Rate-Aocation Strategies for Cosed-Loop MIMO-OFDM Joon Hyun Sung and John R. Barry Schoo of Eectrica and Computer Engineering Georgia Institute of Technoogy, Atanta, Georgia 30332 0250, USA Emai: {jhsung,barry}@ece.gatech.edu Abstract A cosed-oop MIMO-OFDM transmitter can expoit channe knowedge using eigenbeamforming to create MN parae channes, where M is the number of antennas at both ends and N is the number of OFDM tones. We find that the penaty due to a fat-frequency constraint, which forces each tone to convey the same amount of information, becomes negigibe as M grows. We propose ow-compexity bit-aocation strategies by combining the fat-frequency constraint with previousyreported spatia bit-aocation strategies [1]. A fixed (nonadaptive) aocation across space and frequency performs remarkaby we with M as sma as 4. I. INTRODUCTION It is we-known that the soution to the cassica rateaocation probem for a bank of scaar AWGN channes is given by the water-pouring procedure [2]. Parae channes arise in a cosed-oop wideband muti-input muti-output (MIMO) channe when orthogona-frequency division mutipexing (OFDM) creates a bank of narrowband MIMO channes across frequency, and when eigenbeamforming transforms each narrowband MIMO channe into scaar channes across space [3]. On MIMO-OFDM fading channes, the capacity-achieving rate aocation is based on water-pouring over space, frequency, and time [4]. Compexity can be significanty reduced by adopting a power contro strategy, which performs waterpouring in space and frequency but not time [5][6]. With power contro, each OFDM bock has the same fixed tota rate regardess of tempora channe changes. Even with the power-contro strategy, the rate-aocation compexity can sti be high when the number of OFDM tones is arge. To further reduce compexity, we introduce a fatfrequency constraint, in which each narrowband MIMO channe is restricted to have the same rate budget. In other words, with the fat-frequency constraint, water-pouring is performed over space but not frequency and not time. Athough the fatfrequency constraint is grossy suboptima for the case of a singe-input singe-output channe, we show that it is neary optima for the case of a MIMO channe. This paper aso examine bit-aocation probem, where each rate assigned to one of the parae channes is constrained to a discrete and finite set. The best aocation strategy woud enumerate a possibe combinations of bit aocations and choose the one that has the minimum SNR requirement. Unfortunatey the compexity of this exhaustive search is This research was supported by Nationa Science Foundation grants CCR- 0082329 and CCR-0121565, and by the Yamacraw program prohibitivey high when the number of scaar channes is arge in MIMO-OFDM. We propose ow-compexity bit-aocation strategies for MIMO-OFDM by combining the strategies of [1] with the fat-frequency constraint. The reduction of bit-aocation compexity is significant because the number of parae channes can be very arge. We find that even a nonadaptive strategy, which uses a fixed spatia aocation on top of the fat-frequency constraint, performs surprisingy we. The rest of paper is organized as foows. Section II describes the channe mode for MIMO-OFDM. In Section III we introduce the fat-frequency constraint and assess its penaty. In Section IV, we propose ow-compexity bit-aocation strategies and evauate their performance. Finay we concude in Section V. II. SYSTEM MODEL A. Channe Mode We consider a wideband MIMO system with M transmit and M receive antennas, where we assume the same number of antennas at each end for simpicity. A frequency-seective channe is characterized by L significant deayed paths. Let x k be an M 1 compex transmitted signa vector and y k be an M 1 received signa vector in the baseband during the k-th signaing interva. Then the discrete-time baseband mode is: y k = H x k + n k, (1) =0 where H is an M M matrix representing the -th tap of the discrete-time MIMO channe response [7]. The noise n k is an M 1 white Gaussian vector with zero mean and E[n k n k ]= N 0 I M, where the