ECSE413B: COMMUNICATIONS SYSTEMS II Tho Le-Ngoc, Winter Digital Transmission in AWGN Optimum Receiver Probability of Error

Transcription

1 ECE43B: COUICAIO YE II ho Le-go, Wnter 8 Bas Dgtal odulaton ehnques: Dgtal ransmsson n AWG Optmum Reever Proalty of Error Dgtal odulaton ehnques: AK, PK, QA, FK

2 Elements of a dgtal ommunatons ln CODIG CHAEL ODE CHAEL IPU IGAL IPU RADUCER OURCE ECODER X CHAEL ECODER ODULAOR wth CHAEL CODIIOER x AFE Input X { n, n {+, -}} Output X { n, n {+, -} } Ojetve: mnmze Pr{ n n } resoures: POWER & BADWIDH Channels: AWG, II, ULIPAH CHAEL OUPU IGAL OUPU RADUCER OURCE DECODER X CHAEL DECODER DEODULAOR wth CHAEL EQUALIZER Rx AFE DIGIAL RAIIO Bas Dgtal odulaton ehnques ho Le-go PAGE

3 IE-LIIED IGALLIG CHEE I A AWG EVIROE BIARY EQ. { } BI-O- YBOL COVERER {a j } x s(t) r(t) COHERE RECEIVER {ˆ a j } YBOL-O- BI COVERER ˆ } { wth Pr{ wth Pr{ } / } / WG w(t) Input nary sequene: {} }, transmtted n t- /, : t nterval Every m ts: grouped to form symol m possle symols, ymol sequene: {a }, a ja,,,3,, m wth Pr{a ja }/. a j s transmtted n the nterval t-j s s /, s m : symol nterval. RAIER: generates s(t) g (t-j s ) for t-j s s / f a j A e.e., OE-O-OE APPIG AWG CHAEL: r(t) s(t)+w(t) where w(t): whte Gaussan nose, zero-mean, varane o /. Bas Dgtal odulaton ehnques ho Le-go PAGE 3

4 VECOR REPREEAIO OF IE-LIIED IGAL s(t) g (t-j s ) for t-j s s / f a j A : one-to-one orrespondene g (t): sgnalng element g (t) for t > /: tme-lmted and + s s / / g (t) dt E We want to fnd ( ) orthonormal funtons Φ (t),,,, so that g (t) gφ(t) sgnal element: waveform g (t) an e represented y -dmensonal vetor g ( g,, g ) ORHOORAL FUCIO: Φ (t) for t > / and + f j (t)dt Φ (t)φ * j f j Φ * j (t)φ () j j( (t) f Φ ( (t): REAL, {Φ (t), (),,,,},, forms an ORHOORAL BAI of dmensons. waveform g (t) an e represented y -dmensonal vetor g ( g,, g ): g (t) g Φ(t) where + * g g (t).φ (t)dt < + g + / dt / g g Φ (t) Φ (t) Σ WAVEFOR g (t) Φ * (t) Φ * (t) + / dt / g g Φ (t) Φ* (t) + / dt / g I RAIER g (t) g Φ(t) I RECEIVER g (g, g, g ), + * g g (t).φ (t)dt Bas Dgtal odulaton ehnques ho Le-go PAGE 4

5 VECOR CHAEL g j + / g dt j +w w(t) / Φ (t) Φ * (t) s(t) r(t) g + / j Σ g dt j +w Φ (t) Φ * (t) + / g j { g j } { g j } Φ (t) WAVEFOR CHAEL Φ* (t) + / / dt g j +w { g j + w} w(t): Whte Gaussan ose, zero mean and varane σ W /, nose omponents w s,,,, are Gaussan wth zero mean and varane of /. hey are also statstally ndependent. w s ρs+w OPIU RECEIVER VECOR CHAEL ĝ j x p w (x ) exp, p w (x) / pw (x ( π ) x ) exp / ( π ) Bas Dgtal odulaton ehnques ho Le-go, x x PAGE 5

