Types of Communication

Transcription

1 Tps f Cmmunicatin Analg: cntinuus ariabls with nis {rrr 0} 0 (imprfct) Digital: dcisins, discrt chics, quantizd, nis {rrr} 0 (usuall prfct) Mssag S, S, r S M Mdulatr (t) channl; adds nis and distrtin M-ar mssags, whr M can b infinit Rcid mssag S, S, r S M Dmdulatr; hpthsiz H H N (chss n) Lc4.0- /6/0 Th channl can b radi, ptical, acustic, a mmr dic (rcrdr), r thr bjcts f intrst as in radar, snar, lidar, r thr scintific bsratins. A

2 Optimum Dmdulatr fr Binar Mssags Hpthsis: H rbabilit a priri Mssag: S OK ERROR S ERROR OK Dmdulatr dsign E.G. -D cas b V V masurd a b t a V V " H " H " " c Lc4.0- /6/0 Hw t dfin V,V? A

3 Optimum Dmdulatr fr Binar Mssags E.G. -D cas b V V Hw t dfin V,V? masurd Minimiz rrr p { S } d + p { S } a a V V " H " H V V " b " c d t rplac with V { } { } + p S p S d V Lc4.0-3 /6/0 Nt: { } d p { S } d p S + V V A3

4 Optimum Dmdulatr fr Binar Mssags [ p { S } p { S }] + V d { } { } T minimi z, chs V p S > p S rrr Vr gnral slutin [i.., chs maximum a pstriri ( MA stimat)] Lc4.0-4 /6/0 A4

5 S Exampl: Binar Scalar Signal Cas A lts, S π N p { S } ( A) N p { S } O lts, σ n N, Gaussian N π N nis If : { S } p p { S } 0 A/ A Dcisin thrshld if (bias chis tward H and a priri infrmatin) 0 A/ A p { S} Lc4.0-5 /6/0 Thrshld if > H A5

6 Rul Fr Dfining V : (Binar Scalar Cas) { S } { S } Chs V p p > p { S } π N N (binar cas) Liklihd rati A { } { } p S p S > " V " r (quialntl) A ( ) " n A > An V " Fr additi Gaussian nis, Lc4.0-6 /6/0 [ ( ) ] ( )? - - A N + N A A N An ( ) A n A > A + N A n ( ) A N A ( ) chs V i f >, r > + n A A bias A6

7 Binar Vctr Signal Cas Fr bttr prfrmanc, us multipl indpndnt sampls: (t) A p p { S }? > { S } m { S } p { S } Hr,,..., m i i Whr S m π (indpndnt nis sampls) { } p S i i ( S ) i π N S 0 t i N { } p S i ( π N ) m m i ( ) i S i N Lc4.0-7 /6/0 B

8 p { S } i ( π N ) Binar Vctr Signal Cas m Thus th tst bcms: ( S ) m i i i m m? ( ) ( ) i i i i i i n A A S S > n N A S S But S S S + S + S S S S N A { } { } p S p S? > Thrfr S S S S V iff ( S S ) > + N An Bias 0 Bias 0 Lc4.0-8 /6/0 if nrg E E if B

9 V iff Binar Vctr Signal Cas S S S S ( S S ) > + N A n S S m τ i m τ i + - pratr H H Multipl hpthsis gnralizatin: Chs? i i i i S S i + A i > j i H if f S N n all f This matchd filtr rcir minimizs rrr Lc4.0-9 /6/0 B3

10 Graphical Rprsntatin f Rcid Signals 3-D Cas: Arag nrg i i S i 0 S n V V S (t) S S 3 S n t S Lc4.0-0 /6/0 B4

11 Dsign f Signals S i E.G. cnsidr + S S S + s. S 0 Arag nrg S E - S ( ) E E.G. -D spac fr S, S ( S S, S ) Lc4.0- /6/0, S 3, S 4 S 3 : dcisin bundaris S 0 S 4 n b S 3-D spac S ( S, S, S ) S Bttr b a 3 a S 4 rati b S S 3 B5

12 Dsign f Signals S i -D spac: S,..., S 6 S i 9 S i s 6-ar signals r magnitud/phas quilatral triangl slightl lwr arag signal nrg fr sam p{rrr} n-dimnsinal sphr packing ptimizatin unsld Lc4.0- /6/0 B6

