Optimum Receivers for the AWGN channel

Transcription

1 Opiu Receivers for he AWG chael Coes Opiu Receiver... Opiu Deodulaor... Correlaio Deodulaor... Mached filer Deodulaor... 5 Frequecy doai ierpreaio of ached filer... 8 he Opiu Deecor... Miiu Disace Deecio... he Maxiu Lielihood Sequece Deecor... 5 A syol y syol MAP deecor for sigals wih eory Perforace of he Opiu Receiver for Meory less Modulaio... 7 Proailiy of Error for Aipodal Biary Modulaio... 7 For Biary orhogoal sigals... 8 Copariso of Digial Modulaio Mehods... Perforace aalysis for wire-lie ad radio couicaio Syses... Regeeraive repeaers... Li Budge aalysis i radio co. Syses... AWG: Addiive Whie Gaussia oise Ojecive : Receiver desig Perforace evaluaio (eory, o eory) oe: hese oes are preliiary ad are posed y he reques of he sudes. Please, repor o e all he isaes ha you fid i he docue. (uqaiel@fup.edu.sa). his aerial is for he sole purpose of i class usage. Please oserve he copyrigh of he origial auhors for ay coe i hese oes. Dr. Ali Muqaiel Digial Couicaios I

2 Opiu Receiver s () + r() () r () s () + () { s( ),,,..., M} S( f ) W / Hz Ojecive: Upo oservaio of r(), wha is he opiu receiver desig i ers of proailiy of aig error. receiver deodulaor deecor Deodulaor : covers he received wavefor r() io a -diesioal vecor r [ r r r ] where is he diesio of he rasied sigal. Deecor: Is o decide which of he M possile sigals wavefors was rasied ased o r. ypes of deecors. he opiu deecor. Maxiu lielihood sequece deecor 3. Syol y syol MAP Opiu Deodulaor. Correlaors. Mached filers Correlaio Deodulaor Decoposes he received sigal-o-oise raio io -diesioal vecors. Liearly weighed orhogoal asis fucios {f ()}, where {f ()} spas he sigal space u o he oise space. We ca show ha he oise ers ha falls ouside he sigal space is irreleva o he deecio. Dr. Ali Muqaiel Digial Couicaios I

3 f () r ( ) d r () f () ( ) d r o he deecor f () ( ) d r() f () d [ s () + ()] f () d r s +,, where s s () f () d,,, () f () d,,, ' ' () () + () + () () + () ' f () () () r s f f r f r ' () is zero ea Gaussia oise process. I is he urepreseed par of oise ihere. ' () is Gaussia ecause i is he sapled oupu of a liear filer excied y a Gaussia ipu. E( ) E[ ()] f () d for all E( ) E[ ( τ) ( τ)] f ( ) f ( τ) ddτ δ δ δ( τ) f () ( ) f τ ddτ f () f ( τ) d δ, whe ad zero oherwise. 3 Dr. Ali Muqaiel Digial Couicaios I

4 { } are zero ea ucorrelaed rado variale wih coo covariace σ Er [ ] Es [ + ] s σr σ Gaussia ucorrelaed iplies saisically idepede. r [ r r r ] prs ( / ) prs ( / ),,..., M ( r s ) whe p( r/ s ) exp[ ],,,... π he joi codiioal pdf ( r s ) prs ( / ) exp[ ],,,..., M ( π ) / As a fial sep we ca show ha ( r, r,..., r ) are sufficie saisics. o addiioal releva iforaio ca e exraced fro ' (). ' E [ () r ] ucorrelaed proof p 35 Gaussia ad ucorrelaed iplies saisically idepede which iplies igore ' () Exaple M-ary PAM g() a E g () d a d a g PAM oe asis fucio 4 Dr. Ali Muqaiel Digial Couicaios I

