X(z)z"-',zJ—Res[X(Z/,z]=௃ll2z=2z=3(z--)(z-3)(z--)(z-2)ᨬ2ᑮ=0.9x2"-0.5x3",n<0x(n)=0.9x2"-0.5x3",஻<0ᡈx(஻)=0.9(z)"஻(஻)+(0.9x2"-0.5x3")«(-»-1)sᦈ201C᩽Z=3ᙠvᑖ}~᜜0ᑖ⚗Ḅ▤ᑖ⚗Ḅ▤3-஻ு2(0n<0),ᦑ

33x(஻)=—Res[X(z)z-i,3]=T'+D¡_τ£=_5x3",஻<00(z-z)(z-2)'0.9(z)"-0.9x2”,஻20ᨬ2x(஻)=«-0.5x3",n<0x(n)0,9-r-r](n)-0.5x3»(-n-l)=[(MM2.31056¤pᡠ?ᓫᙊ¥¦ᙠᦈᑁ஺ᵫ56Ḅ᩽7=¨ᡠ?ᦈlzl=©஺022Iim"(z)=8,ᦑ@56ªA0«56஺2.32(1)¬஻)=/⚚஻-1)+ᑍ(஻),y(஻)=69(஻)+¥஻-1)W(z)=//¯&)+X(z),W(z)=X(Z\1ABz1+7TY(z)=W(z)+-'W(z)=~^Xiz)Zi-pzᡠ?56wᦪ±ফl+z-'H(z)=⚣᳛´t.(/).0).(0CD,-i(o/2C0S—1(ஹ1+e)e2g2+e2).2HY=----------=---------------------=----------=-jcot—\-Be-J6)-j-j--j-@2e2(e2-e2)2smyফᵫ±஺=¶X(z)iᑏ¸56Ḅ¹ᑖº»y(஻)-fiy(n-1)=x(஻)+x(n-1)11+z-1ব½x(n)ᓫ▤¾^ᑡ¿ÀX(z)=——j■ÂᐭY(Z)=-JX(z)2ᑮ1-zpi1+z-11Y(z)l-z''mᵨÄᑖᑖr:""=”Æz(11)=4+ᓰ")+Ḅ

344__*±£LI_22~\-pz-'2=1B1ᵫX(z)=τஹ…lzl>஻2ᑮp1-pz%(஻)=•?//'஻(஻){-p21ᵫE(z)=-------------plzl>l2ᑮ2%(஻)=="஻(஻)1᜛0156Ḅᓫ▤¾´tB21-0"2l-B"1-/?,,+1/(஻)=%(஻)+/2(஻)=-z~—^'M(n)+-~~-M(/2)=---«(»)+-~~-«(»)=-~~—«(«)+—~~—M(rt)1—/yi—//i-//i-pi—pi—/?2.33(1)¹ᑖº»g]ḄzLᣚy(z)=/?(z)+¡2±(z)+/a)ÌÍ2ᑮ56wᦪz_,"ব1ÎÏ56wᦪḄ᩽-I777H(7ஹ=________=_______=____________-Z2-Z-1(zM)(Z-Ð)ᐸÒ0;᩽?=z(1+Ó)Ô=3(1-6)஺ᵫ1iÕ¸᩽᝞1.9ᡠ஺×Ø560«5601ᦈI¯I

35mᵨÄᑖᑖrÀ56wᦪᑖâZAτ%”ফ==MäZå+“2äZå(z-£I)(Z-/?2)5B\5BIᐸÒA,=^-\=-FI—A=♦I=^&-B\I1ᵫ"1äZåÛÜᓫÝ᪵´t"ä஻å஺0ᦈlzl<£æᦑ"ä஻åç]^ᑡP\-Plz-40Ìᓡ஻1äZ=0ᨵ▲êᦑë2ä஻åìA⌮0«^ᑡ஺mᵨᶇᦪpᳮruvwᦪB'z"T,½n<0¿᩽î=zä1+ªåᙠvᑖ}~᜜ïuvwᦪḄᑖ£P\~Piz—B\ᑖ⚗▤ᦪð¹1-஻+122ä0n<0å,01ᨵ%(஻)=—Res[஻1(z)zi,£J=~P\1ᘤ஻<0A-AA~P\ᵫ“2äóå=5----LÛÜᓫÝ᪵´tô5å஺01ᦈlzl<£,,ᦑë,ä஻åõ]^á!I,£2—£1Z—ö0lim஻2äzå=/4•ᨵ▲êᦑôä“åìA0«^ᑡ஺mᵨᶇᦪpᳮruvwᦪ஺-B^B\”,äzåz"T=S---,½“20¿vᑖ}~ᑁᨵ÷Ḅ᩽J3,=LQ_#`,01ᨵaH2iø஻)=Res¸,(z)z"T,R=z"Tù஻20*\-cPi-P\ᨬ2ᑮúû⚪ý¹ᑖº»ḄC¤pþÿ
ḄḄᓫ᪵/?(஻)=%(஻)+%,(஻)=-1-(M'u(-஻-i)+£'”(஻))Pi-P\2.341ᑖ"#$%ḄZ'ᣚz-'yz--rz+zrz=xz2ᵫ012ᑮ4ᦪ7ফ1Z"ফ=XQ)z-1_|+z(z-2)(z-1)

364ᦪḄ<=2=0>᩽=@=2,p.=1oᓫ᪵Ḅ3ABC⌱E"ᫀ᝞H(IὃK1.10ᡠMḄ᩽N<=K)஺(1)ᦈQR22(n<0),ᨵ12(஻)=——(N+d)஻(N஻-1)=Nz(-2஻+2஻P\~Pi3(4)NA"ᫀsᑖ"#☢ᵫᑖ"#24ᦪ”(z)=———ᦑ⌕-I)zF+Z2NA"ᫀḄ4ᦪᓽB஺(1)e?7x71IH(z)=XQ"N2N")»(n)z-=-Zr(2z-T-(2-'z-')n]=-(ߟߟp-=r)“=-00JJW=-«O31-Zz1-2z_21zஹ_]ফz=31-2Z-1-z-2-'=_>5z--+z"ফ=NT(2"஻(f-1)-2%(஻))"=-'(2)"-(2-௃Nn=-ooDJn=-con=0ஹ“1τ2,,2T)z,1ஹ1—351-27-1-3l-2z-'1-2Z-1_-i_5_z+2(3)

37HY=£N32"-2"(—஻—-"£(2/)"—£dn=—ooJn=—oon=—ooZ(2zT)"-Z(2z)"]=(¡2z)=13«=1“=131-'Zl-2zz¤|+z2.35Z”ফN¦7ফ+§ফ=X(z)1z")=¨=t10,1ஹz-1+z(z-3)("?᩽=3,-oVWᓫᙊᙠᦈQRᑁᓽ¯§¾À,(஻)É%^ᑡ110”,&)=Ë1•ᨵ▲bᦑ"(஻)S
^ᑡ஺fᵨ13ZfO8NÍJ?”11ᶇᦪWᳮjkl4ᦪ஻2(Z)z"T=——m“NOnlᑖuvᑁᨵÐNḄ᩽=0Z~31_i1Ñ஻)=ResÓব*9=U"z¤Ö×஻஺ᨬÙ2ᑮ/?(஻)=4(஻)+62(஻)=NËz3஻஻(N஻N1)-1+3-"஻(஻)Í82.36প᪷Ýᑖ"#BÞßḄ᪾K᝞K1.11ᡠM஺(2)ᑖ"#$%ḄZ'ᣚY(z)-2rcos6>z~'Y(z)+r2z~2Y(z)=X(z)ᵫ012ᑮ4ᦪ

