Consider a Binary antipodal system which produces data of δ (t)

Transcription

1 Modulaion Polem PSK: (inay Phae-hi keying) Conide a inay anipodal yem whih podue daa o δ ( o + δ ( o inay and epeively. Thi daa i paed o pule haping ile and he oupu o he pule haping ile i muliplied y o( π o have a pa-and peum in he hannel (up-onveion). o( π Mappe daa = ± δ ( Pule Shaping File p ( ( daa p( o(π a) valuae ime domain and equeny domain epeenaion o anmied ignal. Why pa-and ommuniaion i equied? Wha i he ignal andwidh a he anmie oupu i Squae-Roo-Raied Coine (SQRC) ile i ued a pule haping ile? ) Wha i he opimum eeive uue? Wha i he ane union o mahed ile? Diu he poailiy o eo o he pa-and yem. ) I he phae o he aie in he anmie and he eeive ae no he ame, wha happen? Can we eove he daa in hi ae? Soluion: Pa a) The anmie i anmiing eihe p( o(π o p( o(π o in ohe wod i i anmiing eihe p( o(π o p( o(π + π ). Thi i why we all hi yem inay phae hi keying (PSK). We eihe end he oine wih phae o o wih a phae oπ. Conide a pule haping ile o p( a hown elow: p ( T Then, he anmie uue and oupu o anmie in ime domain i a hown elow: Conodia Univeiy LC4

2 Modulaion,,,,,,,... + Mappe o [ π ] ( o [ π ] + o[π ] ( = o[π ],,,,,,,... The ignal a he oupu o he anmie in he equeny domain i:.p( + ) +.P( ) Thee ae wo eaon o having paand ommuniaion: - To divide he availale andpa andwidh eween eveal ue. - To have mall ize anenna due o having high equenie. The andwidh o he ignal a he oupu o he paand anmie i: W ignal = ( + β ) T whih i wie he aeand andwidh. Pa ) Opimum eeive i a ollow: o( π o o daa p( o(π x ( Mahed File p( T Reeive z ( Theeoe, he aove i he opimum andpa eeive. T Deeo daa + o(4π daa p( daa p( o(4π x( = daa p( o (π = daa p( = + Sine mahed ile i a law pa ile we have: daa z( = p( * p( T Theeoe, pa and yem i equivalen o aeand yem. The mahed ile ha he ame ane union a pule haping ile and heeoe i i SQRC. ) Conodia Univeiy LC4

3 Modulaion The poailiy o eo o he paand yem i he ame a aeand yem and heeoe he R would e a alulaed eoe: Pa ) Aume a phae dieene oϕ, x( = daa p( o(π o(π + ϕ) R( PSK ) = Q o(4π + ϕ) + o( ϕ) daa p( o( ϕ) daa p( o(4π + ϕ) = daa p( = + Sine mahed ile i a law pa ile we have: daa z ( = p( * p( T o( ϕ) Sine ϕ i a andom nume eween and π, we anno make a deiion uing maximum likelihood deeo. To e ale o eove he daa, we hould i indϕ. Conodia Univeiy 3 LC4

4 Modulaion Two dimenional modulaion heme: o( π o( π = T m Mappe I h ( ( ( m ( Î = T Deiion m ) Q h ( m ( Qˆ Tanmie in( π in( π Re eive andwidh: W= ( ) ( β ) + β = + T The onellaion diagam o i a ollow: R Le onide he yem elow, whee adjaen poin (hoizonally o veially) have equal diane om eah ohe. Deiion oundaie o maximum likelihood deeion ae hown y dahed line. 3 P ( ) = 4( ) Q P ( ) 3 P = = ( ) Q log 4 = 3Q Conodia Univeiy 4 LC4

5 Modulaion I Q The deiion lok wok aed on deiion egion. The onellaion diagam o PSK i a ollow: Le onide he PSK yem elow, whee adjaen poin have equal angula diane. Deiion oundaie o maximum likelihood deeion ae hown y dahed line. d θ θ = π /8 Fo PSK, we an ue Gay oding a hown in he onellaion diagam and heeoe: P 4 4P = = = in( /) = P Q in( π /) Q π Q log R in( π /) Anwe: P 8P = Q in( π / R,,,..., Gay oding i oviou. I 3 ae poin in he onellaion diagam in lok-wie ode, hen Gay oding an e hown in he mappe a ollow: I x.7 y -y -.7 -x - -x -.7 -y y.7 x Q y.7 x x.7 y -y -.7 -x - -x -.7 -y x = o(π / 8) y = in(π / 8) = an The deiion lok i evaluae he phae o he eeived veo ϕ and hen deiion Q I ae made a ollow: ( i ) π ( i + ) π I < ϕ <, deide ha i ha een anmied. (o i =,,3,.., ) Conodia Univeiy LC4

