MAXIMIZI THE THROUHPUT OF CDMA DATA COMMUICATIOS THROUH JOIT ADMISSIO AD POWER COTROL Penna Orensten, Davd oodman and Zory Marantz Department of Electrcal and Computer Engneerng Polytechnc Unversty Brooklyn, Y, USA porenste@poly.edu Abstract - We consder termnals transmttng at the same data rate (), wth the power lmts on the termnals beng P max. We assume that the power of the most dstant termnal (termnal ) s fxed at ts maxmum value, P max and arbtrary nose power s σ. The path gans of the termnals are gven by h, correspondng to the locatons of the termnals. The obectve s to maxmze the aggregate throughput at the base staton, n terms of (a) the optmum number of termnals admtted to the system and whch ones and (b) the power levels of each admtted termnal. We also descrbe the maxmum throughput as a functon of data rate and nose power level. Keywords-power control; rado resources management; power balancng, admsson control I. ITRODUCTIO Early work on uplnk CDMA power control focused on telephone communcatons and determned that to maxmze the number of voce communcatons, all sgnals should arrve at a base staton wth equal power []. Intal studes of power control for data communcatons focused on maxmzng the utlty of each termnal, wth utlty measured as bts delvered per Joule of radated energy [,3]. By contrast ths paper consders maxmzng the aggregate throughput of a CDMA base staton. A dfferent approach to ths problem, s descrbed by Ulukus and reensten [4] who adust data rates and transmtter power levels n order to maxmze network throughput. Lee, Mazumdar, and Shroff [5] adapt data rates and power allocaton for the downlnk, and provde a sub-optmal algorthmc soluton based on prcng. Sung and Wong [6] assume that the termnals data rates are dfferent but fxed, and maxmze a capacty functon. A related strand of recent work [7-0] adusts the power and rate of each termnal to maxmze Σ β T, the aggregate weghted throughput of a base staton. T b/s denotes the throughput of termnal and the weght β admts varous nterpretatons, such as prorty, utlty per bt, or a monetary prce pad by the termnal. Ths research assumes that the number of actve termnals s fxed, ther data rates are contnuous varables to be optmzed, and that the system s nterference lmted (nose s neglgble). In ths paper, we set all the β and take the data rates to be dentcal and fxed, but we vew the number of actve termnals and ther assocated power levels as key varables to be optmzed. We also consder addtve nose to be non-neglgble, whch s essental when out-of-cell nterference s sgnfcant, and ncluded n the nose term. Our recent work reported n [] dealt specfcally wth the case when nose and out-of-cell nterference * are neglgble, and found that the transmtter power levels should be controlled to acheve power balancng. Wth power balancng, all sgnals arrve at the base staton wth equal power. By contrast, n [], we fnd that wth addtve nose (a) power balancng leads to sub-optmal performance and (b) that when one termnal has a maxmum power constrant, the other termnals should am for the same receved power, whch depends on the maxmum SR of the constraned termnal. Here, we brefly revew the man result from [] and then extend the analyss to fnd the number of actve Supported n part by YSTAR through the Wreless Internet Center for Advanced Technology (WICAT) at Polytechnc Unversty and by the atonal Scence Foundaton under rant o. 098 *To be concse we refer to the combnaton of nose and nter-cell nterference smply as nose. The analyss does not dstngush the two mparments. It consders only ther combned power. CISS Prnceton March 6 th -8 th 004
