New Cross-correlation Results for Multi-rate CDMA

Transcription

1 New Cross-correlaton esults for Mult-rate CDMA Erc Hameln, esle A. usch and aul Forter Department of Electrcal and Computer Engneerng Unversté aval, Québec, Canada GK 74 ( , ( fax ABSAC - Support of multple data servces can be accomplshed va code dvson multple access (CDMA wth a constant chp rate, but varable data rates. We present exact euatons for the cross-correlaton of codes n a multple-data rate system, and thus euatons for the multple access nterference (MAI and bt error rate (. revous analyss has found the usng the mean of the MAI assumng codes are truly random seuences. We show that Gold and Kasam seuences have a cross-correlaton wth a dstrbuton closely approxmated by a Gaussan one, and present confdence ntervals to uantfy the performance under the random code assumpton.. Introducton New vdeo and multmeda applcatons reure more bandwdth than classcal applcatons wth whom they must coexst. One popular approach to share bandwdth s va the use of code dvson multple access (CDMA. We examne multrate CDMA systems wth fxed chp rate for all users, but varyng seuence length accordng to the data rate reured. Semnal work on sngle rate systems by ursley and Sarwate examned the cross-correlaton propertes of pseudorandom codes for CDMA [-]. hey consdered the popular bnary shft-regster seuences (m-seuences, Gold codes and Kasam seuences and charactered ther performance va what they defned as an average nterference parameter. As bounds on ths parameter were too loose to be of value, the mean (calculated by modelng the pseudo-random codes as truly random was used to evaluate performance. Ottosson and Svensson [] extended sngle rate results to mult-rate CDMA systems wth constant chp rates usng the random seuence assumpton. As they were only nterested n the mean value of the average nterference parameter, they bypassed calculaton of the exact value of the parameter as a functon of actual code values. In ths work we determne an exact euaton for the average nterference parameter for the mult-rate CDMA system usng actual seuences. For sngle rate systems the average nterference parameter nvolves correlaton over two partally nterferng bts, whle the mult-rate system can have many overlappng nterference bts leadng to much more complcated expressons for the average nterference parameter. Extensve calculaton of the exact nterference parameter among actual code seuences led us to observe a Gaussan dstrbuton. Consder for example a sngle user system wth length 7 codes, and 7 dstnct codes. here are 7 6 dstnct average nterference parameters dependng upon the partcular par of codes consdered. If we consder the choce of actve codes from the set of 7 determnstc codes to be random, the nterference parameter s observed to have a Gaussan dstrbuton, for both sngle-rate or mult-rate systems. Examnaton of several code famles shows that the mean of the Gaussan dstrbuton concdes wth the mean under the random code assumpton. By assumng a Gaussan dstrbuton for the code, we are able to uantfy (va confdence ntervals the value of the random code assumpton.. Mult-rate DS-CDMA System Model We use drect seuence (DS CDMA wth bnary phase shft keyng (BSK. Each user transmts ndependently of other users, and the system can support p dfferent data rates (subsystems. he transmtted sgnal of user k n subsystem s sk ( t bk ( t ak ( t cos ω ct + θk b g ( where ω c s the carrer freuency, θ k s the ntal phase offset (modeled as a seuence of ndependent random varables unformly dstrbuted over [,π], b k (t s the data bt of user k of subsystem at tme t (modeled as a set of ndependent euprobable random varables takng values n {-,, a k (t s the spreadng code of perod N of user k of subsystem (takng values n {-,, and E b / s the average power correspondng to data rate /, where s the bt nterval on subsystem. We assume the spreadng code a k (t conssts of rectangular pulse shapes wth chp pulse duraton c /N. he sgnal s average power s dfferent for each data rate, wth constant energy per bt [],.e.,. he receved sgnal s at any network node s gven by r( t w( t + sk ( t τ k ( k where w(t s the channel nose modeled as ero mean addtve whte Gaussan nose (AWGN, K s the number of smultaneous users wth data rate, and τ k s the relatve delay of user k of subsystem (modeled as a set of ndependent random varables unformly dstrbuted over [, ]. Whle sub-optmal for ths channel, we nonetheless adopt the matched flter detector due to ts smplcty and wdespread use. he recever matched to user l n subsystem has output Z l r ( t a l ( t cos( ω ct dt ( where s the desred sgnal s bt duraton and a l (t s ts spreadng code seuence of the desred sgnal. A seuence has degree n for perod N n -. Gven two seuences of dfferent n n y x lengths ( s the rato between the seuence lengths.

