BENEFITS OF MULTIUSER DIVERSITY WITH LIMITED FEEDBACK. Mathias Johansson

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1 BENEFITS OF MULTIUSER DIVERSITY WITH LIMITED FEEDBACK Mathias Johansson Signals & Systems Grop, Uppsala University, Uppsala, Sweden. ABSTRACT This work stdies the effects of tilizing mltiser diversity in realistic settings with limited feedback and ncertain traffic loads. A simple optimal method for setting rate qantization levels is derived and it is shown that an added benefit of mltiser diversity is that channel feedback can be heavily qantized withot performance loss. In the limit as the nmber of sers goes to infinity, even the most severe rate qantization 2 levels) does not redce throghpt at all from the case with nlimited rate resoltion. Some important implications for the joint tilization of mltiser diversity and mlti-antenna techniqes are addressed. Frther, a throghpt-optimal schedling policy for ncertain and brsty traffic loads is derived. An approximate policy is given with attractive comptational complexity.. INTRODUCTION The main object of this paper is to investigate the effects of limited channel feedback, or eqivalently a small discrete set of transmit rates, on tility-based schedling in the down-link of a mobile celllar commnications system. By tility-based schedling we mean schedling methods that adapt to some given tility fnction. Here we focs on schedling for maximm system throghpt, possibly extended with ser- and time-varying priorities. Schedling transmissions based on the flctating channel qality is referred to as exploiting mltiser diversity. The basic idea is to transmit more data to a ser with high instantaneos channel gain and less to a ser with low channel gain. This reslts in higher individal and total throghpt provided that all sers have independent time-varying channels []. In the following section we investigate the effects of qantized channel feedback, and derive a method for setting qantization levels in a mltiser system. It trns ot that by explicitly sing the fact that tility-based schedling is employed, the optimal qantization levels can be mch fewer This work was partially fnded by Vinnova the Swedish Agency for Innovation Systems) and PCC++. and will also be set differently for a given service degree, as compared to qantizing from a single ser perspective. The effect of restrictive feedback in a system tilizing mltiser diversity is ths less prononced than in a traditional system where each mobile terminal is served in a fixed and eqal order. Indeed, as the nmber of sers goes to infinity, in a tility-based schedling regimen the system throghpt does not degrade at all as the nmber of bits sed for feedback is decreased. In Section 3, a schedler which maximizes the expected system throghpt is derived for ncertain brsty traffic loads and perfect channel knowledge, e.g. obtained by channel prediction and qantized feedback as described in Section 2. A low-complexity approximate soltion is also presented. Then, in Section 4, the performance of schedling with channel feedback qantized sing the proposed method is investigated throgh simlation. In the conclding section, it is noted that the reslts have some important implications for overall system design and deployment of MIMO techniqes. 2. BENEFITS OF MULTIUSER DIVERSITY IN CONNECTION WITH LIMITED FEEDBACK In order to redce the amont of channel feedback in a system employing mltiser diversity, it becomes important to find an efficient means of qantizing the channel information. We here smmarize the channel information in a nmber c t which denotes the nmber of bits that can be transmitted over the channel to ser at time slot t at a ser specific bit error rate, provided that the ser gets exclsive access to the channel. This nmber is henceforth denoted the effective capacity of the channel. Assme that each mobile terminal in the cell has an accrate estimate of the effective channel capacity for the next time slot to be schedled. The qestion now becomes to determine how many levels are needed and how to set the qantization levels. Obviosly, intervals of eqal size may not be the best choice. The aim shold instead be to find a set of levels that redces the total throghpt as little as possible compared to having perfect channel knowledge. A critical observation is that