38Hz=——=---------------——=-------------Xz\-2rcos6z~'+r2z-2z-/?!z-/?2ᐸ¹᩽=஺\=rejt>=rcos3+jsin0`,=re~j0=rcos0-jsin6x஻=a"uMḄZ'ᣚXz=ߟJ,B2ᑮI-azZ2z3rz=------------------—=------------------z-ভz-᜛21-z-a`S
ᦑᦈQRlzl>max[%4a]xᨵy஻=0஻<0஺±²"20,fᵨᶇᦪWᳮjYz⌮Z'ᣚkl4ᦪ?"+2Ylz`z"-'=------------------ᙠlᑖévᑁᨵ3ê᩽=Z\=m*Z=A>ì=஺஺ᨵ2z-4z-&z-a3y஻=ZRe$[yzz"T,zJ/=1_஻+2_஻+2_஻+2___________________τ_|___________________τ_|_____________________τ"z-Aa-«z=p'a-ভ…4N=q+2+☜+ñ+2᜛2-ভyN஺᜛2-1
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39÷øùᦣᏝüý'ᣚþᐸÿ
⚪ᫀὃ3.1P3.1ᡠḄᑡ஻)ᕜ4Ḅᕜឋᑡ஺ᐸᏝᦪḄ"ᦪ#$)஺N-lN-lTNT)nk%&ᐗ())=W>(஻)W+=W>(—஻)W0=E^n)W=X(-k)=X\k)Nn=0n=0n=03.2(1),-஻).ᕜᑡ/01£(஻)ḄᏝᦪᦻ(᝕)ᐳ678Ḅ/ᓽᐗ(k)=X*(-&)஺(2)01;<஻).Ꮤ>ᦪ?/@(᜜B.Ꮤ>ᦪ஺01&(1)

40N-\X(-k)=X^n)W-nkn=0N—lN-lX\-k)=[XD")w/G=2x(n)Wok=X(k)஻=0H=0(2)K<").>ᦪ/ᦑᵫ(1)NᨵX(k)=X*(-k)ᡈQ(ߟA)=X\k)TKU஻)Ꮤ>ᦪ,ᓽ-஻)=£(—஻)/ᡠ-ᨵN-lN—1U(N-1)ᦻ*)=[ᙢ)]"=£^-_`=X[ab"=ᦻ(-k)=X\k)n=0n=0n=03.3P3.3ᡠḄUf.ᦪᕜgh-஻)oᑭᵨDFSḄᱯឋp3.2⚪Ḅqr/stuv
ᐸᏝᦪḄ"ᦪᐗ(᝕)/-wxᔲz஺(1)X(k)=X(k+lO),7{ᡠᨵḄk&(2)X(k)=X(-k),7{ᡠᨵḄk}(3)ᐗ(0)=0}~jk—(4)X(k)e5,7ᡠᨵḄk.>ᦪ஺uP3.3%&(1)z஺K£(஻)UfᕜN=10Ḅᕜᑡ/ᦑT)BUfᕜN=10Ḅᕜᑡ஺(2)sz஺K£(஻)Uf.ᦪᕜᑡ/3.2Ḅ(1)N/T(6ᐳ78Ḅ/ᓽᨵ%(k)=X"(—k),T(᝕)sU.ᦪᑡ஺(3)z஺K£(஻)ᙠUfᕜᑁz᪵Ḅfᦪ᪵Ḅfᦪ/ᡠ-ᨵN-1X(0)=£x(n)=0”=0

41(4)sz஺᪷ᕜᑡḄឋ/T(J=T)7ᕜᑡ£(஻+2),᝞~jk—P3.3ᡠ/¡s.Ꮤᑡ஺ᵫ⚪3.2Ḅ(2)N⍝/X(k)e5s.Ꮤᑡ஺uP3.3_1003.4,x(஻)=&(஻)/x(n)=x(n+6r),¤᱄(᝕)/¦§⊤©(஻)ª@(ᦇ)஺r=-ooN-\5=£®=¯-7ᡂ1-(-1/%&)())=£W஻).ᐔ1.ᑍIn=0஻=0-j—k-i—kn=01U%1-e31-e3@(0)=1X(2)=X(4)=0X(1)=-----J—=l-jV3X(3)=——=11-eµX(5)=——2—^=\+j431-(1+JV3)-“)ª·)Ḅ¸᝞3.4_1ᡠ:-3-2-10123456789

42,X(k)R06.tx(k)t••♦,-ruP3.4_13.5ᙠP3.5⊤»¼fᕜᑡᵨ½஻¾ªᨴ5¾/¼ὅḄᕜÁ6,v
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wᑡᑡḄNÏDFT&(1)x(஻)=3(஻)(2)x(஻)=6\[n-Ñ)]w*RN(஻),0<%

45ÝÞ1uᡠ-wᡠஹT”MC=~XeN+eN2Nsz=oN)(\j_/2yk-m)y_^-jl7t(k+iri)I.24஻ஹ.2TT.ஹZ~2-/—(k-m)-i-(K+//i)U—eN1—e%J]/½1"¾----------eN+--------------------eN-2ᑍ71j-(k+m)-j-Ck-Hii)\eN-eNeN-eNJ,N+\ஹ1sin[(k-m)ÄUᱏ$,sin[(᝕+ḄÄne%~2sin[ᓄ-m)7i/N]sin[ᓄ+?7i1N]—,k=n^Hc=~in20,ᐸÑ3.7P3.7⊤ḄUfᨵ▲çᑡ஻¾/èé᳝½ë¾ªë2½஻¾Ḅ¸஺প᳝(஻)=ì஻-2)1&(஻)(2)x(n)=x[(2-n)]R(n)244.x(n)}—••—----------'—••------n0123uP3.7%&%½஻¾ª%½஻¾Ḅ¸᝞᝞.7ᡠ:x[(n-2)]R(n)x[(2-n)]R(n)4444n01230123uP3.7_1

463.8P3.8⊤Uf4Ïᑡx(஻)஺(1)îéx(஻)x(஻)ḄïឋᔁÃqrḄ¸஺(2)îéx(஻)x(஻)Ḅ4ÏðñᔁÃqrḄ¸஺(3)îéx(஻)x(஻)Ḅ8ÏðñᔁÃqrḄ¸/¦òqr(1)óô/õ1ïឋᔁÃðñᔁÃö÷Ḅᐵ"஺.x(n)2•91•,----------------------—•~•------n0123uP3.8%&(1)P3.81(1)ᡠḄx(஻)x(஻)ḄïឋᔁÃqrḄ¸஺(2)P3.8(2)ᡠḄx(஻)x(஻)Ḅ4ÏðñᔁÃqrḄ¸஺(3)P3.8(3)ᡠḄx(஻)x(஻)Ḅ8ÏðñᔁÃqrḄ¸஺Ç-Èé/x(஻)x(஻)Ḅ8ÏðñᔁÃqrḄ¸(1)x(w)x(஻)ḄïឋᔁÃqrx(n)টx(n)10n9"8.%T01234567(2)

