Submtted to the IEEE Transactons on Wreless Communcatons Optmzaton of CDMA systems wth respect to Transmsson Probablty, Part I: Mutual Informaton Rate Optmzaton Its Bergel and Hagt Messer, fellow, IEEE Abstract Drect sequence code dvson multple access (CDMA) systems that use non-contnuous transmsson were consdered throughout the hstory of spread spectrum systems, but ganed renewed nterest wth the emergence of mpulse rado (IR) technology. Recently, several wors had shown that n non-contnuous CDMA, the transmsson duty cycle (or transmsson probablty) has a sgnfcant effect on system performance. In ths wor we address the optmzaton of the performance of CDMA systems wth adjustable transmsson probabltes. We consder CDMA systems that mplement non-contnuous transmsson by random puncturng, and study the system optmzaton wth respect to both transmsson powers (termed power control) and transmsson probabltes (termed probablty control). We show that the jont optmzaton has a sgnfcant performance advantage over the optmzaton wth respect to transmssons powers only. For some cases we even show that the optmzaton wth respect to transmsson probabltes alone s suffcent to acheve optmal performance. In ths part, we demonstrate the mportance of probablty control, by studyng the case of frequency-flat slow-fadng multple access channel (MAC) and no spreadng. We prove, for ths specal case, that mutual nformaton rate optmzaton s acheved by probablty control only, whle power control s redundant. The theory s supported by smulaton results, whch show that the achevable rates of all users are better n a system that uses probablty control nstead of power control. Index Terms CDMA, UWB, IR, Power Control, Probablty control. Manuscrpt submtted ovember 3, 5. Revsed July, 7. I. Bergel was wth the Department of Electrcal Engneerng Systems, Tel-Avv Unversty, Tel-Avv, 69978, Israel. He s now wth the School of Engneerng, Bar-Ilan Unversty, 59 Ramat-Gan, Israel. (e-mal: bergel@eng.bu.ac.l). H. Messer s wth the Department of Electrcal Engneerng Systems, Tel-Avv Unversty, Tel-Avv, 69978, Israel (e-mal: messer@eng.tau.ac.l).
Submtted to the IEEE Transactons on Wreless Communcatons D I. ITRODUCTIO rect sequence code dvson multple access (CDMA) systems that use non-contnuous transmsson were consdered throughout the hstory of spread spectrum systems, but ganed renewed nterest wth the emergence of mpulse rado (IR) technology [],[]. Whle IR technology had evolved ndependently of the CDMA technology, several authors had ponted out that IR systems can be regarded as a type of CDMA system [3],[4]. The mplementaton of non-contnuous transmsson s usually performed n CDMA systems n one of two methods. The frst method s by ncludng the zero n the chp alphabet [5]. Tang ths method to the extreme, the spreadng sequence may contan only small number of non-zero elements, so that the symbol s actually transmtted on a small number of pulses wth large quet gaps between them, and thus becomng the equvalent of an mpulse rado. The use of ths method was manly consdered for reasons of complexty (e.g. [],[3],[4]), where the transmtter and recever can rest durng the quet perods. The second method for the mplementaton of non-contnuous transmsson s by abortng the transmsson of some of the CDMA symbols. Ths method s commonly used to adapt to varable-rate data sources (mostly encoded speech, e.g., [6],[7]). In such case, f data s not avalable, the CDMA symbols are not transmtted, and other users gan from the reducton n nterference power. In ths wor we use the same method but wth a constant-rate data source. The non-contnuous transmsson s acheved by pseudorandom puncturng of a contnuous CDMA sgnal. Ths s smlar to the puncturng of coded symbols that s commonly used for rate matchng between the coded symbol rate and the output symbol rate [8],[9]. But, n ths wor, we wll not analyze the effect of the puncturng on specfc codes [],[]. Instead, we analyze the effect of the "quet" perods caused by ths puncturng on the nterference n the system and on the achevable performance. Early performance analyss [] of non-contnuous transmsson systems was based on the assumpton of very large number of users. In such case, usng the central lmt theorem, the nterference can be approxmated as a whte Gaussan nose, and the system performance s dentcal to that of contnuous transmsson CDMA system wth the same average transmsson powers. Later publcatons [],[3] had shown that ths s not the usual case, and demonstrated performance dfferences between contnuous transmsson and non-contnuous transmsson CDMA systems. A more recent approach loos at the systems duty cycle (or transmsson probablty when tang a
Submtted to the IEEE Transactons on Wreless Communcatons 3 statstcal approach) as an optmzaton parameter. Thus, the use of non-contnuous transmsson opens a new dmenson for system optmzaton. Intutvely, decreasng the transmsson probablty reduces the amount of collsons between symbols of dfferent users. On the other hand, to acheve the same performance, the transmsson energy must be compensated and therefore ncrease the nterference caused by the transmtted symbols. Ths optmzaton problem had only been addressed for some specfc specal cases. For example, Sadler and Swam [5] used a random spreadng approach, and found an upper bound on the bt error rate (BER) that s ndependent of the transmsson probablty. To ths end, they assumed that each spreadng sequence s composed of dentcally dstrbuted ndependent (d) chps of ternary alphabet, and used, as the optmzaton parameter, the transmsson probablty,.e., the probablty that the chp value s ±. Furthermore, t s nown [4], [5] that at the lmt where the spreadng factor goes to nfnty, the sgnal to nose plus nterference rato (SIR) of a random spreadng CDMA system converge to a constant whch s ndependent of the spreadng sequence dstrbuton (under some normalzatons). Therefore, at ths lmt, user performance depends only on the average symbol power, and the transmsson probablty has no effect. On the other hand, when the spreadng factor s fnte, the abovementoned upper bound s not too tght. Fshler and Poor [6] had shown for the two user broadcast channel n whch both users use the same parameters, that the BER s a non-ncreasng functon of the transmsson probablty. Garby et al [7] had consdered spreadng sequences that nclude varable number of zeros, and found that the mutual nformaton over an addtve whte Gaussan nose (AWG) channel s maxmzed usng only sngle nonzero chp per symbol. For the frst tme, ths study systematcally consders the performance optmzaton of non-contnuous CDMA systems wth respect to the transmsson probablty of each of the users. We show that ths optmzaton can yeld sgnfcant performance mprovement, and n some cases s even more mportant than the commonly used optmzaton wth respect to transmsson powers (usually termed power control). The transmsson probablty manly affects the performance through the nterference caused by each user to the others. As a result, the performance optmzaton must be performed smultaneously for all users n The exact formulaton n the referenced paper dd not use the term "transmsson probablty". Instead, t assumed a transmsson of one out of each chps.
