7th European Sgnal Processng Conference EUSIPCO 29 Glasgow, Scotland, August 24-28, 29 Ergodc Capacty of Block-Fadng Gaussan Broadcast and Mult-access Channels for Sngle-User-Selecton and Constant-Power Mohammad Shaqfeh, Norbert Goertz 2 and John Thompson 3 Department of Electrcal Engneerng, Teas A&M Unversty n Qatar, P.O.Bo 23874, Educaton Cty, Doha, Qatar. E-mal: mohammad.shaqfeh@qatar.tamu.edu 2 Insttute of Communcatons and Rado-Frequency Engneerng, Venna Unversty of Technology, Gusshausstrasse 25/389, A-4 Wen, Austra. E-mal: norbert.goertz@nt.tuwen.ac.at 3 Insttute for Dgtal Communcatons, Jont Research Insttute for Sgnal & Image Processng, School of Engneerng and Electroncs, The Unversty of Ednburgh, Mayfeld Rd., Ednburgh EH9 3JL, Scotland, UK. E-mal: John.Thompson@ed.ac.uk Abstract We consder the ergodc capacty regon of block-fadng Gaussan multuser channels wth channel-state nformaton at both the transmtters and the recevers. We assume a sngle constrant on the total long-term average power used for both broadcast and multaccess channels. In addton to the optmal soluton known from the lterature, we provde analytc epressons some of whch are novel to characterze the boundary surface of the capacty regon under aulary constrants whch nclude sngle-user-selecton per block, constant total transmt-power per block and the combnaton of both. We also provde optmal resource allocaton schemes to acheve the capacty lmts for each case under consderaton. Moreover, we provde numercal eamples to compare the cases. As an llustratve eample, we analyze the two-user case, although the results carry over to the M-user case. I. INTRODUCTION Fadng channels both tme and frequency selectve can be modeled as a famly of parallel Gaussan channels: ths s called a blockfadng channel ]. Each of the parallel Gaussan channel blocks corresponds to a fadng state. In general, the capacty of block-fadng multuser channels wth channel-state-nformaton CSI at both the transmtters and the recevers can be acheved by optmal power allocaton over the channel blocks and optmal resource rate and power allocaton over the users n each of the channel blocks. Ths s applcable to both the broadcast channel BC one-to-many multuser channel 2], and the mult-access channel MAC manyto-one multuser channel 3]. From a practcal communcatons engneerng perspectve, the optmal solutons are n most cases dffcult f not mpractcal to mplement. Thus, sub-optmal solutons whch have close-to-optmum performance and, at the same tme, lend themselves to an easy mplementaton are favorable. The optmal power allocaton scheme over block-fadng Gaussan broadcast and mult-access channels s gven by the water-fllng approach: more power s allocated when the channel s better and, dependng on the desred operatng pont on the capacty regon s boundary surface, some users are assgned hgher average power to meet ther rate demands. As a consequence of ths power allocaton polcy, the total and ndvdual transmsson powers wll vary hugely. Ths wll cause problems when, e.g., the transmtter.e. the base staton n the broadcast case has mamum power constrants n order not to cause too much nterference n adjacent cells. Furthermore, adaptve power control requres addtonal computatonal complety to mantan the average power constrant, and varable transmsson power s also lkely to requre more epensve rado-frequency crcutry. The optmal resource allocaton over a flat-faded channel block nvolves applyng the optmal channel-access scheme, whch s code dvson multple access n MAC or superposton codng n BC wth successve nterference cancellaton SIC at the recevers. Furthermore, the number of users scheduled n a channel block vares dependng on the channel condtons. Superposton codng wth SIC at the recevers can hardly be mplemented n practce, because of the complety nvolved, the necessty to nform all users about the order n whch successve cancellaton has to be conducted ncludng the codng schemes used sgnalng overhead, and dfferent blockszes used for encodng of dfferent users: cancellaton of a user s sgnal s only possble when the whole codeword for ths user has been receved, although the user to be detected due to delay constrants may well have a much shorter although stll long channel codng blocksze. As ths user would have to wat for decodng untl the nterferng user s much longer codeword has been receved, delay constrants are lkely to be volated. II. OBJECTIVES We nvestgate the ergodc capacty lmts and