transmtters,. We show that n order to maxmze base staton throughput wth any power control algorthm, that should be lmted to *, where *, s a functon of background nose and the frame success rate f(), the probablty that a termnal s data packet s receved successfully as a functon of, the receved sgnal-tonterference-plus-nose rato (SIR). The specfc form of f() depends on the detals of the CDMA transmsson system, ncludng the packet sze, modem confguraton, channel codng, antennas, and rado propagaton condtons. Our analyss apples to a wde class of practcal frame success functons, each characterzed by a smooth S-shaped curve [4]. The outlne of the paper s as follows. We frst present the detals of the CDMA transmsson system and a bref statement of the throughput optmzaton problem. We then examne the effects of non-neglgble nose and out-of-cell nterference on both the optmal number of transmtters and the receve-power rato whch ontly maxmze base-staton throughput. We also show how nose reduces the number of actve termnals and llustrate our results wth a numercal example. The mplcatons for admsson control are then brefly dscussed. II. THE OPTIMIZATIO PROBLEM A data source generates packets of length L bts at each termnal of a CDMA system. A forward error correcton encoder, f present, and a cyclc redundancy check (CRC) encoder together expand the packet sze to M bts. The data rate of the coded packets s R s b/s. The dgtal modulator spreads the sgnal at a rate of R c chps/s. The CDMA processng gan s W/R s, where W Hz, the system bandwdth, s proportonal to R c. Termnal also contans a rado modulator and a transmtter radatng P watts. The path gan from transmtter to the base staton s h and the sgnal from termnal arrves at the base staton at a receved power level of Q P h watts. The base staton also receves nose and out-of-cell nterference wth a total power of σ Wη 0 watts, where η 0 s the one-sded power spectral densty of whte nose. The base staton has recevers, each contanng a demodulator, a correlator for despreadng the receved sgnal, and a cyclc redundancy check decoder. Each recever also contans a channel decoder f the transmtter ncludes forward error correcton. In our analyss, the detals of the transmsson system are emboded n a mathematcal functon f(), the probablty that a packet arrves wthout errors at the CRC decoder. The dependent varable, s the receved SIR. For termnal, P h P h + σ Q Q + σ Acknowledgment messages from the recever nform the transmtter of errors detected at the CRC decoder that have not been corrected by the channel decoder. The transmtter employs selectve-repeat retransmsson of packets receved n error. Our earler study [] assumes that ntra-cell nterference domnates the total dstorton and examnes system performance when σ 0. When σ >0, we denote the sgnal-to-nose rato of recever as s Q /σ and rewrte Equaton () as s s, + (). () In cases of practcal nterest f() s a contnuous, ncreasng S-shaped functon of, wth f(0) -M 0 and f( ) [3]. If the probablty of undetected errors at the CRC decoder s neglgble, the throughput of sgnal, defned as the number of nformaton bts per second receved wthout error, s: L T R s f ( ) b/s, (3) M The aggregate throughput, T total, s the sum of the ndvdual throughput measures n Equaton (3). Assumng that L, M, and R s are system constants, we analyze the normalzed throughput of smultaneous transmtters as U defned as M U Ttotal f ( ). (4) LRs U s dmensonless and bounded by 0 U. III. FIRST-ORDER ECESSARY CODITIOS As stated earler, ths paper deals wth the effects of addtve nose and nterference from other cells. The total power n these mparments s σ Wη 0 watts (and we refer to them together as nose to be concse). The nose appears at the recever as an addtonal sgnal that does not contrbute to the overall throughput. The system has CISS Prnceton March 6 th -8 th 004
