北京邮电大学通信原理课件--第3章-随机过程

1()()()cos2cYtXtftπθ=+()Xtθ()Xt0()XRτ()XPf(1)θ()0,2π()Yt(2)θ()Yt(1)(2)()()()cos20cEYtEXtEftπθ=+⎡⎤⎡⎤⎡⎤⎣⎦⎣⎦⎣⎦=()()()()()()()()()()()()()()()()cos2cos22cos2cos221cos2cos422211cos2cos42222YccccXcccXcXccREYtYtEXtftXtftfEXtXtEftftfREfftfRfREftfττπθτπθπτccτπθπθπττπτπθπττπττπθπτ=+⎡⎤⎣⎦=++++⎡⎤⎣⎦=++++⎡⎤⎡⎤⎣⎦⎣⎦=+++⎡⎤⎣⎦=+++⎡⎤⎣⎦(1)θ[)10,2()20elsepθπθπ⎧∈⎪=⎨⎪⎩()()201cos422cos42202ccccEftfftfdππθπτπθπτθπ++=++=⎡⎤⎣⎦∫()()1cos22YXcRRfττπ=τ()()()()()22cos224jfYYXcjfXcXcPfRedRfedPffPffπτπττττπττ∞−−∞∞−−∞==−++=∫∫(2)()()()()11cos2cos42222YXcXccRRfRftfττπττπθπ=++τ+1/15t()Yt()cos4220ccftfπθπτ++=()Yt()()1cos22YXcRRfττπ=τ()()()4XcXYPffPffPf−++=c202N()Hf()Xt()11cos2()0sssTfTfTHfelseπ⎧+≤⎪=⎨⎪⎩()Xt()Xt()()()2202011cos820ssXsNTfTfNPfHfTelseπ⎧+≤⎪==⎨⎪⎩()Xt()()12200131cos88ssTssXsTNTNTPPfdfTfdfπ∞−∞−==+=∫∫3aa()Yt()nt()Hfb()Yt(a)(b)2/15:()YtB()Yt()YPf()Yt()cYt()sYt()cYPf()sYPf()()()()20csYcYcYYBPffPfffPfPfelse⎧++−≤⎪==⎨⎪⎩()()222002()2220cYBNfffNPfHfjfelseππ⎧±≤⎪==⎨⎪⎩()()()2222002220()()2220420csYYcccPfPfBNffNfffelseBNfffelseπππ=⎧++−⎪=⎨⎪⎩⎧+≤⎪=⎨⎪⎩≤()()csYYPfPf=()202cNfπ2B−2Bf4()()110Tnttdtξϕ=∫()()210Tnttdtξϕ=∫()nt02N()1tϕ()2tϕ1ξ2ξ()nt()0Ent=⎡⎤⎣⎦0f=[]()()()()111000TTEEnttdtEnttdtξϕϕ⎡⎤===⎡⎤⎣⎦⎢⎥⎣⎦∫∫3/15[]()()()()222000TTEEnttdtEnttdtξϕϕ⎡⎤===⎡⎤⎣⎦⎢⎥⎣⎦∫∫()nt12,ξξ1ξ2ξ[]120Eξξ=[]()()()()()()()()()()()()()()()()()1212001200120001200012022TTTTTTTTTEEnttdtnttdtEntntttdtdtEntntttdtdtNttttdtdtNttdtξξϕϕϕϕϕϕδϕϕϕϕ⎡⎤=⎢⎥⎣⎦⎡⎤′′′=⎢⎥⎣⎦′′′=⎡⎤⎣⎦′′′=−=∫∫∫∫∫∫∫∫∫1ξ2ξ()()1200Tttdtϕϕ=∫()()12,tϕϕt()1tϕ()2tϕ02N02N3.5()()()cos2sin2ccscXtXtftXtftππ=−02Xσcos2()cAftXtπ+()()()Ytutvt=+()utcos2cAftπ()vt()Xt()cXt()sXt1θ()ut()vt2θ()Xt02σ()ut()vt4/153.5()()cos2cos2cos2ccLPFAutAftftππθ=×+=⎡⎤⎣⎦θ()()()()()()()cos2sin2cos