2 . Analyss he matched flter output Z l reduces to nose plus p K b k ( t k a k ( t k a l ( t cos( k dt τ τ ϕ (4 k where the relatve phase dfferences and delays are ϕ k θ k -ω c τ k. he desred sgnal we have taken to be user l on subsystem, therefore the term kl corresponds to the desred user. Furthermore, we assume coherent detecton and deal tmng synchronaton between the desred user and the recever,.e., θ l τ l hence Z l s b t a l ( l ( t dt + nose K p (5 + b k ( t τ k a k( t τ k a l ( t cos( ϕ k dt ( k l k where the frst ntegral s the desred sgnal, the second term s AWGN and the thrd term s the multple access nterference (MAI. he sgnal porton can be wrtten as S l b t a l l t dt b ( ( ( l (6 ( where b l F G HG s the desred user s data bt. he MAI I l G b k ( t τ k a k ( t τ k a l( t cos( ϕ k J dt (7 k s the sum of all non-desred users transmttng asynchronously at varous rates n the shared medum wth ther assocated power levels, delays and sgnature seuences. In order to determne the bt error rate performance of any CDMA system we must evaluate the statstcs of the MAI. We assume that the nterference has a Gaussan dstrbuton (ustfed under certan condtons by the central lmt theorem, t suffces to determne the frst and second moments of the MAI. For the case of sngle-rate CDMA, exact expressons for the varance of the MAI have been found n [], however due to the dffculty n nterpretng ths expresson for specfc codes, research has focused on approxmatons, modelng codes as truly random seuences of ± []. Under ths assumpton the varance s averaged over the dstrbuton of the random codes, effectvely elmnatng the characterstcs of the partcular type of code n use (e.g., m-seuence, Gold, Kasam, etc.. he only results for mult-rate CDMA wth constant chp rate have employed the random code assumpton []. In the followng secton we determne the exact expresson for the varance of the MAI as a functon of the actual codes n use.. Multple Access Interference (MAI Assumng euprobable bpolar data, the MAI has ero mean and the varance var[i l ] s gven by I J KJ M NM N O O b g b g b g b g (8 Q Q p K E M b k t τ k a k t τ k a l t cos ϕ k dt k M l where the expectaton s taken over the data bts, phase offsets and delays, each of whch forms a set of ndependent random varables whch are also mutually ndependent. Note that cross-terms n the suared expresson are the product of ndependent, ero mean random varables. Averagng over the ntal phase offsets θ k, we can wrte var I l p K k l, E J bk τ k k (9 and defne the cross-correlaton Jk bkbt τ kga kbt τ kga lbtgdt ( We further defne the partal cross-correlaton functon as t k, lbt, tg akbt τ kga lbtgdt ( t he ntegral of ( s defned over, but the delays are defned over, g, ths means that to solve euaton (9 t s necessary to dvde further calculatons nto three dstnct cases:, < and >.. Case hs s the classc sngle rate case [,,], all nterferng sgnals seuence lengths beng eual to the desred sgnal s seuence length (.e., N N. J k can be wrtten as [] Jk bkbt τ kga kbt τ kga lbtgdt ( ( where b k ( and b k are nterferng bts overlappng the de- ( sred sgnal s bt b l. Defne C xy (, the aperodc crosscorrelaton functon, by Cx, y( N N N + yd N < N ( where N s the length of the code seuences and x d s the d th chp of seuence x. It has been shown [,,] that k,τ k Jk c r k (4 he nterference parameter r k s the lnk between the dscrete (chp synchronous aperodc cross-correlaton functon that we can easly evaluate numercally and the partal cross-

