Chaper 6 DEECIO AD EIMAIO: Fundamenal ssues n dgal communcaons are. Deecon and. Esmaon Deecon heory: I deals wh he desgn and evaluaon of decson makng processor ha observes he receved sgnal and guesses whch parcular symbol was ransmed accordng o some se of rules. Esmaon heory: I deals wh he desgn and evaluaon of a processor ha uses nformaon n he receved sgnal o exrac esmaes of physcal parameers or waveforms of neres. he resuls of deecon and esmaon are always subec o errors Model of dgal communcaon sysem Consder a source ha ems one symbol every seconds, wh he symbols belongng o an alphabe of M symbols whch we denoe as m, m,...... m M. We assume ha all M symbols of he alphabe are equally lkely. hen p p m emed M for all
he oupu of he message source s presened o a vecor ransmer producng vecor of real number.,,..., M.. Where he dmenson M. he modulaor hen consrucs a dsnc sgnal s of duraon seconds. he sgnal s s necessarly of fne energy. he Channel s assumed o have wo characerscs: Channel s lnear, wh a bandwdh ha s large enough o accommodae he ransmsson of he modulaor oupu s whou dsoron. he ransmed sgnal s s perurbed by an addve, zero-mean, saonary, whe, Gaussan nose process. such a channel s referred as AWG addve whe Gaussan nose channel GRAM CHMID ORHOGOALIZAIO PROCEDURE: In case of Gram-chmd Orhogonalzaon procedure, any se of M energy sgnals {} can be represened by a lnear combnaon of orhonormal bass funcons where M. ha s we may represen he gven se of real valued energy sgnals,....... M each of duraon seconds n he form................ M.... M M........ M,,... M 6.
Where he Co-effcen of expanson are defned by,,..... M d,,... 6. Vecor he basc funcons,... are orhonormal by whch f d f 6. he co-effcen may be vewed as he h elemen of he dmensonal herefore ' ' ' ' =,,...... M Le Vecor 4 4
Geomerc nerpreaon of sgnal: Usng orhonormal bass funcons we can represen M sgnals as Coeffcens are gven by,,..., M 6.4 d,,..., M,,..., 6.5 Gven he se of coeffcens {s }, =,,. operang as npu we may use he scheme as shown n fga o generae he sgnal s = o M. I consss of a bank of mulplers, wh each mulpler suppled wh s own basc funcon, followed by a summer. fga conversely gven a se of sgnals s = o M operang as npu we may use he scheme shown n fg b o calculae he se of coeffcens {s }, =,,.
fgb.,,..., M.. he vecor s s called sgnal vecor We may vsualze sgnal vecors as a se of M pons n an dmensonal Eucldean space, whch s also called sgnal space he squared-lengh of any vecor s s gven by nner produc or he do produc of s, E Where s are he elemens of s wo vecors are orhogonal f her nner produc s zero he energy of he sgnal s gven by E d subsung he value s from equaon 6. ] [ k [ k k ] d nerchangng he order of summaon and negraon E k k k d
snce forms an orhonormal se, he above equaon reduce o E hs shows ha he energy of he sgnal s s equal o he squared-lengh of he sgnal vecor s he Eucldean dsance beween he pons represened by he sgnal vecors s and s k s k [ ] d Response of bank of correlaors o nosy npu Receved gnal s gven by k k W,,..., M 6.6 where W s AWG wh Zero Mean and PD / Oupu of each correlaor s a random varable defned by o W d,,... 6.7 he frs Componen s deermnsc quany conrbued by he ransmed sgnal, s defned by d 6.8 he second Componen W s a random varable due o he presence of he nose a he npu, s defned by o W W d 6.9
le ' s a new random varable defned as subsung he values of from 6.6 and from 6.7 we ge whch depends only on nose W a he fron end of he recever and no a all on he ransmed sgnal s. hus we may express he receved random process as ow we may characerze he se of correlaor oupu, { }, = o, snce he receved random process s Gaussan, we deduce ha each s a Gaussan random varable. Hence, each s characerzed compleely by s mean and varance. Mean and varance: he nose process W has zero mean, hence he random varable W exraced from W also has zero mean. hus he mean value of he h correlaor oupu depends only on as varance of s gven by 6. ' ' ' W W W W W ' ' W x E W bu E W W E eqn from E m ] [ ] [ ] [ 6.7 ] [ ] [ 6.7 ] [ ] [ ] [ x x x E W equon from E m subsung m E Var
subsung he value of W from eqn 6.9 x x E E W d W u u W W u u E[ W W u] u R w u du d du d du, u d du 6. where R w, u = E[ W W u ] auocorrelaon funcon of he nose process W.cence he nose s saonary, wh psd /,R w,u depends only on he me ference -u and expressed as R w, u u 6. subsung hs value n he equaon 6. we ge x d u u d du cence he have un energy, he above equaon reduce o x hs shows ha all he correlaor oupus { }, = o have a varance equal o he psd o / of he addve nose process W. cence he forms an orhogonal se, hen he are muually uncorrelaed, as shown by for all
