数字通信第14---15章答案68455

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1、CHAPTER15Problem15.1:L
−1g(t)=ejθka(n)p(t−nT)kkcn=0Theunitenergyconstraintis:T∗gk(t)gk(t)dt=10Wealsodefineascross-correlation:T∗ρij(τ)=gi(t)gj(t−τ)dt0(a)Forsynchronoustransmission,thereceivedlowpass-equivalentsignalr(t)isagaingivenby(15-3-9),whilethelog-likelihoodratiois:√2TKΛ(

2、b)=0r(t)−k=1Ekbkgk(t)dtT2√T=

3、r(t)

4、dt+EEbb∗g(t)g∗(t)dt0kjkjjk0jkK√T∗−2Rek=1Ekbk0r(t)gk(t)T2√=0

5、r(t)

6、dt+kjEkEjbjbkρjk(0)√K−2Rek=1EkbkrkT∗whererk=0r(t)gk(t)dt,andweassumethattheinformationsequence{bk}isreal.Hence,thecorrelationmetricscanbeexpressedinasimilarformto(15-3

7、-15):ttC(rk,bk)=2bKRe(rK)−bKRsbKTheonlydifferencefromthereal-valuedcaseofthetextisthatthecorrelationmatrixRsusesthecomplex-valuedcross-correlationsgivenabove:ρ∗(0),i≤jR[ij]=ijsρij(0),i>jandthatthematchedfiltersproducing{rk}employthecomplex-conjugateofthesignaturewaveforms{gk(t)}.301(b)F

8、ollowingthesameprocedureasinpages852-853ofthetext,weseethatthecorrelatoroutputsare:(i+1)T+τk∗rk(i)=r(t)gk(t−iT−τk)dtiT+τkandthatthesecanbeexpressedinmatrixformas:r=RNb+nwherer,b,naregivenby(15-3-20)-(15-3-22)and:Ra(0)Ra(−1)0······Ra(1)Ra(0)Ra(−1)0···........RN=0....⇒....

9、...........0···0Ra(1)Ra(0)HRa(0)Ra(1)0······HRa(1)Ra(0)Ra(1)0···........RN=0...................0···0Ra(1)Ra(0)whereRa(m)isaK×Kmatrixwithelements:∞∗Rkl(m)=gk(t−τk)gl(t+mT−τl)dt−∞andwehaveexploitedthefact(whichholdsinthereal-valuedcase,too)that:∗tHRa(m)=Ra(−m)=

10、Ra(−m)Finally,wenotethatRa(0)=Rs,thecorrelationmatrixofthereal-valuedcase.Problem15.2:ThecapacityperuserCKis:1PCK=Wlog21+,limCK=0KWN0K→∞andthetotalcapacity:PKCK=Wlog21+WN0302whichisindependentofK.Byusingthefactthat:P=CKEbwecanrewritetheaboveequationsas:C=1Wlog1+CKEb⇒KK2WN0KCK=

11、log1+CKEb⇒W2WN0CKE(2K)W−1b=N0CKWwhichistherelationshipbetweentheSNRandthenormalizedcapacityperuser.Therelation-shipbetweenthenormalizedtotalcapacityC=KCKandtheSNRis:nWE2Cn−1b=KN0CnThecorrespondingplotsfortheselasttworelationshipsaregiveninthefollowingfigures:1098765K=143Capacityperuser

12、perHe