The Capacity Gain from Intercell Scheduling in Multi ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 2, FEBRUARY 2008

The Capacity Gain from Intercell Scheduling in Multi-Antenna Systems Wan Choi, Member, IEEE, and Jeffrey G. Andrews, Senior Member, IEEE

Abstract— The capacity and robustness of cellular MIMO systems is very sensitive to other-cell interference, which will in practice necessitate network level interference reduction strategies. As an alternative to traditional static frequency reuse patterns, this paper investigates intercell scheduling among neighboring base stations. We show analytically that cooperatively scheduled transmission, which is well within the capability of present systems, can achieve an expanded multiuser diversity gain in terms of ergodic capacity as well as almost the same amount of interference reduction as conventional frequency reuse.√This capacity gain over conventional frequency reuse√is O(Mt log Ns ) for dirty paper coding and O(min(Mr , Mt ) log Ns ) for time division, where Ns is the number of cooperating base stations employing opportunistic scheduling in an Mt × Mr MIMO system. From a theoretical standpoint, an interesting aspect of this analysis comes from an altered view of multiuser diversity in the context of a multi-cell system. Previously, multiuser diversity capacity gain has been known to grow as O(log log K), from selecting the maximum of K exponentially-distributed powers. Because multicell considerations such as the positions of the users, lognormal shadowing, and pathloss affect √ the multiuser diversity gain, we find instead that the gain is O( 2 log K), from selecting the maximum of a compound lognormal-exponential distribution. Finding the maximum of such a distribution is an additional contribution of the paper. Index Terms— Dirty paper coding, MIMO systems, multiuser diversity, opportunistic scheduling, other-cell interference.

I. I NTRODUCTION ESPITE the enormous promise of multi-antenna technologies (known as MIMO) [1]–[4], so far only a fraction of their potential has been realized in cellular systems. One major reason is that MIMO capacity and bit-error probability are severely degraded in low SINR channels, which are prevalent in cellular systems due to other-cell interference (OCI) [5]–[10]. The mobile stations are generally unable to simultaneously suppress the spatial self-interference (from the home base station) and the multi-dimensional other-cell interference. This necessitates a network-level interference

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Manuscript received August 21, 2006; revised February 8, 2007; accepted June 7, 2007. The associate editor coordinating the review of this paper and approving it for publication was S. Kishore. This work was presented in part at the Conference on Information Science and Systems, NJ, March 2006 and the IEEE International Symposium on Information Theory, Seattle, WA, July 2006. W. Choi was with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Texas, USA. He is now with the School of Engineering, Information and Communications University, Daejeon 305-732, Korea (email: [email protected]). J. G. Andrews is with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Texas 78712 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2008.060615.

management approach, if MIMO is going to be usefully deployed in future broadband cellular systems. Traditionally, frequency reuse has been adopted to achieve an acceptable signal-to-interference-plus-noise ratio (SINR) (especially near cell boundaries) in cellular systems at the cost of system-wide spectral efficiency. Furthermore, frequency reuse introduces a particularly large burden in emerging wideband systems that have a smaller number of available frequency channels. These limitations have motivated research on alternative OCI mitigation schemes, some of which are summarized in the context of MIMO in [11]. A. Intercell Processing or Base Station Coordination Among the alternative OCI mitigation techniques, this paper focuses on intercell processing or base station (BS) coordination1 . Intercell processing requires real-time information exchange among the base stations, but has the potential to significantly reduce their mutual interference [12]–[19]. Dirty paper coding (DPC)2 across neighboring base stations (multicell DPC) is the theoretically optimal solution in a multicellular environment [14]–[19], but requires non-casual knowledge of all the interference signals from the other base stations and a huge amount of information exchange. The high computational burden of successive encoding and interference cancellation is also prohibitive. Multicell DPC provides a useful performance bound, but is not a reasonable design approach. A number of suboptimal BS coordination techniques have recently been proposed including joint beamforming, joint pre-coding, joint spatial multiplexing, and joint space-time coding [13], [22]–[25]. If the coordinated base stations are treated as a virtual MIMO system, conventional multiuser MIMO techniques are applicable. The fundamental difference between base station coordination techniques and conventional multiuser MIMO schemes is the individual power constraints of base stations; conventional co-located MIMO has a single diagonal sum power constraint. Although the aforementioned suboptimal techniques do not require non-casual information, they still generally suffer from the following requirements: (1) Perfect channel state information at the transmitter (CSIT) for all neighboring base-stations, (2) significant overhead for real time intercell coordination, and (3) signal alignment for synchronized reception at the receiver from each base station. These requirements appear daunting. 1 This class of techniques is also referred to as network MIMO, cooperative encoding, or joint multiple cell site processing. 2 DPC is known as the capacity achieving strategy in downlink MIMO multiuser channels, i.e., MIMO broadcast channel (BC) [20], [21]