3 correlaton functon that wll enable us to evaluate the performance of the asynchronous network. N rk { Ck, l ( N + Ck, l ( + N + C ( N C ( + N k, l k, l k, l k, l k, l k, l + C ( C ( + + C ( + C ( + (5.. Case > he frst mult-rate case we examne has the desred user s bt beng longer than the nterferer s bt. We use the defnton of the rato of chp rates to wrte ( as c b g h b g d c b g h Jk bkm ( k, l τ k + m, τ k + m+ m ( ( + bk k, l, τ k + bk k, l τ k +, ( + bk k, l τk +, τk + (6 where Erreur! Source du renvo ntrouvable. llustrates the many overlappng bts contrbuted by each nterferng user to calculate J k. It s necessary to average J k over these bts and ther delays. When suarng J k, the cross-terms are the product of ndependent, ero mean random varables, so the varance of J k s d b g O NM NM e c b g Eτ E k k, l, τ k τ k k, l τ k, τ k Q Eτ m m k d k, lbτk +, τk + ( + NM g O Q m + Eτ k, l τ k +, k NM e d O Q h O Q (7 We dvde the ntegral nto a sum of ntegrals over the chp ntervals. We can express the partal cross-correlatons as b g b g b g c b g h b g b g b g b g cb g h b g x, y, τ x, y, c + τ c x, y, + c x, y, c x, y τ, x, y c, + τ c x, y + c, x, y c, as n the sngle user case []. We defne the aperodc crosscorrelaton functon for > as Cx, y( N > N d y x Nx + yd Nx < Nx b g Nx N y b gn x b g Nx < b g Nx + b g N y b g Nx + b g N y otherwse (8 (9 We note that for N x N y the euaton reduces the sngle user case. Wth ths defnton the euatons for the partal crosscorrelatons become b g b g b g b g b g b g b g b g b g b g x, y, τ ccx, y N + τ c Cx, y + N Cx, y N x, y τ, c Cx, y + τ c Cx, y + Cx, y ( Whle these two expressons are dentcal to those for the sngle user case, we note that the aperodc cross-correlaton functon C xy dffers greatly from the sngle user case. After substtutng ( nto (7 and extensve algebrac manpulaton (verfed va symbolc mathematcs software we can wrte k,τ k Jk c r k where the nterference parameter for ths case s as follows. r C ( N + C ( + N k N k, l k, l + C ( N C ( + N k, l k, l k, l k, l + C ( + mn + C ( + + mn m { { + C ( + mn C ( + + mn k, l k, l.. Case < Usng the same analytcal approach t can be shown that k,τ k Jk where the nterference parameter s c r k N rk { Ck, l ( N + Ck, l ( + N N + Ck, l ( N Ck, l ( + N N + { Ck, l ( N + Ck, l ( + N + Ck, l ( N Ck, l ( + N + Ck, l ( + Ck, l ( + + Ck, l ( Ck, l ( + and the aperodc cross-correlaton functon s Cx, y( N < N d y x ( ( N y N y N x + yd Nx N y Nx ( N y N y Nx otherwse

4 . Sgnal to Nose + Interference ato (SNI We are now n a poston to take the varance of the MAI and wrte an expresson for the sgnal-to-nose-plus-nterference rato (SNI. In all cases of relatve seuence length (, <, > we found herefore we can wrte k,τ k Jk var MAI var I l c r k N N r k 6 k (4 Euaton (4 gves us an expresson of the MAI n terms of the relatve transmsson rates and the nterference parameter r k that s only a functon of the code seuences (of varous lengths. he SNI s gven by SNImultrate N N r k + SN 6 k (5 where SN s the sngle-user sgnal-to-thermal-nose rato for the network. he probablty of error functon for BSK, usng the Gaussan assumpton for the MAI, s e F HG I KJ SNI 5. erfc multrate (6 Wth euaton (6 we have an expresson for the probablty of error ( as a functon of the nterference parameter for specfc code seuences. 4. Calculaton esults As seen n (5, the SNI (and hence the depends on the set of nterference parameters r k for all the actve users. Snce the set of actve users s tme-varyng and we wsh to know overall performance of the system, we must do some type of averagng over the nterference parameter. revous work for both sngle-rate [] and mult-rate [] CDMA has accomplshed ths averagng by modelng the spreadng codes as truly random. he varance of the MAI s then determned by further averagng over the random chp seuences. Snce the nterference parameter appears lnearly n the MAI varance, ths amounts to calculatng the mean of (5, ( or ( under the random code assumpton. We calculate ths to be Erandom codes rk N N > N N < corroboratng the results n [] and []. he mean no longer depends on the subscrpt k, as all codes have the same mean. We calculate the value of all