13 Binar cas: Fr additi Gaussian nis, ptimum is " H " if ( S S ) > Whr S + n Calculati n f p{rrr} S S + N An : N n (t) [ N B kt B N ] W Hz B, s ( ) dubl sidband S p ( S S ) < S S + N A n Lc4.0-3 /6/0 S S p n S S < + N n b ( ) A p { < } B[GRVZM] -b B D

14 Lc4.0-4 /6/0 Dualit f Cntinuus and Sampld Signals S ( ) S p n S S N n S p{ b} < + A < B[ GRVZM] b B Cnrsin t cntinuus signals assuming nquist sampling is hlpful hr, S (t)[0 < t < T] (BT sampls, sampling thrm) T b σ n(t) [ S (t) S(t) ] dt T [ S ] (t) S(t) dt A n( ) E S B BT N [ ] E n ( S S ) j j j j nis -B 0 B [ ] N WHz f D

15 Calculatin f, cntinud σ E B j [ ] E n ( S S ) B E i j BT BT BT n n i j j j j ( )( ) Si S Sj S i j whr E n n i j Nδ ij Lc4.0-5 /6/0 σ B N S N B B S T N T [ S(t) S(t) ] [ S (t) S (t)] dt dt D3

16 σ Calculatin f, cntinud B p () N S N B B S T πσ N T σ [ S(t) S(t) ] [ S (t) S (t)] dt (GRVZM) dt Thrfr: Lc4.0-6 /6/0 S b πσ σ d -b 0 () D4

17 Errr functin Dfinitin f ERFC(A) ERF (A) π A A Cmplmntar rrr functin x dx ERFC (A) - ERF(A) -A A Thn S ERFC ( A ), whr A must b fund σ If w lt x thn A A σ x dx π A πσ A σ σ ERF(A) d σ x whr th nw limits A σ and factr πσ aris as fllws: Lc4.0-7 /6/0 D5

18 Dfinitin f ERFC(A) ERF(A) π A A x dx πσ A σ Aσ σ d whr th nw limits A σ and factr πσ aris as fllws: Sinc x σ, th limit x A σ bcms a limit whr Aσ Als, dx d σ s bcms π πσ Lc4.0-8 /6/0 D6

19 Slutin fr fr Binar Signals ( ) S ERFC( A) ERFC b σ and S + S ( whr th limit b A σ, s A b σ ) Lc4.0-9 /6/0 If, and sinc S S, thn ERFC b σ ( ) T [ ) S (t)] S(t dt 0 ERFC T ( N ) [ S (t) S (t)] dt 0 D7

20 Slutin fr fr Binar Signals T [ S (t) S (t)] dt 0 ERFC T ( N ) [ S (t) S (t)] dt 0 T ERFC 0 [ S (t) S (t ] ) dt N T T If S (t)dt + S (t) dt is fixd fr 0 0 thn T minimiz, lt S ( t) S (t) maximizs T 0 [ S (t) S (t)] dt Lc4.0-0 /6/0 D8

21 Exampls f Binar Cmmunicatins Sstms ERFC T [S(t) S(t)] dt N 0 Assum T and dfin s (t)dt E 0 Mdulatin tp s (t) s (t) OOK (n-ff king) FSK (frqunc-shift king) BSK binar phasshift king) A cs ω t 0 ERFC E 4 N ERFC E ag A cs ω t A cs ω t ERFC E ag N A cs ωt A cs ωt ERFC E ag N Lc4.0- /6/0 D9 N

22 Exampls f Binar Cmmunicatins Sstms Nt: f ( E N ) AVG [J] [ - ] W Hz J Cst f cmmunicatins cst f nrg, Juls pr bit (.g. r lw bit rats impl r lw pwr transmittrs, small antnnas) Lc4.0- /6/0 D0

23 rbabilit f Baud Errr 6 db FSK OOK (chrnt) BSK 3 db FSK nn-chrnt E/N (db) Nn-chrnt FSK: carrir is unsnchrnizd s that bth sin and csin trms admittd, incrasing nis. Such nlp dtctrs ha a diffrnt frm f (E/N ). Lc4.0-3 /6/0 Nt hw rapidl dclins fr E N ~ > 6 db D