2 when exploiting mltiser diversity, fewer levels shold be needed than in the single ser case becase the plasibility that some ser has a high capacity grows with the nmber of sers. 2.. A practical method for rate qantization Let Q be the nmber of bits sed for qantization, and M 2 Q the nmber of qantization levels. We formlate or problem as finding a set of qantization levels q m sch that the probability that at least one ser will have an effective capacity higher than the qantization level is p m, where p m is a design parameter. Let A m be the event that at least one ser has c q m. The event Ā m is then the event that no ser has c q m. This implies that P A m I) P Ām I) ) where I is any information we might have abot the channels, and, letting U denote the nmber of sers in the cell, P Ām I) P c < q m ) c U < q m ) I) 2) P c < q m I) P c U < q m I) 3) U qm P c I)dc 4) assming logical independence between channels. Inserting 4) into ) we obtain P A m I) U qm P c I)dc. 5) If all sers are assigned identical channel distribtions e.g. becase the individal channel characteristics are not known in detail), we have qm ) U P A m I) P c I)dc. 6) It follows that the probability of being able to transmit at the rate q m increases with the nmber of sers, and ths with many sers in the cell we can set the qantization levels q m higher for a fixed desired probability of transmitting at that rate. In smmary, the method consists of deciding how many sers are typically located in the cell, then finding the common probability distribtion P c I), and setting the desired probabilities p m of being able to transmit at each rate q m. The optimal rate levels are then calclated from 6) with p m P A m I). The best choice of p m depends on reqirements on robstness and fairness, as well as on the nmber of bits sed for feedback. A scheme for a two-bit qantization cold for instance be to set p.99, p 2.8, p 3.5 and p 4.2, and solve for each q m in 6) given the effective capacity distribtion and the typical nmber of sers in the cell. Another approach is to set the qantization levels adaptively as the nmber of sers in the cell changes. The method can also be sed to analyze a given qantization and ths give a measre of how efficient the qantization is for different nmbers of sers. An efficient qantization shold have well separated p m.) 2... Examples In Rayleigh fading MIMO systems where the nmber of antennas is large, according to the central limit theorem the theoretic channel capacity is asymptotically Gassian [2]. Using the theoretic capacity as a rogh approximation to the effective capacity gives P A m I) 2 U [ erfc µ q m 2σ )] U p m 7) where erfcx) 2 π x e t2 dt, µ is the mean of the capacity for each ser, and σ 2 the variance. If we for instance set a level q m sch that q m µ + 2σ, the single ser case U gives a probability of.2 of being able to transmit at that rate. For 32 sers, the probability increases to.52, a dramatic increase! A similar probability for the single ser case wold reqire that q m be set below the average individal capacity. By exploiting mltiser diversity, we can se fewer qantization levels, and still transmit at almost maximm rate! It is clear that as the nmber of sers tends to infinity, the probability that at least one of them has a channel qality eqal to the maximm rate tends to. 3. MULTIUSER DIVERSITY WITH UNCERTAIN AND BURSTY TRAFFIC Consider a mobile celllar commnications system with one access point per cell. The access point transmits data to a set of U sers, and schedles its downlink transmissions on a time-slot basis, possibly sing exclsive allocations as in a TDMA system or, in a more flexible setting, splitting the bandwidth with arbitrary resoltion. The schedling horizon is an integer nmber of time slots T and over this interval the th ser receives n bits of data from the fixed network. Each ser has its own transmission bffer at the access point where data is received, and the access point is assmed to have accrate knowledge of the effective channel capacity for each ser in the cell over the schedling horizon T. The aim of the schedler is to allocate the channels so as to maximize system throghpt taking channel qality and

3 bffer levels into accont. The channel can be predicted with high accracy a small nmber of time slots ahead [3], bt the data received in each bffer over the coming T time slots can in general not be predicted de to the brsty natre of typical Internet traffic. The schedler can however be assmed to keep a record of the average incoming amont of data per time slot, and se this information to aid the schedling decisions. In [4] we derived a schedling framework for maximm system throghpt in a general setting for different amonts of channel and data rate information. Here we only recapitlate the main reslts applicable to or crrent problem, and then extend the schedler to the case where the only knowledge of the traffic load is a short record of past rates. The main principle behind the proposed schedler is to avoid nnecessary assmptions by the se of the maximm entropy principle. Briefly, the principle translates prior information I into probability assignments P A I) where A is the ncertain variable of interest) by maximizing the Shannon) entropy sbject to the constraints that I imposes on the probability distribtion. The reslting distribtion is the least biased possible on the given information [5], and correctly describes a state of knowledge while avoiding gratitos assmptions concerning long-rn behavior. A maximm