Chapter 5 Signal-Space Analysis

Transcription

1 Chaper 5 Sgnal-Space Analy Sgnal pace analy provde a mahemacally elegan and hghly nghful ool for he udy of daa ranmon. 5. Inroducon o Sacal model for a genec dgal communcaon yem n eage ource: A pror probable for nformaon ource p P m for,,..., n ranmer: he ranmer ae he meage ource oupu m and en-code no a dnc gnal uable for ranmon over he channel. So: p P m P for,,..., Po-ng Chen@ece.ncu Chaper 5-

2 5. Inroducon o mu be a real-valued energy gnal.e., a gnal wh fne energy wh duraon. E d <. n Channel: he channel aumed n h ex lnear and wh a bandwdh wde enough o pa wh no doron. o A zero-mean addve whe Gauan noe AWG alo aumed o faclae he analy. Po-ng Chen@ece.ncu Chaper A ahemacal odel o We can hen mplfy he prevou yem bloc dagram o: o Upon he recepon of x wh a duraon of, he recever mae he be emae of m. We haven defned wha he be mean. Po-ng Chen@ece.ncu Chaper 5-4

3 5. Creron for he Be Decon o Be nmzaon of he average probably of ymbol error. e P p P mˆ m n I opmum n he mnmum probably of error ene. n Baed on h creron, we can begn o degn he recever ha can gve he be decon. m Po-ng Chen@ece.ncu Chaper Geomerc Repreenaon of Sgnal o Sgnal pace concep n Vecorzaon of he dcree or connuou gnal remove he redundancy n gnal, and provde a compac repreenaon for hem. n Deermnaon of he vecorzaon ba o Gram-Schmd orhogonalzaon procedure Po-ng Chen@ece.ncu Chaper 5-6 3

4 5. Gram-Schmd Orhogonalzaon Procedure Gven v!, v!,, v!, how o fnd an orhonormal ba for hem? ep Le u!! v v!. ep! u '! v! v! u! u. Se! u ep For 3, 4,...,! u '! u '. Le u! ' v! v! u! u! v! u! u! " v! u! u!. Se u!!' u u! '.! ep v hen u, u!,", u! form an orhonormal ba. Po-ng Chen@ece.ncu Chaper 5-7 Propere: vecor : v! v,,v n nner produc :"! v! v n v v orhogonal, f nner produc. v norm : v! v +#+ v n v orhonormal, f nner produc, and ndvual norm. v lnearly ndependen, f none can be repreened a a lnear combnaon of oher. v rangle nequaly : v! + v! v! + v!. Po-ng Chen@ece.ncu Chaper 5-8 4

5 v Cauchy Schwarz nequaly :! v! v! v! v wh equaly holdng f! v a! v. x norm quare :! v +! v! v +! v +! v! v. x Pyhagorean propery : If orhogonal,! v +! v! v +! v. x arx ranformaon w.r.. marx A :! v A! v. x egenvalue w.r.. marx A: [ ]. oluon λ of de A-λI x egenvecor w.r.. egenvalue λ : oluon! v of A! v λ! v. Po-ng Chen@ece.ncu Chaper Sgnal Space Concep for Connuou Funcon Propere for connuou funcon complex valued gnal : z b nner produc : < z, ẑ > zẑ * d. orhogonal, f nner produc. v norm : z a b z d v orhonormal, f nner produc, and ndvual norm. Po-ng Chen@ece.ncu Chaper 5- a v lnearly ndependen vecor, f none can be repreened a a lnear combnaon of oher. v rangle nequaly : z + ẑ z + ẑ. 5

6 v Cauchy Schwarz nequaly : < z, ẑ > z ẑ wh equaly holdng f z a ẑ, where a a complex number. x norm quare : z+ ẑ z + ẑ + < z, ẑ > + < ẑ, z >. x Pyhagorean propery : If orhogonal, z+ ẑ z + ẑ. x ranformaon w.r.. a funcon C,τ : b ẑ C,τ zτ dτ Recall v a n v. a x.a egenvalue and egenfuncon w.r.. a funcon C,τ : oluon λ and {φ } of λ φ C,τ φ τ dτ and C,τ can be repreened a C,τ φ λ φ τ. Po-ng Chen@ece.ncu Chaper 5- n a b x.b Gve a deermnc funcon {, [, }and a e of x.c If orhonormal ba{ ψ } where a ψ d. orhonormal e { ψ } poble ha ˆ < ha can pan. hen a ψ, <, K < K doe no pan he pace, hen a ψ for all choce of { a }. < K Po-ng Chen@ece.ncu Chaper 5-6

