Omum Sgnal Mappng fo Mult-Pha Bnay-CDMA Sytm Yong-Jn So and Yong-Hwan L Shool of Eltal Engnng and INMC Soul Natonal Unvty Kwanak P O Box 34 Soul 5-744 Koa -mal: yl@nuak Atat - Although th CDMA ytm an ffntly uppot multpl u may uff fom pak-to-avag pow ato (PAPR) na a th num of u na A a ult t nd hghly lna pow amplf wth a lag ak off Rntly a nw CDMA hm alld nay CDMA (B-CDMA) ha n popod to allvat th polm y quantzng th nvlop of mult-u CDMA gnal nto a mall num of lvl whl pvng th advantag of CDMA gnalng [] Th pfoman of B-CDMA ytm manly dtmnd y th quantzaton and dtton o Th quantzaton no an mnmzd ung th Lloyd-max algothm [] In th pap th omum gnal dgnd to mnmz th dtton o n mult-pha B-CDMA ytm Fnally th analyt ult a vfd y omput mulaton INTRODUCTION On of mao dawak of mult-od CDMA ytm hgh pak-to-avag pow ato (PAPR) du to th agggaton of multpl padng od A a ult multod CDMA tanmtt qu th u of hghly lna pow amplf wth a lag ak off Bnay CDMA (B- CDMA) a nw modulaton mthod that quantz th gnal ampltud nto a mall num of lvl and mploy -modulaton fo tanmon wth ontant nvlop [] Thu th B-CDMA an allvat th nd of lna pow amplf whl pvng th advantag of CDMA Th wok wa uppotd y Koa Rah Foundaton gnalng uh a th oft apaty and outn to ntfn Howv th pfoman an gnfantly afftd y th quantzaton po Th B-CDMA gnal an gnatd y vaou mthod nludng th pul-wdth (PW) mult pha (MP) and od lton (CS) mthod [] Th PW B-CDMA gnal otand y onvtng th magntud of mult-lvl gnal nto a fnt num of pul wdth Thu th tanmon andwdth of th PW B-CDMA na a th quantzaton lvl na In pat th gnal an quantzd nto two lvl to aommodat th na of tanmon andwdth Th MP B-CDMA gnatd y tanfomng th gnal ampltud nto a fnt num of gnal ontllaton Th CS B-CDMA gnatd y a two-tp po In th ft tp th ut of padng od ltd to du th num of gnal lvl In th ond tp th ltd od modulatd ung th MP B-CDMA hm Omum quantzaton of th gnal ampltud an ahvd y ung th Lloyd-max algothm [] Howv th gnal ontllaton ha not n omzd analytally fo th MP B-CDMA Th manly du to th fat that th man qua o of th hp don o and th ymol o n th ytm annot pntd n a mpl fom Th two gnal pont havng th lagt dtan aft th quantzaton a -mappd o that thy hav th lagt dtan on th gnal ontllaton [3] Fo a gvn gnal ontllaton th don gon an dtmnd o a to mnmz th Bay ot ton [4] Howv t may not omum au th t o at (BER) pfoman an mo afftd y th gnal mappng than th don gon In th pap w omz th mappng pont of th MP B-CDMA gnal n addtv wht Gauan no (AWGN) hannl Sn th
N d = () = N N å d Quantz tunato x y z n Mapp mapp Dmapp Dmapp Tx Channl Rx ˆ H N th total num of u Th agggatd u gnal d quantzd at th hplvl Th output of th quantz an pntd a = f q ( d ) () CS B-CDMA th am a th MP B-CDMA x th od lton lok [5] th analyt dgn an alo appld to omum dgn of th CS B-CDMA gnal Ston d th tutu of th MP B-CDMA ytm In Ston 3 th no du to th quantzaton and hp dtton o analyzd Th mappng pont a omzd to mnmz th hp dtton o ung an tatv mthod Th popod gnal mappng hm valuatd ung omput mulaton n Ston 4 