211cossin22ccsccLPFcsvtXtftXtftftXtXtπππθθθ⎡⎤=−×+⎣⎦=+θ()ut22cos4uAPθ=()vt2222cossin44XXvP24Xσσσθθ=+=1222cosuvXPAPθσ=(2)()vt24XvPσ=()ut22222222coscos442uAAPEedθσθθθθπσ−∞−∞⎡⎤==⎢⎥⎣⎦∫[]()222222222222222cos1cos2211co2212uvXXXXPAEPAEAedAeθσσθσθσs2θθσπσσ−∞−∞−⎡⎤=⎣⎦=+⎡⎤=+⎢⎥⎢⎥⎣⎦=+∫60()nt6010W/Hz2N−=()()dntytdt=B=10Hz()yt()oyt(1)()yt(2)()oyt5/15(1)()2225200223.9510W/Hz2yNPfjfNffππ−===×(2)()23220004220.0263W3oBByyBNBPPfdfNfdfππ−====∫∫7()tξHf2Hf()tξ12,,,nξξξnn()tξ()tξ002N()tξ0HNf()tξ()202012HxNfHpxeNfξπ−=()tξ()020esleHNffPfξ⎧⎪=⎨⎪⎩()()()20sinc2jfHHRPfedfNffπτξξττ∞−∞==∫k()10sinc00kkk=⎧=⎨≠⎩()tξ12Hf12,,,nξξξ()()()()()1212222120,,,1212220,,,2nnnHnnxxxnNfHpxxxpxpxpxNfeξξξξξξπ+++−−==3.8()bt()bt02N1()bt2maxγ36/153.8()()0htbtt=−(1)b(t)0tT=()()htbTt=−2()()()2220022002222NNHfdfHfdfNNhtdtATσ∞∞−∞−∞∞−∞====∫∫∫tT=()()()()20TbThdbTbTdATττττττ∞−∞−=−−=∫∫()222max202ATATNγσ==(3)2byATξ=+ξ0202N2ATσ=y()()2220201yATNATpyeNATπ−−=7/1591X()212112112,0xXxpxexσσ−=≥2X()222222222022,0xAXxAxpxeIxσσσ+−⎛⎞2=≥⎜⎟⎝⎠1,2XX12XX()()()()()()()12121212212212222222222222212,1212121221101221202222022202222022,XXxxxXxxXXxxXxxAxxAPXXpxxdxdxpxpxdxdx22pxpxdxdxxpxedxdxxAxeIedxAxeIdxσσσσσσσσσ∞∞−∞∞+−−∞+−==⎡⎤=⎢⎥⎣⎦⎡⎤=⎢⎥⎢⎥⎣⎦⎛⎞=×⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠∫∫∫∫∫∫∫∫∫20∞∫x22xx=2AA′=()22222221202202AxAexAxPXXeIdxσσσσ′−′+−∞′⎛⎞=⎜⎟⎝⎠∫20,0Aσ()22222022001xAXxApxdxeIdxσσσ+−∞∞⎛⎞==⎜⎟⎝⎠∫∫x()222224121122AAPXXeeσσ′−−==3.1002N()nt()1Hf()2Hf()1tξ()2tξ()1Hf()2Hf()1tξ()2tξ8/153.10()nt()1tξ()2tξ0()()*12,0Ettτξξτ⎡⎤∀+⎣⎦=()()()()()()()()()()()()()()()()()()()**1212*12*12*012*01222EttEhuntuduhvntvdvEhuhvntvntududvhuhvEntvntududvNhuhvuvdudvNhuhuduξξττττδττ∞∞−∞−∞∞∞−∞−∞∞∞−∞−∞∞∞−∞−∞∞−∞⎡⎤⎡⎤+=−+−⎣⎦⎢⎥⎣⎦⎡⎤=+−⎢⎥⎣⎦=+−⎡⎤⎣⎦=+−=+∫∫∫∫∫∫∫∫