Submtted to the IEEE Transactons on Wreless Communcatons 4 the system. Followng the common practce n CDMA systems, we wll assume that the transmtters cannot be synchronzed. A synchronzed networ can reduce the nterference by lettng all transmtters transmt orthogonal sgnals (as for example n tme doman multple access (TDMA) systems [8]). As the debate between CDMA and TDMA systems had been analyzed extensvely (e.g. [9],[],[]), we wll not get nto ths ssue, and adopt the CDMA approach of unsynchronzed transmssons. Another ssue s the capablty of each of the recevers to wor n the presence of nterference. Most of the methods developed for effcent recepton n the presence of mult-user nterference (MUI) requre hgh complexty, whch may be too hgh and prevent ther mplementaton n hgh data rate systems. In ths paper we lmt the dscusson to recevers that employ a sngle user decoder. As wll be dscussed n the squeal, many recevers n ths group stll requre hgh complexty. Yet, ths complexty s much less than that of optmal mult-user detecton (MUD) schemes [],[4]. In order to dfferentate our case from the conventonal contnuous transmsson CDMA systems, we refer to the type of CDMA systems that we analyze as Generalzed CDMA (GCDMA) systems. A GCDMA system s defned as a CDMA system that mplements non-contnuous transmsson by random puncturng. We consder the optmzaton of a GCDMA system performance n a K dmensonal doman (where K s the number of users). For each user we search for the optmal chp energy (transmsson power) and transmsson probablty. The mechansm that optmzes wth respect to transmsson powers s commonly termed power control, and was analyzed extensvely (e.g. [3]-[6]). Usng equvalent termnology, we term by probablty control the second mechansm, whch optmzes wth respect to transmsson probabltes. In the followng we show that the jont optmzaton wth respect to both powers and probabltes has a sgnfcant performance advantage over the optmzaton wth respect to only transmsson powers. In order to show the sgnfcance of the probablty control mechansm, we analyze the jont optmzaton problem and show that n several scenaros the optmzaton wth respect to transmsson probabltes alone s suffcent to acheve optmal performance. In such scenaros, for an optmal operaton, all actve users should transmt ther maxmum allowed power, whle any tradeoffs between users' performance are managed through the users' transmsson probabltes. We defne such systems as Probablty controlled
Submtted to the IEEE Transactons on Wreless Communcatons 5 optmal GCDMA systems. In ths paper we analyze the optmzaton of the users mutual nformaton rates n the smplfed case of frequency-flat slow-fadng multple access channel (MAC) wth no spreadng. The analyss of ths smplfed channel allows us to address the mathematcal subtletes of the optmzaton problem, whle beng able to show that the system s probablty controlled optmal. I.e., we show that n ths case the mutual nformaton rates of all users are maxmzed by probablty control alone, whle all users transmt ther maxmal allowed average power. In the second part of the paper [7] we tae a more practcal approach, and consder the optmzaton of the average sgnal to nose plus nterference rato (ASIR) n a general GCDMA system, over a frequencyselectve channel model and no constrants on the networ topology or spreadng sequences. The exact defnton of a GCDMA system and the system model consdered n ths paper are gven n the next secton. The system optmzaton problem and the defnton of probablty controlled optmal systems are gven n Secton III. Secton IV contans the proof that the consdered system s probablty controlled optmal. Secton V ncludes the dscusson of the results and numercal examples, and secton VI contans our conclusons for ths part. II. GCDMA SYSTEMS AD PROBABILITY COTROL The transmtted sgnal of the th user n a Generalzed CDMA system s defned smlarly to the one n a conventonal CDMA system (e.g. [8], [3]): s = d, d, dv,, s+ c d v=, () s () t E x g c p( t ( d v) T ) where p() t s the transmtted pulse shape, T c s the chp tme, E s the th user chp energy, s ts spreadng factor, x d, s ts d th data symbol and c dv,, s the spreadng sequence value for the v th chp. The non-contnuous (pulsed) transmsson s generated by the d bnary gatng sequence, gd, {,}, whch "punctures" the CDMA sgnal, and determnes whether a symbol s transmtted or not. The probablty p = Pr( g = ) s termed the transmsson probablty. The transmsson probablty determnes the nature, d of the system: CDMA systems use p =, and the case of p < s termed non-contnuous transmsson. At ths stage of the research we do not consder practcal ssues regardng the mplementaton of probablty control. However, some gudelnes must be stated. A practcal probablty control system wll