Conluon a ummazd n Ston 5 SYSTEM MODEL In th MP B-CDMA ytm th um of multpl u data quantzd nto a fnt num of lvl and thn modulatd ung a modulaton hm Fg d th tanv tutu of a aand-quvalnt MP B- CDMA ytm wth quantzaton lvl 4 wh and vly dnot th t and padng od of th -th u and th gnal d dnot th um of multpl u data gvn y Φ Φ Φ Φ Φ Fg MP B-CDMA tanv tutu d quantzaton pont Ω Ω guad pha Ω gnal pont and dtton gon E Fg modulaton of MP B-CDMA 3 4 Ω 3 Ω 4 wh fq ( d ) dnot th quantzaton funton that map th gnal d n th quantzaton gon Φ onto th gnal pont m at th hp-lvl Thn th quantzd gnal mappd a x = () (3) f map wh fmap ( ) dnot th mappng funton that map th gnal onto th ontllaton Fg d th quantzaton and -mappng gon of th MP B-CDMA ytm wh d dnot th dtan twn th quantzd gnal pont and Ω dnot th dtton gon of th -modulatd gnal pont N th num of gnal pont aft th quantzaton Not that th guad pha qud n th MP B-CDMA ytm to du th don o twn th gnal pont havng th lagt dtan [3] Th vd gnal an wttn a = x+ n (4) wh n dnot th AWGN tm Th dmodulato tanfom th pha nfomaton of nto th magntud y = f map () (5) Th dmodulatd gnal y dpadd ung th padng od of th -th u to dtt th u data ˆ a ˆ = f y f y < Lt and th gnal omponnt quantzaton no and no du to hp dtton o at th -th hp vly Thn th dmodulatd gnal y at th -th hp an pntd a (6)
y = + + (7) Tal Mappng pha E Convntonal Exhautv ah Popod o 3 OPTIMUM SIGNAL CONSTELLATION FOR MP B-CDMA Sn th no olaton twn vaan of y an wttn a y and th σ = σ + σ (8) wh σ and σ a th vaan of and vly Aumng that th padng od of ah u ha unt pow th vaan of d opondng to th - th hp of d qual to th num of mult-od N Th vaan of th quantzaton no p ah hp σ an pntd a [6] x = ( x m ) xp dx π N N σ = Φ (9) Th omum m and Φ mnmzng σ an otand ung th Lloyd-max algothm [] Th Lloyd-max algothm an fnd th omum quantzaton lvl n an tatv mann o a to mnmz th quantzaton no pow Th vaan of an alulatd a σ N p( ) p( ) d = = = () wh p ( ) th poalty dnty funton (pdf) of gnal pont gvn y x p () = xp dx Φ π N N () and p( ) th poalty that gnal pont mdttd to [6] n ( ) ( R o( )) / p( ) = R drd () π Ω db 35 9 45 85 46 8 75 3 db 35 9 45 5 9 45 4 8 34 db 35 9 45 7 85 46 9 84 38 H th hp ngy to no pow ato ( E o) and th pha of gnal pont Povdd that hgh nough mot of hp dtton o a aoatd wth th don to th adant gnal pont Thu th vaan of th hp dtton o an appoxmatd a σ p( ) p( ) d = = + = (3) wh and + dnot th adant gnal pont of gnal pont Sn () nvolv nonlna funton w appoxmat p( ) fo a of mathmatal analy It an hown that n ( ) ( R o( )) / p ( ) = R dr d π Ω (4) n ( ) + = d Ω π πo( ) ( o( )) f Sn th ft tm n (4) an gnod at hgh E (4) an appoxmatd a p( ) = Ω Ω Fo a mall π n ( ) π o( ) f( o( )) d (5) (n ( ) ) d π π o( ) f ( o( ) ) ( ) p an futh appoxmatd a
