PROPORTIONAL FAIR SCHEDULING OF UPLINK SINGLE-CARRIER FDMA SYSTEMS

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1 PROPORTIONAL FAIR SCHEDULING OF UPLINK SINGLE-CARRIER SYSTEMS Junsung Lim, Hyung G. Myung, Kyungjin Oh and David J. Goodman Dept. of Eectrica and Computer Engineering, Poytechnic University 5 Metrotech Center, Brooyn, 1101, USA E-mai: {jim01; hmyung01; oh01}@utopia.poy.edu, dgoodman@poy.edu ABSTRACT We appy nove utiity-based scheduing schemes to upin singe carrier frequency division mutipe access (SC- ) systems. Two utiity functions are used for managing two dimensiona resources (time and frequency): user data rate for maximizing system capacity and ogarithmic user data rate for proportiona fairness. To deveop utiity-based scheduing agorithms, we revise channe-dependent scheduing (CDS) schemes derived in our previous wor [1]. The resuts show that proportiona fair scheduing with ogarithmic user data rate can improve the rate-sum capacity up to 100% for ocaized and 30% for intereaved, with the capacity gains equay shared among a users. I. INTRODUCTION High-speed wireess data transmission requires adaptive resource aocation to combat radio impairments. Channes with wide bandwidth may experience time and frequency seective fading. When the channe can be estimated, the transmission scheme can be adapted to changing channe characteristics by means of channe-dependent scheduing, adaptive moduation and coding or power contro. In this paper, we investigate utiity-based channedependent scheduing schemes of upin singe carrier frequency division mutipe access (SC-) systems. SC- is drawing great attention as an attractive aternative to orthogona frequency division mutipe access (O) for upin high speed data services in Long Term Evoution of 3GPP due to the ower pea-to-averagepower (PAPR) in the time domain [1],[],[3]. It can be viewed as DFT-spread O, where time domain data symbos are transformed to frequency domain by a discrete Fourier transform (DFT) before going through O moduation. SC- has two types of sub-carrier mapping: Locaized (L-) and Intereaved (I-). In L-, the scheduer assigns consecutive sub-carriers to convey information from a particuar user. The advantages of frequency seective diversity can be achieved in L- by assigning consecutive sub-carriers to a user with favorabe channe conditions for the assigned sub-carriers. In I-, users This wor is supported by the Nationa Science Foundation and by Wireess Internet Center for Advanced Technoogy at Poytechnic University, Brooyn, New Yor, USA, under grant No /06/$ IEEE are assigned sub-carriers that are distributed over the entire frequency band in order to avoid aocating adjacent subcarriers that are simutaneousy in a deep fade. By seecting users that have favorabe channe conditions over the entire system bandwidth, we obtain muti-user diversity in an I- system. A ey question we woud ie to discuss is how we baance time and frequency resources fairy among users whie achieving muti-user diversity and frequency seective diversity. To do so, we introduce utiity-based scheduing and consider two different utiity functions: aggregate user data rate for maximizing system capacity and aggregate ogarithmic user data rate for maximizing proportiona fairness. In this paper, the channe-dependent scheduing methods proposed in our previous wor [1] are revised to provide tractabe agorithms for utiity-based scheduing. Utiity-based adaptive resource aocation has been studied previousy in [4],[5],[6],[7]. Reference [4] defines a mode of utiity that represents user satisfaction and derives a distributed power contro scheme that maximizes the utiity of each user. References [5,6] formuate a cross-ayer optimization probem in order to maximize a utiity function in a downin O system. They propose an iterative sorting search agorithm for dynamic sub-carrier aocation and adaptive power aocation. The resuts show that fairness is achieved by means of the behavior of a margina utiity function. In [7], a simpified scheduing scheme for proportiona fairness has been proposed using ogarithmic average user data rate. However, the proposed method ony aows singe user transmission at a time which is not a genera case of O. Seeing optima sub-carrier aocation agorithms in upin SC- systems by soving a standard form of optimization probem is extremey compex for two reasons: 1) The objective function is formuated as a compex form dependent on chun aocation and ; ) There is an additiona power constraint for each user. Therefore, finding a tractabe soution in resource aocation is a chaenging tas for upin data transmission. This paper is organized as foows: Section II reviews a measure of system capacity when the minimum mean square error (MMSE) frequency domain equaizer (FDE) is empoyed in the receiver. Section III describes utiity-based scheduing schemes for both I- and L-. The system anaysis incuding the scheduing schemes and per-