c 2008 IEEE 1536-1276/08$25.00

CHOI AND ANDREWS: THE CAPACITY GAIN FROM INTERCELL SCHEDULING IN MULTI-ANTENNA SYSTEMS

B. Intercell Scheduling The burden of the global CSIT requirement for intercell processing can be reduced by several approaches such as channel reciprocity [26] or limited feedback based on high resolution codebooks [27]–[29]. However, the real-time exchange of the gathered channel state information for intercell processing is still difficult in practical implementation. If the information is exchanged with a delay approaching the channel coherence time, the value of base station coordination rapidly diminishes. In this context, BS coordination strategies minimizing the amount of intercell information exchange are preferable, for example passing the information to a neighboring BS [23] or exchanging statistical channel information only [25]. We focus our attention on the gains attainable from practical intercell scheduling. In intercell scheduling, neighboring base stations cooperatively schedule their transmissions to reduce other-cell interference. The scheduling can either be dynamic (and hence require some intercell coordination) or pre-determined based on a universally shared time-hopping sequence. Intercell scheduling is considered the most practical BS coordination technique since the required message passing is comparable to what is required for handoff, which is already a feature of every cellular system. It should be conceded up front that intercell scheduling does not improve the overall network capacity relative to universal frequency reuse. The previously discussed intercell processing techniques attempt to maintain universal frequency reuse and suppress the interference through cooperative encoding; clearly from a capacity standpoint this will be superior to intercell scheduling which suffers from a duty cycle in time just as frequency reuse suffers from a duty cycle in frequency. These schemes remain wholly unproven in a practical setting, though. Intercell scheduling is more aptly compared to the more realistic alternative of frequency reuse, and there are three important advantages of cooperatively scheduled transmission relative to traditional frequency reuse. First, universal frequency reuse can be adopted. Second, frequency planning is eliminated and the total number of required frequency channels is reduced. The third advantage, which is unique to this technique and is the focus of this paper, is that intercell scheduling can achieve an extra multiuser diversity gain if straightforward opportunistic scheduling is employed among neighboring base stations. This gain is quantified in this paper. C. Multiuser Diversity The idea of multiuser diversity was first proposed in [30] and commercial adoption can be found in several systems such as 1xEV-DO and HSDPA [31], [32]. Multiuser diversity exploits the inherent channel variations across users, and the capacity can be improved by serving only the user(s) with the highest instantaneous signal-to-interference-plus-noise ratio (SINR). Previous studies [33]–[36] show that in Rayleigh fading, the multiuser diversity gain grows like log log K where K is the number of users. Although this scaling order is widely cited, it neglects a number of crucial factors. First, diversity techniques like space-time and error correction codes harden the SINR fluctuations due to Rayleigh fading, lowering multiuser diversity gains considerably [37], [38]. Secondly, in

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a multicell environment SINR fluctuations across users depend strongly on the location of users and lognormal shadowing, which are often more significant than the Rayleigh fading fluctuations, and are not affected by other diversity techniques. An additional goal of this paper is to more accurately quantify the multiuser diversity gain in a cellular environment, in particular by introducing a lognormal shadowing component and finding a more realistic scaling law. This additional gain further differentiates intercell scheduling from frequency reuse. D. Contributions The main contributions of this paper are summarized into two aspects. First, we find the capacity of intercell scheduling in the context of a realistic cellular MIMO system reflecting pathloss, shadowing, location of users, and co-channel interference. Through asymptotic analysis and simulations, this paper shows that the multiuser diversity increased by intercell scheduling, √ named expanded √ multiuser diversity gain, α + log Ns − α where α = log KMr scales like β and β = MNtsσ when DPC is employed within each cell3 , and α = log K and β = min(MNts,Mr )σ for TDMA, when cooperatively scheduled transmission among Ns base stations is considered in Mt × Mr MIMO channels. The expanded multiuser diversity gain helps to compensate for the incurred throughput loss due to the alternating transmission by the Ns cooperating base stations. Second, we find how cellular considerations (pathloss, shadowing, location of users, and OCI) affect multiuser diversity gain. The derived ergodic capacity bounds on a general frequency reuse √ system show that the multiuser diversity gain grows like log 2K in a cellular system, whereas it has previously been known to grow only as log log K when idealized short-term fading is considered. This provides new ground for optimism on the utility of multiuser diversity in cellular systems. E. Organization The rest of this paper is organized as follows: Section II derives the capacity bounds and expanded multiuser diversity gain of a TDMA MIMO system with intercell scheduling for a large number of users. When DPC is employed within each cell and the intercell strategy is opportunistic scheduling, the capacity bounds and expanded multiuser diversity gain are analyzed in Section III. In Section IV, the analysis is verified by simulations and some numerical examples are presented. Conclusions are drawn in Section V. II. E XPANDED M ULTIUSER D IVERSITY G AIN OF T IME D IVISION MIMO S YSTEM As previously mentioned but not yet demonstrated, intercell scheduling can achieve expanded multiuser diversity if base stations dynamically and cooperatively schedule their transmissions. In this section, we will provide an analytical 3 DPC is used within each cell while the intercell strategy is opportunistic scheduling. This strategy is distinct from DPC across neighboring base station’s (multicell DPC)

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achievable throughput of an optimal TDMA system with multiple antennas is obtained by [39], [40] Ci = max log |Zk,i max 1≤k≤K Qk,i :Qk,i 0,Tr(Qk,i )≤1 H +Lk,i P Hk,i Qk,i Hk,i − log |Zk,i | (1)

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framework for scrutinizing this hypothesis. We consider opportunistic scheduling where a transmission in each time slot is allocated only to the base station that is able to provide the highest throughput among the base stations consisting of one cluster (7 cells) as shown in Fig. 1. Ideally, intercluster coordination could also be taken into account to minimize OCI. For example, an algorithm that maximizes when possible the distance between the cells selected in neighboring clusters could be adopted. However, intercluster coordination may conflict with intercell coordination and exactly considering both together makes analysis intractable. Therefore, in this paper, we exclude intercluster coordination and assume that scheduling among neighboring base stations is done only within a cluster. Throughout this paper, intercell scheduling refers to intercell coordination with opportunistic scheduling among base stations, excluding intercluster coordination. A. Time Division MIMO System It is known that if both the transmitter and the receivers can perfectly track the fading processes, the sum capacity, defined as the maximum achievable sum of long-term average data rates transmitted to all the users, can be achieved by a simple time division multiple access (TDMA) strategy in a single antenna system: at each fading state, transmit to the user with the strongest channel [30], [33]. Even though this is not the case for a multiple-antenna system [20], [21], TDMA with optimal user selection is still considered a practical option especially for narrow band systems. In this context, we investigate the potential gains of intercell scheduling in MIMO TDMA systems with optimal user selection. In TDMA systems with optimal user selection4 , the user with the highest instantaneous mutual information is selected among the K users in a cell. At a given time slot, the 4 In this paper, a TDMA system with optimal user selection is referred to as optimal TDMA.