∫−−Paserval()()()()()()**01212*2012220jfNEtthuhuduNHfHfedfπτξξττ∞−∞∞−∞⎡⎤+=+⎣⎦==∫∫()()*2120jfHfHfedfπτ∞−∞⎡⎤=⎣⎦∫0()()*12HfHf()()*12,0fHfHf∀=()()12,0fHfHf∀=()Xt()Yt12,tt()()1,2XtXtt()Xt()Yt()Xt()Yt11()()2cjftteπθξ+=θ0,,33ππ⎧⎫−⎨⎬⎩⎭3()tξ()Etξ⎡⎣⎤⎦()()Ettξξτ∗⎡⎤+⎣⎦()tξ9/15()2cjftjteeπθξ=jeθ()223312133ccjjjftjftjftjEtEeeeeeeππππθξ−⎛⎞⎡⎤==++=⎡⎤⎜⎟⎣⎦⎣⎦⎝⎠2cπ()()()()()222cccjftjftjfjjEttEeeeeeπτππτθθξξτ+−∗−⎡⎤⎡⎤+==⎣⎦⎣⎦()tξtt12()()11cos2NiixtitNπθ==+∑,1,2,,iiNθ=iθ[]0,2π(1)()xt()()xmtExt=⎡⎤⎣⎦(2)()xt()()(),xRttExtxtττ+=+⎡⎤⎣⎦(3)N→∞()xt(1)()()()()12011cos211cos220xNiiNimtExtEitNitdNππθπϕϕπ===⎡⎤⎣⎦=+⎡⎤⎣⎦=+=∑∑∫(2)()()()()()()()()()111111,1cos2cos221cos2cos221cos2cos22xNNijijNNijijNNijijRttExtxtEitjtjNEitjtjNEitjtjNττπθππτθπθππτθπθππτθ======+=+⎡⎤⎣⎦⎡⎤⎛⎞⎛⎞=++⎢⎥⎜⎟⎜⎟⎝⎠⎢⎥⎝⎠⎣⎦⎡⎤=+++⎢⎥⎣⎦⎡⎤=+++⎣⎦∑∑∑∑∑∑+iθjθij≠()()()()cos2cos22cos2cos220ijijEitjtjEitEjtjπθππτθπθππτθ⎡⎤+++⎣⎦⎡⎤=+++⎡⎤⎣⎦⎣⎦=10/15ij=()()()[]()1cos2cos22cos2cos42211cos2cos4222cos22iiiEititiEiitiEiEitiiiπθππτθπτππτθπτππτθπτ+++=++⎡⎤⎡⎣⎦⎣=+++⎡⎤⎣⎦=+⎤⎦()()11,cos2NxxiRttiRNτπττ=+=2=∑()xt(3)N→∞()xt0xm=()102xR=()21xpxeπ−=13()()0nnDtagtnTt∞=−∞=−∑−na1±((,nmaanm≠))0t0,Tna()100tTgtelse≤≤⎧=⎨⎩()Dt()()()()()()()[]()(()()00000000,DnmnmnmnmnmnmnRttEDtDtEagtnTtagtmTtEaagtnTtgtmTtEaaEgtnTtgtmTtEgtnTtgtnTtττττττ∞∞=−∞=−∞∞∞=−∞=−∞∞∞=−∞=−∞∞=−∞+=+⎡⎤⎣⎦⎡⎤=−−+−−⎢⎥⎣⎦⎡⎤=−−+−⎢⎥⎣⎦=−−+)−−−⎡⎤⎣⎦=−−+−−⎡⎤⎣⎦∑∑∑∑∑∑∑t()Dt()(),,DDRttRtTtTττ+=+++()Dt11/15()()()(00001,1TDDTnRRttdtT)EgtnTtgtnTtdtTτττ∞=−∞=+⎡⎤=−−+−⎢⎥⎣⎦∫∑∫−)0t()(0ngtnTtgtnTtτ∞=−∞−−+−−∑()()()()()()()()()()()00000000001111110TDntTtnTnTnTREgtnTtgtnTtdTEgtnTtgtnTtTEgxnTgxnTdxTgxnTgxnTdxTgxgxdxTTTelseττττττττ∞=−∞∞+=−∞∞=−∞∞=−∞⎡⎤=−−+−−⎢⎥⎣⎦⎡⎤=−−+−⎢⎥⎣⎦⎡⎤=−+−⎢⎥⎣⎦=−+−=+⎧−≤⎪=⎨⎪⎩∑∫∑∫∑∫∑∫∫tdt−()DRτ()()2sincDPfTfT=1402N2B()tn()