Submtted to the IEEE Transactons on Wreless Communcatons 6 possess a fnte set of allowed probabltes and wll choose the best probablty wthn ths set. For each probablty n the set, the system wll have a predefned pseudo random gatng sequence, and therefore the transmtters and recevers only need to now whch of the sequences s used at any gven tme. In ths paper we analyze the most basc probablty control approach,.e., the approach whch s the easest to analyze but requres the hghest complexty to mplement. We assume that there exsts a central unt whch performs the optmzaton, and nforms all users what s the optmal transmsson powers and what pseudorandom gatng sequence s used by each user. We also assume that each recever estmates the exact channel delay and gan from all transmtters n the system, and therefore can compute the exact nterference power for each symbol. We assume that the channel changes slowly enough so that the algorthm has enough tme to converge, and that the algorthm converges to the system optmal operatng pont. We also do not tae nto account the dscrete set of powers/probabltes allowed by an actual algorthm. Thus, n ths paper we focus on fndng the optmal transmsson power and optmal transmsson probablty for each of the users. ote that the use of pseudo random gatng sequences allows us to analyze the system under the assumpton of random puncturng whle stll assumng that the gatng sequence s nown to the recever. The GCDMA system has all of the advantages of conventonal CDMA systems. For example, t has the same capablty for random multple access,.e., a user can start transmttng usng ts own spreadng sequence wthout a pror notce to other users and wthout requestng for resource allocaton. Also, the bandwdth of a GCDMA system s dentcal to that of a CDMA system wth the same pulse shape. Consequently, the spectral effcency of the GCDMA system, gven by Rb / B W where B W s the sgnal bandwdth and R b s the data rate, s also dentcal to that of CDMA system wth the same data rate. In ths part of the paper we wsh to analyze and show the advantage of the probablty control mechansm n a smplfed scenaro. We focus on the achevable nformaton rate n a frequency-flat slow-fadng MAC wth no spreadng ( s = ). The analyss of a more general system and channel models s left to part II of ths paper [7]. Ths may seem counter ntutve because the puncturng operaton of the probablty control mechansm seems to reduce the spectral effcency. However, the reducton n the spectral effcency s compensated by the reducton n the nterference, and therefore - as long as we acheve the same data rate - the spectral effcency s dentcal.
Submtted to the IEEE Transactons on Wreless Communcatons 7 It s mportant to note that the case of no spreadng gves an upper bound on the achevable data rate n a conventonal sngle user CDMA recever for any spreadng factor 3 (see for example [8], [9]). To see ths we need to recall that the system has two bandwdth expanson mechansms. The frst s the CDMA spreadng that expands the bandwdth by (and produces a spreadng gan of ). The second s the s s codng whch expands the bandwdth by R / (where R s the nformaton data rate n [bts/chp] unts) s and produces a codng gan. Recallng also that the spreadng s a type of (repetton) code whch s a wea code, whle we assume optmal error correcton code for the codng operaton, we conclude that the maxmal gan s acheved f all of the bandwdth expanson s performed by codng and not spreadng 4. To further smplfy the mathematcal analyss, we assume that the chp pulse shape s chosen so that there s no nter chp nterference and that the receved sgnal of all users s chp synchronzed. The chp synchronzaton assumpton s used only for mathematcal smplfcaton. As we assume that the transmtted sgnals of the dfferent transmtters are statstcally ndependent, the chp-synchronzaton assumpton does not reduce the nterference, and the expressons derved hereon wll be good approxmatons to the completely unsynchronzed scenaro. Usng a match flter, the recever output for the d th sample s gven by: K yd = p( t dtc ) hs( t) + n( t) dt =, () K = Ehx g + n =, d, d d where h s the th user channel gan, nt () s the addtve whte Gaussan nose and n d s the nose sample wth zero mean and varance 5 /. Droppng the symbol ndex, the receved symbol can be wrtten as: K, (3) = y = Ehg x + n In ths model we assume that the nput symbols have Gaussan dstrbuton wth zero mean and unt 3 We use the term conventonal sngle user recever for a recever whch consders all of the nterference as whte nose. Other types of recever are descrbed n part II of ths paper. 4 Mathematcally we can descrbe ths by the nequalty ( ρ) ( ρ) log + log + / where ρ s the receved sgnal to nose plus nterference rato. 5 In ths paper we consder base-band (real) transmsson and recepton. However, all results hold for pass-band (complex) transmsson and recepton as well. s s
Submtted to the IEEE Transactons on Wreless Communcatons 8 varance ( x ~ (,)), and the pars { x, x } and { g, g } are statstcally ndependent for whle { x, g } are statstcally ndependent for all, =,,..., K. Ths channel model s demonstrated n Fg.. ote that the average transmtted power s E p / T. c As stated n the ntroducton, for complexty reasons, the recever employs K separate sngle user decoders. However, we do let each decoder use the nowledge of the gatng sequences of all transmtters. The rate of the th user ( R ) s bounded by the mutual nformaton rate: R < I = I( x ; y g,..., g ). (4) K Gven the nowledge about whch of the nterferng users s transmttng, the nose plus nterference term s a Gaussan random varable wth a nown varance. Therefore, the mutual nformaton can be wrtten as an expectaton over the well-nown mutual nformaton of a Gaussan codeboo over an AWG channel [3]: Eh I = E g log +. (5) Eh g + In ths paper we assume that the system uses optmal codes whch acheve rates that are very close to the achevable rates descrbed n (5) (.e. we assume that R = I). III. SYSTEM OPTIMIZATIO We consder the optmzaton of the achevable rate regon,.e., the set of rate vectors R = [ R,..., R ] T K that are achevable under the system constrants. Usng the formulaton ntroduced n secton II, a system operatng pont, ( pe, ), can be defned by two vectors: the users transmsson probabltes vector, = p p K p [,..., ] T, and the users chp energes vector, E = [ E,..., E ] T K. The optmzaton problem s: maxmze the user rates, R = R( p, E ), subject to an average power constrant for each user. The average power constrant s mposed on each user separately, and s defned as: p E / T S av, for all =,...,. c Thus, the optmzaton doman s defned by the set of all allowed operatng ponts: where S av { K av ( pe, ) R : p ; pe S Tc,..., K} S = =, (6) s the average power avalable to the th transmtter. The achevable rate regon s the set of all achevable user rates R = R( S ). Any practcal system needs to fnd ts optmal operatng pont. However, the choce of optmal operatng
