Geometrical Description of Signals GEOMETRICAL DESCRIPTION OF SIGNALS. Geometrical/Vectorial Representation. Coder. { } S i SOURCE CODER RECEIVER

Size: px

Start display at page:

Download "Geometrical Description of Signals GEOMETRICAL DESCRIPTION OF SIGNALS. Geometrical/Vectorial Representation. Coder. { } S i SOURCE CODER RECEIVER"

Damian Lawson
5 years ago
Views:

UNIVERIÀ DEGLI UDI DI CAGLIARI Unversà degl ud d Cglr Corso d Lure Mgsrle n Ingegner Eleronc e delle elecomunczon UNIVERIÀ DEGLI UDI DI CAGLIARI Geomercl Descrpon of gnls n( { } OURCE CODER RECEIVER

In more qunve erms, e recever wll ve o mnmze e verge error probbly. sem d elecomunczone Modulo: Comunczon Elerce A.

1 UNIVERIÀ DEGLI UDI DI CAGLIARI Unversà degl ud d Cglr Corso d Lure Mgsrle n Ingegner Eleronc e delle elecomunczon UNIVERIÀ DEGLI UDI DI CAGLIARI Geomercl Descrpon of gnls n( { } OURCE CODER RECEIVER GEOMERICAL DECRIPION OF IGNAL Gven source, cnnel nd nose n(, e problem s o desgn coder nd recever so e ler, durng me nervl, s ble o decde w symbol ws rnsmed mng s few errors s possble. In more qunve erms, e recever wll ve o mnmze e verge error probbly. sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 UNIVERIÀ DEGLI UDI DI CAGLIARI Coder UNIVERIÀ DEGLI UDI DI CAGLIARI Geomercl/Vecorl Represenon e selecon of e wveform e coder s o rnsm s mporn n order o relze correc deecon. e prevous formul s e me domn represenon. gnl(offneduron For s coce, n communcon sysems, we consder generl represenon of sgnls,.e. e geomercl or vecorl represenon. Ex. : ( e α cosω I s nown perodc sgnls my be represened s lner combnon of snusodl funcons(fourer s seres. c We cn n of generlzng s nd of represenon. sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A

2 UNIVERIÀ DEGLI UDI DI CAGLIARI Bss Funcons We cn represen generc sgnl s: UNIVERIÀ DEGLI UDI DI CAGLIARI Bss Funcons e (mus be oronorml: M ( ( ( ( d (: bss funcons We refer o s decomposon w bss funcons. Once defned e (: ( M M sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A UNIVERIÀ DEGLI UDI DI CAGLIARI Bss Funcons UNIVERIÀ DEGLI UDI DI CAGLIARI Decomposon w Bss Funcon (IGNAL ANALYI Gven e ((bss funcons s srgforwrd o fnd e represenon coeffcens: M ( d ( ( ( d ( ( ( M ( M Vecor / Geomerc represenon s s clled correlor deecors. I my be relzed lso w mced flers. sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A

3 UNIVERIÀ DEGLI UDI DI CAGLIARI gnl Anlyss w Mced Flers UNIVERIÀ DEGLI UDI DI CAGLIARI gnl Anlyss w Mced Flers We cn use flers mced o e bss funcons ( ( ( y( ( y( ( ξ ( + ξ dξ ( fler mced o ( ( oronorml bss funcon (<< mplng y( e nsn, en: ( ( ( ( y( ( y ( ( ξ ( ξ dξ sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 UNIVERIÀ DEGLI UDI DI CAGLIARI gnl Anlyss w Mced Flers UNIVERIÀ DEGLI UDI DI CAGLIARI ysness of e gnl erefore, n e plce of e correlor deecors, we cn relze sceme w one mced fler nd one smpler for ec bss funcon: ( ( o reconsruc e sgnl e recever we wll use e followng sceme: δ ( ( ( ( ( M M M δ ( M ( ( M M sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5