the optmal solutons to acheve these lmts under practcally relevant restrctons that enforce the use of constant total transmsson power per fadng state channel block, sngle-user selecton per fadng state or both. Constant total power per block s to be nterpreted such that n each and every channel block the sum power for all users s constant. For a broadcast channel ths means that the total power used by the base staton for all users s the same n every channel block although the number of users scheduled n every block s varable and subject to In practce, codng for a user wll be spread over as many blocks as possble to obtan long codewords that wll allow for effcent channel codng. EURASIP, 29 784
optmzaton. In the multple-access case, agan the sum of all powers of all users transmtters s assumed to be constant. In our analyss of the mult-access channels MAC, we assume a sngle long-term average sum-transmt-power constrant nstead of ndvdual power constrants for the users that are assumed n prevous work 3]. Ths case s also relevant n practce 4]. Furthermore, t gves a more general soluton wth an etra nformaton cannot be obtaned from 3] about the optmal average powers to be allocated to each user to acheve a certan operatng pont. In 5] the dualty of the MAC and BC channels was dscussed. It was shown that the capacty regon of the MAC channels wth sum-power constrant s dentcal to the capacty regon of the dual 2 BC channels. Thus, n all the cases under consderaton, the equatons characterzng the boundary surface of the capacty regon are applcable to both the BC and the dual MAC channels. Furthermore, there ests a strkng smlarty between the optmal resource allocaton for both channels. Our objectve, n ths paper, s to study how much we wll lose n terms of system capacty when we apply one or both of the specfed aulary constrants. In order to answer ths queston, we gve closed-form epressons whch are novel contrbutons that characterze the capacty lmts and descrbe resource allocaton schemes for BC and MAC agan wth some novel contrbutons to acheve these lmts for the followng four cases wth dfferent constrants: OPT: optmal soluton, wthout any aulary constrants new analytcal results that complement the orgnal work n the lterature,.e. 2] for the BC case and 3] for the MAC case, presented n ths paper. CP: constant sum power of all users n every channel state new analytcal results presented n ths paper. SU: selecton of a sngle-user only n every channel state. CP-SU: constant sum power and sngle-user selecton n every channel state. We provde numercal results n whch we compare the four cases. To vsualze the capacty lmts, we consder the two-user case, wth the assumpton of dfferent long-term average channel qualtes of the users. Qualtatvely, the results carry over to the M-user case. We perform analyss for a hgher number of users as well by selectng a specfc operatng pont ma. sum-throughput for symmetrc channels. III. CHANNEL MODEL The block-fadng channel s used to model tme and frequency selectve fadng channels. The fadng channels are dvded nto a famly of parallel Gaussan constant channels, each corresponds to a flat fadng state. These constant channels are called blocks. A channel block could last for several tme slots as long as the channel qualty s almost constant dependent on fadng statstcs. The M-user Gaussan block-fadng broadcast channel BC conssts of a sngle transmtter and M recevers. In channel block k, the transmtter broadcasts a sgnal k], and the receved sgnals are y k] = h k]k] + n k], =,, M where h k] > s the constant channel qualty.e. power gan 3 between the transmtter and the -th recever at channel-block k, 2 BC and MAC channels are dual f they have the same channel vector h.e. h of recever n the BC equals h of transmtter n the MAC. 3 In ths paper we assume, wthout loss of generalty, that the channel gan h s real and representng the power gan of the lnk. h does not have magnary part snce we assume perfect phase nformaton at the recevers. and n k] s Gaussan nose wth zero mean of that recever. The noses n k] are statstcally ndependent, and are assumed to have a common varance. The M-user Gaussan block-fadng mult-access channel MAC conssts of a sngle recever and M transmtters. At channel block k, each transmtter transmts a sgnal k], and the recever receves the composte sgnal yk] = hk] k] + nk] = where h k] > s the constant channel qualty between the -th transmtter and the recever at channel-block