possble nterference parameters r k for varous code sets and examne the hstograms of these calculatons to gauge the accuracy of the random code assumpton, and determne the probablty densty functon (pdf of the nterference parameter. Knowledge of the pdf of the nterference parameter allows us to go beyond smply usng the of the SNI to calculate the. Wth the pdf we can also generate confdence ntervals for the calculatons. able Sample mean and varance of the average nterference parameter (r k for varous seuences Case Desred user Interferer r k ype N ype N mean var Gold 7 Gold Gold 5 Gold Gold 5 Kasam Gold 7 Gold Gold 7 Kasam In able we present the sample mean and varance of the nterference parameter for several code lengths and code types. Case s a mult-rate ( > system wth Gold codes of length 7 and 5. Case s a mult-rate ( > system wth Gold codes of length 5 and Kasam codes of length 55. Case 4 has Gold codes of length 7 and 5, as n case, however the desred user now has code length 7,.e., <. Case 5 uses the same codes as case, however the desred user's code s length 7,.e., <. In Fgure we plot the hstograms of the nterference parameter for a sngle-rate ( system (case of Gold codes of length 7. Occurrences theoretcal mean lower bound µ.99 σ.79 Gaussan ft upper bound.5.5 normaled value Fgure Average nterference parameter (r k for Gold seuences; N N 7 In Fgure as well as n able, we have normaled the nterference parameter to ts mean under the random code assumpton. hat s, a sample mean of µ. corresponds to perfect concdence wth the theoretcal mean. he sample

5 mean was seen to vary between.99 and.. he hstogram n Fgure also shows that prevous lower and upper bounds are ute wde compared wth the lower/upper lmts of the dstrbutons. We observed that the pdf of the nterference parameter closely resembled that of the Gaussan dstrbuton. In Fgure we nclude a Gaussan pdf usng the sample mean and varance. Havng determned by calculaton the pdf of the nterference parameter, we next calculate the confdence ntervals for the determned usng the random code assumpton,.e., usng the mean of the nterference parameter. ecall that for a length 7 code and a code set of 7 codes, there are 6 7 nterference parameters. For example, for twelve actve users only twelve of these nterference parameters would contrbute to the MAI varance, and ther values could vary wdely from the mean. he pdf shows how large the devaton s lkely to be. We now assume that our nterference parameters r k are ndeed well modeled as a set of ndependent Gaussan random varables. hat s, for a gven value of, for a randomly chosen set of actve users, ther nterference parameters are ndependent and dentcally dstrbuted. We defne xk rk N rk N > rk N N < (7 whch wll nomnally have unty mean. he observed means and varances of these Gaussan random varables are determned by the prevous exhaustve calculatons and are ndcated n able for the partcular code seuences studed. Usng euaton (7 we can defne X rk k 6N N xk N k xk > N k xk < N k l (8 We assume the parameters x k are uncorrelated, so that ther moments are easly calculated. Gven euaton (6 we can plot confdence ntervals for the probablty of error usng the dstrbuton of X, F HG e 5. erfc mean X ± std X + SN b I gkj (9 where X s replaced by ts mean, then by ts mean plus one standard devaton, then by ts mean mnus one standard devaton. hs gves a confdence nterval of 68%. For we add/subtract twce the standard devaton Sngle rate case, hgh data rate user detected 68% confdence KSmultaneous users Fgure Sngle rate system, hgh data rate user detected, case wo dual rate cases, low data rate user detected low rate ; K- hgh rate (case low rate ; K- med. rate (case - 68% confdence KSmultaneous users Fgure Dual rate systems, low data rate user detected, cases &, one low data rate user, K- other rate users wo dual rate cases, hgh data rate user detected hgh rate ; K- med. rate (case 5 hgh rate ; K- low rate (case 4-68% confdence KSmultaneous users Fgure 4 Dual rate systems, hgh data rate user detected, cases 4 & 5, one hgh data rate user, K- other rate users