47(3)uP3.8_13.9x(஻)UfçúNḄᑡ/û01x[(—஻)]N=x[(N—஻)],v஺01&KX[(—஻)],vᵫx(஻)ᕜឋýþÿᑮḄᕜᑡᦑ⊤x[(—஻)]"=X[(—஻+/W)]Nr=l,ᓽ—("஻)9=M(N-&஺3.10,-ᑡ.஻)=/'஻(஻),01஺11஺2ᙠ4ᐸZ7ᣚᙠᓫ:ᙊ<=N>ᑖ᪵AX(k)=X(z)L%"Eᨵ▲HᑡḄIDFT»NOᙠzP☢Ḅᓫ:ᙊḄNR>ST4z7ᣚ<=᪵UVWXYZ[ᑡḄᕜ\]\]ᕜN,ᓽᡠEᨵ▲HᑡḄIDFT8opnx(n)=Zx(n+rN)=Z+rN)=———=0,1,…N-1pr=—«r=—oo1-஺3.11aHNḄᨵ▲Hᑡx(஻)bc▣ᑡx(஻)=/?w(஻)஺(1)EZ[x(஻)],ghijᐸ"kTᑖlm஺(2)E⚣pX(eJ"),ghistIX(e"")lḄuᦪwx஺(3)Ex(஻)ḄDFTḄy⊤gzXIு஺)4᯿஺NOপooN-\1_-NফZ=",n=—oo«=0-ZN-\N-\N-\jdrna-w/)na-)…)=4'=^0______________________________________"ZNT(Z-1)—Z=(Z-1)"᩽T:z0=0(N—l▤)kTOz=e^,k=l,2,...,N-lpkmP3.111(1)b᩽-kTᑖlm஺

48.(N\.N.N-J-(DJ-CO-j—msin——co.NTe2(e2e2)(2J-J-------CD(2)x(i)=x(z)L/e2jJ-"7/j-(o.CDe22e2)sin"2,Nsin——co(2N—lzIX(-3)1==----CD.CDsin—2mP3.11(2)ᡠḄb⚣pstIX(஻&)IḄuᦪwx஺N—l]_U7Nk\ᡂবX(k)=ERg)Wz=={0,i=l,2…,N-\«=01"%]_/{a-Nk2X(k)>X(e"")ᙠNR>◞⚣᳛T®=—(Z:=0,1,2,...,N—l)Ḅ᪵A஺N3.12ᙠmP3.12hiᨵ▲Hᑡx(஻)hiᑡx[(-஻Ḅᶍm஺•x(n)*------«-----•------------------------•-----•-----•----•-►n-2-10123456uP3.12N:

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50NOmP3.151ᡠḄbᵨáR8TDFTᩭÀÁ"R16TDFTḄåæm஺XXXXX(((XXXXX3.16çx(஻)={O,l,O,l,l,l},24x(஻)<=⚣pᑖ᪆஺hiFFTḄåæmFFTÁÆë⌱஺gÀÁiíî≈ðñÁḄò¼஺NOmP3.16_1ᡠḄÒ[ó⌱8TFFTḄåæmÏíî≈ðñÁḄò¼஺x(0)0X(0)4x(4)1x(2)0Xফ1-jx(6)0X(3)(-l+l/ô-Tx(1)1X(4)-2x(5)1X(5)(-l+l/Y)x(3)1X(6)1+jx(7)0X(-1-1/3)+/஺uP3.163.17᪷öÝᦟᩞm3.27ᡠḄåæmẆøù2⚣᳛ú⌱FFTÁÆ஺çN2Ḅëû᦮ᦪýþÿ
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52টᵨÒ92FFTF^EFî(ñḄ16qIDFT,òêᑮx(")P"஻)ḄÔឋᔁ×y(஻)஺(2)3.20oZôÞᑡ(஻)P4(஻)ḄõᦣᏝHøùᣚᑖÌX1(Z)PX*k)஺Þᑡg(஻)g(஻)=(஻)+J/5)ᐸõᦣᏝHøùᣚG(Q஺þG°R(%),GER(k),G(k),G(6ᑖÌ⊤ÿ0lEIG(᜜ḄḄ᜻ᦪᑖᦪḄᏔᦪᑖᦪḄ᜻ᦪᑖᦪḄᏔᦪᑖ஺ᵨG°R(%),6(k),G(k),GH(k)ᩭ⊤X|(᜜*2(᝕)஺ol!GoR(k)=%[GR(k)-GR(-k)],G{k}=^-[G(Z:)+G(-Z:)]ERRsᦑGR(᝕)=G°R(k)+GER*)./ᨵG/(%)=GO//)+GE/(Z:)!12J4ᵨGoR(k),GER(k),Go](k),GE](k)⊤G(k)G(A)=GR(A)+▀(A)=[G°R(Z)+GERa)]+j[G஺,(9+G஺(A)]জ;<=☢ᵫ@g(஻)=%](஻)+J%25)ᦑᨵG-)=X1য+/2(G)ঝI!m(஻)X2(஻)KLMᑡᦑOযXgḄKLᏔPQMᑡKL᜻PQMRU!1TUজV᦮ᳮᡂZᑡ[VG(k)=2\](6+jG,(ᦇ)]+j[G(k)-jG°R(%)]ঞ0ElP᯿VঝVঞabcᑮX[(k)=G(k)+jG(k)ER0lX2(k)=GEH)-jG°R(k)3.21gឋi⚣ZkᣚmnḄ

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-”)=e~/h(n)(e(------n)-e2)஻=o2%=e2Z/2/i5)sin[⚚------n)]n=02%=e22^2/z(n)sin[d;(----n)]n=02Nᵨ^--஻ᣚn,2NH(e஻°)=e22£2h(——n)sm[a)(n—)]஻=i22!ᘤḄ⚣᳛&'⊤)*”(஻3)="(o)e./ᡠ1N—1TC᜛(0)=2জ+2N71H3)=>d(n)sin[co(n——)]92ᐸ;NNd(஻)=2஻(n),n=l,2,•••,224.7ABCD!ᘤḄE⌴Gᦪ*3s+2I(s)2s2+3S+1JᑖLᵨMN&'OPQRSTឋPᣚQVWXᣚᡂᦪZ!ᘤḄ[\GᦪH(z)]^T=0.5o

63`a1bMN&'OPQVa5bcdᡂeᑖᑖf“/ஹ3s+23s+2H(s)=-o------=------------"2?+35+1(25+1)(54-1)=J'2s+1s+1.3s+2=-----=11~2s+1s=-\kl”,(s)=m-+m-'2.V+15+1pfজq⌮stPᣚ]h(t)=(0.5e-°-5'+e-')u(t)auf;ਮt=nT,h(஻)=h(nT)=(0.5e~°-5nT+e-nT)u(n)apufqh(n)ḄZPᣚ]“(z)=£஻(஻)""=£(0.5e-5"+e-")zTn=-

6414+4zT—10z145-62k+2114.8^4⊤)CD!ᘤḄMN&'t>0»{Q't<0ᵨMN&'OPQVlCD!ᘤXᣚᡂᦪZ!ᘤ஺Tᦪ]T*]ᦪZ!ᘤḄ]l!ᘤ*!ᘤ¡¢!ᘤ஺`ᙠ⚪ḄMN&'⊤);]¤1=9,-0.9nTen>0h(஻)=h(nT)'n<0qh(n)ḄZPᣚ]ᦪZ!ᘤḄ[\Gᦪ"(z)=£h(n)z"=_J_9rn=-con=01©Zᵫ§[\GᦪḄ᩽©*2=6-"?ª«T*ឤᨵ®="091>1,ᓽ᩽©O°±ᙠᓫ³ᙊᑁ஺¶·]O¸¹TឋºOP[\ḄᐙᑖR¼⌕ᩩ¿஺ᡠ1ÀᦪZ!ᘤOḄ஺¤Z=*oᵫ[\Gᦪ!ᘤḄ⚣᳛ᱯឋ1H(eJt0)=——….kl]!ᘤḄÂÃ&'*kᙠ(0-Ä)ÅÆ]ÇḼ஺ḄÉÊ]VËÌ]ᦑÀ!ᘤ*!ᘤ஺4.9AB•CD[\ḄXºGᦪ*S+Q““(s)=(S+Q)2