7 o Problem : How o mnmze he energy of e ˆ? o elec he coeffcen{ a } ha mnmze e d a K [ a ψ ] d a a ψ ψ d + ψ d + a K K e d, [ ψ ] a d ψ ψ d. a ψ d. Po-ng Chen@ece.ncu Chaper 5-3 a, ψ a e ˆ, ψ ψ ψ Hence, e, ˆ e ˆ d o Inerpreaon n a he proecon of ono he Ψ -ax. n a he energy-proecon of ono he Ψ -ax. Po-ng Chen@ece.ncu Chaper 5-4 7

8 a, ψ a e ˆ, ψ ψ ψ e d e [ ˆ ] d e d d d d K K a a K e d e d a ψ d ψ d e ˆ d [ ˆ ] d K a oably, ˆ d. Po-ng Chen@ece.ncu Chaper Sgnal Space Concep for Connuou Funcon o Compleene n If every fne energy gnal afe { ψ } K n Example. Fourer ere defned over [, ]. d a complee orhonormal e. Po-ng Chen@ece.ncu Chaper 5-6 K π π co, n a complee orhonormal e for gnal a, 8

9 5. Gram-Schmd Orhogonalzaon Procedure Gven v, v,, v, how o fnd an orhonormal ba for hem? ep Le u v v. ' ep u v v,u u. Se u u ' ' u. ep For 3, 4,..., Le u ' v v,u u! v,u u. Se u u ' u '. form an orhonormal ba. ep v hen u,u,!,u Po-ng Chen@ece.ncu Chaper Geomerc Repreenaon of Sgnal f, <,,,..., { f } orhonormal f d,,,...,,,..., Po-ng Chen@ece.ncu Chaper 5-8 9

10 5. Geomerc Repreenaon of Sgnal o hrough he gnal pace concep, where can be unambguouly repreened by an -dmenonal gnal vecor,,, over an -dmenonal gnal pace. o he degn of ranmer become he elecon of pon over he gnal pace, and he recever mae a gue abou whch of he pon wa ranmed. o In he -dmenonal gnal pace, n Lengh quare of he vecor energy of he gnal n angle beween vecor energy correlaon beween gnal, co θ d Po-ng Chen@ece.ncu Chaper Geomerc Repreenaon of Sgnal n he angle beween vecor ndependen of he ba ued. o From h vew, n he ranmer may be vewed a a ynhezer, whch ynheze he ranmed gnal by a ban of mulpler. n he recever may be vewed a an analyzer, whch correlae produc-negrae he common npu no ndvdual nformaonal gnal. Po-ng Chen@ece.ncu Chaper 5-

11 5. Geomerc Repreenaon of Energy Sgnal o Illuraon he geomerc repreenaon of gnal for he cae of and 3 Po-ng Chen@ece.ncu Chaper 5-5. Eucldean Dance o Afer vecorzaon, we can hen calculae he Eucldean dance beween wo gnal, whch he quared roo of: d, Kronecer dela funcon : δ, Applcaon : We may ay ha orhonormaly mean φ, φ δ. Po-ng Chen@ece.ncu Chaper 5-

12 Example 5. Schwarz Inequaly o Cauchy-Schwarz nequaly and angle beween gnal n Cauchy-Schwarz nequaly ad ha, wh equaly holdng f c. n Alo, he angle beween gnal gve ha, co θ n Hence, Cauchy-Schwarz nequaly can be equvalenly aed a: coθ wh equaly holdng f θ or π Po-ng Chen@ece.ncu Chaper Ba o he complee orhonormal ba for a gnal pace no unque! n So he ynhezer and analyzer for he ranmon of he ame nformaonal meage are no unque! o One way o deermne a e of orhonormal ba he Gram-Schmd orhogonalzaon procedure. o ry and pracce Example 5. yourelf! Po-ng Chen@ece.ncu Chaper 5-4

