ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate s one whch only transmts to the user wth best channel recepton []. We now present a smlar result for uplnk (mult-access channels [2]. Consder the sngle-cell multuser uplnk scenaro where the sgnal receved by the base staton s as follows: y = h x + z, = where K s the number of users n the cell, h and x are gan and nformaton for user, respectvely. We assume that the nformaton sources x s are zero-mean, have unt energy, and are mutually uncorrelated. The nose z s a zero-mean Gaussan random varable wth varance N 0. If the channel gans h s are determnstc, then ths s smply a Gaussan mult-access channel whose capacty regon s defned as follows [3]. ( R < 2 log + γ, S {,2,...,K}, S S where R and γ = h 2 /N 0 are the nformaton rate and receved SNR of the user, respectvely. Therefore, the sum-rate s R < ( 2 log + = γ. Now, f we assume frequency flat Raylegh fadng, then h has a Raylegh dstrbuton, and n turn, γ has the followng exponental dstrbuton p γ (x = ( x exp γ s where γ s s the average receved SNR for user. γ s =2 I x 0,
2 We would lke to fnd a power control law µ (γ for user, wth γ = [γ,γ 2,...,γ K ], whch takes the nstantaneous receved SNR of other users nto account. Ths can be done by ntroducng some feedback between the base staton and users. The goal s to maxmze the sum-rate C sum = ( log + µ (γγ p(γdγ 2 subject to the constrants that the average power s one unt,.e., µ (γp(γdγ =, ( and µ (γ 0. Ths s a standard convex optmzaton. Introducng the Lagrange multplers, λ, for each constrant, we obtan the followng Lagrangan form: ( max L = log + µ γ p(γdγ s.t. µ 0,. The partal dervatve s = = ( λ = µ p(γdγ Thus, the optmalty condton yelds L µ = γ + K = µ γ λ. γ + K = µ λ γ = 0 f µ > 0, γ + K = µ λ γ < 0 else, and λ s chosen so that the average power condton ( s satsfed. The above optmalty condton can be rewrtten as: Assume that all + = µ j = 0, j and consequently, µ γ γ λ, wth equalty ff µ > 0. (2 γ λ s are dfferent. Then we can see from the condton (2 that f µ > 0, then γ λ > γ j λ j, j. That s, the only user allowed to transmt at any gven tme s the one wth the largest γ λ. Now, suppose that all users have the same average receved power. By symmetry, all λ s are equal. Then the above result can be nterpreted as follows: only the user wth largest nstantaneous receved power s allowed to transmt, and the others must reman quet untl one of them becomes the strongest user.
3 The power allocaton for user s a water-fllng soluton wth the form µ (γ = λ γ 0 else where λ s chosen such that the condton ( s satsfed. f γ > λ, γ > λ λ j γ j, j II. MORE ABOUT PROPORTIONAL FAIR SCHEDULING (PFS A. Addtonal property of PFS It has been shown n [4] that, for PFS, f the channel processes have the form h (t = a b (t where a s a constant and the b (t processes are..d, then n the long run the fractons of slots allocated to each user are equal. As a specal case, n the statc envronment, PFS wll become the equal-tme schedulng. B. Proportonal far wth mnmum/maxmum rate constrants Note that the PFS algorthm does not provde any guarantees on the servce rate provded to any user. For some cases, we may need to ntroduce the mnmum as well as maxmum bandwdth to each user. Suppose that for each user we have a mnmum rate T mn average throughput T to satsfy T mn T T max. Therefore, the optmzaton problem becomes and a maxmum rate T max and we want the max log T (3 s.t. T mn T T max, {T } C where C denotes the capacty regon of the system. An algorthm for ths problem has been consdered n [5]. The dea s to mantan a token counter Q (t for each user, and t s updated accordng to the followng rule: where T token = T mn Q (t + = f Q (t 0 and T token s the same as n the orgnal PFS: T (t + = Q (t + T token Q (t + T token = T max R (t f user s served otherwse f Q (t < 0. Note that the update rule of T (t ( W T (t + W R (t f user s served ( W T (t otherwse.