2 formance measures is discussed in section IV. Section V presents concusions. BI C P I P I N ch, (,, ) = og 1 + γ (,, ) ch ch (bps). (4) II. SC- SYSTEMS We consider time synchronized upin SC- transmission with system bandwidth B Hz. The time axis is divided into transmit time intervas (TTIs) as a basic unit of time scheduing (e.g. 0.5msec). The tota bandwidth is partitioned into L sub-carriers. A set of sub-carriers comprises a chun, and one or mutipe chuns can be aocated to each user in each TTI. The number of subcarriers per chun is M= L/N, where N denotes the number of chuns. Thus, the number of sub-carriers in a chun is regarded as a minimum resource unit for sub-carrier aocation in the frequency domain. There are two types of sub-carrier mapping: L- and I-. A chun in an L- system consists of M consecutive sub-carriers. Sub-carriers in a chun of I- are distributed over the entire bandwidth with equi-distance frequency spacing. Fig. 1 shows an exampe of chun structures for I- and L-, where there are 16 sub-carriers and 4 chuns. We assume that the base station has perfect nowedge of the channe gains of a users in the time and frequency domains. The data consteations of the aocated users are aso determined in the base station, and transmitted to the terminas via downin contro signas. In genera, different transmit powers or different bit consteations can be aocated to the different chuns when a user occupies mutipe chuns. Simiar to O, chun-based greedy power and bit oading can be empoyed with SC-. However, the improvement in throughput may not be significant enough to justify the added compexity. As a reaistic soution, we consider equa-bit-equa-power (EBEP) aocation for each chun. Thus, we assume that the power assigned to each sub-carrier is determined as P (sub) =P / I sub,, where P is the tota transmit power of user, I sub, is the sub-carrier index set assigned to user, and I sub, is the number of sub-carriers assigned to user. I ch, is the assigned chun set of user and I (n) sub denotes the set of sub-carriers in chun n. ( n) I = I (1) sub, n Ich, Then, the SNR for the data deivered by chuns in I ch, can be derived as () when minimum mean square error frequency domain equaization is impemented in the receiver to mitigate inter-symbo interference [1]. ( sub) 1, P Hi, γ ( P, Ich, ) 1 γ = i, = (,3) 1 γ σ i i, Isub, i I γ,, 1 sub i+ where σ i is the noise power of sub-carrier i, and H i, is the channe gain of sub-carrier i for user. γ i, is the SNR of sub-carrier i for user. Using Shannon s formua, the achievabe data rate of user with chuns I ch,, has upper bound sub 1 III. UTILITY-BASED SCHEDULING: CHUNK ALLOCATION Utiity is an economic concept representing eve of satisfaction, and it is used for baancing the efficiency and fairness among users [4]. Since user data rate is a ey parameter to determine user satisfaction in wireess communications, utiity can be defined as a monotonicay increasing function of user data rate. Our objective is to find utiity-based EBEP chun aocation. Thus, a genera optimization probem is formuated as (5), where the goa is to find an optimum index set of the assigned chuns I ch,, for a users in order to maximize the sum of user utiity at T-th TTI. A superscript T is added in the equations to denote time index. ( T) ( T) ( P Ich, ) K U R (5) = 1 max, To cacuate data rate for a given SNR, we use the upper bound of achievabe data rate in (4). Therefore, the instantaneous data rate of user at t-th TTI is represented as C (t) (P,I (t) ch, ). Then, the average data rate of user over T TTIs is cacuated as (6). T ( T) ( T) 1 ( t) ( t) R P, I ch, = C ( P, Ich, ) (bps) (6) T t = 1 The orthogonaity of the users stems from the fact that each user occupies different sub-carriers: If n I ch, j, then n I ch, for j, { 1,,,K}, j (7) For EBEP aocation, the transmit power of each sub-carrier P (sub) is determined by (8). ( sub) Pmax P = Pmax P = (8) Isub, If the user data rate is regarded as a utiity function, the resource aocation with the objective function in (5) with constraints in (7) and (8) maximizes rate-sum capacity ignoring fairness among users. Therefore, ony some users near the base station may occupy most of the resources. On the other hand, ogarithmic user data rate as a utiity function provides proportiona fairness as shown in [7]. The optimum soution entais combinatoria comparisons with high compexity since the optimization probem has a noninear objective function with noninear and discrete constraints. Instead of soving the optimization probem, we provide a chun aocation scheme improving the margina utiity using the foowing procedures. A. Locaized We define the margina utiity as the difference between the utiity obtained when chun n is aocated to user and the utiity of user in the absence of a chun aocation at the current TTI. In order to shorten the description, we omit the superscript of time index hereafter. ( {}) { } ( ) Λ = U R I = n U R I = φ (9) n, ch, ch,