k. The fadng processes of all users are ndependent of each other, are statonary and have contnuous probablty densty functons, f h. In the numercal eamples through the paper, we assume the fadng processes have the Raylegh 4 dstrbuton. The cumulatve dstrbuton functons of the fadng processes are denoted by F h =. f h h dh. We use the notaton P k] and R k] to ndcate the power 5 and the rate bts/sec/hz respectvely that are allocated to user n channel block k. The long-term average rate that s allocated to user s denoted as R. The long-term average sum-power constrant s denoted as P. IV. PROBLEM FORMULATION The ergodc capacty regon s defned as the set of all achevable rate vectors R such that the long-term average power constrant P over all channel blocks s not eceeded. The optmum ponts wthn the capacty regon are those that are located on the boundary surface. The latter can be characterzed as the closure of the parametrcally defned surface { Rµ : µ R M +, µ = } where for every weghtng factor vector µ, the rate vector Rµ can be obtaned by solvng the optmzaton problem: ma K K k= = µ R k] subject to K K k= = P k] = P 2 where K s the total number of channel blocks, and M s the number of actve users. We assume w.l.o.g. that all channel blocks have dentcal frequency bandwdth and tme duraton. The two aulary constrants that are consdered n ths work to be added to the problem defnton n 2 are: Constant sum power per channel block: Pk] = P k] = P 3 = Sngle-user selecton per channel block: Rk] has a mamum number of one non-zero element. V. CHARACTERIZATION OF THE BOUNDARY OF THE CAPACITY REGION In ths secton, we provde characterzaton of the boundary of the capacty regon of block-fadng BC and MAC channels for the cases 4 f h = h ep h, h s the average channel qualty 5 When we use the notaton P, we mean the transmt power P T. The receved power s ndcated as P R. 785
under consderaton 6. Ths ncludes descrbng resource allocaton schemes to acheve the capacty boundary lmts of BC and MAC channels, and gvng closed-form epressons that characterze the capacty lmts the same epressons are applcable to both BC and dual MAC for a gven weghtng vector µ defnng one pont n the boundary surface. A. OPT: Optmal Case No Aulary Constrants As dscussed n ], problem 2 can be solved by frst applyng the Lagrangan characterzaton n order to defne the problem n an unconstraned format. The resultng optmzaton problem s: K M ma µ R k] P k] 4 {Pk]} k= Ths s equvalent to K k= ma Pk] = = M µ R k] Pk] = where s selected such that K Pk] = K P 6 k= Thus, the man optmzaton problem s decomposed nto a famly of optmzaton problems, one for each channel block, and an equaton to control the power prce n order to mantan the longterm average power constrant. Followng the procedure descrbed n ] by defnng margnal utlty functons, and by etendng these results to the MAC case, we provde a summary of the soluton: Power allocaton over the channel blocks: The total power transmtted n a block k s dentcal n BC ] and MAC channels wth + = ma,: P sumk] = ma µ h k] + ] 5. 7 Resource allocaton n each channel block: The optmal resource allocaton over a flat-faded channel block nvolves applyng the optmal channel-access scheme, whch s code dvson multple access n MAC or superposton codng n BC wth successve nterference cancellaton SIC at the recevers. The SIC at the recevers of BC channels s n order of decreasng µ. Each recever decodes the sgnals sent to users of hgher µ before decodng ts own sgnal. However, n MAC channels, the recever performs SIC n order of ncreasng µ 5]. We provde a summary of the greedy algorthm procedure to compute the power allocated to each user n channel block k, for both BC ] and MAC novel contrbuton by etendng results of BC to MAC wth sum power constrant channels: Margnal utltes functons rate revenue mnus power cost are defned for each channel block k: BC: u z µ h k] + z, z 8 MAC: u z µ + z h k], z 9 Then based on the margnal utltes whch are dependent on the 6 The detaled proofs are omtted due to paper length restrcton. We assume here that the reader s aware of the orgnal papers n ths topc manly ] as our work complements these results. The proofs wll, however, be gven n a full journal paper verson of ths work 6]. channel qualtes vector hk], the ntervals A are obtaned: A {z, : u z > u jz j and u z > } Snce u z s monotoncally decreasng and u z, u jz j cross each other at mamum once, the nterval A s contnuous. The power allocaton s calculated