asterisk denotes the Hermitian transpose and I M is an M M identity matrix. The eements of H are possiby correated, which is represented by a correation matrix R.IfR = R 1/2 R 1/2,the -th channe matrix can be written as: H = R 1/2 Q, =0, 1,...,L 1, (2) where Q is an uncorreated M M matrix with i.i.d. compexvaued eements. In (2), the deterministic matrix R 1/2 modes the spatia fading correation at the receiver. If there is no spatia correation at the receiver, R is simpy an identity matrix. The uncorreated fading happens when there are many scatterers around the receiver providing sufficient scattering from a directions. When there exists spatia correation, we use the correation mode in [7], in which the deay spread 0-7803-7954-3/03/$17.00 2003 IEEE. 483
channe is represented by L significant scatterer custers at the transmitter side. In this case the eement at the m-th row and n-th coumn of correation matrix can be approximated as: [R ] m,n σ 2 e j2π(m n) cos(θ) e 1 2 (2π(m n) sin(θ )σ θ ) 2, (3) where is the antenna spacing reative to waveength. The two parameters, θ and σθ 2, denote the average arriva ange and the variance of custer ange spread, respectivey, for the -th custer. The path gains {σ 2 } are basicay dependent on the power deay profie and the shaping fiter [3]. In fact the approximation in (3) is accurate ony for sma custer ange spread, but it provides the correct trend for arge spread. The rank of R criticay impacts the maximum achievabe rate of MIMO-OFDM. Note that R coapses to a rank-1 matrix when σ θ =0, that is, when there is no custer ange spread. In such case a arge increase in capacity is expected as L grows [7]. In this paper we ony consider Rayeigh fading, where each eement of Q is circuary-symmetric compex Gaussian with zero mean and unit variance. We assume that =0 trace(r ) = M regardess of L for the sake of normaization. The path gains σ in (3) are equa for a. B. MIMO-OFDM An OFDM system converts a wideband MIMO channe into a bank of parae narrowband MIMO channes, avoiding the need for time-domain equaization at the receiver. Let N denote the number of tones (subcarriers) in OFDM. Then the narrowband MIMO channe at the n-th tone is given by: G n = H e j2πn/n. (4) =0 Let G n = U n diag{s 1/2 n }Vn be a singuar-vaue decomposition of G n, where U n and V n are unitary, and where eements of s n =[s 0,n,...,s,n ] are rea and nonnegative eigenvaues of G n G n that are ordered from argest to smaest. When the eigenbeamforming transmitter and receiver fiter by V n and U n for each n, respectivey, a bank of scaar channes is created across space and frequency [3]: z m,n = a m,n + w m,n, n =0, 1,...,N 1 m =0, 1,...,M 1, (5) where a m,n is the data symbo across m-th spatia channe of n-th tone, and z m,n is the corresponding received signa affected by the noise w m,n, which is white in space and frequency. III. FLAT-FREQUENCY CONSTRAINT In this section, We investigate the power-contro strategy in MIMO-OFDM using the channe mode of (5). The powercontro strategy fixes the tota rate per OFDM bock, and rates are distributed to the parae channes of (5) such that the transmitter power requirement for Shannon s error-free communications is minimized. The rate-aocation probem for Fig. 1. An exampe of rate aocations in MIMO-OFDM with M = 2 antennas and N = 8 tones for R = 32 bits per OFDM bock: (a) with fat-frequency constraint; and (b) without fat-frequency constraint. the power-contro strategy is soved by water-pouring over space and frequency. The power-contro strategy has ower compexity than the capacity-achieving water-pouring strategy over space, frequency, and time, but it is not optima as it does not consider tempora water-pouring. On MIMO channes, however, its penaty is known to be very sma, especiay for arge antenna arrays [1][6]. Let R denote the tota rate per OFDM bock. The rateaocation probem for the power-contro strategy is soved by the foowing strategy: Strategy 1 (water-pouring in space and frequency): Choose the set of rates {r m,n } so as to minimize the instantaneous SNR requirement: 1 N N 1 m=0 n=0 