6 Note that under the random code assumpton, the mean of the nterference parameter s the same for all codes, therefore the s a functon of the number of actve users, K. For our observaton of a Gaussan ft to the nterference parameter, the depends on the number of users at each data rate, K. We wll plot cases for systems wth two and three data rates, usng the calculaton for the Gaussan pdf presented n able. For Fgures -7 we fx the chp rate and the sngle-user SN (.e., the sgnal to thermal nose rato and plot the versus the number of smultaneous users K, wth all data taken from able. A farly strong SN of 8 db was used to stress condtons under whch the MAI s the most mportant nose source. Usng the Gaussan ft of the nterference parameter pdf, we calculate the bounds on the parameter for and 68% confdence, and plot the for the, and the extreme values for these confdence ntervals. In Fgure we see a sngle rate system, and the confdence ntervals are very tght, confrmng the mean s a good estmator of the nterference parameter for sngle rate systems. In Fgure and Fgure 4 we present dual data rate systems, but wth the desred user the only user at one of the two data rates. Here too the confdence ntervals are reasonably tght, although we begn to see some dvergence n Fgure 5. In Fgure 6 we take one of the dual rate systems and let the occupaton of actve users at each data rate vary. As we let the number of hgh rate users ncrease from the sngle desred user to half the actve users, we see that the confdence nterval can become sgnfcantly wde. For nstance, for a of - the number of users can vary from 8 to. In Fgure 6 and Fgure 7 we see smlar behavor n a three-rate system wth varous occupances at each rate. 5. Conclusons Our smulatons demonstrate a near Gaussan dstrbuton of the values of r k when all combnatons of seuences and delays are consdered. he devaton from the mean value can be uantfed by the use of a confdence nterval n the evaluaton of the of a mult-rate CDMA system. he results of our calculatons confrm that n sngle rate systems the random seuence assumpton s a good estmator of the mean value of r k. On the other hand, some dscrepances can be seen n the mult-user case,.e., sgnfcant varaton from the mean. 6. eferences. M. B. ursley, erformance Evaluaton for hase-coded Spread Spectrum Multple Access Communcaton-art I: System Analyss, IEEE ransactons on Communcatons, vol. 5, no. 8, pp , August M. B. ursley and D. V. Sarwate, erformance Evaluaton for hase-coded Spread Spectrum Multple Access Communcaton-art II: Code Seuence Analyss, IEEE ransactons on Communcatons, vol. 5, no. 8, pp. 8-8, August Ottoson and A. Svensson, Mult-rate Schemes n DS/CDMA Systems, roceedngs IEEE VC 95, Chcago, pp. 6-, July Dual data rates, hgh data rate user detected 5% hgh rate ; 5% low rate 5% hgh rate ; 75% low rate % hgh rate ; 9% low rate hgh rate ; K- low rate - 68% confdence KSmultaneous users Fgure 5 Dual rate system, hgh data rate user detected, case 4, varous rate occupances System wth three data rates, hgh data rate user detected hgh rate ; 5% med. rate ; 75% low rate hgh rate ; 5% low rate ; 5% med. rate - 68% confdence KSmultaneous users Fgure 6 System wth three data rates, hgh data rate user detected, case 4, one hgh rate user, varous rate occupances

7 - System wth three data rates, hgh data rate user detected -4 % hgh rate ; % med. rate ; % low rate -6-8 % hgh rate ; % med. rate ; 6% low rate - 68% confdence KSmultaneous users 7. Fgure 7 System wth three data rates, hgh data rate user detected, case 4, varous rate occupances