5 f() / f() g () oherwise a he oupu of he correlaor r r() f () d r() d he correlaor ecoes a siple iegraor whe f() is recagular. r { [ s() ()} d [ s() d () d] + + r s + E [ ] σ E[ ( ) ( τ) ddτ] E[ ( ) ( τ)] ddτ δ( τ) ddτ pdf of he sapled oupu ( r s) prs ( / ) exp[ ] π Mached filer Deodulaor We use filers f ( ) h () oherwise y () r( τ) h ( τ) dτ r( τ) f ( + τ) dτ,,... If we saple a he ed he period 5 Dr. Ali Muqaiel Digial Couicaios I

6 y ( ) r( τ) f ( τ) dτ r,,... Mached filer: A filer whose ipulse respose h()s(-) or s() where s() is assued o e cofied o he ie ierval. s() y() A f ( ) r () f ( ) f ( ) Propery of Mached filer: If a sigal s() is corruped y AWG, he filer wih a ipulse respose ached o s() axiizes he oupu sigal o oise raio (SR). Proof : 6 Dr. Ali Muqaiel Digial Couicaios I

7 y ( ) r( τ) h ( τ) dτ s( τ) h ( τ) dτ + ( τ) h ( τ) dτ a y ( ) s( τ) h ( τ) dτ + ( τ) h ( τ) dτ SR y ( ) + y ( ) s s Ey y ( ) [ ( )] [ ( )] var [ ( )] [ ( ) ( )] ( ) ( ) where E y oise iace E y Eτ h τ h ddτ SR δ( τ) h ( τ) h ( dd ) τ h ( ) d [ s( τ) h( τ) dτ] [ h( τ) s( τ) dτ] h ( ) d h ( ) d Ca we axiize he ueraor while he deoiaor is held cosa. Cauchy-Schwarz iequaliy: [ g () g () d] g () d g () d Wih equaliy whe g () c g () g () h(), g () s(-) ore whe h() c s(-), c drop fro he ueraor ad deoiaor E SR s () d Propery: he oupu SR fro he ached filer depeds o he eergy of he wavefor s() u o o he deails (shape) of s(). 7 Dr. Ali Muqaiel Digial Couicaios I

8 Frequecy doai ierpreaio of ached filer Y( f) S( f) e jπ f jπ f jπ f jπ f Y ( ) Y ( f ) e df S( f ) e e df s a Y ( ) S( f ) df s () d E Pasaval ' s relaio s oise a he oupu of he ached filer Φ ( f) H( f) ( ) ( ) ( ) P Φ f df H f df S f df E Ps E E Ps Ys ( ) > SR P E Mached filer ad correlaor a equivale a u ached filer is iue o ie jiers. Exaple 5.. M4 i-orhogoal sigals f () f () h() f ( ) h () f ( ) y () s y () s A A oe he respose o s () is evaluaed a 8 Dr. Ali Muqaiel Digial Couicaios I

9 r [ r r ] [ E + ] ( E) E SR Addiioal Exaple (Mached Filers) Cosider he sigal s() A A a) Deerie he ipulse respose of a filered ached o his sigal ad sech i. h()s(-) A A ) Deerie he ached filer oupu as fucio of ie. c) his is doe y covoluio, we ay as aou he values a he pea (figure) A /4 E A 4 A A A pair of pulses ha are orhogoal o each oher over he ierval [,], are used for wo diesioal ached filer A A ipu Filered ach o s() Filered ach o s() Oupu Oupu A A 4 Wha is he oupu of he ached filer a? 9 Dr. Ali Muqaiel Digial Couicaios I

10 a) Whe s () is applied o he wo raches. ) Whe s () is applied o he wo raches. Firs rach is show i prole 4. Secod rach is zero [ geeralize o all orhogoal sigals]. Addiioal Proles o Mached filers. Aoher ehod for approxiaig realizaio of ached filer is he (RC) low pass filer [iegraor]. R ipu C oupu H( f), f + j f π RC f he ipu sigal is recagular of pulse apliude A ad duraio. Ojecive: Opiize he selecio of 3-dB cuoff frequecy f of he filer. So ha he pea oupu SR is axiized. Show ha f./ is he opiu. Copared o ached filer db loss. s() A[ exp( π f)] he pea value of he oupu power is Pou A[ exp( π f )] f is he 3-dB cuoff frequecy of he RC filer. ou π f df ( f ) + f he correspodig value for he SR A ( SR) ou [ exp( π f )] π f Differeiaig wih respec o (f ) ad seig he resul equal o zero. he axiu value of (SR) ou is a f./. Dr. Ali Muqaiel Digial Couicaios I

