Chapter 6 DETECTION AND ESTIMATION: Model of digital communication system. Fundamental issues in digital communications are

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1 Chaper 6 DCIO AD IMAIO: Fndaenal sses n dgal concaons are. Deecon and. saon Deecon heory: I deals wh he desgn and evalaon of decson ang processor ha observes he receved sgnal and gesses whch parclar sybol was ransed accordng o soe se of rles. saon heory: I deals wh he desgn and evalaon of a processor ha ses nforaon n he receved sgnal o erac esaes of physcal paraeers or wavefors of neres. he resls of deecon and esaon are always sbec o errors Model of dgal concaon syse Consder a sorce ha es one sybol every seconds, wh he sybols belongng o an alphabe of M sybols whch we denoe as,, M. e asse ha all M sybols of he alphabe are eqally lely. hen p p eed for all M

2 he op of he essage sorce s presened o a vecor ranser prodcng vecor of real nber.,,..., M.. here he denson M. he odlaor hen consrcs a dsnc sgnal s of draon seconds. he sgnal s s necessarly of fne energy. he Channel s assed o have wo characerscs: Channel s lnear, wh a bandwdh ha s large enogh o accoodae he ranssson of he odlaor op s who dsoron. he ransed sgnal s s perrbed by an addve, zero-ean, saonary, whe, Gassan nose process. sch a channel s referred as AG addve whe Gassan nose channel GRAM CHMID ORHOGOALIZAIO PROCDUR: In case of Gra-chd Orhogonalzaon procedre, any se of M energy sgnals {} can be represened by a lnear cobnaon of orhonoral bass fncons where M. ha s we ay represen he gven se of real valed energy sgnals, M each of draon seconds n he for M M M M,,3... M 6. here he Co-effcen of epanson are defned by

3 ,, M d,, he basc fncons,... are orhonoral by whch f d f 6.3 he co-effcen ay be vewed as he h eleen of he densonal Vecor ' herefore ',, M ' ' Le 4 3 Vecor 3 4 Geoerc nerpreaon of sgnal: Usng orhonoral bass fncons we can represen M sgnals as,,..., M 6.4

4 Coeffcens are gven by d,,..., M,,..., 6.5 Gven he se of coeffcens {s },,,. operang as np we ay se he schee as shown n fga o generae he sgnal s o M. I consss of a ban of lplers, wh each lpler sppled wh s own basc fncon, followed by a ser. fga conversely gven a se of sgnals s o M operang as np we ay se he schee shown n fg b o calclae he se of coeffcens {s },,,.

5 fgb he vecor s s called sgnal vecor.,,..., M.. e ay vsalze sgnal vecors as a se of M pons n an densonal cldean space, whch s also called sgnal space he sqared-lengh of any vecor s s gven by nner prodc or he do prodc of s, here s are he eleens of s wo vecors are orhogonal f her nner prodc s zero he energy of he sgnal s gven by d sbsng he vale s fro eqaon 6. d nerchangng he order of saon and negraon d

6 snce fors an orhonoral se, he above eqaon redce o hs shows ha he energy of he sgnal s s eqal o he sqared-lengh of he sgnal vecor s he cldean dsance beween he pons represened by he sgnal vecors s and s s d Response of ban of correlaors o nosy np Receved gnal s gven by,,3..., M 6.6 where s AG wh Zero Mean and PD / Op of each correlaor s a rando varable defned by d o,, he frs Coponen s deernsc qany conrbed by he ransed sgnal, s defned by d 6.8 he second Coponen s a rando varable de o he presence of he nose a he np, s defned by o d 6.9 le ' s a new rando varable defned as

7 sbsng he vales of fro 6.6 and fro 6.7 we ge whch depends only on nose a he fron end of he recever and no a all on he ransed sgnal s. hs we ay epress he receved rando process as ow we ay characerze he se of correlaor op, { }, o, snce he receved rando process s Gassan, we dedce ha each s a Gassan rando varable. Hence, each s characerzed copleely by s ean and varance. he nose process has zero ean, hence he rando varable eraced fro also has zero ean. hs he ean vale of he h correlaor op depends only on as Mean and varance: varance of s gven by 6. ' ' ' ' ' b eqn fro eqon fro sbsng Var σ

8 σ σ sbsng he vale of fro eqn 6.9 d d R w d d d d, d d 6. σ where R w, aocorrelaon fncon of he nose process.cence he nose s saonary, wh psd /,R w, depends only on he e dfference - and epressed as R w, δ 6. sbsng hs vale n he eqaon 6. we ge δ d d d cence he have n energy, he above eqaon redce o σ hs shows ha all he correlaor ops { }, o have a varance eqal o he psd o / of he addve nose process. cence he fors an orhogonal se, hen he are ally ncorrelaed, as shown by for all

9 nce he are Gassan rando varables, fro he above eqaon s pled ha hey are also sascally ndependen. d d d d d R d d Cov w, δ