Submtted to the IEEE Transactons on Wreless Communcatons 9 pont depends on the system preferences for the tradeoffs between users' rates. For example, a system may decde that all users must use an equal rate, and try to maxmze ths equal user rate. Another example s to maxmze the weghted sum-rate of all users, where more "mportant" users wll use hgher weghts. In ths paper we wsh to characterze the optmal solutons wthout specfyng the users' preferences. For ths purpose we turn to mult-objectve optmzaton. In mult-objectve optmzaton we try to smultaneously optmze the rates of all users. Typcally, no pont n the optmzaton doman wll maxmze the rates of all users smultaneously, and therefore there s no sngle optmal operatng pont. Instead we use the Pareto optmalty concept, whch was orgnally ntroduced n economc theory [3]. We frst defne the strct domnaton relaton: Defnton Strct domnaton: a vector R strctly domnates a vector R f: R > R, =,,..., K. (7) In other words, a rate vector R strctly domnates another rate vector R f all users get hgher rates. Thus, f a system can between rates R and R t wll always prefer R, regardless of the user preferences. The concept of Pareto optmalty uses the domnaton relaton, and fnds the operatng ponts that can be optmal under some system preferences: Defnton Wea Pareto optmal operatng pont: An operatng pont ( p, E ) s sad to be wea Pareto optmal f there exsts no other operatng pont ( pe, ) such that RpE (, ) strctly domnates Rp (, E ). In smpler words, an operatng pont s Wea Pareto optmal f there exsts a set of systems preferences for whch ths pont s the optmal operatng pont. In the rest of the secton we focus on the wea Pareto optmal operatng ponts. In general, a system has many wea Pareto optmal operatng ponts. We do not try to sort between these ponts, but rather mae clams that hold for any wea Pareto optmal operatng pont. The mportance of such clams s that they wll hold for any optmal operatng pont, regardless of the system preferences. We denote by P the set of wea Pareto optmal operatng ponts. The correspondng domnatng rates set D = R( P ) s the set of rate vectors that strctly domnates all other rates. (More exactly, the set D = R( P ) s the set of rates that are not strctly domnated by any other rate. We wll show n Proposton that n the analyzed problem ths set actually domnates all of the achevable rates.) As an llustraton, Fg. demonstrates the wea Pareto optmal operatng ponts of a two-user system.
Submtted to the IEEE Transactons on Wreless Communcatons The two axes dsplay the rate acheved by each of the users, and the gray area represents all rate pars that are achevable n the system. The bold lne s the set of all non domnated ponts ( D ), whch are acheved by wea Pareto optmal operatng ponts. Obvously, for any system preferences, an optmzed system wll always wor n one of the wea Pareto optmal operatng ponts. In the next secton we show that the analyzed system s probablty control optmal,.e., that any optmal soluton requres only probablty control. For ths purpose, we defne the maxmal allowed chp energy as: av / max S Tc p p > E ( p) =. (8) p = Ths chp energy s the maxmum allowed one, gven the user transmsson probablty. It s used for a formal defnton of a probablty controlled optmal system. Defnton 3: A GCDMA system s termed Probablty controlled optmal system f n any wea Pareto optmal operatng pont and for all users ( =,..., K ), the chp energy satsfes: E = E max ( p ). Thus, n a probablty controlled optmal GCDMA system, each optmal operatng pont s completely defned by the users transmsson probablty, mang the power control redundant. Also, n such a system, f a user s actve (that s, pe > ) then t always transmts ts maxmum allowed average power. IV. PROPERTIES OF THE PARETO OPTIMAL OPERATIG POITS The followng proposton shows that all nterestng operatng ponts are n the wea Pareto optmal set. The proposton guarantees both the exstence of a domnatng set, and that ths domnatng set ( D = R( P )) domnates all achevable rates. Proposton : Any achevable rate R R s ether n the domnatng rate set ( R D ), or there exsts a rate R D so that R strctly domnates R. Proof of Proposton : See Appendx I. Thus, f an operatng pont s not a wea Pareto optmal one, there exsts a wea Pareto optmal operatng pont n whch all users acheve hgher rates. Snce such an operatng pont wll be preferred by any optmzaton crteron, we can state that f an operatng pont ( p, e ) s optmal under any optmzaton crteron, then ths operatng pont s a wea Pareto optmal one ( ( p, e) P ). Snce any system prefers to wor n a wea Pareto optmal operatng pont, t s desrable to characterze