4 UNIVERIÀ DEGLI UDI DI CAGLIARI Properes of e Vecorl Represenon Dsnce beween wo sgnls: Dsnce of sgnl from e orgn: ( ( j [ ( ( d + j E d UNIVERIÀ DEGLI UDI DI CAGLIARI Properes of e Vecorl Represenon E s e energy of (sgnl, s equl o e squred dsnce of from e orgn: E [ ( d ( d ( d ( sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A UNIVERIÀ DEGLI UDI DI CAGLIARI Number of Bss Funcons UNIVERIÀ DEGLI UDI DI CAGLIARI Number of Bss Funcons If s e source me, B s e bndwd of e sgnl o be rnsmed, en e number M of bss funcons necessry o represen, s: Ms lso e dmenson of vecorlspce M B W: M N Demonsron: π If c s e smplng me, en e funcons snc c recproclly sfed nervls K c, re orogonl. From e smplng eorem, we ve: c B erefore, durng e me of e sgnl, e number of bss funcons s: M B c Ignorng e edge effecs of e snc funcon, e vlue MB represens e spce dmenson of e sgnls only f B>>. sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A

5 UNIVERIÀ DEGLI UDI DI CAGLIARI e Grm-cmd Procedure I s sequenl meod used o clcule e bss funcons. ( ( ( ( d [ ( ( ( + ( ( d ( ( ( ( ( d ( d [ ( ( ; E d UNIVERIÀ DEGLI UDI DI CAGLIARI e Grm-cmd Procedure Gong ed: ( ( ( ( ( + ( + ( ( We fnd n nlogous wy s, so we could denfy φ (. ( ( d ; ( ( ( d [ ( ( ( d d sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A UNIVERIÀ DEGLI UDI DI CAGLIARI e Grm-cmd Procedure UNIVERIÀ DEGLI UDI DI CAGLIARI e Grm-cmd Procedure en we go for w: Connue n s mnner unl ll Nsgnls ve been consdered. If en φ (s useless! (cn be expressed s lner combnon of only φ nd φ ( + + ( Fnlly, we ll ve Mbss funcons w M N ( ( ( ( ( ( ( d ; ( ( d sem d elecomunczone Modulo: Comunczon Elerce A.A sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5

6 UNIVERIÀ DEGLI UDI DI CAGLIARI Vecorl Represenon UNIVERIÀ DEGLI UDI DI CAGLIARI Vecorl Represenon Exmple: snusodl sgnls Exmple: sep sgnls cosω cosω If e perod s neger mulple of e perods nd e bss wveforms re orogonl. ey cn be expressed by mens of smll recngles, no overlpped, n order o cover e enre nervl. Prcclly,wen >>, cos ω nd cos ω re consdered orogonl. s fxed by e wd of recngles composng e seps. sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 UNIVERIÀ DEGLI UDI DI CAGLIARI Vecorl Represenon UNIVERIÀ DEGLI UDI DI CAGLIARI Vecorl Descrpon Benefs M e decson process s mced w dsnce mesures. / Δ / Δ / Δ ere s smple geomerc nerpreon for e funconng of e recever. Wors cse: f ( nd N>/ M/ e pproxmon w recngles s bss funcons cn be successfully used for ll sgnls ( smll. Aenon: smll ncreses e bss funcon number nd e bndwd. sem d elecomunczone Modulo: Comunczon Elerce A.A. 4-5 sem d elecomunczone Modulo: Comunczon Elerce A.A

7 UNIVERIÀ DEGLI UDI DI CAGLIARI W s bss funcon? F.A.Q How s e mnmum vlue of useful bss funcon n order o represen N sgnls? Gven bss funcon, ow o obn e represenon coeffens? sem d elecomunczone Modulo: Comunczon Elerce A.A

Brand Selection and its Matrix Structure -Expansion to the Second Order Lag-

Brand Selection and its Matrix Structure -Expansion to the Second Order Lag- Journl of Compuons & Modellng vol.3 no.3 3 87-99 ISS: 79-765 (prn) 79-885 (onlne) Scenpress Ld 3 Brnd Selecon nd s Mrx Srucure -Expnson o he Second Order Lg- Kuhro Tkeysu nd Yuk Hguch src Focusng h consumers