13 5.3 Converon of he Connuou AWG Channel no a Vecor Channel o Influence of he AWG noe o he gnal pace concep x + w where w zero-mean AWG wh PSD /. o Afer he correlaor a he recever, we oban: x x, f, f w, f + n Or equvalenly, x + w. n oably, here no nformaon lo by he gnal pace repreenaon.,, f f x x Po-ng Chen@ece.ncu Chaper Converon of he Connuou AWG Channel no a Vecor Channel o Sac of {w } x w!! +! x w o Snce { } deermnc, he drbuon of x a meanhf of he drbuon of w. o Oberve ha w Gauan drbued becaue w AWG. he drbuon of w can herefore be deermned by mean vecor and covarance marx. Po-ng Chen@ece.ncu Chaper 5-6 3

14 4 Po-ng Chaper Converon of he Connuou AWG Channel no a Vecor Channel o ean o Covarance [ ] ] [ ] [ d f w E d f w E w E [ ] d f f dd f f dd f f w w E d f w d f w E w w E δ δ ] [ ] [ Po-ng Chen@ece.ncu Chaper Converon of he Connuou AWG Channel no a Vecor Channel o A a reul, [w, w,, w ] are zero-mean..d. Gauan drbued wh varance /. o h how ha x ndependen Gauan drbued wh common varance / and mean vecor [,,, ]. Equvalenly, x f exp π x

15 5.3 Converon of he Connuou AWG Channel no a Vecor Channel o Remander erm n noe n I poble ha w' w w f n However, can be hown ha a an error erm w orhogonal o for. Hence, w wll no affec he decon error rae on meage. w', wh probably. Po-ng Chen@ece.ncu Chaper Lelhood Funcon o An equvalen gnal-pace channel model m m ˆ, c m x + w m d x { m,..., m } o he be decon funcon d ha mnmze he decon error : d x m, f P{ m arg m x} P{ m max P{ m x} { m,..., } m x}for all n h he maxmum a poeror probably AP decon rule. Po-ng Chen@ece.ncu Chaper 5-3 5

16 5.4 Lelhood Funcon o Wh equal pror probable, dx arg max P{m x} m {m,...,m } { } argmax P{m x}, P{m x},..., P{m x} argmax P{m x} f x, P{m x} f x,..., P{m x} f x Pm Pm Pm argmax P{m x} f x, P{m x} f x,..., P{m x} f x / / / { } argmax f x m, f x m,..., f x m fx m named he lelhood funcon gven m ranmed Hence, he above rule named he maxmum-lelhood L decon rule. Po-ng Chen@ece.ncu Chaper Lelhood Funcon o AP rule L rule, f equal pror probably aumed. o In pracce, more convenen o wor on he loglelhood funcon, defned by d x arg max arg max { f x m }, f x m,..., f x m { log f x m,log f x m,...,log f x m } o Why log-lelhood funcon are more convenen? he decon funcon become um of quared Eucldean dance n AWG channel. Po-ng Chen@ece.ncu Chaper 5-3 6

17 dx argmax argmax argmax argmn log f x m argmax log f x log exp x π logπ x x argmn x argmn x Upon he recepon of receved gnal pon x, fnd he gnal pon ha cloe n Eucldean dance o x. Po-ng Chen@ece.ncu Chaper Coheren Deecon of Sgnal n oe: axmum Lelhood Decodng o Sgnal conellaon n he e of gnal pon n he gnal pace o Example. Sgnal conellaon for BQ code decon regon for decon regon for decon regon for 3 decon regon for 4 Po-ng Chen@ece.ncu Chaper

18 5.5 Coheren Deecon of Sgnal n oe: axmum Lelhood Decodng o Decon regon for and 4 Po-ng Chen@ece.ncu Chaper Coheren Deecon of Sgnal n oe: axmum Lelhood Decodng o Uually,,,, are named he meage pon. o he receved gnal pon x hen wander abou he ranmed meage pon n a Gauan-drbued random fahon. Po-ng Chen@ece.ncu Chaper