() (6) σ = (8) ntgand of p( ) 8 6 4 4 6 8 4 6 8 It an hown that pntd a a funton of = f ( ) (9) It an aly hown that σ a onvx funton of and t ha a unqu gloal mnmum W an fnd ung an tatv mthod wth an atay ntal valu Not that th omum a th funton of E Th mpl that th omum mappng pont a aoatd wth th valu of E Fg 3 Appoxmaton of p( ) ( ) p( ) f d π (6) π Ω A na th dffn twn th ntgand of () and (6) na Howv th dffn twn th two ntgal nglgl Fg 3 ompa th ntgand of () and (6) whn E = db N = 3 = and padng fato N SF = 8 Numal ult ndat that th appoxmatd p( ) qut vald n nomnal opatng ondton Lt Ω th mdtton gon of gnal pont to and th opondng ang fom ω to ω ' n (6) wh ω ' = ω + and ω ' + = ω Bau ntgand n (6) da auly a na and not pod ω ' an t to nfnt Thu th vaan of an wttn a 4 PERFORMANCE EVALUATION To vfy th pfoman mpovmnt w valuat th pfoman of th MP B-CDMA ytm wth N = 7 ( 8-) and on guad pha n AWGN hannl ung omput mulaton Th MP B-CDMA u an xtndd PN qun wth N SF = 8 and N = 3 a th padng od Fom (8) th omum pha dtmnd y 7 6 5 3 Ω 7 = 7 Ω 6 = 6 Ω 5 = 3 5 ( ) f d ω = = π + σ = π Φ π N x xp N dx d (7) = d 4 d 3 d Ω E 3 4 Ω 3 Ω 4 Ω W an fnd th omum mappng pont ( N ) that mnmz σ ung an tatv mthod Th omum pha an found y tunaton pont gnal pont and d dtton gon Fg 4 Popod 8- gnal ontllaton of th MP B-CDMA
4π = 3 4 d7 p () 3ln + + π + π d p() + d p() d3 p() + d3 p (3) 4log + 3 d p () + d p () = ( 3) d34 p(3) + d34 p(4) 4log + d3 p () + d3 p (3) 3 = () Th popod mappng pont { ˆ ˆ ˆ 3 } an found y tatvly ahng () Convntonal mappng pont hav an qual dtan on th gnal ontllaton wth on guad pha that = 35 = 9 3 = 45 Altnatvly th omum pont an alo found y xhautv ah Tal ummaz th onvntonal xhautvly ahd omum and popod mappng pont Th popod pont ˆ mla to xhautvly ahd omum pont ut ˆ and ˆ 3 a qut dffnt fom th omum pont Fg 5 d th BER pfoman of MP B-CDMA wth th mappng pont It an n that th popod mappng pont an povd th BER pfoman ompaal to th xhautvly ahd omum on although ˆ and ˆ 3 a dffnt fom th omum pont Th may mply that ˆ wth th lagt quantzaton lvl BER E-3 E-4 E-5 E/No Convntonal Popod Exhautv (omum) th domnant fato on th BER pfoman Fg 6 d th pfoman of th MP B-CDMA ytm wth th mappng pont omzd fo ah E It an n that th mappng pont omzd und a nomnal E ondton an ud fo tan vaaton of E wthout notal pfoman vaaton 5 CONCLUSION In th pap w hav analytally dtmnd th omum mappng pont of th MP B-CDMA n AWGN hannl Th mappng pont omzd y mnmzng th man qua o du to hp dtton o Th omum mappng pont and otand ung an tatv mthod ung an analyt xpon Th mulaton ult how that th popod gnal ontllaton an povd na omum pfoman yldng a BER pfoman gan of aout 4dB ov th onvntonal on REFERENCE [] H S An S M Ryu and S W Na Intoduton to Bnay CDMA Po of JCCI Ap [] T M Cov Infomaton Thoy Wly-Intn 99 [3] E K Hong M G Ahn W M L and S M Ryu "Dgn of Sgnal Contllaton fo MP/CDMA" Po of JCCI Ap [4] W M L E K Hong and Y S Pak Dgn of Omum Thhold fo Chp-Lvl Mult Phad MC- CDMA Sytm wth Nonlna Po Noth-Eat Aa IT Sympoum [5] S P Km M J Km and H S An and S M Ryu A Contant Ampltud Codng fo CS-CDMA Sytm Po of JCCI Ap [6] J Y Ko and Y H L Pfoman Analy of Bnay-CDMA Tanv n AWGN Channl JCCI 3 Ch VI-C- pp -4 Ap 3 Fg 5 BER pfoman du to dffnt pha mappng