65J᪷϶Ð[\q¸¹ËᑡÒᩩ¿ḄÓᦣ[\Ḅ[\GᦪH(z)஺(1)MNOPᩩ¿]Õ·//(஻)=/?(nT)a(2)▤×OPᩩ¿]Õ·s(஻)=s“(஻T)ᐸ;s(஻)=s(஻T)S”=(7)dra`পV஻஻(s)cdᡂeᑖᑖfs+a_A+————(s+a)2+b2s+(a+jb)s+(o-jb)s+aA==0.5s+(a+jb)s=-(a+jb)ᐸ;s+a4=0.5s+(a+jb)s--(a-jb)0.50.5ᡠ1”“G)=---------------------1---------------------s+(a+jb)s+(a—jb)q⌮stPᣚ]h=0.5[e-(a+;fc),+e-(a-jby,]u(t)a(l)uf;¤t=nT,%)=0.5[e-("+Ù'pufqZPᣚ][\Gᦪ"(z)=£Ú(஻)%-"n=011=0.5[Ûe-aTe-ibTz-'+\+e-arejbT\-e-aT(.cosbT)z~'l-2e~aT(cosbT)z~'+e~2aTz~2(2)CD!ᘤḄ▤×&'Ü(])ÜMN&'ᙳᨵ1Ëᐵ[▤×&'ḄstPᣚÜMN&'I஺)ḄstPᣚᓽEÞGᦪ”“(s)ßÆᨵ1Ëᐵ[

66C/ஹ1s+aklS“ম=/(s+a>+஻VS/s)cdᡂeᑖᑖf1S+QSS(S+஻)2+஺2_A),44——--------------1-------------ss+(஻+jh)5+(a-jb)ᐸ;s+aaA)(s+a>+b25=0a2+b2s+a-1-a+jb4=s[s+(a-jb)]s=—(a+jb)2(a+å-2(/+/)s+a-1-a-jbA,çs[s+(a+å]s=—(a-jb)2(஺-å2(1+/)ᡠ1aஹ1/-a+]஺ஹ1/A”jbஹ1~?7)—(s-----------H(;z-)----------cr+b~s2(a~+b~)s+(a+jb)2(a2+Z?2)s+(a-jb)qstPᣚ]ஹপ=(ê)+(ᜩìr…,+(írf'uf;¤l=nT,▤×&'5“(?)Ḅî᪵ðᑡs(஻)=s“(஻7)s(஻)=s"(஻T)a={(*^M…-a-jb+(-)e«-Ù"T}“(஻)2(/+/)pufqZPᣚ]▤×&'S"(f)Ḅî᪵ðᑡs(஻)ḄZPᣚS(z),

67S(z)=£s(஻)z-"/J=-00al1ò(—"᝞ra2+b2l--'a2+b-\-e-uTe-jhT-}1"—ôZZa___1___1a2+b21-z-1+a2+b2-[(-a+jb)(l-e-aTejbT-')+(-a——e-aTe-jbTz'')]Zl-e-ilT(e-jhT^ejhT)-l^e-2ilTz-2JZa11a2+b2l-z~]a2+Z?21[-2a+ae-aT(eihT+e-jbT)z~'-jbe-aT(eibT—"᝞õöl-e-aT(e-jbT+ejbT)z-'+e-2uTz-2a11cos(/?r)+bsin(hT)]e~aTa2+b2ߟ-+a2+b2l-2cos(bT)e-aTz-'+e-2aTz-2ᵫ§▤×&'s“(f)Ḅî᪵ðᑡs(஻)ḄZPᣚS(z)ÜMN&'h(n)ḄZPᣚᓽ[\GᦪH(z)ßÆᨵ1Ëᐵ[S(z)=—^H(z)ᡈH(z)=(l-z-')S(z)l-zᡠ1ᨬù[\Gᦪal-z~["(z)=-a+[acos(")+bsin(bT)ú"çl-2cos(bT)e~aTz~'+e~2aTz~2

684.10•ûü*rḄᳮþ▲
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ᑖᘤḄ+@(/),ᵨᵫRSᑮ"(஻)஺N",(஻)ḄᏝVWᣚᵨ⊤<ᑣᨵHজ3)=£3(஻)]=)^_(ᓩ+b6n=—QO-*k=E1defgH„(jQ)=0,|Q|>Q.=7r/TᡠpHd)JtH“(j=*(e,q

69Aj——

704.12ᵨN¢Aᦪ£ᱯ¤¥¦§ᘤ஺EᘤḄ¨©ᙠ§«¬⚣᳛=0.26137ᜐḄ⊦°ᜧ²0.7518,ᙠµ«¬⚣᳛¶=0.4018·ᜐḄ⊦°¸²20dB஺£M}1ᘤḄ▤ᦪN1¾0.1x0.751Iτ/஺26131ஹlg(04018.)0.18851¿~^~—1.3602727771g0.6503-0.1863uN=8(3)ᘤḄ3dB«¬⚣᳛_|__QW_1Å

715=5*=0.9111/ìi=-0.7575-jO.5062525=så=0.911le"ìí,=-0.5062-jO.757565=s0*=0.9111J?*=-0.1778-J0.89637(4)îïðᦪ”.(s)091118Ha3=ñAò--------=:"ᑓ"tlGFDU(f)Dk=00.4748A~ù,--(Q.+SB)S+SQJ|k=0__________________________________0.4748________________________________-(s2-0.3556.v+0.83)Gv2-1.01245+0.83)(52-1.5155+0.83)(52-1.7872.V+0.83)___________________________________0.4748_________________________________-s8+4.67?+10.90556+16.536?+17.7/+13.715?+7.515?+2.67B+0.475(5)ᵨû⊤îïðᦪ”“(s)᪷ý⊤Sᑮ8▤þAᓄᱯᘤḄᦪ஻G)஻,1a0—d+5.1258s'13.1371/+21.8462s$+25.6884/+21.8462s?+5.1258s+1&஻'(s))sᵨ+=-./012H”(s)஺10.91110Hব=________________________________04748________________________________“5-58+4.675710.905.V6+16.536.V5+17.7,V4+13.71+7.515.v2+2.67l.v+0.475;(4)<=>?஺4.13@ᵨABឋDᣚFGH.IᱯJKᘤ஺ᎷM/᪵⚣᳛£=1.5*"zᜐ⊦WXY12dB஺[&\1•^fr_ᣚᡂᦪa⚣᳛%^lOy=2X\1=2^x1000=2000஻=2eᑖ=2%xl500=3000e=10-4A10x10,