11 he Opiu Deecor r Deecor assue o eory Ojecive: Maxiize he proailiy of correc decisios. Poserior proailiies. P(sigal s was rasied r ) Ps ( r ) where,,..m Hece he ae axiu a poseriori proailiy (MAP) Pr ( s) Ps ( ) Usig Baye s rule: Ps ( r) Pr ( ) Ps ( ) a priori proailiy of he h sigal. M Pr ( ) Pr ( s) Ps ( ) Whe he M-sigals are equally proale Ps ( ) /M for all. he sae rule ha axiizes Ps ( r) is equivale o axiizig Pr ( s ) Lielihood fucio: is he codiioal PDF Pr ( s ) or ay oooic fucio of i. Maxiu lielihood (ML) crierio. MAP ML if { s } is equi-proale. For AWG, he lielihood fucio is give y 5.. ( r s ) Pr ( s) exp / ( π ) or l Pr ( s) l( π ) ( r s ) Maxiizig l Pr ( s ) is equivale o iiizig Dr ( s ) ( r s ) where Dr ( s ) is Euclidea disace ad,, M Dr. Ali Muqaiel Digial Couicaios I

12 Miiu Disace Deecio Aoher ierpreaio of ML crierio. + D( r, s ) r rs s r rs. + s where,,3 M D ' (, rs) rs. + s ' Le Crs (, ) D(, rs) o ge rid of he ius. ow, we axiize C raher ha iiize D. Crs (, ). rs. s s ca e eliiaed if eergy is fixed for all,, M, u cao if sigals have uequal eergy (PAM). C(, r s ) r() s () d E,,,,M A aleraive realizaio of he opiu AWG receiver. s () E ( ) d r () s () ( ) d E Selec he Larges s () ( ) d E Suary: Opiu ML Drs ) copue (, ) or D ' (, rs ) disace erics ad chooses he salles or Dr. Ali Muqaiel Digial Couicaios I

13 Crs ) copue (, ) correlaor erics ad choose he larges 3) ML MAP if eqiproale { s }, oherwise PM (, r s) P( r s) P( s) Exaple: Le us assue ha we are sedig oe of wo levels eiher or 6 as show i he figure. o illusrae he iporace of suracig he eergy of he syol. We will cosider wo cases. he firs oe will assue ha he received apliude is 4 ad he we will cosider he apliude o e 3. s () 6 s () Crs (, ). rs. s For he iddle case wih apliude 4 we ge a fair copariso afer reovig he ias 4 r() Crs (, ). rs. s Cr (, s ).(4).() Crs (, ).(4).(6) For he case ha he received sigal wih apliude of 3 we ge he wrog decisio uless we reove he ias Crs (, ).(3).() Crs (, ).(3).(6) r() Exaple: I iary PAM s -s E Prior proailiies P(s ) ρ, P(s ) - ρ Deerie he erics of he opiu MAP for AWG. 3 Dr. Ali Muqaiel Digial Couicaios I

14 ( ) r ± E + y zero ea ad σ oe: he variace of he sapled oise is /. I geeral he oise power, P B, Accordig o yquis B/(), Whe looig a he eergy we EP ( r E ) Pr ( s ) exp πσ σ ( r+ E ) Pr ( s) exp πσ σ ρ PM ( r, s ) ρ p( r s ) exp ( r E ) πσ σ ρ PM ( r, s ) ( ρ) p( r s ) exp PM ( r, s ) > PM ( r, s ) he choose s ( r+ E ) πσ σ If PM ( r, s ) PM ( r, s ) s > < s ( r+ E ) ( r E ) > exp < PM ( r, s ) ρ PM ( r, s ) ρ σ s s ( r+ E) ( r E) > ρ l σ < ρ Er E s > < s s s ρ ρ σ l l ρ 4 ρ E R s R τ s h ) If ρ ½, τ h ) If ρ ½, owledge of or If he M sigals are equi-proale. E is required for opial deecio. he axiu Lielihood (ML) iiize P(correc decisio) 4 Dr. Ali Muqaiel Digial Couicaios I