where | · | denotes the matrix determinant and Lk,i denotes propagation pathloss including lognormal shadowing from the base station i to user k. The log term in this paper denote the natural loge . The MIMO channels of all the users in the cell i, {Hk,i }, are assumed to be static during a transmission slot and have an independently and identically distributed complex Gaussian distribution ∼ CN (0, 1). The matrix Qk,i is the normalized covariance matrix of the transmitted signal with a transmit power constraint of Tr(Qk,i ) ≤ 1. We assume that all users employ Gaussian codebooks since this is the optimum channel input in terms of MIMO capacity [2][41, Theorem 9.6.5]. The matrix Zk,i denoting the covariance matrix of interference plus noise perceived by user k in the cell i is given by Lk,j P 2 Hk,j HH (2) Zk,i = k,j + σn IMr Mt j∈IB

where the set IB contains the indices of interfering base stations in other clusters and σn2 is the variance of additive white Gaussian noise (AWGN). Note that we assume even power allocation across antennas (P/Mt ) in interfering base stations for analytical simplicity. Although OCI is likely to be colored, we assume spatially white Gaussian interference for analytical tractability. Spatially white OCI considered the worst case so yields a lower bound on actual capacity [42]. Under the spatially white interference assumption, Zk,i is approximated by ⎡ ⎤ Lk,j P 2 ⎦ Hk,j HH (3) Zk,i = EH ⎣ k,j + σn IMr Mt j∈IB 2 = Lk,j P IMr + σn2 IMr = σk,i I Mr (4) j∈IB

2 where = j∈IB Lk,j P + σn , reflecting the random characteristics of interference plus noise power perceived at the user k at cell i because {Lk,j } depend on the location of user k and lognormal shadowing. The OCI model can be further justified from the fact that OCI is mainly determined by large scale fading rather than small scale fading (or shortterm fading). If intercell scheduling is applied to TDMA systems with optimal user selection, the user with the highest instantaneous mutual information is selected for transmission among all the users in the cells involved. Therefore, the per-cell ergodic capacity of this system is given by

1 E max max max E [Ccoop ] = 1≤i≤Ns 1≤k≤K Ns Qk,i :Qk,i 0,Tr(Qk,i )≤1 Lk,i P H (5) log IMr + 2 Hk,i Qk,i Hk,i ) σk,i 2 σk,i


where dk,i is the distance from a base station to mobile station k in cell i and l is the pathloss exponent. d0 is the received power reference point and the term χk represents lognormal shadowing. The standard deviation of lognormal shadowing is assumed to be 10dB and l = 3.5, which are typcial values in an urban cellular environment. Fig. 2(a) coop by Monte Carlo simulation. shows the distribution of log γk,i coop The distribution of γk,i by simulation is well modeled by lognormal distribution because the sum of lognormal random variables is well approximated by a lognormal random variable [43] and the ratio of lognormal random variables is also a lognormal random variable [44]. Then, the per-cell ergodic capacity given in (5) can be represented by 1 E max E [Ccoop ]= 1≤k≤Ns K Ns log IMr +γkcoop Hk Qk HH max (7) k Qk :Qk 0,Tr(Qk )≤1

where is γkcoop an independent and identically distributed (i.i.d.) lognormal random variable with mean μcoop and vari2 . ance σcoop Similarly, the per-cell ergodic capacity of TDMA systems with conventional frequency reuse with reuse factor 1/Ns is obtained by 1 E [Creuse ]= E max 1≤k≤K Ns reuse H log IMr +γk Hk Qk Hk (8) max Qk :Qk 0,Tr(Qk )≤1

γkreuse

is an i.i.d. lognormal random variable with where 2 . The distributions by Monte mean μreuse and variance σreuse Carlo simulation and approximation for γkreuse are shown in Fig. 2(b) when the frequency reuse factor is 1/7 (Ns = 7). It should be noted that γkcoop and γkreuse have similar statistics as in Fig. 2. Correspondingly, intercell scheduling and frequency reuse have almost the same amount of interference 5 Asymptotic

in the sense of a large number of users, i.e. K → ∞.

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where Ns is the total number of cooperating cells and the factor 1/Ns reflects the loss in spectral efficiency by the transmission duty cycle. In order to investigate the exact capacity of intercell scheduling, we could rely on computer simulations but asymptotic5 analysis can provide insights into the nature of the expanded multiuser diversity gain in intercell scheduling. To do this end, we first investigate the distributions of the L P coop = σk,i by Monte Carlo large scale SINR in a cluster γk,i 2 k,i simulations reflecting uniformly distributed Ns K users in a cluster. At each Monte Carlo trial, a cell is randomly selected in each interfering cluster and users are randomly distributed. Lognormal shadowing also changes at each Monte Carlo trial. The total number of cells involved in intercell scheduling NS is 7. In the considered system configuration, 48 cells in 7 clusters are considered as OCI sources. The propagation pathloss and lognormal shadowing are given by −l dk,i χk,i d ≥ d0 (6) Lk,i = d0

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reduction in terms of average interference. On the other hand, if we compare (7) and (8), we can find that the cardinality of the selection pool of the intercell scheduling scheme is Ns times larger than that of frequency reuse (Ns K vs. K). As the cardinality of the selection pool increases, the multiuser diversity gain correspondingly increases. This effect can be interpreted as expanded multiuser diversity.

B. Extreme Value Theory For the asymptotic analysis, now we provide some known theorems and lemmas on the asymptotic behavior of the maximum of n i.i.d. random variables when n is sufficiently large, i.e., Extreme Value Theory. The theorems and lemmas provided in this section will be used for asymptotic analysis of the expanded multiuser diversity gain. Theorem 1 (Unified Extremal Types Theorem [45]–[48]): Let x1 , x2 , · · · , xn be a sequence of i.i.d. random variables with cumulative distribution function F (x) and Mn = max{x1 , x2 , · · · , xn }. If there exist a sequence of

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constants an > 0 and bn such that as n → ∞, M n − bn ≤ x → G(x) Pr an

(9)

for some non-degenerate distribution6 G, then G is of the generalized extreme value distribution type G(x) = exp[−(1 +

−1/ξ ξx)+ ]

(10)

for some ξ, and we can say that F is in the domain of attraction of G, where the notation x+ denotes max(0, x).

Because the error function erf(x) can be represented for x 1 by 2 1 3 e−x erf(x) = 1 − √ x−1 − x−3 + x−5 + · · · , (16) 2 4 π the hazard function can be given by ⎛ ⎞ 3 √ √ √ 2σx x ⎝ 2σx 2σx 1 h(x) = − + ···⎠. 2 log x − μx 2 log x − μx

Proof: Refer to [45]–[48]. A pseudo proof can be found in [45, pp. 37-38] and a rigorous proof can be found in [48].