Submtted to the IEEE Transactons on Wreless Communcatons these ponts. The followng proposton states that n all wea Pareto optmal operatng ponts, each actve user uses ts maxmum allowed average power. Proposton : At any wea Pareto optmal operatng pont ( pe, ) P, the average power, pe / T c, for each user, =,..., K, equals ether zero or S av. Proof of Proposton : We frst need to remove the nactve users. Assume that there are M nactve users, these users have no effect on the actve users. Therefore we consder the subsystem wth only K = K M actve users,.e., pe / T c > for =,..., K. ext we use the Kuhn-Tucer condton to prove the proposton: Consder a general optmzaton problem - maxmze the vector functon fx ( ) = [ f(x),..., f ] T n (x), under the constrant g( x) = [ g ( x),..., g ( )] T m x. Accordng to the Kuhn-Tucer condton [3], a necessary condton for a pont x to be wea Pareto optmal, s that there exsts no vector h such that: for all =,..., n, and: f T ( x) h >, (9) for any {,..., m} for whch g ( ) = T g ( x) h, () x (where [ /, /,..., / ] = x x x n s the gradent operator). ote that ths condton s vald under the Kuhn-Tucer constrant qualfcaton [3], whch s easly valdated for our case. Usng the Kuhn-Tucer condton, and the system constrants ( p, pe S av Tc ), a necessary condton for an operatng pont ( pe, ) to be a wea Pareto optmal one, s that there exsts no par of vectors γ = [ γ,..., γ ] T K, η = [ η,..., η ] T K, such that for all =,..., K : where: T T ER( pe, ) γ + pr( pe, ) η >, ()a ηl, for any l { l: pl = }, ()b p γ Eη, for any l { l: p E = S av T}, ()c l l l l l l l [ ] T [ ] R ( pe, ) R ( pe, ) E,..., R ( pe, ) E E K pr( pe, ) R( pe, ) p,..., R( pe, ) pk We use these condtons to prove that n any optmal operatng pont, p E c T T. () = S av T for all =,..., K. ote that equaton ()b follows from the constrant pl, and s vald only f the constrant s actve,.e., f the c
Submtted to the IEEE Transactons on Wreless Communcatons tested operatng pont use p l =. Smlarly, equaton ()c follows from the maxmal power constrant and s vald only for users that use ther maxmal allowed average power. Assume that the operatng pont ( pe, ) s a wea Pareto optmal pont. Accordng to the Kuhn-Tucer condton, there exsts no par of vectors γ, η that satsfes all the condtons n (). Consder the vectors where only the parameters of one user are changed: γ = [,...,, γ,,...,] T, η = [,..., η,...,] T, (wth η < as n ()b). For these vectors, condton ()a becomes: R( pe, ) R( pe, ) γ + η >, =,..., K. (3) E p The followng lemma proves that wth the approprate choce of γ and η these vectors satsfy condton ()a. Lemma : Settng the values of η, γ to be: wth suffcently small ε > Proof of Lemma : See Appendx II. R ( pe, ) η = = ph E E + Eh + geh, (4) R( pe, ) Eh γ = + ε = E log + ε p + geh +, satsfes condton ()a for any operatng pont. Snce we assume that the pont ( pe, ) s wea Pareto optmal, and snce the vectors γ, η satsfy condtons ()a and ()b, then accordng to the Kuhn-Tucer condton these vectors must contradct condton ()c. For l, condton ()c s satsfed, snce plγ l = elη l =. Therefore the vectors γ, must contradct condton ()c for l =,.e., we must have pγ > Eη. But, ths s not enough. In order η for the nequalty to be contradcted t must frst be vald,.e., we must have p E = S av T. Ths last c requrement s the mportant one for the current proof and we conclude that f ( pe, ) s a wea Pareto optmal pont then p E = S av T. Ths proof can be repeated for any =,..., K, so that each user must c satsfy p E = S av T. That s, n any wea Pareto optmal operatng pont, all actve users must deploy all of c
Submtted to the IEEE Transactons on Wreless Communcatons 3 the allowed average power. Based on Proposton we conclude that the GCDMA system over a frequency-flat slow-fadng MAC, s a probablty controlled optmal system. V. UMERICAL EXAMPLES The result of the prevous secton demonstrates the mportance of probablty control as a complement to the power control mechansm. ote that we do not suggest that the power control mechansm s redundant n CDMA systems. In general, the soluton of the optmzaton problem n more complcated scenaros wll requre both power control and probablty control. However, the case dscussed n ths paper s a frst ndcaton that n some GCDMA systems the probablty control s as mportant as power control. Another nterestng consequence of our result s that the analyzed system can use addtonal power of any of the users to the beneft of all users. Assume that a system wors n the wea Pareto optmal operatng pont ( p, E ), and acheves a rate R R. ow, consder another system n whch the th user can ncrease ts power to av S > S av. Accordng to our results, the operatng pont ( p, E ) s not wea Pareto ' optmal any more, and there exsts a rate R R that strctly domnates R. In other words, n ths new scenaro all users can acheve hgher rates. ote that even f the user that has addtonal power does not requre a hgher rate, t s best for all users that t wll use ts maxmal power. The extra power allows the user to acheve the same rate whle reducng ts transmsson probablty, and therefore allow other users to acheve hgher rates. Ths s n contrast to the conventonal power control mechansm that dctates the users to use less than ther maxmal power, so the achevable rates are not as hgh as they can be. To demonstrate ths performance advantage, consder the followng numercal example. Fg. 3 depcts the achevable rate regons n a two users system wth and wthout probablty control, when the maxmal power allowed to each of the users can acheve a sgnal-to-nose-rato (SR) of 4dB av av ( T S h / = T S h / =.5). The system wthout probablty control uses contnuous transmsson c c and power control. As expected, the system that employs probablty control unformly outperforms the contnuous transmsson system. The fgure also allows a quanttatve comparson. For example, when user # acheves a rate of bt per chp, user # can acheve 3 tmes more usng probablty control (.3 bts per chp) than n a contnuous transmsson system wth power control only (. bts per chp).