15 M P() c p( r s ) dr ML M M P() c p( s r ) p() r dr MAP M R R R is he regio for correc decisio. Explai he Uio oud Cocep of QPSK P(e)-P(c) he Maxiu Lielihood Sequece Deecor If o eory (syol-y-syol deecor) is opial (iiu proailiy of error) Meory > successive syols are ierdepede. he axiu lielihood (ML) sequece deecor: searches for he iiu Euclidea disace pah hrough he rellis ha characerizes he eory i he rasied sequece. Exaple RZI (PAM) (zero : lie efore, oe : flip) s -s E r ± E + ( r E ) Pr ( s ) exp πσ σ ( r + E ) Pr ( s) exp πσ σ K ( ) ( ) ( ) K ( r s ) Pr (, r,... r s ) Pr ( s ) exp πσ σ Maxiize he aove proailiy. By aig he logarih ad cosider oly hose releva er. ( ) K ( ) Dr (, s ) ( r s ) For iary we eed o search sequeces, where is he sequece legh. Vieri algorih: is a sequeial rellis search algorih for perforig ML sequece deecio Used for decodig covoluioal codes Assue iiial sae s he exaple of Biary RZI 5 Dr. Ali Muqaiel Digial Couicaios I

16 / E / E / E / E / E / E / E / E 3 A ad so o here are wo arrows eerig he sae, we choose he iiu disace survivor. D (,) ( r + E ) + ( r + E ) For sae we do siilar D (,) ( r + E ) + ( r E ) D (, ) ( r E ) + ( r E ) A 3 Suppose he survivors are (,) ad (,) D (,,) D (,) + ( r + E ) 3 D(,,) D(,) + ( r3+ E ) ad D (,,) D (,) + ( r E ) 3 D (,,) D (,) + ( r E ) 3 D (,) ( r E ) + ( r + E ) Usig Vieri i his exaple he uer of pah searched is reduced y a facor of wo a each sage. he eory legh is L (L i he previous exaple) You ae a decisio whe he survivor pah agree. Variale delay egaive?? Pracically a 5L ad he ae decisio a eve he efore -5L is alos ideical. Mae a decisio. 6 Dr. Ali Muqaiel Digial Couicaios I

17 Exaple (i he previous RZI) Le E ad he deodulaed sequece is (.9,-.8,.3,-.,.,.5,-.7) D (, ) (.9 + ) + (.8 + ) D (,) (. 9 + ) (.8 + ) (.) D (,)(.9+) +(-.8-) D (,)(.9-) +(-.8-) D D (,, ) D(,) (.3 ) (,,) D(, ) (.3 ) D D (,,) D (,) (.3 ) (,, ) D (, ) + (.3 + ) A syol y syol MAP deecor for sigals wih eory. Opiu (iiize syol error) If M is large > large copuaioal coplexiy. Used for covoluio codes ad uro codig. Beyod he scope. Perforace of he Opiu Receiver for Meory less Modulaio Proailiy of Error for Aipodal Biary Modulaio E E R s R τ s h PAM aipodal s () -s () r s + E + AWG 7 Dr. Ali Muqaiel Digial Couicaios I

18 Pr ( s) Pr ( s) π π e e ( r E ) / ( r+ E ) / ( r+ E ) / P ( e s ) P ( r s ) dr e dr π r E By replacig variales x /, dx dr > dr dx x, > r π E / E / r > x x / E x / E e dx Q π x / Q( x) e dx π x e dx Assuig s ad s are equiproale E P Pe ( / s) + Pe ( / s) Q d E > E d E 4 oe P e depeds o he raio d P Q For Biary orhogoal sigals s E E E s 8 Dr. Ali Muqaiel Digial Couicaios I