(17) Therefore, h (x) → 0 as x → ∞, and by Theorem 2, the lognormal distribution belongs to the MDA of a Gumbel distribution.

Theorem 2 (Gumbel distribution [45]–[47]): Define reciprocal hazard function h(x) as

the

Lemma 2: For i.i.d. lognormal random variables x1 , x2 , · · · , xn with logarithmic mean μx and variance σx2 ,

(11)

Pr[bn − an log log n ≤ max xi ≤ bn + an log log n] 1 ≥1−O (18) log n and an = where bn = exp (2 log n)1/2 σx + μx bn σx /(2 log n)1/2

h(x) =

1 − F (x) f (x)

where f (x) is the probability density function (p.d.f.) corresponding to F (x). Then, the shape parameter (or tail index) that determines the type of the limit distribution is obtained by dh(x) h (x) = → ξ as x → ∞. (12) dx Particulary, the limit distribution with ξ = 0 is referred to as a type I extreme value distribution or Gumbel distribution, i.e. G0 (x) = exp[− exp(−x)]. That is, if h (x) → 0, then F (x) belongs to the domain of attraction G0 (x), and we can pick an and bn by: 1 , an = h(bn ). (13) n Proof: Refer to [45, pp. 46]. Theorem 1 presents the unified limiting distribution for the cumulative distribution of the maximum of n i.i.d. random variables and Theorem 2 explains how the shaper parameter ξ can be determined and is related to the class of the limiting distributions. From Theorem 1 and Theorem 2, the following lemmas can be easily derived, which explain the asymptotic behavior of the maximum of i.i.d. lognormal random variables. Because we have shown that γkcoop and γkreuse can be well modeled by lognormal random variables, the following lemmas will be directly used in the asymptotic analysis of the expanded multiuser diversity gain. 1 − FX (bn ) =

Lemma 1: A lognormal distribution with logarithmic mean μx and variance σx2 belongs to the maximum domain of attraction (MDA) of a Gumbel distribution. Proof: The p.d.f. and c.d.f. of a lognormal distribution are given, respectively, by 2 2 1 e−(log x−μx ) /2σx σx (2π)x log x − μx 1 √ . FX (x)= 1 + erf 2 2σx

fX (x)=

(14) (15)

6 A degenerated distribution is the probability distribution of a random variable that always has a value of unity, while a non-degenerated distribution is the complement of the degenerated distribution

Proof: By Lemma 1, the lognormal distribution belongs to the MDA of a Gumbel distribution. Then, there exist constant an and bn from Theorem 2 such that as n → ∞, M n − bn ≤ x → G0 (x) = exp[− exp(−x)] (19) Pr an where Mn = max{x1 , x2 , · · · , xn }. Letting x = log log n in (19) and then x = − log log n in (19), this lemma can be easily shown. We can also obtain bn and an by 1 , an = h(bn ). (20) n The detailed derivation of bn and an are given in Appendix A. 1 − FX (bn ) =

C. Scale of Expanded Multiuser Diversity Gain Now we are ready to analyze the per-cell ergodic capacity and expanded multiuser diversity gain for a large number of users. We will state a theorem that gives bounds on the achievable per-cell ergodic capacity for large K, for both intercell scheduling and traditional frequency reuse. After proving the theorem, we summarize the insights in 2 key results. Theorem 3: When Mt , Mr , and P are fixed, the capacity of optimal TDMA with intercell scheduling for large K is asymptotically given by lim

K→∞ 1 Ns

E [Ccoop ] √ = 1. min(Mr , Mt )σcoop 2 log Ns K

(21)

Similarly, the capacity of the optimal TDMA with frequency reuse for large K is asymptotically given by lim

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E [Creuse ] √ = 1. min(Mr , Mt )σreuse 2 log K

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Proof: We first assume that Mt ≥ Mr and derive an upper bound on the per-cell ergodic capacity of optimal TDMA with intercell scheduling for large K. The case that Mr ≥ Mt can be derived similarly. The capacity is maximized by choosing the transmit covariance matrix Qk to be along the eigenvectors of the channel matrix Hk HH k and by choosing the eigenvalues according to the water-filling procedure [2]. Therefore, the ergodic capacity can be given by 1 max max E [Ccoop ] = EH Eγ Ns 1≤k≤Ns K Qk,i :Qk,i 0,Tr(Qk,i )≤1 log IMr + γkcoop Hk Qk HH (23) k 1 = EH Eγ Ns

max

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· Pr

≥ A max Tr Hk HH k

1≤k≤Ns K

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(29)

where A = logNs K + Mr Mt log log Ns K. has a chi-squared distribution ∼ Because Tr Hk HH k 2 χ (2Mr Mt ), it is known that [35] Pr log Ns K − Mr Mt log log Ns K ≤ ! max Tr Hk HH ≤ log Ns K + Mr Mt log log Ns K k 1≤k≤Ns K 1 ≥1−O . (30) log Ns K We also know from Lemma 1 that maxk γkcoop − bcoop ≤ x → G(x) = exp(− exp(−x)) Pr acoop

m=1

(24)

where ρk,m is the transmitted power through the mth eigenmode of the channel of the kth user and λk,m is the mth eigenvalue of Hk HH k . The transmitted power ρk,m is given by 1 ρk,m = η − coop (25) γk λk,m + where x+ = max(0, x) and η is the water-filling level. Then, the ergodic capacity is upper bounded by 1 max Mr log E [Ccoop ] ≤ EH Eγ Ns 1≤k≤Ns K Mr Mr coop γk 1+ ρk,m λk,m (26) Mr m=1 m=1 1 max Mr log = EH Eγ Ns 1≤k≤Ns K coop γ 1 + k Tr Hk HH (27) k Mr Eγ [maxk γkcoop ] Mr log 1 + ≤ EH Ns M r · max Tr Hk HH (28) k 1≤k≤Ns K