3 Using (9), a necessary rue for optimaity is derived as foows. Property 1. If A is a set of users with chun aocations that maximize the sum in (5), the seected user for each chun ies in the set of N-th best users with respect to the margina utiity derived in (9) with the chun. A = [ a1, a,, an ] (10) an Sch, n = bn,1, bn,,, bn, N for n () b = arg max i Λ (11) ni, n, [ 1,,, K] In (10) and (11), a n is the user aocated chun n. S ch,n is a set of best users (N users) where each eement b n,i, denotes the i-th highest user with respect to the margina utiity with chun n. This property can be proven by contradiction of the converse. Greedy chun aocation based on margina utiity From property 1, we pic N best users per chun with respect to the margina utiity in (9), and add them into the set of avaiabe users. Initiaization: Add a chuns to the set of avaiabe chuns I avai_chun and N best users per each chun are regarded as the candidates to assign the chun. Iavai _ chun = { 1,,, N}, Sch, n = bn,1, bn,,, b n, N, n (1) Step 1 (Chun seection): Find a chun which has the highest margina utiity defined in (9) among a avaiabe chuns and users. For each avaiabe user j and chun n, find the pair, where n, j = arg max Λ (13) n, j n Iavai _ chun, j Sch, n Step (Chun aocation): Find a user () who can maximize the margina utiity when chun n is additionay aocated to a user and the utiity without chun n. Then, aocate chun n to user as foows. I { ch, = Ich, n } for S (14) ch, n ( max ch, ) ( max ch, ) { } = (15) arg max U R P, I U R P, I Iavai _ user I = I n (16) ch, ch, Step 3: Deete the chun from the set of avaiabe chuns i.e. I avai_chun =I avai_chun -{n }. Repeat steps 1, and 3, unti a chuns are aocated. The steps above assign each chun to the user obtaining highest margina utiity for a given chun. However, it is neither necessary nor sufficient for optimaity. B. Intereaved Since each chun in I- consists of distributed subcarriers, the channe quaity is simiar for a chuns. Therefore, chun aocations simiar to the approach taen with L- do not provide significant improvements. Instead, the scheduing for I- aims to aocate a chun to the user that can obtain highest margina utiity. At first, we define a representative channe gain to noise ratio for user over the entire set of sub-carriers: 1 L H, Ω (17) L = 1 σ Then, we estimate SNR of the data deivered with N ch, chuns as P γ ( P, Nch, ) = Ω, (18) MN ch, which is derived from () by repacing each H, /σ with Ω. Using (18), the instantaneous data rate of user is estimated as (19) and the estimate of average data rate of user at T- th TTI is updated as (0). Superscript (t) is added into the equations to represent the time index. () t BN () t () t ch, () t C ( P, Nch, ) = og 1 + γ ( P, Nch, ) N (bps) (19) T 1 ( T) ( T) 1 ( t) ( T) ( T) R P, N ch, = { C } + C ( P, Nch, ) T (bps) (0) t= 1 (t) C is the actua and instantaneous data rate transmitted over t-th TTI. We omit the superscript of time index hereafter. Simiar to the case of L-, we define an estimate of margina utiity as Λ (, 1, ) (, 0 = U R P Nch = U R P Nch, = ) (1) Using (1), we identify the N best users as the candidates for chun aocation. () I best = { a1, a,, a N }, a arg max i i = Λ () where a i is the user index of i-th best user with respect toλ. Next, the objective is to find the number of assigned chuns for each user in set I best and Fig. 3 shows the procedures of chun assignment using greedy aocations. In Fig. 3, we have two operations: Greedy aocation and fooring. Greedy aoc.[n, I avai_user, N ch ] aocates N additiona chuns to I avai_user when a number of chuns are aready aocated to each user. N ch is a set of number of chuns aocated to each user in the set of I avai_user. We repeat the foowing procedures unti a N chuns are newy aocated. = arg max U( R P, Nch, + 1 ) U( R P, Nch, ) (3) I avai _ user N = N + 1 (4) ch, ch, Fooring: There is a restriction to chun aocation where the number of sub-carriers assigned to each user shoud be a power of in order to maintain equidistant sub-carrier mapping and ower PAPR. It foows that the number of assigned chuns per user is aso a power of if the number of sub-carriers per chun is a power of. Thus, the number of assigned chuns derived in greedy aocation has to be foored to the nearest integer x, x {1,,,og N} but the integer shoud be