entropy distribtion is simply the one which can arise in the greatest nmber of ways while agreeing with the given information [6]. Denote the mean of the nmber of bits received in each bffer by < n >. The maximm entropy distribtion for n sbject to that the average is known, is [4], [7] P n I) < n > n P n I) 8) n < n > ) n. 9) < n > + < n > + Maximizing the system throghpt is eqivalent to minimizing the total nmber of bits in the U transmission bffers, given by L gs + n x ), ) where gα) α if α >, else gα), S is the nmber of bits in stock in the :th bffer, and x T t c tρ t is the otflow where ρ t is the channel fraction a nmber between and ) assigned to ser at time t. The schedler minimizes the expectation of the bffer levels after T time slots: < L > < L > ) where < L > P n I)gS + n x ) 2) n x S <n < n > <n >+) S + < n > x, x S. 3) The minimization is achieved by adjsting ρ t nder the constraint U ρ t. A detailed derivation can be fond in [4]. A limitation of this schedler is that the average data rates are assmed well-determined, which is not the case when a new ser arrives. We now extend the schedler to inclde the effects from having only a short data record of previos amonts of incoming data for each ser. Denoting the probability assignment 9) by P n µ, I) where µ < n > for compact notation) to make explicit the dependence on the expectation, we now wish to find the probability conditioned on a short data record. Let y... y N } be the respective nmber of bits received at bffer each of the N previos periods. For clarity of expression we briefly drop all notational dependencies on. We then wish to derive the probability P n y}, I): P n y}, I) P n µ, y}, I)P µ y}, I)dµ. 4) Since we have no reason to sppose correlations, P n µ, y}, I) P n µ, I). In order to determine P µ y}, I) we se Bayes theorem to obtain: P µ y}, I) P y} µ, I)P µ I). 5) Note that in order to flly specify the distribtion given merely past data, the only new element consists of the prior P µ I). Since all that is known abot the expected amont of incoming data is that it is non-negative, the prior that best describes or state of ncertainty is Jeffrey s prior [8]: P µ I) µ, 6) which assigns niform probability for log µ. Let Y y y N be the sm of all past amonts of incoming data in bffer. Using the Jeffrey s prior 6), then inserting 5) into 4) and normalizing to nity, we obtain n + Y )!N + Y )! P n y}, I) N 7) Y )!N + n + Y )! N N i N i Y + i). 8) Y + n + i)

4 Interestingly, the expected vale trns ot not to be the arithmetic mean, bt < n > np n y}, I) n Y N, 9) which reflects the fact that the interval is open to the right side while bonded from the left. This means that the best estimate as to the next vale from an independent non-negative seqence is slightly larger than the arithmetic mean. Note that the se of a niform prior wold make the estimate Y even larger N 2 ) as it gives mch higher initial probability to a large vale. Using 8) instead of 9), the expected loss contribtion the expected nmber of bits remaining in the corresponding bffer) of ser becomes x S +Y )!N +Y )! N < L > )Y )!N +x S +Y )! S + Y N x, x S 2) which is to be minimized by adjsting ρ t as before. This schedler has a comptational disadvantage since it involves factorial fnctions. Simply setting < n > Y N in the basic expected loss contribtion per ser 3) is an excellent approximation of 2) see the appendix) with better comptational properties. Conseqently, a near-optimal schedling policy for ncertain traffic loads and known effective channel capacities is to minimize where < L > < L > < L > 2) x S µ µ µ +) S + µ x, x S, 22) x T t c tρ t and µ Y N. Using exclsive allocations, i.e. restricting ρ t to being or, and setting T, it is readily seen that an approximate throghpt-optimal allocation consists of simply picking the ser with maximm L diff < L ρ t ) > < L ρ t ) > 23) with < L ρ t ) > given by 22). It shold be noted that exclsive allocations are not generally optimal. The performance penalty for restricting the decisions to exclsive allocations is however small [4]. 4. SIMULATIONS A set of simlations was performed in order to illstrate the performance gains obtained by employing mltiser diversity with properly chosen qantization levels. The nmber of sers was held at a constant nmber of U 32. Each ser received data generated by independent bt identical Poisson distribtions. The sorce rates were set higher than what the channels cold spport so that there was always data to send. The effective channel capacities were generated from a Gassian distribtion with standard deviation eqal to a qarter of the channel mean vale. This choice was motivated prely from a standpoint of providing a simple qalitative analysis. Inclding more realistic models is a topic for frther research bt will not change the qalitative reslts obtained here. For techniqes were compared sing the same data sets:. Rond Robin schedling sing bit for feedback, set sing the here proposed method bt optimized for U ser. The levels were chosen as p q ) and p 2.9 q 2 µ.3σ) 2. The here proposed approximate schedler sing exclsive allocations 23), and the same qantization as in the preceding method. 