Cov[ k ] E[ E[ E m E[ W W ] W k x k k d u R, u d du k k w u u d du u d m k k x k k W u u du nce he are Gaussan random varables, from he above equaon s mpled ha hey are also sascally ndependen. ] ] k Deecon of known sgnals n nose Assume ha n each me slo of duraon seconds, one of he M possble sgnals,....... M s ransmed wh equal probably of /M. hen for an AWG channel a possble realzaon of sample funcon x, of he receved random process s gven by x w,,,..., M where w s sample funcon of he whe Gaussan nose process W, wh zero mean and PD /. he recever has o observe he sgnal x and make a bes esmae of he ransmed sgnal s or equvalenly symbol m he ransmed sgnal s, = o M, s appled o a bank of correlaors, wh a common npu and suppled wh an approprae se of orhonormal basc funcons, he resulng correlaor oupus defne he sgnal vecor. knowng s as good as knowng he ransmed sgnal self, and vce versa. We may represens s by a pon n a Eucldean space of dmensons M.. uch a pon s referred as ransmed sgnal pon or message pon. he collecon of M message pons n he Eucldean space s called a sgnal consellaon. When he receved sgnal x s appled o he bank o correlaors, he oupu of he correlaor defne a new vecor x called observaon vecor. hs vecor x fers from he sgnal vecor s by a random nose vecor w x w,,,..., M
he vecors x and w are sampled values of he random vecors and W respecvely. he nose vecor w represens ha poron of he nose w whch wll nerfere wh he deeced process. Based on he observaon vecor x, we represen he receved sgnal sby a pon n he same Eucldean space, we refer hs pon as receved sgnal pon. he relaon beween hem s as shown n he fg Fg: Illusrang he effec of nose perurbaon on locaon of he receved sgnal pon In he deecon problem, he observaon vecor x s gven, we have o perform a mappng from x o an esmae of he ransmed symbol, n away ha would mnmze he average probably of symbol error n he decson. he maxmum lkelhood deecor provdes soluon o hs problem. Opmum ransmer & recever Probably of error depends on sgnal o nose rao As he R ncreases he probably of error decreases An opmum ransmer and recever s one whch maxmze he R and mnmze he probably of error.
Correlave recever Observaon Vecor x For an AWG channel and for he case when he ransmed sgnals are equally lkely, he opmum recever consss of wo subsysems.recever consss of a bank of M produc-negraor or correlaors Φ,Φ.Φ M orhonormal funcon he bank of correlaor operae on he receved sgnal x o produce observaon vecor x
. Implemened n he form of maxmum lkelhood deecor ha operaes on observaon vecor x o produce an esmae of he ransmed symbol m = o M, n a way ha would mnmze he average probably of symbol error. he elemens of he observaon vecor x are frs mulpled by he correspondng elemens of each of he M sgnal vecors s, s s M, and he resulng producs are successvely summed n accumulaor o form he correspondng se of Inner producs {x, s k } k=,..m. he nner producs are correced for he fac ha he ransmed sgnal energes may be unequal. Fnally, he larges n he resulng se of numbers s seleced and a correspondng decson on he ransmed message made. he opmum recever s commonly referred as a correlaon recever MACHED FILER cence each of he orhonormal basc funcons are Φ,Φ.Φ M s assumed o be zero ousde he nerval. we can desgn a lnear fler wh mpulse response h, wh he receved sgnal x he fer oupu s gven by he convoluon negral y x h uppose he mpulse response of he sysem s h hen he fler oupu s y d x d samplng hs oupu a me =, we ge y x d Φ s zero ousde he nerval,we ge y x d y = x where x s he h correlaor oupu produced by he receved sgnal x. A fler whose mpulse response s me-reversed and delayed verson of he npu sgnal s sad o be mached o. correspondngly, he opmum recever based on hs s referred as he mached fler recever. For a mached fler operang n real me o be physcally realzable, mus be causal.
For causal sysem h causaly condon s sasfed provded ha he sgnal s zero ousde he nerval Maxmzaon of oupu R n mached fler Le x = npu sgnal o he mached fler h = mpulse response of he mached fler w =whe nose wh power specral densy o / = known sgnal Inpu o he mached fler s gven by x w scence he fler s lnear, he resulng oupu y s gven by y n where and n are produced by he sgnal and nose componens of he npu x.
