2. Grundlegende Verfahren zur Übertragung digitaler Signale (Zusammenfassung) Informationstechnik Universität Ulm

Transcription

1 . Grundlgnd Vrfahrn zur Übrtragung dgtalr Sgnal (Zusammnfassung) wt Dc. 5

2 Transmsson of Dgtal Sourc Sgnals Sourc COD SC COD MOD MOD CC dg RF s rado transmsson mdum Snk DC SC DC CC DM dg DM RF g physcal channl analog sgnal modm channl analog sgnal dscrt channl, sourc-snk dgtal sgnals dscrt channl, sourc codng dgtal sgnals dscrt channl, channl codng dgtal sgnals Sourc COD COD MOD MOD SC CC dg RF dgtal sourc sourc codng channl codng modulaton, dgtal transmsson modulaton, rf transmttr Snk DC DC DM DM SD CD dg RF dgtal snk sourc dcodng channal dcodng dmodulaton, dgtal transmsson dmodulaton, RF rcvr wt Dc. 5

3 Transmsson wth M Basc Wavforms: Transmttr xampl: 4PSK (M4) slcton by nformaton sourc t t s ntrfrnc transmsson channl g SLCTION t 3 t M 4 df frnt wavforms wt Dc. 5 3

4 Transmsson wth M Basc Wavforms: Rcvr xampl: 4PSK (M4) procssng: comparson wth dcson varabls y dcson DCISION g wth wth y y max. ndx of transmttd wavform ( bt) wth 3 y 3 wt Dc. 5 4

5 Bnary Transmsson wt Dc. 5 5

6 Bnary Transmsson ovr an AWGN Channl slcton by nformaton sourc compar wth ( t ) dcson WGN n, N P s + g P max. No. of most probably transmttd wavform transmttr AWGN channl compar wth ( t ) rcvr wt Dc. 5 6

7 Optmum Dtctor () Startng-pont: Probablty thory Sgnal spac rprsntaton of sgnals Knowldg at th dtctor: Rcvd sgnal g( t) (vnt ) ( t) + n( t) Statstcs of th addtv nos procss (wht Gaussan nos, WGN) Hypothss: : ( t ) has bn transmttd : ( t ) has bn transmttd Condtonal probablts: P Prob{ ( t) g( t) } { ( t) g( )} P Prob t wt Dc. 5 7

8 Optmum Dtctor () Maxmum A Postror (MAP) dcson rul: Dcd n favor of that hypothss or whch s th most probabl caus for vnt Usng Bays rul w obtan for th most probabl caus for vnt f and ar mutually xclusv vnts: k arg { [ ]} max Prob( ( t)) Prob(g ( t) Maxmum Lklhood (ML) dcson rul: ( ) Most lkly caus for vnt : k arg { [ ]} max Prob(g ( t) ( ) wt Dc. 5 8

9 Bnary Tx: Rprsntaton of Condtonal PDF () Sgnal vctor spac rprsntaton: 3D Plot D Gaussan dstrbutons Basc wavforms, rx sgnal g, and nos sampl functon n ar rprsntd by D vctors, g,and n p g ( x ) p g ( x ) p g ( x ) p n ( x ) N x xp π ( N f ) ( N f g ) g N-dmnsonal condtonal jont probablty dnsty functon wt Dc. 5 9

10 Bnary Tx: Rprsntaton of Condtonal PDF () D Contour Plot Dcson Boundary Mnmum Dstanc Rul (drvd from ML-prncpl): Calculat dstanc btwn rx sgnal g and th possbl tx sgnals and. Dcd n favor of mnmum dstanc. wt Dc. 5

11 Mnmum Dstanc Dcson Rul Calculaton of dstancs n sgnal vctor spac: g, ( g g ) ( g, g) + (, ) ( g, ) (, g) R{ ( )} g g, Modfd mnmum dstanc dcson rul : k arg max R {( g, )} basc wavform k has most lkly bn tx wt Dc. 5

12 Calculaton of th Dcson Varabls Intgrat and dump : ( g, ) g( t) ( t) dt Corrlaton fltr or Matchd Fltr (MF) ( g, ) ϕ () ϕ () g( t) ( t) g g wt Dc. 5

13 D Probablty Dnsty Functon at Output of MF transmttd transmttd Dcson boundary wt Dc. 5 3

14 wt Dc. 5 4 Informatonstchnk Unvrstät Ulm Bt rror Probablts for Bnary Transmsson () () 8 ) ( rfc ) ( N y y c P P y smlar: P P Avrag total symbol rror rat P s : Avrag total bt rror rat (BR) P b for bnary tx: 8 ) ( rfc N y y P P s b.5.5 s ) ( Prob( ) ( ( Prob P t P t P +