The inequality (26) is due to the log inequality coming from the concavity of the log function, while the equality (27) follows from the normalized transmit power constraint ) = of Tr(Qk ) ≤ 1 and the property of Tr(Hk HH k m λk,m . Inequality (28) can be obtained from Jensen’s inequality and the fact that the log function is a non-decreasing function. Utilizing the fact that the sum rate capacity is bounded by the capacity of a MIMO single user with Mt transmit and Ns KMr receive antennas, then (28) is again bounded by Eγ [maxk γkcoop ] Mr log 1 + · E [Ccoop ]≤EH Ns Mr H max ≤ A Tr H H max Tr Hk HH k k k 1≤k≤Ns K 1≤k≤Ns K · Pr max Tr Hk HH k ≤A 1≤k≤Ns K Mt coop + log 1 + Eγ max γk M r Ns K k Ns

and the mean value of the Gumbel distribution G(x) becomes the Euler-Mascheroni constant γe = 0.57721566 · · · [49]. Therefore, E[maxk γkcoop ] = bcoop + γe acoop where bcoop = exp (2 log Ns K)1/2 σcoop + μcoop and acoop = bcoop σcoop /(2 log Ns K)1/2 and (29) is bounded by E [Ccoop ] Mr bcoop + γe acoop log Ns K ≤ log 1 + Ns Mr Mt log +Mr Mt log log Ns K + Ns 1 1 + (bcoop + γe acoop )Mr Ns K O (31) log Ns K bcoop + γe acoop Mr log Ns K log 1 + ≤ Ns Mr Mt log 1 + bcoop Mr Ns K +Mr Mt log log Ns K + Ns 1 + log 1 + γe acoop Mr Ns K O (32) log Ns K bcoop + γe acoop Mr log Ns K log 1 + = Ns Mr 1 +Mr Mt log log Ns K + O √ + O(1).(33) log Ns K The non-decreasing property of a log function results in the inequality (32) and finally, (33) is obtained by substituting bcoop = exp (2 log Ns K)1/2 σcoop + μcoop and acoop = bcoop σcoop /(2 log Ns K)1/2 . Now, we derive a lower bound with the assumption of Mt ≥ Mr . Based on (24), the ergodic capacity is bounded by E [Ccoop ]

Mr γkcoop λk,m 1 max log 1 + ≥E (34) Ns 1≤k≤Ns K m=1 Mt

γkcoop λmin Hk HH 1 k max Mr log 1 + ≥E (35) Ns 1≤k≤Ns K Mt where λmin (A) denotes the minimum eigenvalue of A. The inequality (34) follows from the fact that capacity with waterfilling is higher than that with equal power allocation.

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Because γkcoop is an i.i.d. lognormal random variable, the ergodic capacity is bounded from lemma 2 by

through intercell scheduling since DPC is known as the capacity-achieving downlink strategy [20], [21]. Therefore, in this subsection we consider the scenario where each base E [Ccoop ] station adopts DPC to simultaneously serve all the users in coop its cell, which cooperatively scheduling transmissions among ≥ EH Eγ Ccoop max γk > bcoop − acoop log log Ns K k adjacent cells. That is, DPC is used within each cell while intercell strategy is opportunistic scheduling. Compared to coop > bcoop −acoop log log Ns K (36) · Pr max γk multicell DPC or DPC across neighboring base stations, this k scenario is able to avoid a huge amount of information Mr (bcoop − acoop log logNs K) ≥ EH log 1 + exchange among base stations for multicell DPC encoding. Ns Mt It is more practical than multicell DPC in this context. 1 Using the duality between the broadcast channel and MAC · max λmin Hk HH (37) 1−O k k log Ns K [51], the corresponding achievable per-cell capacity with DPC and intercell scheduling can be given by γkcoop λmin (Hk HH k ) 1 where Ccoop = Ns max1≤k≤Ns K Mr log 1+ .

Mt 1 H E max max E [Ccoop ] = It can be shown that Mr λmin Hk Hk is exponentially K 1≤i≤Ns Ns k=1 Tr(Qk,i )≤1 satisfies distributed [50] and maxk Mr λmin Hk HH k K L P k,i H log IMt (t) + (Hk,i ) Qk,i Hk,i Pr log Ns K ≤ max Mr λmin Hk HH 2 k σk,i k k=1 ! 1 1 ≤ log Ns K + log log Ns K ≥ 1 − (38) E max Ccoop,i (41) log Ns K 1≤i≤Ns Ns Therefore, the lower bound on the ergodic capacity is finally where P is the total transmit power of each cell, Qk,i is the obtained by diagonal matrix representing the optimal power allocation for the kth user in the ith cell with Qk,i 0 ∀k, i. The large scale Mr bcoop −acoop log log Ns K coop 2 log 1+ log Ns K E [Ccoop ] ≥ SINR in a cluster γk,i = Lk,i P/σk,i can be approximated Ns M M r t as a lognormal random variable. Upper and lower bounds on 1 1 · 1 −O 1− (39) per-cell ergodic capacity Ccoop can be obtained for large K log Ns K log Ns K by the following theorem. Upper and lower bounds on the capacity of optimal TDMA Theorem 4: When Mt , Mr , and P are fixed, the capacity systems with frequency reuse for large K can be derived of intercell scheduling in a dirty paper coded system for large similarly. K is asymptotically given by Key Result 1: Theorem 3 indicates that the capacE [Ccoop ] ity of TDMA systems with intercell scheduling scales lim = 1. (42) √ √ 1 K→∞ Mt σcoop 2 log Ns Mr K like Ns min(Mr , Mt )σcoop 2 log Ns K whereas the capacNs ity of TDMA systems√ with frequency reuse scales like Similarly, the capacity of a dirty paper coded system with 1 frequency reuse for large K is asymptotically given by Ns min(Mr , Mt )σreuse 2 log K. Therefore, as K goes to infinity, the net capacity gain from expanded multiuser diversity E [Creuse ] in TDMA systems (difference of the capacities) scales like = 1. (43) lim √ K→∞ Mt σreuse 2 log Ns Mr K Ns GT DM A lim = 1. √ √ Proof: The proof can be done by an extension of the K→∞ min(Mr ,Mt ) σ coop 2 log Ns K −σreuse 2 log K Ns proof of lemma 3 in [35]. We first derive the upper bound on (40) the capacity of intercell scheduling in a dirty paper coded sysKey Result 2: The derived capacity bounds on a TDMA sysM tem for large K. From the inequalities |A| ≤ (Tr(A)/M) tem with frequency reuse in Theorem 3 shows that the multiH where A is an M × M matrix and Tr Hk,i Qk,i Hk,i ≤ user diversity √ gain in time division MIMO system grows like (j) (j) (j) min(Mr , Mt ) 2 log K when the geometry of mobile stations, max1≤j≤Mr hk,i (hk,i )H Tr(Qk,i ) where hk,i is the jth row lognormal shadowing, and OCI are considered, while it has of the matrix Hk,i , Ccoop,i is upper-bounded by [35] been previously known to grow like min(Mr , Mt ) log log K 1 when only the short-term Rayleigh fading is considered [33]– max Mt log 1 + max max Ccoop,i ≤ K Mt 1≤k≤K 1≤j≤Mr [36]. k=1 Tr(Qk,i )≤1 K (j) H coop (j) γk,i hk,i (hk,i ) Tr(Qk,i ) III. E XPANDED M ULTIUSER D IVERSITY G AIN OF D IRTY k=1 PAPER C ODED MIMO S YSTEM 1 coop = Mt log 1 + max γl,i κl (44) Although the implementation of dirty paper coding (DPC) Mt 1≤l≤KMr even within a single cell appears difficult or impossible in practical systems, asymptotic analysis of dirty paper coded where {κl } are i.i.d. random variables with χ2 (2Mt ) distribcoop systems can provide insight into the ultimate capacity possible ution and the large scale SINR in a cluster {γl,i } are i.i.d.