Submtted to the IEEE Transactons on Wreless Communcatons 4 ote that the achevable rate regons depcted n Fg. 3 are not convex. Therefore, one can acheve better performance by tme sharng,.e., worng a porton of the tme n one operatng pont, and the rest of the tme n another operatng pont. Employng such strategy, the achevable rate regon wll expand to the complex hull of the regons plotted [3]. However, as stated above, we lmt ths dscusson to the random unsynchronzed multple access scheme commonly used n CDMA systems, and leave for further research the performance mprovements achevable by tme sharng. Another example s shown n Fg., n whch the performance of two users n an asymmetrc channel s av depcted. Here user # has more power than user # ( TS h / =.5, c TS h / =.5). In ths case av c the asymmetry between the users emphaszes the mportance of the control mechansm, and we see even larger dfference between the power controlled system and the probablty controlled system. In the last example we consder a system wth dentcal users. As t s mpossble to plot the achevable rate regon n a dmensonal space, we requre all users to wor n equal rates. We chec what s the maxmal achevable rate n a system wth power control and a system wth probablty control under the constrant that all users acheve the same rate. For ths example we choose the channel gans to be h = db, h = 4dB,, h = 36dB. Fg. 4 shows the achevable rate as a functon of the frst user sgnal to nose rato TS h av c /. As can be seen, the probablty control mechansm has a small advantage n low sgnal to nose rato. In ths case the system s power lmted, the nterference s not domnant, and the role of the nterference management mechansm s small. But, for large sgnal to nose rato the advantage of probablty control over power control s much more sgnfcant. Moreover, the power controlled system s lmted to a maxmal rate of ( K ).5log + /( ) =.76 bts per chp. (The well nown equvalent result for CDMA systems wth sngle user recever s ρ < /( K ) where K s the number of dentcal users and ρ s the user sgnal to nose plus nterference rato. ormalzng by the spreadng factor and calculatng the achevable rate we have ( ) equal to.5log( /( K ) ) s s.5log + /( K ) / whch s smaller or +.) On the other hand, the achevable rate n the probablty controlled system s not lmted, and s an ncreasng functon of the transmtted power. These examples demonstrate the advantage of a non-contnuous communcaton system that can employ both power control and probablty control, over a contnuous transmsson system that can use only power s
Submtted to the IEEE Transactons on Wreless Communcatons 5 control. VI. COCLUSIOS In ths paper we ntroduce the concept of probablty control and show ts mportance to GCDMA system optmzaton. We do so by provng that a GCDMA system wth no spreadng over the frequency-flat slowfadng MAC s probablty controlled optmal,.e., ths system acheves maxmal user rates when usng probablty control, whle the power control s redundant. In other words, n any optmal system operatng pont, all users should transmt ther maxmal allowed average power, whle user preferences and tradeoffs between user rates are managed by optmzng the users' transmsson probabltes. The mportance of probablty control s further emphaszed n part II of ths paper [7], where we show smlar results for the optmzaton of the ASIR n any GCDMA system. APPEDIX I - EXISTECE OF DOMIATIG RATES We assume that R s not a strctly domnatng pont ( R domnatng pont D ) and prove the exstence of a strctly R D. The proof wll be based on Hartleys theorem whch guarantees the exstence of such domnatng pont n any by D-compact set. As a frst stage we assume that all elements of R are strctly postve, and fnd a compact subset of R, by fndng a compact subset of S (we wll later consder the case n whch some elements of R are zero). S s not a compact set because the users pea power s not bounded. We therefore need to fnd a subset of S whch s bounded and contans all ponts of nterest. For ths purpose we defne K reference functons: av p S Th c f ( p) = log +, (5) p for any =,..., K and < p. Each of these reference functons taes values n the range < f ( p) log( + S Th / ) av c, and s a monotonc ncreasng functon of p. ext, consder R th element of R. Comparng (5) and (5) and recallng that E / S av Tc p, t s clear that: av S Tch R f( p) log +. (6) Therefore, we can defne a mnmal probablty p = f ( R ) >, so that for every operatng pont ( pe, ), the
Submtted to the IEEE Transactons on Wreless Communcatons 6 n whch p < p, the th user rate satsfes R( pe, ) < R. Consder the set: {( pe, ) : p p {,..., K} S = S. (7) In contrast wth the set S, the set S s bounded (snce E S av T /,..., c p = K ) and therefore t s a compact set. ext we defne the set R = R( S ), whch s also compact because the rate functon s contnuous and S s compact. In addton, the constructon of the set nsures that R R, and that S ncludes all ponts of nterest,.e., there s no pont n R\ R that strctly domnates R. ow that we have a compact set n R, we use t to defne a D-compact set. Defnton 4: Followng [3], a set Y s D-compact f for any y Y, the ntersecton K { x R : x y {,..., K} } Y s compact. Usng Defnton 4, the followng s a non-empty D-compact set: R s not empty, because we assumed that R = R >. (8) { r : r {,..., } R K R D, and therefore there exsts at least one pont, R, that strctly domnates R and therefore R R. Usng Hartleys theorem ([3] pp 5) there exsts a rate vector R R whch s not strctly domnated by any other vector n R. From the defnton of R and R, the rate vector R s also not strctly domnated by any vector n R, nor by any vector n R, and therefore R D. Recallng the defnton of R we conclude that R strctly domnates R. Fnally we need to consder the case n whch some of the elements of R are zero. Snce exsts R that strctly domnates R (and therefore all elements of R are strctly posstve). If R D there R D the proof s completed. If R D we repeat the frst part of the proof for R and fnd R D that strctly domnates R. otng that R strctly domnates R we conclude that R also strctly domnates R whch completes the proof. APPEDIX II - PROOF OF LEMMA In ths appendx we prove that (4) satsfes condton ()a. ote that by choosng the vectors ηρ, wth only one non-zero element, condton ()a becomes equaton (3).