4 ess than the number of assigned chuns required to be foored as (5). F Nch, = Nch,, N, x {1,,,og x, N} (5) x ch To maintain equidistant sub-carrier mapping, chuns in I- have a tree structure as iustrated in Fig.. As shown in Fig, we can group the chuns so that the subcarriers in the group are paced with equidistance as in Fig. 1 (b). If one set in a eve is aocated to a user, the descendent sets can t be assigned to other users. Then, the foowing procedures referred from [8] are used for equidistant chun/sub-carrier mapping. Chun/Equidistant sub-carrier mapping 1) Find tree obeying the equidistance rue ) Choose a user with the highest number of assigned chuns and seect a set of chuns in the eve which equas the number of assigned chuns. Eiminate a descendent sets in the tree. 3) Choose an avaiabe set for the user with the second highest number of assigned chuns and eiminate a descendent sets in the tree. 4) Repeat for a assigned users Once the index of chuns assigned to each user is determined from the procedures above, we use (), (3), and (4) to cacuate the instantaneous data rate transmitted at the TTI. Thus, user data rate is updated as (6) with the instantaneous data rate. IV. PERFORMANCE EVALUATIONS We have simuated the utiity-based scheduing schemes derived in the previous section. We compare them with the performance of a round-robin scheduing scheme. We have simuated frequency seective fading of K users at each TTI using (6) and (7), and coected the sum of average user data rate after a system time (00 TTIs), where the time duration of 1 TTI is 0.5 msec. The simuation is repeated over many times to obtain a statistica average. Path-oss and shadowing are generated randomy but they are assumed to be constant during the system time. Thus, each user is assumed to be stationary or sowy moving. We aso assume that the muti-path fading component is time invariant over a TTI but changes independenty at each TTI. Path-oss is modeed by (6) where the distance d (m) between user and the base station is randomy generated with the density function f d (d)=d/d. D denotes the ce radius which is set to 1 m. Loss,, db = og10d + ξ (6) where ξ is a shadowing parameter modeed by a normay distributed random variabe with standard deviation 8 db. We consider the muti-path fading channe in which a discrete time channe impuse response mode is used as (7) where T s (=1/B) denotes the samping time of mutipath components. Zeros are padded into the time domain so that the tota ength of v(n) = L(tota number of subcarriers). A Pre ( τ ) w( τ ), if nts = τ v( n) = (7) 0, Otherwise where w(τ ) is a zero mean compex Gaussian noise process and τ is the propagation deay of path. A is a normaized parameter such that the average power E[ v(n) ] =1 watt. P re (τ ) is a reative power of path. We consider a typica urban area propagation mode with 6 tap setting specified in [13], as isted in tabe 1. TABLE I RELATIVE POWERS OF DELAY PROFILE [13] Deay (µsec) Re.Pwr.(dB) The channe gain of sub-carrier i for user can be expressed as (8) where L oss, denotes a inear scae of path-oss and shadowing of user assumed to be constant over a subcarriers and the system time. V () i L Hi, =, π i V() i = ()exp L v (8,9) oss, = 1 Ts The simuation resuts in Figures 4-8 use the foowing abbreviations: R-L-: L- with N users seected by roundrobin. S-L-: The proposed CDS method of L-. R-I-: I- with N users seected by roundrobin. S-I-: The proposed CDS method of I-. Fig. 4 and 5 show the rate-sum capacity where the sum of user data rates (Fig. 4) and the sum of the ogarithmic user data rates (Fig. 5) are the utiity functions. Fig. 6 and 7 show the average user data rate as a function of user distance from the base station. In Fig. 4 we see that the scheduing gain of the case of rate utiity increases with increasing number of users. This is because the scheduer seects the coser users which can transmit with higher data rate in addition to the gain of seecting users in exceent channe condition. If there are more users, the possibiity of ocating some users at coser distance to the base station increases. As a resut of this, the schedued transmissions achieve significant improvements for both I- and L-. For the case of ogarithmic rate utiity in Fig. 5, the scheduing gain stops increasing beyond approximatey 3 users. With 3 users, maximizing ogarithmic rate utiity can increase system capacity by a factor of 1.8 for L- and 1.6 for I-. Comparing Fig. 6 and 7, we see that the scheduing scheme based on the ogarithmic user data rate provides proportiona fairness whose gains are shared among a users, whie the gains of CDS are concentrated to the users near the base station when the user data rate is considered as the utiity function. Fig. 8 shows the outage probabiity which is defined as P r (user data rate<minimum required