19 5.5 Coheren Deecon of Sgnal n oe: axmum Lelhood Decodng o Conan-energy gnal conellaon n In uch cae, he L decon rule can be reduced o an nner-produc. d x arg mn x arg mn arg mn arg max x, x x, + x, + E, f E conan. Po-ng Chen@ece.ncu Chaper Correlaon Recever o If gnal do no have equal energy, we can ue d x arg max x, E. o mplemen he L rule. n he recever coheren becaue he recever requre o be n perfec ynchronzaon wh he ranmer more pecfcally, he negraon mu begn a exacly he rgh me nance. Po-ng Chen@ece.ncu Chaper

20 producnegraor or correlaor x x Correlaon recever x demodulaor or deecor decon maer Po-ng Chen@ece.ncu Chaper 5-39 producnegraor or correlaor x x x mached fler decon maer Po-ng Chen@ece.ncu Chaper 5-4

21 Po-ng Chaper Equvalence of Correlaon and ached Fler Recever o he correlaor and mached fler can be made equvalen. o Specfcally, τ τ τ φ d h x d x x f h ϕ and mplcly ϕ zero oude. Po-ng Chen@ece.ncu Chaper Probably of Symbol Error o Average probably of ymbol error { } Z c e d f m m m d P m m d P m P P P, ranmed mn Pr ranmed ranmed x x x x x x { }. mn : where, Z x x x R

22 5.7 Invarance of Probably of Symbol Error o Probably of ymbol error nvaran wh repec o ba change.e., roaon and ranlaon of he gnal pace. o Specfcally, he ymbol error rae SER only depend on he relave Eucldean dance beween he meage pon. { x mn x m ranmed} P Pr e, Po-ng Chen@ece.ncu Chaper Invarance of Probably of Symbol Error o Specfcally, f Q a reverble ranform marx, uch a roaon, hen { x R : x mn x }, { x R : Qx Q mn } Qx Q, n If a gnal conellaon roaed by an orhonormal ranformaon, where Q an orhonormal marx, hen he probably of ymbol error P e ncurred n maxmum lelhood gnal deecon over an AWG channel compleely unchanged. Po-ng Chen@ece.ncu Chaper 5-44

23 5.7 Invarance of Probably of Symbol Error o A par of gnal conellaon for llurang he prncple of roaonal nvarance. Po-ng Chen@ece.ncu Chaper Invarance of Probably of Symbol Error o he nvarance n SER for ranlaon can be lewe proved. n I he ranmon power he ame for boh conellaon? Po-ng Chen@ece.ncu Chaper

24 5.7 nmum Energy Sgnal o Snce SER nvaran o roaon and ranlaon, we may roae and ranlae he gnal conellaon o mnmze he ranmon power whou affecng SER. E g p Fnd a and Q uch ha E a, Q p g Q a n Snce Q doe no change he norm.e., ranmon power, we only need o deermne he rgh a. mnmzed. Po-ng Chen@ece.ncu Chaper nmum Energy Sgnal o Deermne he opmal a. E a g p p a a + a p p a opmal a and E a g p + a opmal p p Po-ng Chen@ece.ncu Chaper

25 5.7 nmum Energy Sgnal o So ubfgure a below ha mnmum average energy. Po-ng Chen@ece.ncu Chaper Unon Bound on Probably of Error o Unon bound P e Pr Pr, P A B P A + P B { x mn x m ranmed} x x " "! x x x > x " "! x > x Pr Pr, { x > x m ranmed} Po-ng Chen@ece.ncu Chaper 5-5 m ranmed m ranmed 5

26 5.7 Unon Bound on Probably of Error x > x x > x 3 x > x 4 { x > x } { } x > x { x > x } 3 4 Po-ng Chen@ece.ncu Chaper Unon Bound on Probably of Error where P, Pr oably, gven m Pe P,, { x > x m ranmed }. ranmed, x Gauan drbued wh mean. w Po-ng Chen@ece.ncu Chaper 5-5 6