72co=TQp=10^X2000%=0.2%pa>=m=10-4x3000%=0.3eTr(2)hᘤḄ▤ᦪN^ᱯjkJKᘤḄ3dBlm⚣᳛Q,/T=l,&ᦪa⚣᳛4^<⚜ᶚD12⚜ᶚDpḄ஺஺^஺r2জ“0.2஻Q=—tan-y-=2tan'-=2tan(O.l^)=0.649841/?<__Q=I'tan^-=2tan=2tan(0.15])=1.01905377rs1jkJKᘤḄᢣ᪗v201g|//(JQ)|=201g|//(j0.649841)|>-1.8ap(1<জ201g|/f(jQ|=201g|W(jl.0190537)|<-12ar)aᵫᱯᘤḄxyz{ᦪ2201g|//(jQ)|=-101gl+A2wঝu&}ঝ0ᐭ}জ12201g(jC/=T(Hg1+(°•841>,ঞ18201gH”(JQr)|=_101g]+(L°0537)2Nট',.ὶh[}ঞ^}ট12•g[-0.7305515N>3.7389421/0.649841ஹ-0.1953899lg(-----------)1.0190537/N=4஺&N=4ᑖ0ᐭ}ঞ^}ট12Q.>0.649841(10(),xl-8-l)=Q=0.7063__1<1.0190537(10°1X,2-1)-^=0.7274/Qc=—2(ஹQc௃,+QC2l.)'=0.7168oᵨ}ᩭhN^1ABឋDᣚFᩭ1})ḄQ,,^⚜ᶚDpḄ஺(3)h”“(s)Ḅ᩽

73s*=Q,ej(2N•1—)j(81—4k)N2=0.7168/,k=0,l,…,7ᐸ)1¥¦sz☢Ḅ᩽v,5So=0.7168J?=—0.2743+j0.6623»=0.7168e'S+N=—0.6623+j0.2743s=*=-0.6623-'0.2743251S3=%*=-0.2743-.6623(4)hᦪ”"(s)0.716843=——-------]a(s-s«)(s-s°)n(Aஹ)G-4)k=0k=00.264k=0___________________0.264___________________02+0.5486s+0.5139)(.1+1.3246s+0.5139)_________________0.264_________________d+1.8732J+1.7545s2+0.9626s+0.264(5)ᵨ³⊤Fhᦪ”஻(s)᪷¶⊤2ᑮ4▤¸-ᱯᘤḄᦪ”(s)1H(s)=-----------'----------'-------------¹54+2.613153+3.4142.r+2.61315+1&“஻(5))Ḅsᵨ+=—^/012"Q,0.7168Q4H(s)=________________________-_______________________°s,+2.613+3.41420*2+2.613lQs+3'__________________________________0.264-54+2.6131X0.716853+3.4142X0.7168253+2.6231X0.71683S+0.71684___________________0.264_________________..+1.8731/+1.7542/+0.9624s+0.264;(4)Ḅ<=»¼>½஺4.14ᵨABឋDᣚFGH.Iᦪaᑗ¿ÀÁJKᘤ1ᔜᢣÃ;⚪4.13>?஺[

744.15KÅ⚣᳛DᣚFGH.Iᦪaᑗ¿ÀÁÆKᘤ1ÇjkᑮᦪaḄ_ᣚÈᵨABឋDᣚF1ᎷG/᪵⚣᳛v2.4%/஻1ᙠ⚣᳛160஻zᜐÊWXᜧY3dB,ᙠ40஻zᜐ⊦WXY48dB஺[Ì1Í&ÆKᦪaᘤḄ⚣᳛ᢣ᪗^fᢚᔠᡂᦪa⚣᳛T,2/rxl602ᐔco,,==------=-----------=——"*f240015sFC2ᐔfr2ex4071Ð=TO=—0=---------=——X240030GT=2,ᢥ᯿ABឋDᣚF1&ÆKᦪaᘤḄᦪaÕ⚣᳛_ᣚvÆKjkᘤḄ⚣᳛Q=—tan—=tan(—x—)=0.2126pT2215O,=—tan—=tan(—x—)=0.0524T2230&ÆjkᘤḄ⚣᳛ᢣ᪗᧜ÙᡂjkJ?ᘤḄ⚣᳛ᢣ᪗%=4.70400.21261_1=19.0840ὡߟ00524Ì2Í᪷¶jkJ?ᘤḄᢣ᪗h£,£1‘,^N/=10°\.1=10஺[*3-1=09953ᡈ£=0.9977Q=%=4.7040,4.7040…!<=—£■=-------=0.2465Q19.0840r10,,Je-11100lx3-l1,d=[---^--]2=[^-75-]2=3.9717x10-310°-\-1100Ix48-l1arch-archN>____d_3.9717x10-arch——10.24651.7814—299414.0568/N=3஺Ì3ÍhjkJKᘤḄz{xyᦪ

75äx=2Q&ᐸ0ᐭ3▤ᑗ¿ÀÁå⚗}Ḅz{)Q4.7040=0.21260ᓃ2(X)=[x(4x2-3)]2=16x6-24/+9x2=16(0.2126Q)6-24(0.2126Q)4+9(0.2126Q)2=0.00148Q6-0.049"+0.4068Q2rs13▤ᑗ¿ÀÁjkJKᘤḄz{xyᦪvè(ᑘ=l+e2^2(Q/Q)r_______________________1_____________________-1+0.9953(0.00148Q6-0.049Q4+0.4068Q2)___________________1__________________-0.00147,.0.0488"+0.4048Q2+1(4)hjkJKᘤḄᦪ&C=—/s0ᐭ|ì()஺)°12/মî(_s)=|஻0(JQ)°S,2=-JS___________________1_________________--0.00147?-0.0488/-0.4048?+1ᵫ+}hïH(s)H(~s)Ḅ᩽ua5=-0.701+J4.25170=-1.4047515=5*=-0.701-74.2517205=0.701-J4.25143S4=1.4047$5=$3*=0.701+74.2517ᐸ)s0,1^52¥¦sz☢Ḅ3I᩽1ðñò᪀ᡂ.IôMḄ3▤ᑗ¿ÀÁjkJKᘤ1ᐸᦪvB““G)=(5-51)(5-S)(5-S)02

76________B________(s-S|)G-%)G-So*)____________B____________(s-S|)[°—(s+s஺)s+ss]0oo______________B______________-(5+1,4047)[52+1,402.v+18.5684]rN=3v᜻ᦪ1ᡠ““(0)=1,rsB=H(0)x1.4047x18.5684=26.083uᨬp2____________26.083____________"S—(s+1.4047)6+1.402s+18.5684]=____________26.083____________"?+2.806752+20.53855+26.0831jkJKᘤḄᦪᙠ¥¦sz☢Ḅ3I᩽ᵨø}hï:s=—ofl.sin(---1---k)+bQ.cos(----1--k)>k=0,l,...,2N-l?L02NN°2NNᐸ)þÿa
bᵨᑡa=£~'+J—+]=0.9977-1+7O.9977-2+1=2.4182i1_11a=-(aN-a=0.5(2.41823-2.41823=0.5(1.3422-0.7451)=0.298611-1b=-(aN+aN)=0.5(1.3422+0.7451)=1.0437aQ<=0.2986x4.7040=1.4046«Q=1.0437x4.7040=4.9096(%aQ,
AO,Ḅ*+ᐭ᩽.Ḅ/012s4☢Ḅ᩽.᝞:7TTT5=-1.4O46sin(y^)+/4.9096cos(y^)o=-0.7023+74.25185,=-1.4046sin(>+-)+74.9096cos(-^-+-)=-1.40462x332x33&="=-0.7023-A4.2518