to use some of ts power and bandwdth resources to overcome the effects of the nose. The effects of nose depend on the power lmts of practcal termnals. Wth unlmted power, we would ncrease all the receved powers Q ndefntely untl the effect of the nose s neglgble. To account for the power lmts, let P,max denote the power of the strongest possble sgnal transmtted by termnal and Q,max P,max h, the power of the correspondng receved sgnal. The maxmum sgnal-to-nose rato of termnal s s,max Q,max /σ. In our analyss, we order the labels of the termnals such that Q,max Q,max Q,max. In many stuatons ths orderng mples that termnal s closest to the base staton and termnal s most dstant. We would lke to determne the optmum recevepower vector when there are termnals, where the receve power of the weakest termnal () s fxed. A. eneralzed case: termnals The throughput equaton for termnals transmttng to the base staton, assumng that z s /s,max, for,,-, z s /s,,max, and /s,max, can be expressed as U z f ( ) f () z + We begn by wrtng the - frst-order condtons that need to be satsfed smultaneously: z f ' ( ) f '( ),.., z Ths system of equatons can be reduced to - equatons of the form f '( )( + ) z,.., (3) f ' ( + ) z + ( ) + + + By applyng a smlar analyss to case of 3, see [,3] for detals, we can reduce ths expresson further and obtan '( )( ( ) f + ) ( + + ),.., + f ' + ( + + ) + ( + ) (4) A soluton to equaton (4) concdes wth ~ + for,,-, from whch t follows that z z + for (),,-. We demonstrate the unqueness of ths soluton n [3]. We conclude that a soluton to the frst-order condtons s z zs/s,max for,,-. Ths greatly smplfes our problem snce the optmal soluton now depends on one power rato z, rather than power levels. We can therefore reduce the throughput equaton gven n equaton () as follows. For each s s, (,,-), we wrte z ; and ( ) z + + ( ) z + Then, the throughput equaton can be expressed as U ( z) ( ) f ( ) + f ( ) (5) Usng the same approach as before, we can dfferentate wth respect to z and obtan (( + ) f '( )) f '( ) (6) z ow equaton (6) can be smplfed f we perform the followng algebrac manpulatons. Let ( ) z + + + ( z) (7) and then express as z (8) / + ( z) We can further reduce condton (8) as z ( + ) (9) ( + ) Substtutng (9) nto condton (6) results n ( + ) ( + ) f '( ) ( + ) f ' ( ) (0) whch yelds the followng crtcal ponts at 0 : a), or b) 0 or ( + ) c) ( + ) f ' ( ) f ' ( ) ( + ) () We can classfy each of these crtcal ponts by examnng the second-order condtons, [3]. In ths CISS Prnceton March 6 th -8 th 004 3
paper, we summarze the key results as follows: () wth no nose, (0), throughput s maxmzed when all sgnals are receved wth equal power ( ) provdng <*. Otherwse some termnals should use P0; () wth addtve nose (>0), equaton (c) s not satsfed wth and throughput s maxmzed when (a) one termnal has a maxmum power constrant (P max ), and (b) the other - termnals am for the same receved power, whch depends on the maxmum SR of the constraned termnal. We would lke to determne *, the optmum number of termnals whch can be smultaneously supported when there s addtve nose and both (a) and (b) are satsfed. We begn by defnng the largest value of whch satsfes equaton (5), say *, as the maxmum number of actve termnals whch ensure that (a) the throughput s maxmzed and (b) that the power vector be an nteror soluton. By ths we mean that 0, snce t s possble to fnd >*, such that the aggregate throughput s maxmzed (> ), however, the power of the non-fxed termnals wll approach nfnty. A. Optmal umber of Actve Transmtters * If we optmze wth respect to, we obtan ( z) f ( ) z ( ) f ' ( ) f ' ( ) () For a crtcal pont, we need to set equaton () to zero. We can then substtute the expresson obtaned n (6), to obtan f ( ) z + + z ( ) f ' ( ) (3) We can reduce equaton (3) by wrtng z z as ( ) z + + z whch can ( ) z + + then be reduced to ( ) z + + + (4) z All that remans s to smplfy equaton (3) usng (4) whch results n ( ) f f '( ) ( + / ) (5) One can therefore solve equaton (5) to fnd the optmal number of actve transmtters. Ths s equvalent to maxmzng wth respect to. Further, t can be seen that as (equvalently, condton a), that maxmzng U wth respect to s equvalent to maxmzng f()/. For the class of functons f beng consdered, f(x)/x has a unque maxmum at the pont where a lne from the orgn s tangent to f(x), [4]. We use the notaton * for the sgnal-to-nterference rato that maxmzes f(x)/x. * s