6 OPIU RECEIVER nmze Pr{ â j A l a j A, l} mnmze Pr{ ŝ s l s g, l}, or maxmze Pr{ â j A a j A } maxmze Pr{ ŝ g s g } maxmze Pr{ ŝ s ρ g +w} optmum maxmum a posteror proalty (AP) reever: ax Pr{ ŝ s ρ g +w}: Proedure: For a reever vetor ρ, alulate all Pr{ g ρ},,, elet the ndex orrespondng to the AXIU value Pr{ g ρ} and delare Usng the Bayes rule: Pr{A B}.p(B)p(B A).Pr{A} for A: dsrete, B: ontnuous p(b): proalty l densty funton (pdf) of B p{ρ g}.pr{g} Pr{ g ρ} ax Pr{ g ρ} ax p{ρ g}.pr{g} p{ρ} wrtg wrtg s ˆ g If Pr{ g } }/,,,, wrtg ax Pr{ g ρ} wrtg ax p{ρ g} optmum reever maxmum lelhood (L) reever For g,,,,, alulate all p{ρ g }. elet orrespondng to the largest p{ρ L reever selets g, the most lely sgnalng vetor n produng ρ g } Bas Dgtal odulaton ehnques ho Le-go PAGE 6

7 AP and L n AWG hannel ransmtter: sends s g wth a pror proalty of Pr{g }. Reever: From the reeved sample r s +w where w: Gaussan (, o /), guess s AWG(, o / )CHAEL: f g sent ρ g+ w or, p( ρ g) exp [ π o ] r-g ln [ p( ρ g) ].5 5 ln [ π o] o r-g o axmum A Posteror (A P): Choose g orrespondng to max Pr{ g ρ} max p( ρ g ) Pr{ g } r-g max ln[ p( ρ g)pr{ g} ] max[ ln p( ρ g) + lnpr{ g} ] mn + (.5ln[ π o] lnpr{ g} ) o [ p ] axmum Lelhood (L): Choose ln Pr{ g }orrespondng to max p( ρ g ) max ln ( ρ g ) mn r-g e.., For general, elet orrespondng to the mnmum Euldean dstane r-g among all r-g m when Pr{ g } / (AP): maxln Pr { g ρ} mn r-g ( L) Bas Dgtal odulaton ehnques ho Le-go PAGE 7

8 AP and L: example of nary ase () ransmtter: From nary sequene { }, or wth a pror proalty Pr{ } and Pr{ }, respetvely, sends s g f or s g f. Reever: From the reeved sample r s +w where w: Gaussan (, o /), guess axmum A Posteror (AP): Choose $ f Pr{ r} > Pr{ r}, otherwse hoose $ P{ Pr{ } f, ( ) ( r r Λ AP r > ΛA P ) ln $ Pr{ r }, f Λ AP ( r ) < p( r )Pr{ }/ p( r) p( r ) Pr{ } From Bayes rule, ΛAP ( r ) ln ln + ln p ( r )Pr{ }/ pr ( ) pr ( ) Pr{ } ( ) Pr{ } ( ) ( pr Λ AP r ΛL ) +ΛP( ), ΛL ( r ) ln, ΛP( ) ln pr ( ) Pr{ } axmum Lelhood (L): Choose $ f ( ) ( ), otherwse hoose p r > p r $, f Λ L ( r ) > $, f Λ L ( r ) < when Pr{ } Pr{ } Λ ( ) Λ ( r) Λ ( r) P AP L Bas Dgtal odulaton ehnques ho Le-go PAGE 8

9 AP and L: example of nary ase () n AWG ( r AWG(, o / ): or, p( r ) exp π o ( r g ) ln ( ).5ln [ pr ] [ π ] o o p( r ) ( r g) ( r g) ΛL ( r ) ln, pr ( ) o Λ AP Pr{ r} ( r g) ( r ) ln Pr{ r} g o ( r g ) Pr{ } +ΛP( ), ΛP( ) ln o Pr{ } ) axmum Lelhood (L): Choose $ f p( r ) > p( r ), otherwse hoose $, f ( r g ) > ( r g ) $, f ( r g ) < ( r g ) For gene ral, elet orrespondng to the mnmum Euldean dstane r-g among all r-g m Bas Dgtal odulaton ehnques ho Le-go PAGE 9