lognormal random variables. By Jensen’s inequality, E [Ccoop ] is upper-bounded by

1 Eκ max Mt log 1 + E [Ccoop ] ≤ 1≤i≤Ns Ns ! coop Eγ max1≤l≤KMr γl,i max κl (45) 1≤l≤KMr Mt Mt = Eκ log 1 + Ns Eγ [max1≤l≤Ns KMr γlcoop ] max κl (46) . 1≤l≤Ns KMr Mt We already know that Eγ [max1≤l≤Ns KMr γlcoop ] = bcoop + γe acoop because the {γlcoop } are i.i.d. lognormal random variables and the maximum of Ns KMt i.i.d. random variables with χ2 (2Mt ) distribution satisfies max κl Pr log Ns KMr − Mr Mt log log Ns KMr ≤ 1≤k≤Ns KMr ≤ log Ns KMr + Mr Mt log log Ns KMr 1 . (47) ≥1−O log Ns KMr Therefore, (46) is upper bounded by Mt bcoop + γe acoop log Ns KMr log 1 + E [Ccoop ] ≤ Ns Mt +Mr Mt log log Ns KMr + O (1) . (48) A lower bound on the capacity for large K can be obtained from the scheduling algorithm with random beamforming proposed in [34] since that algorithm is a suboptimal scheme. If we consider the system with the scheduling algorithm proposed in [34] (within each cell) and intercell scheduling (among adjacent cells), then the ergodic capacity E [Ccoop ] is lower-bounded by E [Ccoop ] M

t 1 coop (m) max log 1+ max γk,i βk,i (49) ≥E 1≤k≤K Ns 1≤i≤Ns m=1

Mt 1 coop (m) =E log 1 + max γk βk (50) 1≤k≤Ns K Ns m=1 (m)

where βk,i represents the short term signal-to-interference ratio of the mth random beam perceived at the kth user in the ith cell. Equality (50) results from the i.i.d. properties of (m) coop {γk,i } and {βk,i } for the index i. Because γkcoop is an i.i.d. lognormal random variable, the ergodic capacity E [Ccoop ] is bounded from lemma 2 by E [Ccoop ] Mt 1 (m) log 1 + max γkcoop βk ≥ Eβ Eγ A 1≤k≤Ns K Ns m=1

·

1−O

1 log Ns K

721

(52)

≥ bcoop − where A = {max1≤k≤Ns K γkcoop acoop log log Ns K}. Finally, using the result of [34], we can obtain the lower bound as Mt log 1 + bcoop − acoop log log Ns K E [CDP C ] ≥ Ns 1 · ξcoop − log log Ns K 1−O log Ns K 1 (53) · 1−O log Ns K where ξcoop = log Ns K − (Mt − 1) log log Ns K. The upper and lower bounds on the capacity of a dirty paper coded system with frequency reuse for large K can be derived similarly, but this derivation is omitted here for brevity. Key Result 3: Based on Theorem 4, the capacity of intercell scheduling in dirty paper coded systems grows √ t like M Ns σcoop 2 log Ns Mr K, whereas the capacity of dirty paper coded √ systems with frequency reuse grows like Mt Ns σreuse 2 log Mr K. Therefore, as K goes to infinity the net capacity gain from expanded multiuser diversity in dirty paper coded systems scales like GDP C = 1. √ √ σcoop 2 log Ns Mr K − σreuse 2 log Mr K (54) Key Result 4: Theorem 4 also indicates that the multiuser diversity√gain in dirty paper coded MIMO system grows like Mt log Mr K when the geometry of mobile stations and lognormal shadowing are reflected, whereas it has been previously known to grow like Mt log log Mr K when only the short-term fading is considered [34], [35]. lim

K→∞ Mt Ns

Even though DPC is significantly more sophisticated than TDMA, it can be readily noted that both TDMA and DPC have expanded multiuser diversity gain that scales like lim

K→∞

G √ √ =1 β α + log Ns − α

(55)

where α = log KMr and β = MNtsσ for DPC, and α = log K and β = min(MNrs,Mt )σ for TDMA because σcoop ≈ σreuse ≡ σ. So it is natural to expect that most MIMO multiple access schemes will have a similar scaling trend in Ns . The main advantage of dirty paper coding is that the gain is O(Mt ) rather than min(Mr , Mt ), which is potentially significant since it is reasonable to assume that Mt > Mr due to mobile size and cost restrictions. IV. N UMERICAL R ESULTS