he sgnal o nose rao a he oupu of he mached fler a = s R E[ n ] am s o fnd he condon whch maxmze he R le f h H f 6. are he Fourer ransform pars, hence he oupu sgnal H f f exp f s gven by oupu a = s H f f exp f 6.4 For he recever npu nose wh psd o / he recever oupu nose psd s gven by f H f 6.5 and he nose power s gven by E[ n ] f H f 6.6 R subsung he values of eqns 6.4 & 6.5 n 6. we ge H f f exp f H f 6.7
usng chwarz s nequaly Eqn 6.6 s equal when f = k * f le f = Hf & f = f exp f under equaly condon * Hf = K f exp f 6.9 hus subsung n 6.6 we ge he value f f f f 6.8 H f f exp f H f f subsung n eqn 6,7 and smplfyng R f Usng Raylegh s energy heorem d f E, energy of he sgnal R E, max 6. Under maxmum R condon, he ransfer funcon s gven by k=, eqn 6.9 H op f * f exp f he mpulse response n me doman s gven by h op f exp[ f ]exp f hus he mpulse response s folded and shfed verson of he npu sgnal
MACHED FILER Φ = npu sgnal h = mpulse response W =whe nose he mpulse response of he mached fler s me-reversed and delayed verson of he npu sgnal h For causal sysem h Mached fler properes PROPERY he specrum of he oupu sgnal of a mached fler wh he mached sgnal as npu s, excep for a me delay facor, proporonal o he energy specral densy of he npu sgnal. le denoes he Fourer ransform of he fler oupu, hence f f H op f f * f f exp f f susung exp f 6. from 6.9
PROPERY he oupu sgnal of a Mached Fler s proporonal o a shfed verson of he auocorrelaon funcon of he npu sgnal o whch he fler s mached. he auocorrelaon funcon and energy specral densy of a sgnal forms he Fourer ransform par, hus akng nverse Fourer ransform for eqn 6. R A me = R E where E s energy of he sgnal PROPERY he oupu gnal o ose Rao of a Mached fler depends only on he rao of he sgnal energy o he power specral densy of he whe nose a he fler npu. R a he oupu of mached fler s eqn 6. R E[ n ] 6. oupu of mached fler s H f f exp f sgnal power a = H f f exp f nose psd a he oupu of recever s f H f
nose power s E[ n ] subsung he values n 6. we ge usng chwarz s nequaly f Eqn 6.4 s equal when f = k * f le f = Hf & f = f exp f under equaly condon * Hf = K f exp f 6.5 hus subsung n 6.4 we ge he value H f H f f exp f R 6. H f f f f f 6.4 H f f exp f H f f subsung n eqn 6, and smplfyng R f Usng Raylegh s energy heorem d f E, energy of he sgnal R E,max 6.6
PROPERY 4 he Mached Flerng operaon may be separaed no wo machng condons; namely specral phase machng ha produces he desred oupu peak a me, and he specral amplude machng ha gves hs peak value s opmum sgnal o nose densy rao. In polar form he specrum of he sgnal beng mached may be expressed as f f exp f where f s magnude specrum and f s phase specrum of he sgnal. he fler s sad o be specral phase mached o he sgnal f he ransfer funcon of he fler s defned by he oupu of such a fler s H f H f exp f f ' H f f exp f H f f exp[ f ] he produc H f f s real and non negave. he specral phase machng ensures ha all he specral componens of he oupu add consrucvely a =, here by causng he oupu o aan s maxmum value. ' ' f H f For specral amplude machng H f f
Problem-: Consder he four sgnals s, s, s and s 4 as shown n he fg-p.. Use Gram-chmd Orhogonalzaon Procedure o fnd he orhonormal bass for hs se of sgnals. Also express he sgnals n erms of he bass funcons. Fg-P.: gnals for he problem -. oluon: Gven se s no lnearly ndependen because s 4 = s + s ep-: Energy of he sgnal s E s d s E Frs bass funcon for ep-: Coeffcen s s s d Energy of s E s d
econd Bass funcon for s E s s ep-: Coeffcen s : d s s Coeffcen s d s s Inermedae funcon g = s - s Φ - s Φ g = for / < < / hrd Bass funcon for d g g he correspondng orhonormal funcons are shown n he fgure-p.. Fg-P.: Orhonormal funcons for he Problem- Represenaon of he sgnals 4
PROBLEM-: Consder he HREE sgnals s, s and s as shown n he fg P.. Use Gram-chmd Orhogonalzaon Procedure o fnd he orhonormal bass for hs se of sgnals. Also express he sgnals n erms of he bass funcons. Fg-P.: gnals for he problem -. oluon: he bass funcons are shown n fg-p.. Fg-P.: Orhonormal funcons for he Problem- Correspondngly he represenaon of he sgnals are: 4 4
PROBLEM-: Consder he sgnal s n fg-p. a Deermne he mpulse response of a fler mached o hs sgnal and skech as a funcon of me. b Plo he mached fler oupu as a funcon of me. c Wha s Peak value of he oupu? oluon: Fg P. he mpulse response of he mached fler s me-reversed and delayed verson of he npu sgnal, h = s- and he oupu of he fler, y = x * h. Gven s = + for < <.5 - for.5 < <. a Wh =, he mpulse response h s h = - for < <.5 + for.5 < <.
Fg. P. b he oupu of he fler y s obaned by convolvng he npu s and he mpulse response h. he correspondng oupu s shown n he fg. P.. c he peak value of he oupu s. un. Assgnmen Problem: Fg. P. pecfy a mached fler for he sgnal shown n Fg.-P4. kech he oupu of he fler mached o he sgnal s appled o he fler npu. Fg P4.