10 PROBABILIY OF ERROR For g sent, the L reever maes an error f t dedes ρ s not mnmum.e. ρ g > ρ g for some n g n sˆ g,l l. hs event ours f and only f In the -dmensonal oservaton spae Z, the optmum reever estalshes dsjont zones Z as follows U Z Z Z I Z j O / for j Z {ρ : ρ g s mnmum} For an oservaton vetor ρ f ρ Z then the L reever delares that herefore the average proalty of error s Pe Pr{ρ Z g p y a ˆ A A }Pr{ A } was sent. for Pr{A },Pe Pr{ρ Z A } [ Pr{ρ Z A }] Pr{ρ Z A } P Pr{orret deson} ρ g where Pr{ρ Z A } p (ρ g)dρ πν exp dρ w / ( ) ρ g ρ g s mnmum s mnmum Bas Dgtal odulaton ehnques ho Le-go PAGE

11 VECOR REPREEAIO FOR A GEERAL BIARY IE-LIIED IGALIG CHEE In the nterval t-n /, g ( t n) f an, wth a pror proalty of / st () g ( t n ) f an, wth a pror proalty of / For a general nary sgnalng sheme wth tme-lmted, fnte-energy elements, g (t) and g (t), for a smple -D reever desgn, we an selet the orthonormal ass wth Φ (t) g (t) - g(t), E E Δ Δ g (t) - g (t) dt g (t) dt + g (t) dt g (t).g E Δ E + E E, E g(t).g(t)dt for g(t),g(t):real- valued d E + E E E ( γ ) where E ( E + E) : average energy per t E γ : orrelaton oeffent etween g, g (g ( t ) & g ( t )) and γ : E (E E )g (t) + (E E )g (t) a E)g(t) dt E hen, Φ (t), E (E E )g (t) + (E a (t)dt g )/ BOUDARY:(g +g φ (t) g g (t) g φ (t)+ g φ (t) and g (t) g φ (t)+ g φ (t), g (E -E )/d, g -(E -E )/d g g (E E -E / )/E a d E Δ E +E -E g - g g g φ (t) Bas Dgtal odulaton ehnques ho Le-go PAGE

12 PROBABILIY OF ERROR OF BIARY RAIIO I A AWG EVIROE For antpodal sgnalng: g (t)- g (t), -E E E, d 4E φ (t) g (t)/e / φ (t) : one dmenson, g - g E / * (t).g For orthogonal sgnalng: g (t)dt E, d E +E g E /[E +E ] / g -E /[E +E ] / g g [E E /(E +E )] / For g transmtted ( or ), reeve r g +n, where AWG n(w,w ) w,w : ndependent Gaussan wth zero mean and varane: o /. For g transmtted, error f w <-d/. For g transmtted, error f w >d/. w and hene r are rrelevant. he Rx onsders only r n deteton. P e Pr{ error g sent }Pr{ g sent } + Pr{ error g sent }Pr{ g sent } d d P e erf erf P e E erf ( γ ) d/ d/ x x Pr{ error g sent} Pr{ w d } ( ) exp exp pw x dx dx dx d / π π d x + d exp( v ) dv erf for v, dv dx, x d v, x + v + π Bas Dgtal odulaton ehnques ho Le-go PAGE

13 ERROR FUCIO x X s alled normalzed zero-meangaussan (Random) varale X: fx ( x) e π + x Q-funton: Q( y) Pr{ X > y} e dx, y π + + x v omplmentary error funton: erf( u) e dv e dx Q( u ), v x / u u π π x lower ounds: x, e Q( x) < π x x x / e upper ounds: x, Q( x) <, x or tghter: Q( x) < e π x Bas Dgtal odulaton ehnques ho Le-go PAGE 3

14 ELECIO OF g (t), g (t) erf s monotone-dereasng, erf() to redue P e we have to maxmze ( γ ) A. For the same average energy per t to nose power densty E /, we an mnmze P e y mnmze γ γ, mn γ when g ( t) g ( t) : AIPODAL IGALLIG P e erf E : the BE A send g( t) Example: RZ antpodal send g ( t) g ( t for t ) elsewhere / E B. WOR CAE: γ when g (t) g (t),.e. send the same sgnal for oth ases. P e /, 5% orret, 5% wrong / C. Orthogonal sgnal: γ, g( t) g( t) dt P E erf e, orthogonal / worse than antpodal sgnallng Bas Dgtal odulaton ehnques ho Le-go PAGE 4