3. The here proposed approximate schedler sing exclsive allocations 23), and feedback bit with levels matched for 32 sers, the corresponding levels becoming p q ) and p 2.9 q 2 µ +.46σ). 4. The here proposed approximate schedler sing exclsive allocations 23), and 2 feedback bits with levels matched for 32 sers, the corresponding levels becoming p q ), p 2.9 q 2 µ +.46σ), p 3.5 q 3 µ + 2.2σ), and p 4.2 q 4 µ σ). The system throghpt obtained from averaging simlations are reported in Table. The throghpt has been normalized so that the Rond Robin schedler obtained a throghpt of nity. The reslts clearly indicate the importance of choosing qantization levels according to the nmber of sers in the cell. Comparing the reslts from Rond Robin schedling and tility-based schedling, it is evident that the mltiser advantage is not prononced when qantization levels are few and selected from a single ser perspective. By proper adjstment of a single non-trivial qantization level however, the throghpt is dobled! This is a dramatic performance gain which comes at virtally no cost. Using 2 bits for qantization does not significantly increase throghpt, and it can be noted that going from bit to infinite resoltion only increases throghpt by 7%. 5. CONCLUSIONS Previos work [2] has sggested that for MIMO systems with a small nmber of receive and transmit antennas, the

5 Table. Normalized system throghpt from schedling a congested system with 32 sers. RR -bit -ser qantization -bit -ser qantization. -bit 32-ser qantization 2. 2-bit 32-ser qantization 2.2 nqantized channel 2.34 se of mltiser diversity techniqes reqires sbstantial amonts of channel feedback. In this work a qite different conclsion is drawn; employing mltiser diversity significantly redces the overall need for channel feedback, and sggests the se of MIMO techniqes operating withot frther need for feedback than the information stating the proper modlation level for all antennas. Instead, the capability and sitability for rate adaptation is of greater importance. No more than for rate levels shold be needed.) Spatial mltiplexing [9] might ths be a viable MIMO techniqe in combination with mltiser schedling. The wellknown disadvantages of spatial mltiplexing in low-rank channels are alleviated in a system tilizing mltiser diversity. In order to ensre adeqate performance also in sitations with few sers or for flfilling fairness reqirements) the system cold switch between spatial mltiplexing and space-time block coding, both techniqes operating withot channel knowledge. Appendix Here we show that x S + Y )!N + Y )! 24) N )Y )!N + x S + Y )! ) x S µ µ. 25) µ + Recognizing that the first term can be rewritten as x S + Y )!N + Y )! N )Y )!N + x S + Y )! Y N N and the second term as ) x S µ µ Y µ + N + x S Y +i + N Y y ) x S 26) 27) 28) the ineqality 25) is an ineqality only in the denominators and we rewrite the relation in a more compact form, M + α ) + M ) α, 29) y + i y with α >, y > and M >. Taking the logarithm of the left side we obtain M log + α ) M α y + i y + i 3) α with eqality for y+i. Inserting this into 29) we frther find that ) M exp α exp [α lny + M) lny + ))] 3) y + i ) α ) α y + M + M/y + M/y) α 32) y + + /y where the ineqality in 3) tends to eqality as M grows to infinity. This concldes the proof, and it is seen that the approximation 25) gains in accracy for large N and large Y. It is easily verified by simlation that the approximation is excellent even for small vales of N and Y. References [] R. Knopp, P.A. Hmblet, Information capacity and power control in single-cell mltiser commnications, Proc. IEEE ICC 95, Jne 995. [2] B. M. Hochwald, T. L. Marzetta, V. Tarokh Mltiantenna channel-hardening and its implications for rate feedback and schedling, Sbmitted, May 22. [3] T. Ekman Prediction of mobile radio channels: modeling and design, Ph.D. Thesis, Signals & Systems Grop, Uppsala University, Oct 22. [4] M. Johansson, M. Sternad Resorce allocation nder ncertainty sing the maximm entropy principle, Sbmitted, March 22. [5] E. T. Jaynes, Information theory and statistical mechanics, The Physical Review, vol. 6, no. 4, pp , May [6] E. T. Jaynes, On the rationale of maximm-entropy methods, Proceedings of IEEE, 7, pp , 982. [7] E. T. Jaynes, New engineering applications of information theory, Engineering Uses of Random Fnction Theory and Probability, Bogdanoff and Kozin eds.), Wiley, New York, pp 63-23, 963. [8] Sir H. Jeffreys, Theory of Probability, Third edition, Oxford University Press, 96. [9] G. J. Foschini, Layered space-time architectre for wireless commnication in a fading environment when sing mltiple antennas, Bell Labs Technical Jornal, vol., no. 2, 4-59, 996.