as: BC: P k] = A dz MAC: P k] = σ2 dz h k] A To derve equatons to characterze the boundary surface of the capacty regon, we complement the work n ] to get the followng equatons to compute Rµ n. We use the assumpton that the fadng processes of all users are statonary wth contnuous probablty densty functons and ndependent of each other: for each user =,..., M R OPT = or equvalently R OPT = µ + z +z µ z f h F hj α ddz 2 µ j + z f h F hj β ddz 3 where n 2, 3 s computed based on 6 whch, n our case of ndependent fadng processes, s equvalent to: f h F hj ζ µ ] d = P 4 µ j There are two other equvalent forms to compute : +z f h F hj α ddz = P 5 j µ α, β, ζ are gven by: µ µ z β = f h j α = + µ +z j µ + z µ ζ = + µ F hj β ddz = P 6 7 8 9 The notaton n 2, 3, 4 s defned as 7]: =. { f. 2 + f < B. CP: Constant Sum-Power per Block In ths case, the man optmzaton problem 2 becomes equvalent to optmzng over each channel block k: ma µ R k] subject to = P k] = P 2 Thus, the problem s to allocate the resources over the users n each channel block. The power allocated to each user n each channel block can be obtaned usng the same greedy procedure of the optmal case, but wth the replacement of the global power prce n 8, 9 by = 786
block-dependent power prce k], whch s obtaned as: k] = ma µ h k] + P 22 The channel access scheme and the order of SIC s dentcal to the OPT case. The boundary surface s characterzed by the equaton: R CP = P + z f h F hj β ddz 23 j where β defned n 8, and the notaton ] n 2. C. SU: Mamum of Sngle-User Selecton per Block In ths case, the soluton of the optmzaton problems over each channel block becomes a user selecton strategy, where the user to be scheduled s the one who mamzes the selecton argument polcy. The only user m scheduled to transmt MAC or receve BC n block k, and the power allocated to ths user are calculated accordng to: m = arg ma µ R k] Pk] 24 where P k] s calculated accordng to: P k] = µ ] + 25 h k] The boundary surface s characterzed by the equaton 8]: R SU µ = f h F hj γ log d 26 µ j where n 26 s computed accordng to f h F hj γ µ j γ n 26, 27 s gven by: γ = W µ µ ] d = P 27 µ ep ] 28 µ wth W the Lambert functon 9] nverse of f = e. D. CP-SU: Both Constrants In ths case, the only user m scheduled to transmt MAC or receve BC n block k, and the power allocated to ths user are calculated accordng to: m = arg ma µ R k] and P k] = P 29 R k] n 24 and 29 s the Shannon capacty of AWGN channel: R k] = log + hk]pk] The boundary surface s characterzed by 8]: R CP-SU = where η s defned as: f h j F hj η log η = + P µ + P d 3 3 P VI. NUMERICAL EXAMPLES A. Comparson of Two-User Case We provde a numercal eample of applyng the equatons to characterze the boundary surface of the capacty regon for the four cases under consderaton n a scenaro of two users. The two-user case s selected because t s possble to vsualze the capacty regons and compare the dfferent cases. Qualtatvely, the results carry over the general M-user case. Second USer Average Rate bts/sec/hz.8.6.4.2 OPT SU CP CP SU.5.5 2 2.5 3 Frst User Average Rate bts/sec/hz Fg.. Boundares of the ergodc capacty regons for the two-user case. The users are Raylegh-faded wth db dfference n average channel qualtes. In Fg. we show the capacty regons wth the assumpton that the users channels are fadng ndependently and wth Raylegh dstrbuton. The frst user channel has db better long-term average channel qualty over the second user channel. Any specfc pont n the capacty boundary can be acheved by adjustng the weghtng factors µ. We selected a relevant case n whch the network average spectral effcency can range between and 3 bts/sec/hz. The man conclusons we obtan from the results are: Power control s more mportant when the operatng pont of the system has overall low spectral effcency to serve weakchannel users. For hgh spectral effcences, usng constant power per block s justfed and has mnor detrmental effects to the capacty of the system. For constant transmt power systems, applyng superposton codng provdes neglgble mprovements to the achevable rates. Thus, usng sngle-user selecton scheme n such systems s justfed. On the other hand, for systems applyng optmal power control, superposton codng s useful for a range of operatng ponts. B. Sum-Throughput Comparson for Symmetrc Channels In ths eample, we compare the capacty dfference between systems applyng optmal power control and constant power per block systems for varous number of users. Snce t s not possble to vsualze the capacty regons for systems wth more than 3 users, we use nstead a specfc operatng pont wthn the capacty boundary surface. We select the mamum sum-throughput capacty and make the analyss wth assumpton of symmetrc users channels. 787