27 5.7 Unon Bound on Probably of Error Snce x + w when wa ranmed, we have Pr x Pr Pr Pr > x + w w w Pr Pr n < where n ranmed > + w > w + > w + w < ranmed ranmed + w ranmed ranmed ranmed w. Chaper 5-53 Po-ng Chen@ece.ncu 5.7 Unon Bound on Probably of Error Oberve ha w zero-mean Gauan drbued wh covarance marx E[ww ] I, where I he deny marx. Hence, n w Gauan drbued wh E[n] E w E[n ] E E w E[w] ww w E ww I. h mple ha w n/ Gauan drbued Chaper 5-54 wh mean zero and varance /. 7

28 5.7 Unon Bound on Probably of Error A a reul, Pr n< ranmed Pr w< ranmed Pr w< ranmed Pr w> ranmed, where he la equaly vald becaue he probably deny funcon of a zero-mean Gauan random varable ymmerc wh repec o w hence, Pr[w >a]pr[w< a] for any a>. Po-ng Chen@ece.ncu Chaper Unon Bound on Probably of Error o Hence, n For, P, Pr { x > x m ranmed} Pr w > d / π d erfc v exp dv, where d, where erfcu π u exp z dz. Po-ng Chen@ece.ncu Chaper

29 5.7 Unon Bound on Probably of Error erfc d, where erfcu exp z dz. u π n he ame formula vald for any. w n For, P, Pr { x > x m ranmed} Pr w > d and w d / exp v dv π don' care, where d Po-ng Chen@ece.ncu Chaper Unon Bound on Probably of Error o Conequenly, he unon bound for ymbol error rae : Pe P, erfc,, o he above bound can be furher mplfed when addonal condon gven. n For example, f he gnal conellaon crcularly ymmerc n he ene ha {d, d,, d } a permuaon of {d, d,, d } for, hen Pe erfc Po-ng Chen@ece.ncu, Chaper 5-58 d d 9

30 Pe 5.7 Unon Bound on Probably of Error o Anoher mplfcaon of unon bound n Defne he mnmum dance of a gnal conellaon a: d mn d mn,, hen by he rc decreang propery of erfc funcon,, d erfc d erfc Po-ng Chen@ece.ncu Chaper 5-59 erfc d mn dmn erfc erfc mn, d 5.7 Unon Bound on Probably of Error o We may ue he bound for erfc funcon o realze he relaon beween SER and d mn. exp u erfc u for u >.683 π d mn d mn P e erfc exp, f dmn > π n Concluon: SER decreae exponenally a he quared mnmum dance grow. Po-ng Chen@ece.ncu Chaper 5-6 3

31 5.7 Relaon beween BER and SER o he nformaon b are ranmed n group of log b o form an -ary ymbol. o h gve he reul ha a large ymbol error rae SER may no caue a large b error rae BER. n For example, a ymbol error for large may be due o only b error. n Opmcally, f every ymbol error due o a ngle b error, hen aumng ha n ymbol are ranmed n SER SER SER BER. n log log In general, BER. log Po-ng Chen@ece.ncu Chaper Relaon beween BER and SER n Pemcally, f every ymbol error caue log b error, hen aumng ha n ymbol are ranmed n log SER BER n log SER. In general, BER SER. n Summary: SER BER SER log Po-ng Chen@ece.ncu Chaper 5-6 3

32 5.7 Relaon beween BER and SER o If he ac for number of b error paern ha caue one ymbol error nown, we can hen deermne he exac relaon beween BER and SER. n SER # b P b BER n log where #b number of ' n b, and b repreen a bnary permuaon of log b paern. Here, a n b mean a b error occurred n he correpondng poon; hence, all-zero paern excluded becaue repreen no ymbol error. Po-ng Chen@ece.ncu Chaper Relaon beween BER and SER o Example. If all b error paer are equally lely, hen n SER # b P b SER BER n log log SER log u log u u log log SER log / SER # b / log oe u u u. Po-ng Chen@ece.ncu Chaper