5 data rate). Considering user capacity at 1% outage probabiity and minimum required rate of 144 Kbps, we can say that round-robin scheduing supports ess than 0 users but our proposed schemes can support 4 users for I- and 48 users for L-. Tabe II compares round-robin scheduing and utiity-based scheduing with ogarithmic user data rate, with respect to system capacity, and fairness. TABLE II COMPARISONS OF UTLITY-BASED SCHEDULING AND ROUND- ROBIN SCHEDULING (LOGARITHMIC RATE UTILIITY) Type Rate-sum capacity (3 users) Fairness (3 users) User capacity S-L- R-L- S-I- 18 Mbps 10 Mbps 1 Mbps R-I Mbps Less Less 48 users 4 users than 0 than 0 Fairness = average user data rate of users at the ce boundary (900m- 1m) / average user data rate. User capacity: Number of users aowed 1% outage probabiity when the minimum rate equas to 144 Kbps [10] J. Jang, and K. B. Lee, Transmit power adaptation for mutiuser OFDM systems, IEEE J. Se. Areas Commun., vo. 1, pp , Feb. 003 [11] T. Shi, et a., Capacity of singe carrier systems with frequencydomain equaization, in Proc. IEEE CASSET 04, vo., pp.49-43, June 004 [1] 3GPP R , Simuation methodoogy for EUTRA UL: IF- DMA and DFT-spread-O, Sept. 005 [13] 3rd Generation Partnership Project (3GPP); Technica specification group radio access networ; Radio transmission and reception (Reease 7) [14] A. Godsmith and S. Chua, Variabe-rate varabe-power MQAM for fading channes, IEEE Trans. Commun., vo.45, pp , Oct Chun #1 # #3 #4 Sub-carriers Sub-carriers Fig. 1 An exampe of chun structures (L:16,M:4) (a) L- (b) I- V. CONCLUSIONS In this paper, we deveoped utiity-based channe dependent scheduing (CDS) schemes and showed that the proposed methods with ogarithmic user data rate as a utiity function provided proportiona fairness with the gains of CDS shared among a users. L- with CDS is more desirabe than I- because L- expoits frequency seective scheduing. Our schemes can be utiized to design efficient radio access networs for upin SC- systems. A simiar method can be appied to chun-based upin O systems as we. REFERENCES [1] J. Lim et a., Channe dependent scheduing of upin singe carrier systems, Submitted to IEEE VTC 06 fa [] H. Myung et a., Pea-to-average power ratio of singe carrier signas with puse shaping, Submitted to IEEE PIMRC 06 [3] 3rd Generation Partnership Project (3GPP); Technica specification group radio access networ; Physica ayer aspects for evoved UTRA (Reease 7) [4] D. J. Goodman, and N. Mandayam, Power contro for wireess data, IEEE Persona Commun., vo. 7, pp , Apri 000 [5] G. Song, and Y. Li, Cross-ayer optimization for O wireess networs-part I: Theoretica framewor, IEEE wireess commun., vo. 4, pp , Mar. 005 [6] G. Song, and Y. Li, Cross-ayer optimization for O wireess networs-part II: Agorithm deveopment, IEEE wireess commun., vo. 4, pp , Mar. 005 [7] H. Kim et a., A proportiona fair scheduing for muticarrier transmission systems, IEEE Commun. Letters, Vo. 9, pp. 10-1, Mar. 005 [8] R. Dinis, et a., A mutipe access scheme for the upin of broadband wireess systems, in Proc. IEEE GLOBECOM 04, vo.6, pp , Dec. 004 [9] K. Kim et a., Joint subcarrier and power aocation in upin OF- DMA systems, IEEE Commun. Letters, vo. 9, pp , June 005 Group : Leve 4 {1,,3,4} Group : Leve {1,3} {,4} Chun {1} : Leve 1 {} Fig. An exampe of chun tree structure (16 sub-carriers, 4 chuns) I {3} {4} avai _ user = Ibest, Nch, = 0, for Iavai _ user Greedy aoc.,, N I avai _ user N ch For each (from 1st to N th best users) Fooring: Nch, = Nch, x Deete: Iavai _ user = Iavai _ user { } Greedy aoc. N N, I, N ch, ch, x avai _ user ch Obtained Nch = Nch,1, Nch,,, Nch, N Fig. 3 Assignment of number of chun for I-