19 s [ E ] s [ E ] d E If s is rasied r [ E + ] Cr (, s ) rs. s Cr (, s).[ E + ].[ E] E () Cr (, s).[ E + ].[ E ] E () () Ca e furher siplified o E E ad () Ca e furher siplified o E + E E E + E Proailiy of error Cr (, s) > Cr (, s) Pe [ s] PCr [ (, s) > Cr (, s)] E + E E < E E E + ( ) E < > E Pe ( s) P [ > E ] ad are zero ea Gaussia rado variales x - is zero ea Gaussia rado variale wih variace σ / x x / E P( > E ) e dx e dx Q π E π / E Because of he syery s is he sae. γ SR/i Copare orhogoal wih aipodal (facor of icrease i eergy) log 3dB d E for orhogoal d 4E for aipodal 9 Dr. Ali Muqaiel Digial Couicaios I

20 Figure 5..4 ( ) P Q γ Proailiy of error P ( ) P Q γ ρ r ρ r SR per i, γ ( db ) Explai he cocep of Uio oud I addiio o he aove aipodal ad orhogoal exaples, we ca exed he aalsys o oher odulaio echiques. For exaple, May orhogoal sigals Bi-orhogoal Siplex M-ary PAM ( M ) d E g PM Q M ( M ) 6(log M) E av Q M ( M ) 6 Because deg P av M M-ary PSK QAM Copariso of Digial Modulaio Mehods P 6-9. Power specral Efficiecy! Readig Maerial Quiz Dr. Ali Muqaiel Digial Couicaios I

21 Perforace aalysis for wire-lie ad radio couicaio Syses Regeeraive repeaers s() r() α () Maheaical odel for chael wih aeuaio ad addiive oise is r () αs () + () E P Q PAM Biary. Repeaers assuig sigle error a a ie. E P Q a repeaer.. Why o exac equal sig? (error cacellaio) Aalog: Required E / reduced y E P Q A decisio receiver oe receiver is coeced + repeaer. Exaple repeaer -5-5 E Q > -7 E Q > SR.3dB -5 E Q > SR 9.6 db db Li Budge aalysis i radio co. Syses Microwave lie-of-sigh rasissio Li udge aalysis PG AR PR () 4π d he desig should specify P, size of aea (rasi ad receiver) Dr. Ali Muqaiel Digial Couicaios I

22 SR required o achieve a give perforace ad daa rae. PG G : aea rasi gai, G for isoropic aea. 4π d PG : Effecive radiaed power. ERP or EIRP copared wih isoropic aea A R : effecive area of he aea, c λ f, c 3X 8 /s Susiue () i () PG GR PR (4 πd / λ) A R GRλ 4π λ Free space pah loss facor L s 4π d Addiioal losses aospheric L a. P PG G LL R R s a () Calculaio of he aea gai is aea specific ad depeds o (diesios) diaeer D, Illuiaio efficiecy facor Effecive area A R Area A Bea widh θ B Dish, hor ec (P R ) db (P ) db + (G ) db + (G R ) db + (L s ) db + (L o ) db Exaple A geosychroous saellie ori (36, ) Power radiaed is W db aove W -- dbw rasi aea gai 7dB ERP 37 dbw 3- paraolic aea a 4 GHz dowli η.5 efficiecy facor π D G R η 39dB λ λ Ls 95.6dB 4π d (P R ) db dbw P R. X - W is his low or high? Dr. Ali Muqaiel Digial Couicaios I

23 Wha aer is he SR. oise is fla for up o - Hz B W/Hz K B : is Bolza s cosa.38x -3 oal oise W E P R PR Perforace is depede o R P R E R Req Exaple for he sae previous exaple. P R. X - W (-9.6 dbw) 4. X - W/Hz, P r W K B W -3.9 dbw/hz P R E db Hz SR is db R db db wih respec o i/sec 6.9 Mps 4 PCM (64 ps) he iroduced safey argi R db P R E M db req (P ) dbw + (G ) db + (G R ) db + (L a ) db + (L s ) db E M db req 3 Dr. Ali Muqaiel Digial Couicaios I