the unque soluton to the equaton ( ) f f ' ( ) (6) Further, we demonstrated how maxmzng f()/ was equvalent to maxmzng the number of users wthout nose (0, ). For the class of functon beng consdered, the unque soluton was gven by *0.75, and the correspondng optmal number of transmtters was found to be */*+, []. One can vew ths result as a specal case of equaton (5). In other words, unbounded s equvalent to treatng nose and out-of-cell nterference neglgble. Thus, s the same as havng 0 and (z). Ths s ntutve, snce unlmted bandwdth allows as many users as possble to transmt. However, n practcal systems, bandwdth s a fnte resource and () the mpact of nose s non-neglgble and () the effect on the number of transmtters needs to be ascertaned. IV. UMERICAL RESULTS So far, we have provded a theoretcal framework to explan the effects of nose, both n terms of the optmal number of transmtters as well as the correspondng optmum receve-power vector. In ths secton, we llustrate the analyss wth numercal examples and then dscuss the mplcatons for admsson control. Throughout, we wll assume a fxed value for Q,max and use ths to ascertan z, the rato of to. Fgure I shows a graphcal plot of the left sde equaton (5) for dfferent values of. One can observe the no-nose case as 0,, (alternatvely ). Ths s the upper bound on the system and corresponds to the frst-order condton gven by (a). Fgure I also dsplays f () as a functon of. The ntersecton of ths curve wth one of the other curves n Fgure I occurs at, the optmum sgnal-to-nterference-plus-nose rato. For example, when 6, *., and smlarly when 40, *.354 and when 8, *0.94. ote that the crtcal value of * approaches 0.75 as the system bandwdth ncreases. CISS Prnceton March 6 th -8 th 004 4
We can fnd * usng the unque value of obtaned from the soluton to equaton (5). We shall refer to ths value as *, the crtcal SR for the *- termnals. But, by defnton, * must satsfy z * ( ) z + + It follows that upon dvdng through by z (<z ) and substtutng equaton (9) we can obtan an expresson for * as ( + )( + *) * + (7) * ( + ) * ote that when 0 and expresson (7) reduces to * + whch was found to be the optmum number * of termnals for the no nose case, []. Ths condton can be vewed as the upper bound on the system. It follows that condton (7) s a more general form and can be used to ascertan the optmum number of actve transmtters (*) whch can be admtted gven the level of background nose and * whch s determned va equaton (5). do not appear to be sgnfcant. We demonstrate ths usng the numercal example n Fgure II. Wth, *, and the throughput, U, s maxmum at / z.76>. However, one can observe that (z) s a close approxmaton. There may be stuatons (dependng on ) where ths dfference mght be more sgnfcant, however n ths example one can see that U s nearly the same for * as t s for >. Further, snce, the maxmum number of transmtters s one less than the case wthout nose and f >, there s no soluton except at z. The graph also shows that f <*, then throughput can be maxmzed at or z. Based on ths observaton, we then plot equaton (5) and observe *, the optmal number of transmtters as shown n Fgure III. If we evaluate equaton (7) at we obtan a bound on *: Thus * + (8) * Assumng that the sub-optmal soluton s a good approxmaton, t follows that equaton (8) relates the optmal number of termnals * to the background nose level. It can be observed that nose acts as nterferng termnals. An example s shown n Fgure III. Fgure I: Effect of bandwdth on *: Fnte values of ncrease * beyond *0.75, the optmal * for the nonose case (equvalently ). ext, we examne the relatonshp between the sub-optmal soluton ( )and the optmal soluton found n [], whch occurs when >. In the next example, we show that the sub-optmal soluton s a good approxmaton and that the gans obtaned by settng > Fgure II: Wth 8 and, we fnd that U s largest when * and maxmzed for / z.76. At (z), U s close to optmal. Other optmal power rato allocatons are possble for <*. CISS Prnceton March 6 th -8 th 004 5