In this section, we show the capacity gains of intercell scheduling over conventional frequency reuse through computer simulations, and compare with the derived asymptotic · Pr [ A ] (51) results. We consider various Rayleigh MIMO−lchannels and M propagation pathloss given by Lk = (dk /d0 ) where d is 1 t (m) the distance from a base station to mobile station k and the ≥ Eβ log 1+ bcoop −acoop log log Ns K max βk k Nsm=1 pathloss exponent l = 3.5. The total number of cells involved

722


12

12 Cooperatively scheduled transmission (Mr =4, Mt =4) DPC with coop. scheduled transmission (Mr=2, Mt=4)

10

10

Frequency reuse (Mr =4, Mt=4)

8

Capacity (baud/Hz)

Capacity (baud/Hz)

8

Cooperatively scheduled transmission (Mr =2, Mt =2) 6

4

DPC with frequency reuse (Mr=2, Mt=4)

TDMA with coop. scheduled transmission (Mr=2, Mt=4)

6

TDMA with frequency reuse (Mr=2, Mt=4)

4

Frequency reuse (Mr =2, Mt=2)

TDMA with coop. scheduled transmission (Mr=1, Mt=4)

2

2 TDMA with frequency reuse (Mr=1, Mt=4)

Upper bound for large K Simulatio n Lower bound for large K

Upper bound for large K Simulation Lower bound for large K 0 100

200

300

400

500 600 Number of Users, K

700

800

900

0

1000

Fig. 3. Ergodic capacity of 2 × 2 and 4 × 4 MIMO TDMA systems with intercell scheduling and with frequency reuse. The upper and lower bounds agree with simulations for large K.

0

50

100

150

200

250

300

350

400

450

500

Number of Users, K

Fig. 4. Capacities of 4 × 1 and 4 × 2 systems for both TDMA and dirty paper coding, also comparing intercell scheduling and with frequency reuse.

2.6

A. Asymptotic Bounds vs. Simulations

Cooperatively scheduled transmission (Mr = 1, Mt = 2) 2.4

2.2

2 Capacity (baud/Hz)

in cooperative scheduling is Ns = 7 with the 48 nearest cells treated as OCI sources. Correspondingly, for comparison with traditional frequency reuse systems, f = 7. The K users in each cell are randomly placed according to a uniform distribution.

1.8

1.6 Frequency reuse (Mr = 1, Mt = 2)

First, we will see how the ergodic capacity is affected by the number of antennas (Mt , Mr ), and how much intercell scheduling helps relative to traditional frequency reuse. Fig. 3 shows both the bounds and simulation results for the capacities of 2 × 2 and 4 × 4 MIMO TDMA systems. As expected, intercell scheduling exploits achieves higher capacity than frequency reuse, with a large user gain of about 1 bits/Hz for 2 × 2 MIMO and about 2 bits/Hz for a 4 × 4 MIMO system. The upper and lower bounds are accurate for large K, but optimistic for smaller K. The intercell scheduling bounds converge faster because the effective number of users is Ns K, compared to just K in traditional frequency reuse. Practical MIMO systems are likely to have more antennas at the base station than at the mobile, so Mt > Mr . In Fig. 4, some permutations of TDMA and DPC are considered for 4×2 and 4 × 1 MIMO, both with and without intercell scheduling. Due to the enormous computational complexity inherent in DPC, only bounds are given for the DPC case. One notable observation is that increased transmit antennas do not buy much in terms of multiuser diversity in a TDMA system: the 2×2 capacity of Fig. 3 is comparable to the 4×2 capacity due to the dominance of the min(Mt , Mr ) term in the bounds of Theorem 3. On the other hand, DPC does not suffer from this limitation, so its theoretical advantage is most pronounced in an unbalanced antenna configuration. From Fig. 3 and Fig. 4, it is also observed that the capacity of TDMA converges faster as min(Mt , Mr ) decreases because in the bounds of Theorem 3 the convergence speed of the log function depends on the 1/ min(Mt , Mr ) term in it. Since the analysis is based on a large number of users, it Fig. 5 examines the validity of the bounds for K ∈ (0, 50) in a 2 × 1 TDMA system. Although the capacity “bounds" are not

1.4

1.2 Upper bound for large K Simulation Lower bound for large K 1 5

10

15

20

25 30 Number of users, K

35

40

45

50

Fig. 5. Capacity of 2 × 1 TDMA with intercell scheduling and with frequency reuse for a small number of users. The simulations do not conform to analytical bounds in this regime, but the scaling trends predicted by the large K analysis are preserved.

accurate in the small K regime, it can be seen that the scaling behavior with K is still consistent between the bounds and simulations, and that the bounds are about equally optimistic for both intercell scheduling and static frequency reuse. Again, it can be seen that the intercell scheduling bounds become accurate more quickly with K due to the effective increased cardinality of the selection set. B. Comments on the Transmit Duty Cycle, Ns As previously stated, the larger the number of cooperating base station (Ns ), the greater the OCI reduction and the greater the expanded multiuser diversity gain. However, a larger Ns suffers considerably due to the fact that only 1 out of every Ns base stations is transmitting at each time. Therefore, unless the amount of OCI reduction and multiuser diversity gain is high enough to compensate for the nominal spectral efficiency decrease, the per-cell ergodic capacity will decrease with Ns . Simulation results of the capacities for Ns = {3, 4, 7} is shown in Fig. 6 for 2 × 2 MIMO TDMA systems. Although the capacity decisively decreases with Ns , this result does not


TABLE I: A BRIEF COMPARISON BETWEEN BS COORDINATION TECHNIQUES

Multicell dirty paper coding

Suboptimal p BS coordination techniques

Spectral Efficiency

Real time information Exchange

Provides a theoretical upper bound

Provide high g spectral p efficiency by significantly reducing mutual interference Superior to intercell sched ling scheduling

Computation Complexity

10

Notes

N =3 s

9

Very high Channel state information and noncausal transmit data should be exchanged.

Very high for successive encoding and decoding

Requires non-casual knowledge for prepre cancellation.

High g Channel state information should be exchanged.