Submtted to the IEEE Transactons on Wreless Communcatons 7 Wthout loss of generalty we demonstrate ths for =, =,. The mutual nformaton of the frst two users s gven by: where K = 3 p Eh Eh R = E plog + ( p) log + + + Z + Eh + Z (9) p Eh Eh R = E plog + ( p) log + +. + Z + Eh + Z () Z g Eh s the sum of other users nterference. The dervatves are gven by: R Eh Eh = E plog + ( p) log p + + + Z + Eh + Z R ph = E p + ( p ) E + Z + Eh + Eh + Z+ Eh R p Eh Eh = E log + log p + + Z + Eh + Z R pph Eh = E. (4) E Z Eh Eh Z Eh + + + + + In ths appendx we demonstrate that for ε = the left hand sde of equaton (3) s equal to for =, and s larger than for. The extenson for suffcently small ε > s straght forward (snce R/ E >, and R E / < ). Frst, for =, Substtutng the dervatves ()-(4) and (4) nto (3), we get: p ext, for =, ε =, equaton (3) can be wrtten as: () () (3) R R ρ + η =. (5) E
Submtted to the IEEE Transactons on Wreless Communcatons 8 pph Eh V = E 4 + Z + Eh + Eh + Z + Eh Eh Eh E plog + ( p)log + + + Z + E Z h +. (6) pph Eh Eh E log + log + 4 + Z + Eh + Z p p E + > + Z + Eh + Eh + Z + Eh Expandng and collectng the terms of V n (6), t can be rewrtten as: 4V Eh = E E log + pph + Z + Eh + Eh + Z. (7) Eh E E log + + Z + Eh + Z + Eh We defne ntermedate varables q Eh = + Z + Eh, β = Eh Eh, and the functon: and rewrte (7) as: q q W( β ) = E log E βq, (8) + + βq 4V Eh = E E W ( ) W( ) pph β. (9) + Z + Eh + Z + Eh + Eh ow, t s obvous that n order to prove that V >, t s suffcent to prove that dw ( β ) dβ > for all β. Wrtng ths dervatve, we have:
Submtted to the IEEE Transactons on Wreless Communcatons 9 q q W( β ) + β q q E = E E + β q β + βq q + βq. (3) q q + E log E + β q ( + β q) Agan, we defne ntermedate varables Q = q ( + βq) ( < Q < ), and: Q V Q = E E{ Q} + E Q log ( Q) E Q and rewrte the dervatve as: { } { } { }, (3) W ( β ) V ( Q) =. (3) β E Q ow, for the last stage of the proof, we need to show that V ( Q ) >. Usng the nequalty ( ) log Q > Q ( Q) ( < Q < ), we can wrte: Q Q V( Q) > V3( Q) = E E{ Q} E E{ Q }. (33) Q Q We prove that V 3 ( Q) by wrtng t as the determnant of the expectaton of a non-negatve defnte (n.n.d.) matrx. Defne the matrx: then, we can wrte V ( Q) = det 3 ( E [ W ]) (where ( ) Q Q W = Q Q, (34) Q Q det s the determnant operaton). The egenvalues of W are λ =, λ = Q/( Q), and therefore W s an n.n.d. matrx. Snce the expectaton over n.n.d. matrces s a postve matrx, and the determnant of a postve matrx s postve, ths concludes the proof that V ( ) 3 Q, and therefore the nequalty (3) s satsfed for =. BIBLIOGRAPHY [] M. Z. Wn and R. A. Scholtz. Impulse rado: how t wors, IEEE Communcatons Letters, vol, pp 36-38, Feb. 998.
Submtted to the IEEE Transactons on Wreless Communcatons [] S. S. Kolenchery, J. K. Townsend and J. A. Freebersyser, A novel mpulse rado networ for tactcal mltary wreless communcatons, In Proceedngs of the IEEE Mltary Communcatons Conference, (MILCOM 98), vol, pp 59-65, Boston, MA, Oct. 998. [3] E. Fshler and H. V. Poor, Low-complexty multuser detectors for tme-hoppng mpulse-rado systems, IEEE Transactons on Sgnal Processng, vol 5, pp 56-57, Sept. 4. [4] C. J. Le Martret and G. B. Gannas. All-dgtal mpulse rado wth multuser detecton for wreless cellular systems, IEEE Transactons on Communcatons, vol COM-5, pp 44-45, Sept. [5] B. M. Sadler and A. Swam, On the performance of epsodc UWB and drect-sequence communcaton systems, IEEE Transactons on Wreless Communcatons, vol 3, pp 46 55, ov. 4. [6] K. S. Glhousen, I. M. Jacobs, R. Padovan, A. J. Vterb, L. A. Weaver and C. E. Wheatley, "On the capacty of a cellular CDMA system" IEEE Transactons on Vehcular Technology, vol 4, pp 33-3, May 99. [7] D. Ayyagar, D. and A. Ephremdes, A., "Cellular multcode CDMA capacty for ntegrated (voce and data) servces," IEEE Journal on Selected Areas n Communcatons, vol 7, pp 98-938, May 999. [8] TIA/EIA/IS-95-A, Moble Staton-Base Staton Compatblty Standard for Dual-Mode Wdeband Spread Spectrum Cellular System, Telecommuncatons Industry Assocaton, Washngton, D.C., May 995. [9] L. Tomba, "Outage probablty n CDMA cellular systems wth dscontnuous transmsson," n Proceedngs of the IEEE 4th Internatonal Symposum on Spread Spectrum Technques and Applcatons, vol, pp 48 485, Sept. 996. [] P. K. Frenger, P. orten, T. ottoson and A.B Svenson, Rate-compatble convolutonal Codes for multrate DS-CDMA Systems, IEEE Transactons on communcatons, vol 47, pp 88-836, June. 999. [] H. Moon and D. C. Cox, "Performance of CDMA forward channels wth random puncturng," In Proceedngs of the 57th IEEE Semannual Vehcular Technology Conference (VTC 3-Sprng), vol 4, pp 47-475, Aprl 3. [] G. Durs and G. Romano, "On the valdty of Gaussan approxmaton to characterze the multuser capacty of UWB TH PPM," In Proceedngs of the IEEE Conference on Ultra Wdeband Systems and Technologes (UWBST), pp 57-6, May. [3] K. A. Hamd and G. Xunye, "On the valdty of the Gaussan approxmaton for performance analyss of TH- CDMA/OOK mpulse rado networs," In Proceedng of the 57th IEEE Semannual Vehcular Technology Conference (VTC 3-Sprng), vol 4, pp 5, Aprl 3. [4] S. Verdu and S. Shama, "Spectral Effcency of CDMA wth Random Spreadng," IEEE Transactons on Informaton Theory, vol 45, pp 6-64, Mar. 999. [5] D.. C. Tse and S. V. Hanley, Lnear Multuser Recevers: Effectve Interference, Effectve bandwdth and User Capacty, IEEE Transactons on Informaton Theory, vol 45, pp 64-657, Mar. 999. [6] E. Fshler and H. V. Poor, "On the Tradeoff Between Two Types of Processng Gans," IEEE Transactons on Communcatons, vol 53, pp 744 753, Oct. 5. [7] M. Garby, T. Garby and R. Zamr, "The Most Favorable Impulsve Interference for Ternary CDMA," In Proceedngs of the IEEE Internatonal Symposum on Informaton Theory 6 (ISIT-6), pp 94-946, Seattle, WA, July 6.