that wth nose throughput s not mathematcally maxmzed usng power-balancng; however, for practcal purposes, power-balancng s nearly optmum. Further, we te two strands of research together: the no-nose case descrbed n [] and the equvalent system where nose s explctly consdered. We show that the former s a specal case of the latter and can be vewed as an upper bound on the system. We also quantfy how nose reduces the optmal number of actve termnals. REFERECES Fgure III: The effect of nose on throughput and the optmum number of admtted termnals: when 0 (no nose), *3. Further ncreases n reduce the optmal number of actve termnals, *. Our results can be used as part of a smple admsson control scheme. ven, the nose-to-sgnal rato of the weakest termnal, where > - > >, and the data rate, we can determne *, the optmum number of actve termnals and ther power levels such that the aggregate throughput at the base-staton s maxmzed. We frst apply equaton (8) wth the par (, ) to fnd the optmal number of termnals whch can be admtted under the near-optmum power-balancng assumpton ( ). We then calculate the correspondng throughput gven by equaton (5). We then elmnate the weakest termnal and apply the algorthm n a recursve manner. When the algorthm termnates, we wll have a vector of values for * and a correspondng vector for U *. The obectve s to fnd a value of * as close as possble to * +, the upper bound on the system, whch also maxmzes U *. Under certan physcal assumptons, ths algorthm wll help decde how many termnals to admt from the populaton and whch ones. Ths analyss s descrbed n detal n [3]. [] R.D. Yates, A framework for uplnk power control n cellular rado systems'', IEEE Journal on Selected Areas n Communcatons, vol. 3, no. 7, pp. 34--347, September 995. [] C. U. Saraydar,. B. Mandayam, and D. J. oodman, Effcent power control va prcng n wreless data networks, IEEE Transactons on Communcatons, vol. 50, no., pp. 9-303, Feb.00. [3] D. oodman and. Mandayam, etwork asssted power control for wreless data, Moble etworks and Applcatons, vol. 6, no. 5, pp. 409-48, Sept, 00. [4] S. Ulukus. and L.J. reensten "Throughput maxmzaton n CDMA uplnks usng adaptve spreadng and power control" IEEE ISSSTA, vol., pp: 565-569, 000 [5] J.W. Lee, R.R. Mazumdar and.b. Shroff, "Jont power and data rate allocaton for the downlnk n mult-class CDMA wreless networks", Proc. of 40th Allerton Conf. on Comm., Control and Comp., Oct. 00. [6] C. W. Sung and W. S. Wong, "Power control and rate management for wreless multmeda CDMA systems", IEEE Trans. Commun., vol. 49, no. 7, pp. 5-6, July 00. [7] V. Rodrguez, "Robust modelng and analyss for wreless data resource management" IEEE WCC, Vol., pp. 77 -, March 003 [8] V Rodrguez and D.J. oodman "Prortzed throughput maxmzaton va rate and power control for 3 CDMA: the termnal scenaro", Proc. of 40th Allerton Conf. on Comm., Control and Comp., Oct. 00 [9] V. Rodrguez, and D.J. oodman, "Power and data rate assgnment for maxmal weghted throughput n 3 CDMA" IEEE WCC, vol., pp. 55-3, March 003 [0] V Rodrguez, and D.J. oodman, "Power and Data Rate Assgnment for Maxmal Weghted Throughput: A Dual-Class 3 CDMA Scenaro" IEEE ICC, vol., pp. 397-40, May 003 [] D.J. oodman, Z. Marantz, P. Orensten, and V. Rodrguez, Maxmzng the throughput of CDMA data communcatons, IEEE VTC, Orlando FL, Oct 6-9, 003 [] P.Orensten, D.J. oodman, Z. Marantz and V. Rodrguez, Effects of Addtve ose on the Throughput of CDMA Data Communcatons, submtted to IEEE ICC 04, Pars, France, June 004. [3] P.Orensten, D.J. oodman, and Z. Marantz, Maxmzng the Throughput of Wreless CDMA Data Communcatons through Jont Power and Admsson Control, (extended verson). COCLUSIOS In ths paper, we examned the effects of non-neglgble nose and out-of-cell nterference on both the optmal number of transmtters and the receve-power rato whch ontly maxmze base-staton throughput. We showed CISS Prnceton March 6 th -8 th 004 6