Lower than multicell DPC, but much higher than intercell scheduling

Promising g in the longg term Examples: joint beamforming, joint precoding, joint spatial m ltiple ing and joint multiplexing, space time coding.

8

7

Capacity (baud/Hz )

Technique

723

6

N =4 s

5

4

Ns = 7

3

Intercell scheduling

Higher than frequency Comparable to handoffs reuse Reduced OCI, multiuser diversity gain relative to frequency reuse

Low A small amount of computation for scheduling

Low spectral efficiency

Low

Practically feasible in the short-term A practical alternative to frequency reuse

2 Cooperatively scheduled transmission Frequency reuse 1

0

50

100

150

200

250

300

350

400

450

500

Number of users, K

Frequency reuse

Not required

Not promising as a long-term solution

suggest that a lower Ns is preferable. In a real communication system, the frequency reuse factor f or the transmit duty cycle Ns is determined by required SINR levels and outage probabilities, which are dominated by the areas near the cell boundaries. A reasonable way to view the results in this figure is that once f or Ns has been determined based on the required SINR level, intercell scheduling achieves about an extra 1 bit/Hz higher capacity than frequency reuse due to the expanded multiuser diversity gain demonstrated in this paper. C. A Brief Comparison Between Intecell Scheduling and BS Coordination Techniques Clearly from a capacity standpoint, multicell DPC provides theoretically achivable capacity by BS coordination, and the suboptimal BS coordination techniques aforementioned in Introduction will be also superior to intercell scheduling which suffers from a duty cycle in time. However, the amount of real time information exchange required for the optimal and suboptimal BS coordinations is much higher than that for intercell scheduling. High computatoinal complexity is also a big burden of the BS coordination techniques. Intercell scheduling is considered a practical alternative of frequency reuse in the short-term. A brief comparison between intercell scheduling and BS coordnation techniques is given in Table I. V. C ONCLUSION In this paper, we have theoretically derived capacity bounds for multicell MIMO systems with intercell scheduling and frequency reuse. Through the asymptotic analysis for large K and simulations, it has been shown that intercell scheduling achieves an expanded multiuser diversity gain as well as a crucial reduction in other-cell interference. Although this interference reduction comes with a theoretical capacity price, the capacity relative to conventional frequency reuse systems is notably increased. The capacity of intercell scheduling is considered an achievable lower bound on attainable capacity through BS coordination. A theoretical contribution of this paper can be found in the provided analytical framework for multiuser diversity gain in a multicell MIMO system. This analysis found that the multiuser diversity gain when √ neighboring base stations cooperate is 2 log K when the

Fig. 6. The multiuser diversity gains versus Ns in 2 × 2 MIMO TDMA. A small value of Ns is preferable in terms of capacity, but Ns is usually fixed by outage and SINR requirements. For a given Ns , intercell scheduling achieves a substantial gain.

geometry of mobile stations, shadow fading, and OCI are taken into account, whereas previously results have concluded that the maximum gain scales like log log K in single cell systems. ACKNOWLEDGEMENT The authors would like to thank Dr. Sriram Vishwanath for his helpful comments. A PPENDIX A D ERIVATION OF an AND bn IN THE P ROOF OF LEMMA 2 From the c.d.f. in (15) and error function in (16), bn can be obtained by solving following equation: 1 − FX (bn ) √ 2 2 σx 2 1 = √ e−(log bn −μx ) /2σx log bn − μx 2 π 3 √ σx 2 1 − + ··· 2 log bn − μx 1 . (A.1) n After applying log, if bn > e, this equation approximately reduces to √ (log bn − μx )2 + log σ 2 − log(log bn − μx ) − x 2σx2 √ (A.2) = log 2 π − log n. =

varies more√rapidly than log(log bn −μx ) Since (log bn −μx )2 √ and log n log σx 2 − log 2 π for large n, the equation becomes, (A.3) (log bn − μx )2 = 2σx2 log n. Therefore, bn is finally given by bn = eσx (2 log n)

1/2

+μx

.

(A.4)

Since an = h(bn ) and the hazard function h(x) is given in (11), an is obtained by σ x bn an = . (A.5) (2 log n)1/2

724


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Wan Choi (S’03-M’07) received the B.S. and M.S. in Electrical Engineering from Seoul National University in 1996 and 1998, respectively, and the Ph.D. in Electrical and Computer Engineering at the University of Texas at Austin in 2006. He is now an Assistant Professor at the School of Engineering, Information and Communications University (ICU), Daejeon, Korea. He was a Senior Member of the Technical Staff at the R&D Division of Korea Telecom (KT) Freetel from 1998 to 2003, where he researched 3G CDMA systems. He also researched at Freescale Semiconductor and Intel Corporation during the summers of 2005 and 2006, respectively, where he collaborated on practical wireless communication issues. He is the recipient of the IEEE Vehicular Technology Society Best Paper Award (Jack Neubauer Memorial Award) in 2002 and the IEEE Vehicular Technology Society Dan Noble Fellowship Award in 2006. While at the University of Texas at Austin, he was the recipient of the William S. Livingston Graduate Fellowship. His research interests include the areas of wireless communications, including spread spectrum communications, OFDM communications, MIMO communications, and advanced communications systems.

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Jeffrey G. Andrews (S’98-M’02-SM’06) received the B.S. in Engineering with High Distinction from Harvey Mudd College in 1995, and the M.S. and Ph.D. in Electrical Engineering from Stanford University in 1999 and 2002, respectively. He is an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin, and the Director of the Wireless Networking and Communications Group (WNCG), a research center of 15 faculty, 100 students, and 10 industrial affiliates. He developed Code Division Multiple Access (CDMA) systems as an engineer at Qualcomm from 1995 to 1997, and has consulted for the WiMAX Forum, Microsoft, Palm, Ricoh, ADC, and NASA. Dr. Andrews is a Senior Member of the IEEE, and serves as an associate editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS. He is co-author of the popular book Fundamentals of WiMAX (Prentice-Hall, 2007). He received the National Science Foundation CAREER award in 2007 and is the Principal Investigator of an eight university team of 13 faculty in DARPA’s Information Theory for Mobile Ad Hoc Networks program.