15 -ARY IGALLIG CHEE: Unon Bound on the Proalty of Error g,,,..., wth Pr{ g sent} / P Pr{ error g sent} e For the optmum reever, Pr { error g sent} Pr{ ρ g < ρ g g sent} Pr{ ρ s loser to at least g than to g g sent} ε denotes the event that ρ s loser to g than to (ote that: P{ A B C} Pr{ A} + Pr{ B} + Pr{ C} ) Consder any par of g, d Pr{ε } erf Pr{ error g g : { } Pr error g sent Pr Pr{ } U ε ε,, g as the nary transmsson ase prevously analyzed. We have otaned sent}, d g g d d : Euldean dstane etween g, g erf P e erf,, he aove nequalty shows that P e s domnated y the term havng the smallest dstane d,mn. In desgnng the set of sgnalng elements, to mnmze P e we should am for the largest mnmum Euldean dstane. d dmn dmn dmn mn d, ( ), erf erf P e erf d Bas Dgtal odulaton ehnques ho Le-go PAGE 5

16 Dgtal odulaton: General odulaton: Proess y whh some haraterst of a arrer, (t), s vared n aordane wth a modulatng wave. (IEEE tandard D. of E&E terms) In the th symol lnterval, t- /, send one of possle tme-lmted sgnalng elements g (t- ) wth a pror proalty of / symol m ts, g (t): modulated sgnal,,,, m, wth fnte energy E Average energy per symol: E, Average energy per t: E, E me + ( t) osωt, E ( t) dt E + g ( t) dt E E AK (Ampltude hft Keyng), PK (Phase K), FK (Frequeny K), APK, QA Ojetves n desgnng a modulaton sheme: Bandwdth effeny: max data rate (f ) n a mnmum hannel andwdth (BW) Power effeny: mnmum pro. of error for mnmum transmtted power (or n terms of E /, E / ), maxmum resstane to nterferng sgnals. Easy mplementaton: mn rut omplexty ome of these goals pose onfltng requrements: ompromsng the desgn for a ertan applaton. Bas Dgtal odulaton ehnques ho Le-go PAGE 6

17 AK (Ampltude odulaton) os ωt t g() t aφ (), t a [ ( + ) ] d /, Φ () t, Energy E a for ω >> or ω π elsewhere / / d d d ( ) E a ( ) ( ) Example of AK sgnal wth 4 when ω, we have aseand PA gnal onstellaton for,4,8 Φ () t () t osω C t Baseand AK (PA), {a } Bas Dgtal odulaton ehnques ho Le-go passand AK PAGE 7

18 AK: Performane ODULAOR AK {a } osω t AW wth σ / osω t dt r selet f r I DEODULAOR ρa +n, n: Gaussan nose wth σ / f a sent, orret deson f ρ I orret deson, f d/ n d/, for, f n d/, for f n -d/, for Pr{orret} (/)[Pr{n d/}+pr{n -d/} +(-)Pr{ d/ n d/}] I I I 3... I I a a a 3 a - a Φ(t) d d 3 E Unon ound: Pe ( ) erf (/ )( ) erf, d E ( ) 3 E s 3 E s Exat Analyss: P e erf for erf >> Bas Dgtal odulaton ehnques ho Le-go PAGE 8

19 AK: Performane Analyss d Pr{n d / } Pr{ n d / } erf d Pr{n d/ } Pr{ n d/ } erf p d Pr{ d/ n d/ } [ Pr{ n d/ } + Pr{ n d / } ] p where p erf Pr{ orret} [ ( p) + ( )( p) ] p ( )p d P e Pr{ orret} erf 3 E s 3 E s Pe erf erf for >> Bas Dgtal odulaton ehnques ho Le-go PAGE 9

20 Bnary Phase hft Keyng (BPK) Can e vewed as BPK or APK: Eah t s enoded n the phase of the arrer wth frequeny f : o o for a and 8 for a In the th symol nterval, t- /, send one of possle tme-lmted sgnalng elements g (t- ) wth a pror proalty of / E E E t () os ω t g() t os ωtg, () t osωt os ( ω t + π ) gnal onstellaton of BPK 3 5 t () osω t d E Pr{t error}: P erf erf, d E Bas Dgtal odulaton ehnques ho Le-go PAGE