Submtted to the IEEE Transactons on Wreless Communcatons [8] J. G. Proas, Dgtal Communcatons. ew-yor, Y: McGraw Hll,. [9] A. J. Vterb, "Wreless dgtal communcaton: a vew based on three lessons learned," IEEE Communcatons Magazne, vol 9, pp 33 36, Sept. 99. [] P. Jung, P. W. Baer and A. Stel, "Advantages of CDMA and spread spectrum technques over FDMA and TDMA n cellular moble rado applcatons," IEEE Transactons on Vehcular Technology, vol 4, pp 357-364, Aug. 993. []. D. Wlson, R. Ganesh, K. Joseph and D. Raychaudhur, "Pacet CDMA versus dynamc TDMA for multple access n an ntegrated voce/data PC," IEEE Journal on Selected Areas n Communcatons, vol, pp 87-884, Aug. 993. [] R. Lupas and S. Verdu. Mnmum probablty of error for asynchronous Gaussan multple access channels. IEEE Transacton on Informaton Theory, vol IT-3, pp 85-96, Jan 986. [3] S. Verdu, Multuser Detecton. Cambrdge, UK: Cambrdge unversty press, 998. [4] S. Glsc and B. Vucetc, Spread Spectrum CDMA Systems for Wreless Communcatons. orwood, MA: Artech House, 997. [5] L. F. Chang and S. Aryavstaul, "Performance of power control method for CDMA rado communcatons system" Electroncs Letters, vol 7, pp 9 9, May 99. [6] J. Zander, "Performance of optmum transmtter power control n cellular rado systems," IEEE Transactons on Vehcular Technology, vol 4, pp 57 6, Feb. 99. [7] I. Bergel and H. Messer, "Optmzaton of CDMA systems wth respect to Transmsson Probablty, Part II: Sgnal to ose plus Interference Rato Optmzaton," Submtted for publcaton n the IEEE Transactons on Wreless Communcatons. [8] A. J. Vterb, "Very low rate convoluton codes for maxmum theoretcal performance of spread-spectrum multpleaccess channels", IEEE Journal on Selected Areas n Communcatons, vol 8, pp 64 649, May 99. [9] P. Frenger, P. Orten and T. Ottosson, "Code-spread CDMA usng maxmum free dstance low-rate convolutonal codes", IEEE Transactons on Communcatons, vol 48, pp 35 44, Jan.. [3] T. M. Cover and J. A. Thomas, Element of Informaton Theory. ew-yor, Y: John Wley & Sons, 99. [3] Y. Sawarag, H. aayama and T. Tanno, Theory of Multobjectve Optmzaton. Orlando, FL: Academc Press, 985. [3] H. W. Kuhn and A. W. Tucer. onlnear programmng. In Proceedngs of the Second Bereley Symposum on Mathematcal statstcs and probablty, pp 48-49, Bereley, CA, 95.
Submtted to the IEEE Transactons on Wreless Communcatons x E h g y x K E K h n g K Fg.. Smplfed GCDMA over frequency-flat slow-fadng multple access channel model.
Submtted to the IEEE Transactons on Wreless Communcatons 3.4. Achevable rate regon Domnatng set Wthout probablty control R [bts/chp].8.6.4..5.5.5 R [bts/chp] Fg.. Achevable rate regon of a two users GCDMA system and ts domnatng set, over a frequency-flat slow-fadng MAC. TS h / =.5, av c TS h / =.5. Also shown n dashed lne s the achevable rate regon wthout av c probablty control.
Submtted to the IEEE Transactons on Wreless Communcatons 4.4. Wth probablty control wthout probablty control R [bts/chp].8.6.4...4.6.8..4 R [bts/chp] Fg. 3. Achevable rate regons of a two users GCDMA system over a frequency-flat slow-fadng MAC wth and wthout av av probablty control. T S h / = T S h / =.5. c c
Submtted to the IEEE Transactons on Wreless Communcatons 5 Probablty control Power control Rate [bts/chp] - - 3 4 log (T c S av h / ) Fg. 4. Achevable rate n a system wth dentcal users as a functon of the frst user sgnal to nose rato. Channel gans are h = db, h = 4dB,, h = 36dB.