Fig. 4 Rate-sum capacity of shared users (utiity: user data

7 Average user data rate between user distances (utiity:

(utiity: ogarithmic user data 8 Outage probabiity (utiity:

6 Fig. 4 Rate-sum capacity of shared users (utiity: user data rate, L=56, N=8, B=5MHz, Noise power per Hz=-160dBm) Fig. 7 Average user data rate between user distances (utiity: ogarithmic user data rate, L=56, N=8, B=5MHz, Noise power per Hz=-160dBm) Fig. 5 Rate-sum capacity of shared users (utiity: ogarithmic user data rate, L=56, N=8, B=5MHz, Noise power per Hz=-160dBm) Fig. 8 Outage probabiity (utiity: ogarithmic user data rate, minimum required data rate=144kbps, L=56, N=8, B=5MHz, Noise power per Hz=-160dBm) Fig. 6 Average user data rate between user distances (utiity: user data rate, L=56, N=8, B=5MHz, Noise power per Hz=-160dBm)

Rateless Codes for the Gaussian Multiple Access Channel

Rateless Codes for the Gaussian Multiple Access Channel Rateess Codes for the Gaussian Mutipe Access Channe Urs Niesen Emai: uniesen@mitedu Uri Erez Dept EE, Te Aviv University Te Aviv, Israe Emai: uri@engtauaci Devavrat Shah Emai: devavrat@mitedu Gregory W