21 Phase hft Keyng (PK) General PK: g () t E s os[ π f t+ θ ] I ϕ () t + Q ϕ (), t t orthonormal ass funton ϕ ( t) os π f t & ϕ ( t) sn π ft elsewhere nary data Aos(θ ) nary-tosymol {θ } -Asn(θ ) osω t π I Aos θ, Q Asn θ, A Es, θ ( ),,,..., snωt PK QPK: 4, an e vewed as a lnear omnaton of an n- phase and 3 4 I + E / "" E / "" E / "" + E / "" s s s s Q + E / "" + E / "" E / "" E / "" s s s s E s E an e vewed as a sum of BAK (or BPK) modulated sgnals wth n-phase and quadrature arrers. Bas Dgtal odulaton ehnques ho Le-go 3 4 gnal onstellaton of QPK PAGE

22 PK Performane nary data Aos(θ ) nary-to- symol {θ } Asn(θ ) osω t PK sgnal PK WG osω t dt dt X Y ρx+jy I elet f ρ symol-tonary nary data snω t sn ω t Average energy/symol: E A dmn Asn E.sn Unon Bound for -ary PK: P E BPK,, P P erf, E E e e π π E erf sn π Bas Dgtal odulaton ehnques ho Le-go PAGE

23 PK Performane: Exat analyss n ρ g + n, n ( n, n): d Gaussan (, o/), θn tan n E + π π Pr{orret g } Pr { ρ I } Pr - θ n,for all,,...,,, E E E E pθ ( x) exp + os xexp sn x erf os x π E E for hgh E / and x < π /, erf os x exp os x π E osx ( ) E E os.exp sn pθ x x x π π / π E s P{ Pr{ orret } pθ ( x ) dx, P e P{ Pr{ orret } erf sn π / o Bas Dgtal odulaton ehnques ho Le-go PAGE 3

24 quare Quadrature Ampltude odulaton QA One ase of AP()K: Quadrature AK a os ω t + sn ω t g ( t ) Choose a d a ( 6, L : nteger d a, where 6, L d a ( + L ) a and are ndependent a, an tae any OE of L possle values / d E ( ) d ), d E m/, m/ for t elsewhere : nteger,,..., L AK {a } { } dt / â L os ω t sn ω t hoose m f â I Reever: -ary QA reat t as ndependent nphase & quadrature AK. WG os ω t os ω t sn ω t dt ˆ hoose l f ˆ I d Eah AK has P ( ) [ ] eak erf Pe aryqa PeAK For large E /, P eak << d 3 E Pe, aryqa PeAK erf erf ( ) otes: For 4, L, 4QA s 4PK Bas Dgtal odulaton ehnques ho Le-go PAGE 4

25 Proalty of t error (P ) vs Proalty of symol error and Gray Codng Examples of Gray odng: PA QA PK When a symol error ours, t s lely that the reever taes the adjaent symol (the symol losest to the rght one). herefore, t s desrale to ode the m-t symol n suh a way that adjaent symols dffer y only one t. In ths way, the average proalty of t error P s P P Pe m log Bas Dgtal odulaton ehnques ho Le-go PAGE 5

26 PROBABILIY OF YBOL ERROR -QA, -PK: BW-effent ut not power-effent For >8, -QA outperforms -PK QA PK E / o Bas Dgtal odulaton ehnques ho Le-go E / o PAGE 6

27 Frequeny hft Keyng (FK) -ary orthogonal FK sgnalng shemes are power-effent effent ut not andwdth-effent. A os ωt t g () t, elsewhere m,,...,, E E A ORHOORAL -ary FK: g ( t). g ( t) dt for j j ote: Gray odng annot e used for orthogonal FK ( ω ω ) π or ( f f ) /, : nteger j j d g g A A j j E : onstant Pe ( ) erf E E Pe ( ) erf ( log ) Bas Dgtal odulaton ehnques ho Le-go E / o PAGE 7