77BCḄDEFG☢Ḅᦪ*IJKL஺(5)%NOPQRSᘤUᣚᡂNOXQRSᘤᵨ1/s+ᣚNOPQRSᘤḄYZ[ᦪ\Ḅs,0ᑮNOXQRSᘤḄYZ[ᦪ23.083?H«G)=1+2.8067.V+20.538552+26.083?(6)ᵨ`aឋcᣚd%NOXQRSᘤ᧜fᡂᦪgXQRSᘤhT=2஺%5=ᓃk+ᐭNOXQRSᘤḄYZ[ᦪ/01+Z-126.083(^—^)3=---------------------------------------1+2.8067—J+20.5385(—•>+26.083(—l+z1+Z1+Z0.5172(1-3Z~1+3Z~2-Z-3)o1-1.0293/+L1482Z-2—0.1458q4.16h%(஻)touᏔwxyᑡ/N=8,z{P4.16(a)}஻)t}஻)Ḅ~.,ᓽo(஻)=%((஻-4)),(஻)(1)4(஻)ḄDFTF%(஻)ḄDFTḄᐵ/ᓽNFH(k)K4(k)F%*)2Ḅᐵ஺(2)ᵫ"()
"(஻)᪀ᡂuFIRᦪgRSᘤ/aឋKᦪgRSᘤᔩ¢£¤¢¥¦§¨¢TT(3)᝞E"(஻)w©ouª«⚣᳛,ḄPQRSᘤ/᝞{P4.16(b)ᡠ¯/°¤±²2(஻)³w©.TTuª«⚣᳛oḄPQRSᘤᔠᳮᔩ¢£¤¢2

78{P4.16¶(1)·¸(஻)=஻1(%-1o஻)
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᜜(஻)tᐹᨵwxឋÛ/ᡠtaឋKᦪgRSᘤ஺¥¦n=(7V-1)/2=3.5(3)ᵫ(1)ḄDEÜ⍝/"()
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794.18ᵨïðæhouaឋKFIRRSᘤ,ᐸ\e-j஻)?@ᦪABJTC஻6=0.54-046cos2ᔊ஻60

80a஺—=0.00254648%=஺19=0.00256375஺2=஺18=°஻3=Ḅ=0.008669364—u=0.02110671%=092430854஻6="14=°a=a=0.06079995713%=v2=014517283a=a=0.220011659n“io—0.25xyz5.1{|}~ḄᙳḄឋឋᓽf(5.16)f(5.17)஺{E(x“+y,“)=££(x+y)p(x,n,y,m)dxdyXniymLLXp^>^X,n,y,m)dxdy+ᐸ=_dLxp%%a஻)'My=%᳛p%%(%஻ߟaLypg«஻p”.(஻y“)ᵫNPx“(X,n,8,tn)=x„

81ᨬ«¬E[x+y“J=£xp(x,n)dx+[,)P%(y,m)dy=E[xJ+E[y]nxnm9®☢ᨵE[ax]=[axp(x,ri)dx=afxp(x,n)dx=aE[x]J-CO-nJ-00”ᦑ|}~Ḅᙳᐹᨵឋឋ஺5.2±²x(n)y(n)?³ᐵḄ´µ|}¶·¸¹Ḅ®ºᑖ¼?cr½<7½,w(n)=x(n)+y(n)Ḅ®ºcr¿஺᪷Á|}¶·ḄᙳḄឋឋ¬ᑮm--£[(஻)]=£[x(n)]+£[>(஻)]=m+tnxxy.=E[(a)(n)-m)2]=E{[x(n)+y(n)-m-m]2}axy=E{[x(஻)+y(஻)]2-2[x(n)+y(n)](m+m)+(m+m)2}xvxr■ᵫNx(n)=E[\x(n)+}-2(m+m)E[x(/j)+y(஻)]+(m+m.yxvxy=E[x2(n)]+El/(஻)]2E[x(n)y(n)]-2(m+m){E[x(n)]+E[y(n)]}+(m+m)2+xyxyy(n)?³ᐵḄ´µ|}¶·ᡠᙠTᑡᨬ«9µfË=E[x(n)y(n)]=E[x(n)]E[y(n)]=mmxyZÌ.=E[x2(n)]+E[y2(n)]+2mm-2(%+m)2+{m+m)2xyyxy=E[x2(n)]+E[y

82)]-mo-m2y=E[x2(n)-m^]+E\y2(n)-mj]=E[(x(஻)=஻4Í]+E[(y(n)-m)2]v22=G+b;5.3ᨵ3µÑÒÓÔᑡx[(n)ஹX2(n)X3O1),¸¹ᑖ¼ᙠÕ(-q,0)ஹ(-q/2,q/2)(0,2^)TᕈᙳᒴᑖÙ஺(1)Úi¸¹Ḅᭆ᳛Û@ᦪÜÝ஺(2)h¸¹ᔜßḄᙳ஺(3)h¸¹ᔜßḄ®º஺àপâ(஻)ஹ஽(஻)ᓃ(஻)Ḅᭆ᳛Û@ᦪᑖ¼1,ஹ-,-q

83——,0

84=E[{x-mJ-]=(x-m)2(x)dx2Xz=-P/2x2Jx=—qL஻23q-q'212a¿=E[(x-m)2]=_(x-mJ2p(x)dx3X3X}=½JJ(x—8)òdx=½(x—^Í=g27t*6TT35.4±²|}¶·x(n)=cos(gn+“)ᐸ=ó⚣᳛?õᦪö஺?ᙠÕ(0,2^)ᙳᒴᑖÙḄ|}~஺x(n)ḄᙳßᐵÔᑡ÷ᑨ¼x(n)?ᔲúBûü|}ýþ஺x(n)Ḅᙳm=E[x(n)]v=J/(஻)pe(e)4e1A=——cos(Go஻z+e)cos(0om+e)de=o2%J)n)=E[x(m)x(n)]=]3(஻(஺)᝝1-——Icos(ry7?2+0)cos(69m+0)d30024(1p”1=——I—{cos[<2>(/t-m)]4-cos[6?(/?4-771)+20]}dO002"12=2COS[0o(஻6஻2)]7gcos[g

85-m\]9:;ஹ=??ஹ*(஻?,஻)@ABCDnEᐵG/?ஹ.(஻?,஻)I@ABJln-mlᨵᐵGᡠNx(n)OPQRST;UV஺5.5XY6Z[\T;]^@6Z@ᐸ`aᐵḄ1cdeᑡaghGiᡂ:6Z1cdeᑡ஺k[\T;]^:x(n),1cdeᑡ:e(n),lmḄgny(n)=x(n)e(n),1cdeᑡe(n)Ḅoaᐵeᑡ:R(n,n+tn)-E[e(n)e

86+m)-ceᐸpGcr2O1cdeᑡe(n)ḄqJGrs)Oᓫuv᪵eᑡ஺y(n)Ḅoaᐵeᑡ:/?ஹஹ.(஻,஻+m)=E[y(n)y*(n+7n)]=E[x(஻)e(஻)x*(஻+஻?)e*(஻+஻?)]9:e(n)=x(n)`aᐵGᡠNxyzᑏᡂ

87R(஻,஻+|)=E[x(n)x*(n+஻?)]E[e(஻)e*(n+m)]=R(m)R(m)=cr2R“(஻?)(T(ni)9y(n)Oyyee1cdeᑡ஺5.6ᔊឋUV6ORSḄGRST;UV6OᔊឋḄGZᔩ:5.7ᨵ,T;ix,lḄᙳ:mஹqJ:cr2஺xḄNZXj(i=(),…,N-l)O`aᐵḄGv1N-\ḄRᙳG=-5-y%,.:v᪵ᙳ஺NᙠG(1)¢G£Ḅ¤¥஺(2)¢6,ḄqJ஺1N-15.8§x⚪஺©ª=xZ(£6u®¯v᪵qJ஺N,=o(1)°v᪵qJḄ¤¥E16±஺¤¥1N-1²E]=EjZ(X(஻)6³)2]N=0n1N-\=-£{Ed(஻)]+E[mo]-2Nx(஻)³]}NMiN-\iN-\N-i9NTN-I=—Xµ/(஻)]+QZZE[x(i)x(·-QZZ²x(i)x(/)]Nn0Ni=oj=0Noj=o=i=iN-1iN-\N-\=¸E£f%2(஻)1-77rEEE[x(i)x(/)]Nn=0Ni=ojQ=iNTiNTN7NT=ᯅX²/(஻)]6Q{X(01+ZZ©x(i)x(j)]}Nn=0Ni=oz=0j=0jjᑴ,1,N2-N=£[x2(n)]--E[x2(i)]———E[x(i)x(j)]NNN-T,N-I,=——E[x2(i)]--—(£[x()/])2NNN—12=-----