4 At tme t, the algorthm serves the user (t = arg max R (t T (t eαq(t, where α s a parameter that determnes the tmescale over whch the rate constrants are satsfed. The basc dea of the token counter s that f the average servce rate to user s less than R mn Q (t s postve and then we are more lkely to serve user. Also, f the average servce rate to user s larger than R max then Q (t s negatve and then we are less lkely to serve user. Fnally, t has been shown n [5] that the algorthm s asymptotcally optmal wth respect to the optmzaton (3 when W goes to nfnty. then C. Other questons What s the ntuton behnd the throughput updatng rule n PFS? We can thnk of T (t as an estmate of the actual throughput for user. The larger the wndow sze W, the more accurate the estmate T (t. Note that we cannot calculate the exact throughput on-the-fly: throughput s a long-term quantty by defnton. So ths s one way to estmate t. h 2 How about the schedulng algorthms based on some functon of h and E[ h 2 ], e.g., to schedule the user only f h s larger than some threshold? Ths s related to the queston I have durng the presentaton: what f we do not fx the power and fnd the optmal power allocaton across tme, e.g., a water-fllng soluton. I do not have an answer for ths queston. 3 What s the rate regon achevable usng the PFS scheme? How does t compare wth max sum-rate mechansm? We have only one rate regon (or capacty regon whch s the regon achevable by any schedulng scheme. The dfference between PFS scheme and max-sum-rate scheme s that they get to dfferent ponts n the capacty regon: the max-sum-rate scheme gves us the pont whch has maxmum total rate T, whle the PFS scheme gves the pont whch maxmzes log T. 4 When s PFS better and when s max-sum-rate s better to use? I guess t depends on the goal of the system operator. In the user s pont of vew, PFS s better because t s farer (at least t asymptotcally acheves the proportonal far allocaton. In the system s pont of vew, max-sum-rate mght be better snce t gves the maxmum total throughput of the system.
5 III. OPPORTUNISTIC BEAMFORMING [6] A. Opportunstc beamformng versus space-tme codes The dea of opportunstc beamformng s motvated by the multuser system pont of vew: the larger the dynamc range of channel fluctuatons, the hgher the channel peaks. Hence, n opportunstc beamformng, we use multple transmt antennas to nduce more randomness to channels. On the other hand, the spacetme codes also use multple transmt antennas n the pont-to-pont scenaro. So we would lke to compare between the opportunstc beamformng scheme and the space-tme codes n a multuser system. Let us consder a multuser downlnk system wth two transmt antennas at the base staton. The best known space-tme code for ths scenaro s the Alamout scheme. We assume that the PFS schedulng scheme s used on top of t. Consder the slow fadng scenaro. The Alamout scheme essentally creates a sngle channel wth effectve SNR for user k gven by P( h k 2 + h 2k 2 2N 0 where P s the total transmt power. Ths effectve channel does not change wth tme n a slow fadng envronment, and hence, the PFS schedulng becomes the equal-tme schedulng (see Secton II-A. On the other hand, t has been shown n [6] that under opportunstc beamformng wth PFS, for large number of users, the users are also allocated equal tme wth the followng effectve SNR: P( h k 2 + h 2k 2 N 0. That s, the opportunstc beamformng has 3-dB gan more than the Alamout scheme. Furthermore, both schemes yeld a dversty gan of 2. Thus, n a multuser system wth enough users under PFS, the opportunstc beamformng scheme outperforms the Alamout scheme. Fnally, the Alamout scheme requres separate plots for each transmt antenna and that recevers need to track the channels from both transmt antennas. However, the opportunstc beamformng does not requre any of those. The same sgnal (plot and data goes through both transmt antennas, and the recevers only need to track the overall channel. B. Other questons If perfect CSI s assumed, the BS can beamform to the drecton of the channel of the user to be scheduled, and ths strategy seems to perform better than opportunstc beamformng. So what s the advantage of opportunstc beamformng?
6 The pont of opportunstc beamformng s to ntroduce more randomness to the channels by addng more dumb transmt antennas. Ths addton s transparent to recevers. Comparng to the orgnal system wthout opportunstc beamformng, the effort to get the channel state nformaton wth opportunstc beamformng s the same. But opportunstc beamformng yelds better performance. So the perfect CSI assumpton s for schedulng and beamformng the sgnal before t comes to the dumb antennas. REFERENCES [] D. Tse, Optmal power allocaton over parallel gaussan broadcast channels, n Proceedngs of Internatonal Symposum on Informaton Theory, Ulm, Germany, June 997. [2] R. Knopp and P. Humblet, Informaton capacty and power control n sngle-cell multuser communcatons, n Proceedngs of the IEEE Internatonal Conference on Communcatons (ICC, Seattle, WA, June 995. [3] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. John Wley & Sons, 99. [4] J. Holtzman, CDMA forward lnk water-fllng power control, n Proceedngs of the IEEE Semannual Vehcular Technology Conference (VTC2000-Sprng, Tokyo, Japan, May 2000, pp. 663 667. [5] M. Andrews, L. Qan, and A. Stolyar, Optmal utlty based mult-user throughput allocaton subject to throughput constrants, n Proceedngs of IEEE INFOCOM, Mam, IL, March 2005. [6] P. Vswanath, D. Tse, and R. Laroa, Opportunstc beamformng usng dumb antennas, IEEE Transactons on Informaton Theory, vol. 48, no. 6, pp. 277 294, June 2002.