Furthermore, wth the assumpton of Raylegh fadng channels, we can derve close-form epressons for the capactes as a functon of the number of users M. For the constant power system, we obtan: R sum = M σ 2 σ 2 ep h P E h P = where E s the eponental ntegral functon E ep u du u 32 For the system applyng optmal power control, we obtan: R sum = M E 33 = where s adjusted so that the power constrant s acheved: M h P σ = ep ] M E 2 = 34 characterzaton of the capacty regon n these cases s mportant n order to be able to compare the performance of the system under the practcal constrants. We have provded numercal eamples to compare the cases under consderaton. Addtonally, we have descrbed the optmal resource allocaton schemes to operate at the capacty lmts. Ths topc was studed n the lterature for the optmal case. However, we have etended the results n order to nclude the cases of the aulary constrants. ACKNOWLEDGEMENTS The work reported n ths paper has formed part of the Core 4 Research Program of the Vrtual Centre of Ecellence n Moble and Personal Communcatons, Moble VCE, www.moblevce.com, whose fundng support, ncludng that of EPSRC, s gratefully acknowledged. Fully detaled techncal reports on ths research are avalable to Industral Members of Moble VCE. The authors would also lke to thank for the support from the Scottsh Fundng Councl for the Jont Research Insttute wth the Herot-Watt Unversty whch s a part of the Ednburgh Research Partnershp. M= M=5 M=2 M= Spectral Effcency bts/sec/hz 2 2 Average SNR at recevers Fg. 2. Dfferences n spectral effcency between optmal power control sold lnes and constant power per block dashed lnes for dfferent number of users M. Raylegh fadng channels wth dentcal long-term average channel qualtes. Mamum sum-throughput s consdered. From the results n Fg. 2, we can fnd rough estmates of system spectral effcences over whch the applcaton of the constant power constrant s justfed. As the number of users n the system ncreases, the rate level, over whch the constant power system approaches the optmal power control system, decreases. For eample, n a sngle user system, usng constant power whle operatng above 4 bts/sec/hz s very close to the optmal case. Whle a value of 3 bts/sec/hz s applcable n two users system, and appromately.5 bts/sec/hz for M =. REFERENCES ] D. Tse, Optmal power allocaton over parallel Gaussan broadcast channels, unpublshed, avalable at www.eecs.berkeley.edu/ dtse/broadcast2.pdf. 2] L. L and A. Goldsmth, Capacty and optmal resource allocaton for fadng broadcast channels Part : Ergodc capacty, IEEE Transactons on Informaton Theory, vol. 47, no. 3, pp. 83 2, Mar. 2. 3] D. Tse and S. Hanly, Multaccess fadng channels Part : Polymatrod structure, optmal resource allocaton and throughput capactes, IEEE Transactons on Informaton Theory, vol. 44, no. 7, pp. 2796 285, Nov. 998. 4] G. Gupta and S. Toumps, Power allocaton over parallel Gaussan multple access and broadcast channels, IEEE Transactons on Informaton Theory, vol. 52, no. 7, pp. 3274 3282, July 26. 5] N. Jndal, S. Vshwanath, and A. Goldsmth, On the dualty of Gaussan multple-access and broadcast channels, IEEE Transactons on Informaton Theory, vol. 5, no. 5, pp. 768 783, May 24. 6] M. Shaqfeh and N. Goertz, Ergodc capacty of block-fadng Gaussan broadcast and mult-access channels for sngle-user-selecton and constant-power, submtted to EURASIP Journal on Wreless Communcatons and Networkng, Apr. 29. 7] M. Shaqfeh and N. Goertz, Comments on the boundary of the capacty regon of multaccess fadng channels, to appear n IEEE Transactons on Informaton Theory. 8] M. Shaqfeh and N. Goertz, A new generc framework for comparson of fleble schedulers for delay-tolerant wreless applcatons, submtted to IEEE Transactons on Communcatons, Apr. 29. 9] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, On the Lambert W functon, Advances n Computatonal Mathematcs, vol. 5, pp. 329 359, 996. VII. CONCLUSIONS We have derved novel closed-form equatons to characterze the boundary surface of the ergodc capacty regon of BC channels and MAC channels wth sum-power constrants under practcal aulary constrants on power per block and user selecton per block. The 788