6 Modulaion Polem: Deign o Communiaion Syem A eam o digial daa wih daa ae o M/ i o e anmied in an addiive whie Gauian noie hannel wih powe peal deniy o Wa/Hz whee he availale andwidh i 4 MHz. The equied yem peomane i a i eo ae o. Conide M o MPSK and deign he e paand anmie and eeive y deemining all he paamee o he deign (inluding he eeived powe) and dawing a yem lok diagam. Soluion: = M ( ) R ( ) aume.3 DS R = + = +.3 =.3 log M log M 4 Hz log M 3.9, o we an hooe M = and oeponding heme ae - o -PSK. 3 = P ( ) = 4( ) Q 3Q P ( ) 3 P = = ( ) Q log 4 P ( ) = 3 = Q 4 = Q.33 4 = = Fom Q-union ale x 4. 7 and heeoe: 9 =. 9 P = R =. =.7mW π P ( PSK ) = P ( M ) Q in and M P P P = = π Q in log = Fom Q-union ale: π x = in = 4. P R R P log M = ( Gay oded ) = 8 = =.. 8 = = = =.8mW log M log Theeoe,- i ee, ine he eeived powe i 3 o -PSK. Conodia Univeiy LC4

7 Modulaion Polem (Analog Communiaion) The polem o he uue o polem. i he ue o ideal ile. We would like o deign anohe anmie whih ue no ile. Thi kind o modulao i alled Quadaue-Ampliude-Modulao (). Daw he uue o hi new anmie o anmi he inomaion aou m ( and m ( uh ha he oupu ignal x _ ou( ha he ame ene equeny and he ame andwidh a o polem.. Show ha you deign wok y howing he uue o he eeive and explaining he eovey o oh ignal m ( ) and m ( ). Daw he equeny peum o all he ignal in he anmie and he eeive a well. Soluion: Single ideand ignal ae had o geneae ine a hown in pa a o hi polem ideal ile ae equied. To anmi wo ignal m ( ) and m ( ), we an ue Quadaue Ampliude Modulaion () whih do no equie any ile a all and heeoe i i paial. Tanmie Reeive = m ( o π m ( in π and he aie ( + The anmie oupu i ϕ ( ) ( ) equeny i = 4 khz. To ind he andwidh and ene equeny o ϕ ( examine wo ignal m ( o( π 4 ) and m ( in( π 4 ), we hould. The ae-and andwidh o m ( and m ( i khz and a we know he andwidh o eah o he Doule Side-and modulaed m ( o 4 m ( in π 4 would e doule o ha whih i khz. The ignal ( π ) and ( ) ene equeny o eah o hee ignal would e he aie equeny o 4kHz. ϕ = m ( o π 4 + m ( in π will have ( 4 Theeoe, he anmied ignal ( ) ( ) ene equeny o 4kHz wih andwidh o khz. To eove m ( and m (, oheen eeive a hown aove ould e ued. Thi an e poved y alulaing x ( and x ( aed on he uue o he eeive. x = ϕ ( o(π 4 = m ( o(π 4 + m ( in π 4 o(π 4 [ ( )] ) ( + m ( in( [ m ( o(π 4 + m ( in( π 4 )] in( 4 ) = m ( + m ( o(π 9 π 9 x = ϕ ( in(π 4 = π ( ( = m m ( o(π 9 + m ( in π 9 ( Conodia Univeiy 7 LC4

8 The la wo em in oh equaion ae andpa ignal eneed a = 9kHz whih ae uppeed y he lowpa ile and heeoe, he oupu o hee ile will eove ( ) oe ha hee low-pa ile ae no ideal. Fequeny peum o all he ignal ae povided elow: M ( ) Modulaion m and m ( ). - [khz ] M ( ) - Re / [ ( )] Φ -4 4 Im / [ ( )] Φ -4 4 We noe ha he anmied equeny peum i omplex. Theeoe, Φ [ Φ ( )] + j Im[ Φ ( )] ( ) = Re Conideing ha low pa ile in he eeive have ane union o H( ) and H ( ), we an how he equeny peum o all he ignal in he eeive and alo ome up wih ane union o hee wo ile. Conodia Univeiy 8 LC4

9 [ X ( )] Re Modulaion [ X ( )] Im / H ( ) ˆ M ( ) [ X ( )] Re [ X ( )] Im / H ( ) M ( ) ˆ aed on he aove wo lowpa ile ae idenial wih pa and o o Kz, op-and o lage han 8 KHz and aniion and o KHz o 8 KHz. The hal ampliude u-o equeny i 4 KHz. oe ha uing hee ile and auming ha hee i no noie in he hannel, we have: M ˆ ˆ ( ) = M ( ) m ( = m ( M ˆ ( ) = M ( ) mˆ ( = m ( ) Conodia Univeiy 9 LC4