15 Optmum Rx for Bnary Tx: Spcal Cass () Antpodal transmsson t ( t) ( t) ( t) ( t) g (-t) k or k Unpolar transmsson t ( t) ( t) ( t) g (-t) dcson k or k thrshold : nrgy of wt Dc. 5 5

16 Optmum Rx for Bnary Tx: Spcal Cass () Orthogonal transmsson (M) ( t) ( t) t g Δ ( t ) k or k Δ( t) ( t) ( t) dcson as for ant-podal transmsson wt Dc. 5 6

17 Sgnal-to-Nos Ratos and Bt rror Probablty antpodal on-off orthogonal SNR a N N N b N N N N rfc N rfc 8 N rfc 4 N P b rfc SNR a rfc 4 SNRa rfc SNRa rfc N b rfc 4 N b rfc 4 N b wt Dc. 5 7

18 Bt rror Probablts for Bnary Transmsson unpolar & orthogonal on-off kyng orthogonal antpodal bpolar wt Dc. 5 8

19 Contnuous Transmsson () δ -samplr basc wavform matchd fltr samplr dcson kts (-t) x (k) s g y x~ (k) x(k) kts squnc of tx symbols tx sgnal rx sgnal squnc of stmatd symbols squnc of dtctd symbols unpolar & antpodal transmsson wt Dc. 5 9

20 Contnuous Transmsson () Symbol ntrval: Symbol rat: T s r s T s In gnral not dntcal wth duraton T of basc wavform! Transmt sgnal: wth: s ( t) x( k) ( t kt s ) x( k) x( k) k {, + } for antpodal {, + } for unpolar tx tx Sgnal at output of matchd fltr (n): y( t) n t) [ x( ) ( t T )] ( ) ( s t ϕ ( t T ) s wt Dc. 5

21 Contnuous Transmsson (3) Dcson varabl (squnc of stmatd symbols): ~ x ( k) y( kts ) n( t) x( k) ϕ() + x( ) ( kts T ) k n( t) ϕ s usfull trm ntrsymbol ntrfrnc (ISI). Nyqust crtron for unpolar and antpodal transmsson: ( s ϕ kt ) for k for k ~ x ( k) n( t ) x( k) - ISI avodd - sam dcson varabl as for sngl tx wt Dc. 5

22 Frst Nyqust Crtron: xampl y pattrn: - ISI, but - y stll opn nos margn tolranc wth rspct to dvatons from th optmum samplng tm wt Dc. 5

23 M-ary Transmsson wt Dc. 5 3

24 M-Orthogonal Transmsson: Optmum Rcvr Matchd fltr bank Samplng t Dcson vctor Dcson (-t) g y y t for,..., M (-t) M- (-t) y y M- t y y M- max No. of most probably transmttd wavform wt Dc. 5 4

25 M-Orthogonal Transmsson: Basc Wavforms wal(,t) c wal(,t) c t xampl for M-orthogonal tx wth M4 basc wavforms (Walsh-Hadamard-Cods) t wal(,t) c t wal(3,t) c t T wt Dc. 5 5

26 M-Orth. Tx: Jont Probablty Dnsty Functons wt Dc. 5 6

27 M-Orthogonal Tx: rror Probablts Avrag symbol rror probablty: P πn xp ( a ) N rfc a N M da no closd form soluton for ntgral Avrag bt rror probablty: P M ( M ) b P s mappng bts->symbols has no nflunc on P b wt Dc. 5 7

28 M-Orthogonal Tx: Bt rror Probablty () Shannonlmt lmt for M powr ffcnt transmsson 64 M ln wt Dc. 5 8

29 Gray Codng xampl: Borthogonal Tx wth M4 Gray codng: bt combnaton of nghbourng symbols dffr n only on poston - -.Bt.Bt wt Dc. 5 9

30 Contnuous M-Orthogonal Transmsson Transmt sgnal: s { ( t), ( t),..., ( t) } ( t) ( k )( t kts ) wth ( k )( t) M k Sgnal at output of matchd fltr m : y m Gnralzd frst Nyqust crtron: ( l Ts ) ϕ () + ϕ ( l Ts k Ts ) m m ( k ) m k l usfull trm ntrsymbol ntrfrnc (ISI) ϕ l ( kt ) δ ( k) δ ( l) for, l,,..., M s wt Dc. 5 3

31 Gnralzd Frst Nyqust Crtron Gnralzd frst Nyqust crtron n frquncy doman T s ( f ) * l ( f ) δ ( l) pr Ts l xampl: l ( f ) ( f ) : l T S. "Nyqust slops", must b oddsymmtrc wth rgard to cntr f wt Dc. 5 3