885.9x(n)OÎᙳT;UVḄv᪵eᑡGd(n)=x(n+l)-x(n):Jᑖeᑡ஺kÐT;UVḄÑ᳛ÒOÓÔḄᓽSxx(e஻")=Õ‘᜜ØᎷkT;UVoaᐵeᑡḄÚZRu(O)=0,a)<1Ü<Ýc(1)¢JᑖeᑡḄᙳqE[d2(n)].JᑖeᑡḄᙳqE[J2(n)]=E[(x(n+l)-x(«))2]=E[x~(஻+1)]-2E[x(n+1)x(஻)]+£[x2(n)]=R£0)-24y1)+ß(0)=2[/?„.(0)-/?„.(1)](2)XYE[d2(n)]>^-coscofS(ejM)dcocxxxyp—cosgýS"(e"")d3=coscoR(0)=(l-2sin2þ)R“(0)>(1-2(þ>)7?(0)9ÿclcu2
R22

89প“-(஺)ᓽ2(0)-প(0)ᡈE“஻)]4d©/(஻)]5.10x(n)ᙳbḄ!"#$%&'()*+ᵨ-./01h(n)Ḅ2ឋ45678Ḅ9ᐭ;<ᑮ9>&'()y(n)஺(1)?x(n)@y(n)ḄABᐵDᑡᙠGHIJm=0IḄK᪵R?(0)஺M2ឋ45678Ḅ9>y(n)9ᐭx(n)ᨵ᝞Pᐵ700y(஻)=Eh(k)x(஻-k)k=-QRR*(0)=E[x(n)y(n)]=E[x(n)£h(k)x(n-k)]CO8=h(k)E[x(n)x(n-fc)]=£h(k)R*(k)k=-ook=-ᐸTRqU)ᙳஹbWḄ!"#$%&'()x(n)ḄXBᐵDᑡᨵR()y(n)ḄbW஺MQx(n)Ḅᙳᡠy(n)Ḅᙳ[QR

900000bo=E\y

91)}=Ef(£"k)x(i—\))(h(k)x(j-A))]k=-oojt=—X=£rয^/(஻`a]+££h\k)E[x{i-kMj-k)]A=-00j=-00j=-0),ᡠo(2)ᡂa2h(0),m=0E[x(n)u(n+m)]=<1ব0,m>0|⚪ᑖl?>78ᦪ

921H
z=p1+f%"k=]᪷ᳮᨵ]o=lim"z=lim=14PZ—>ooz—>oo1+fo4ᐭo3,0o5ᐭo1,ᨬH0A=1¥¦x(n)ḄXBᐵDᑡ&“(஻?)1~Ḅlo஺2sin*")5.12ᨵ2ឋ45678ᐸᓫª./01h(n)/------¬`஻஺,ᙠ®789ᐭ;+ᵨ`i¯ᦣ7tn0,“=0&'()x(n),ᑣᙠ789>;<ᑮy(n)஺(1)?9>()y(n)ḄXBᐵDᑡ஺(2)?9ᐭ()x(n)@9>()y(n)ḄABᐵDᑡ஺(3)³´(2)ᡠµ<ḄABᐵDᑡ᜻Dᑡ஺(4)ᵨx(n)·y(n)᪀ᡂ`i¹Dᑡ஺(n),a>(n)=x(n)+jy(n)¼½µ6;(n)ḄXBᐵDᑡ஺(5)?ᵨᗂ(n)ḄÁ᳛Ã஺M(1)✌ᐜ?h(n)ḄᏝÇÈ6ᣚ8H
*=ZË஻`''n=-oosin“s/2
Íτ£$Í2s/2
“᝞௃ᑍ/:=—oc஻/»=1஻2Òl-cosO
-Ó+yl-cosm?
e.]7y2n''^2n~o1[Ô1-cosseᑠ+J1-cosse_¦1n=\—஻஻=1஻

93]00_j×+ej(ai=`-cos(s)]`ᐔញn=£[(—ØÙ]஻=1஻=[_ÔಔÜsin(m)_ysin(^)])“=1஻n=l஻5T/1ஹ஻-1sin(ᵫ)CD..ᐸTy(-i)----=—.\co\<7i஻=1271—CD_-----,0

94(2)x(n)@y(n)ḄABᐵDᑡ8R(m)=E[x(n)y(n+m)]=E[x(n)>h(k)x(n+m-k)]xyJl=-00s=+æ`%)]k=-co=£h(k)RবQx(n)éDᑡᨵh(n)᜻Dᑡᓽh(n)=-h(—n)ᡠ&,(`m)=£h(k)R—m—k)=(a+A)k=-cok=-co=Nh(-k)R_k)=_ᐗh(k)R(n+m)]=E[(x(n)+jy(n))(x(n+tn)-jy(n+m))]=E[x(n)x(n+m)]+E[y(n)y(n+m)]+jE[y(n)x(n+m)-jx(n)y(n+m)]=&/᪷)+R(m)+j[R(m)]-j[R(m)]vvyxxyপ·(3)Ḅí}ᓽR»(஻z)=R(î·R4(`')=`=R).(ߟ஻஺ᐭðo

95ᑭᵨ(2)Ḅí}ᨵS(ej(0)=H(eja))S(eja))xyxxᦑ59(2)=25ý஻஺)11—j"(")]0,0W2(஻)=£@(/B(஻/)/=—<503BC(")E(஻)Ḅᐵ!ᑡ8COELx(஻)y(/w)]=E{(X%(AM(஻-&))(^/z(/)x(m-/))}222k=—xI=—oo0000=z4—2(/)FGi(஻-2)%2(I[)]=—00/=—coᵫMN(஻)O2(஻)ᐵ3BPQRᨵE[(n-k)x(m-Z)]=E[x(n-k)]E[x(m—I)]2(2ᡠ

965.14ᨵ4ឋ789:ᐸᓫ᪵h(n),)ᵫᓫ᪵ᑖS(n)h(n)2Ḅ]9:^ὶ`ᡂ஺b9:
ᐭᵨᨵcde!ᑡx(n),f1]9:Ḅ
1y(n),f2]9:Ḅ
(01᦮^ὶ9:Ḅ
)1w(n)஺bx(n)Ḅᙳijklb(஺(1)

97op88ᔬ=2§যR,v+EZẑ᝕©v(i-ªফk=0i=0j=0¯jᱏi¬ᯠx(n)1cde®1f1^ὶ]9:(஻)Ḅ
ᓽf2^ὶ]900:¯(஻)Ḅ
ᐭy(n)=Z%Wx(஻—᝕)ᙠ³´µn1cde3BᙠQ(2)RA=0ḄR(0)R(i-j)ᐹᨵ

98(l-a2)(l-fe2)ᑖᵨ⚪R¹ḄQ(c)Q(b)ÊËcr{ᑮḄ¡¢Q(3)Q(4)ÍΣ<Ü)*1Ḅ஺ÝMÞ☢ß'ḄᳮᵫᵨQ(c)ÊËᑮḄ¡¢ᓽQ(3)1qrḄ஺5.15bx“(t)1ᙳijḄâã$¿)Ḅ.ᐵäᦪæ᳛èᑖéê=E[x஺(t)x{(t+r)]S2(JQ)=ÖRᲝᐸRP᪗(a)⊤¼┐/âã$¿(ᡈñò){óô$¿71âã8õ{Q1ñòö⚣᳛஺ø<ᕜúT/x“পÏ¿◞᪵ᑮþᦣ
x(n)-x(nT),n᦮ᦪ஺x(n)Ḅᐵᑡfl+8᳛ᑖR(m)=E[x(n)x*(n+m)]Sxx(e"")=£R(m)exvxx-00(1)/Rஹ(m)1R2)(3)4
Ḅᐵ5஺67ᢥ᯿2ᐵᑡR“w(m)&(m)=E[x(n)x*(n+m)]=E[(nT)x7(nT+mT)]=1m<=S஺*m)>?@ᕜBTCRo«)ḄD
◞᪵஺(2)/SM(eI)1SJ(JQ)4
Ḅᐵ5஺67᪷N(1)ḄOPS2(/Q)Ḅ2ᨵST஻?)=W(஻ᶪ)=Y1Sᨌ¶w·(1)Z[3☢2᪷NSw(e23)Ḅ2ᨵ=(2)^_(1)⊤aᡂc▲efᑖ42ᐸhiefᑖjᙠl2m/TḄn
Yop2ᓽ4m(m)=1TK2k+l“/Tst—>fSRC)CdQ(3)2m24yz{ᣚ7QnQ+3Y2_(3)ᓄTR“x"(m)=y-t¦TS¹jQ+j^k)wº…dQ(4)Fᐔk=—/

99ᣚ/1fᑖḄ2ὃ⇋ᑮCᡠᨵ᦮ᦪkmᨵ/2"*=1,<=_4ᓄRe஺஺"*1஺¼02¯k`௃e¾Q(5)^஺=ᜐᐭ_5,T=1&£SᐸÀ+᳛kl*'ds6^_61_2C᯿2182ᡈ^O^=-E5uOn+j—8/—0013ᜫᙢᵫSxx
TIJS,/Q,SJQ¡¢£ᩩ¥¦277,67§_7ᡈ_8©ª«255t3?52/஺ḄᕜB¬2ᕜB®஺<=2¯¡ᩩ¥S24=0,1஺1±²/7ᑣ//஺´µᕜB¬¶·¸¹º»ᜫ2<¼½¾ᜫᙢᵫ5¿/3ᑮ/À2ᓽᨵSxl="2Á23.m5.16ÃÄÅÆ
ÇÈX.পḄ᳛᝞ÌP5.16ᡠa2@ᕜBTCx“tD
◞᪵ᑮÏᦣ
ÇÈxn=xnTowÒ(0)

100(1)/x(n)ḄᐵᑡRw(m)஺6:=J_Ô"Õ7Q=_L_sin(Qo"zT)2m7imT(2)Öx(n)?×⁐ÇÈᑡ2᝞Ù⌱ÛT?TT67¯7=2%,k?Ü᦮ᦪ2ᑣY_ᓄ0,\Qosin(஻ÝÞ)Q&(”)——:——onmk7rnà?×⁐ÇÈᑡḄᐵᑡḄá_஺<=2â⌕᪵ᕜBT?%/Q஺Ḅ᦮ᦪå2æ£x(n)>?×⁐ÇÈᑡ஺5.17ᎷèÅÆ
ÇÈx஺(t)Ḅ᳛᝞ÌP5.17ᡠa2éê5.16⚪஺67x(n)ḄᐵᑡPR*m)=E[x(n)x(n+m)]=E[x(n)(-^ax(n+m-Z:)+w(H+m)]kk=lp=aE[x(n)x(n+ì[%)]+E[x{n)u(n+m)](1)kk=]p=ak^xx^m-k)+E[x(n+m)]A=1_(1)íîïð⚗00E[x(n)u(n+m)]=E{f^h(l)u(n-l)]u(n+Â}/=o00OP=h(l)E[u(n-l)u(n+in)]-^/2(/)cr2^(7/24-/)(2)1=01=0=a2h(-m)ᵫh(n)?0),ᡠ@_(2)ᡂ

101cr2/2(O),m=0E[x(n)u(nm)]=<0,õு0ᵫ⚪÷øᑖ3ù©/«5òúᦪpl+fk=\᪷NIûᳮ2ᨵ/i(0)=lim஻(z)=lim^_(4)ᐭ_(3),a2,m-0E[x[n}u{n+ni)\=<0,m>0^_(5)ᐭ_(1),ᨬ
,m=0R*m)+-k)=,0,m>0x(n)ḄᐵᑡR(m)Ḅ஺xx5.18⚪5.16"⚪5.17$%ᑮ'()*஺+,-./0123X”পḄ7᳛9:S<)(t),>x஻(t)@A.//◞T஺D᪵
ᑮFᦣ./01ᑡx(n)஺:Hx(n)I⁐01ᑡ

102pqᵨh(n)⊤x(n)ḄᐵᑡR(m)=E[x(n)x(n+m)Jxx00co=E[>h(k)u(n-k)>hYu(n+-r)]Jt="cor=—co008=>h(k)<'k)u(஻+—r)]it=-oor=-(2)rᏝ¹ºjᣚ஻½(-3)²h(-k)ḄᏝ¹ºjᣚ஻M4(e-W)Ḅ¼´஺¿᪵<>(2)rᏝ¹ºjᣚ<À
ᑮSÁ*")=ÆÇ/)I2(3)q-m/2᝛<+”=°<1<”,Ç(4)ÃÄᐵᑡXᑖRu.(m)=A=00,m>(7+1qpqÉH(Z)=£b/k|@r
MAk=0h(k)=%[“(A)-w(A-q-1)]¹u(k)⊤ᓫ▤Íᑡ஺

103q-maVb.b,^.m=0,1/**,qZ஻(᝕)%(+%)={2"t(4)J[o,1ᑗ+1(4)Óᐭ(2),
q-m„(ஹ/Z᝛+”,<=01<…

104&x(m)=E[x{n}x{n+/??)]p=%(஻+/%—%)+u(n+m)]}k=TP='+஻?-k)l++)1k=\=A£%È(ἔAk)+R£m)hl+klḄᓫD᪵:h(n),ᑣklḄcm:00x(n)=2å(,)஻(஻'/)/=0<(3)Ḅ3“():7?(m)=E[x(n)u(n+m)]AM8=E{^h(l)u(n-l)u(n+m)}1=0CO=£h(l)E[u(n-l)u(n+m)](4)/=o00=Z/?(/)cr2b(+/)1=0-a2h(-m)ᵫ`klèkl<ᨵh(m)=0(mv0)ᡈh(-m)=0(m>0),ᡠ@(4)|ᑏᡂa2h(0),m=0%“()=,(5)0,m>0ᐸh(0)|ᵨêUâᳮr
ì=]//(0)=limH{z}-limARz-^>Z->001+£Ü*k=\<(5)ᓄ:cr2,m=0¢,(M=<(6)0,஻R>0(6)Óᐭ(3),ᨬ
p'2%ᜧð('%)+02,ñ=0k=\R£m)=,PaR(m-k),m>0kxx

105q£bk"k5.21+fឋhijklḄbcdᦪ:HAMA£Z¤=-------òó⚪5.20,1-᧡Ëk=1