subject to m n r m,n = R., (6) We now introduce a fat-frequency constraint, where each tone is forced to have the same rate budget. In other words, each tone must satisfy m=0 r m,n = R/N for a n, so that the tota rate per tone is fat in frequency. For instance, Fig. 1 iustrates rate aocations with and without a fatfrequency constraint for the case of M =2antennas (space) and N =8OFDM tones (frequency) when R =32. Notice that each coumn sums up to R/N =4with the fat-frequency constraint, which is the rate budget for each tone. The rateaocation probem with the fat-frequency constraint is soved by the foowing strategy: Strategy 2 (water-pouring in space ony): For each tone, choose the set of rates {r m,n } so as to minimize the instantaneous SNR requirement: m=0, (7) subject to the fat-frequency constraint that m r m,n = R/N for a n. This strategy ignores the variabiity of the channe frequency response and performs water-pouring ony over space. With the fat-frequency constraint, water-pouring for arge (MN) parae channes is repaced by N repetitions of water-pouring for rather sma (M) parae channes. In a singe-antenna system, Strategy 2 eads to an equa aocation, that is, r n is identica for a n, and suffers significant performance degradation. In contrast, on MIMO channes, the water-pouring over space heps decrease the degradation. In this paper, we 0-7803-7954-3/03/$17.00 2003 IEEE. 484
Fig. 2. An exampe of rate-aocation resuts of 1 1 and 4 4 systems for N =64, where the soid ines are for Strategy 1 and the dotted ines are for Strategy 2 with the fat-frequency constraint. investigate how sma this penaty can be reduced on MIMO channes. The penaty due to the fat-frequency constraint can be easiy measured by comparing the average SNR requirement: E N 0 = E [ 1 N N 1 m=0 n=0 ], (8) where r m,n are determined by either Strategy 1 or Strategy 2, and where the expectation is over the random channe. We define the average SNR penaty of Strategy 2 reative to Strategy 1 as average SNR penaty = E/N 0 for Strategy 2 E/N 0 for Strategy 1. (9) Monte-Caro simuations were performed by generating 10,000 independent sets of channes {H } for both spatiay uncorreated and correated cases. A. Spatiay Uncorreated Channes We first consider the case when there is no spatia correation, so that {R } are identity matrices. Spatiay uncorreated channes might not be a very practica mode for some appications, but it provides insight into how Strategy 2 performs for various parameters, such as the number of antennas (M) or the number of channe taps (L). As it wi be mentioned ater, this uncorreated case serves as an upper bound of the penaty in (9). For the singe-antenna system (M =1), there is ony one channe per tone if the fat-frequency constraint is appied. Then each tone inevitaby has the same rate for a tones. In fact the eements of channe matrices G n for each n are compex Gaussian since they are just inear combinations of Gaussian random variabes of Q as shown in (4). Therefore the average SNR requirement in (8) wi be infinite on 1 1 Rayeigh fading channes [4]. Fig. 3. Achievabe rates of Strategy 1 and Strategy 2 for M {1, 2, 4, 6} antennas at each end when L =4and N =64on spatiay uncorreated Rayeigh-fading channes. As M increases, however, each tone has more degrees of freedom (spatia channes) over which to aocate the assigned rate budget, and the water-pouring over space heps decrease the average SNR penaty in (9). For instance, Fig. 2 shows one exampe of rate-aocation resuts for M =1and M =4with N = 64 and R = 640. The soid ines represent aocated rates by Strategy 1, whereas the dotted ines correspond to Strategy 2. Obviousy, as shown in Fig. 2a, Strategy 2 resuts in a fat aocation across frequency when M =1. In contrast, for M =4in Fig. 2b, the aocated rates of Strategy 2 are no onger fat, even though the rates sum up to R/N =10at each tone. The rates for Strategy 2 are not far from the rate aocations by Strategy 1. Fig. 3 pots the achievabe rate for Strategy 1 and Strategy 2 by measuring (8) for M {1, 2, 4, 6} when L = 4 and N =64. As the number of antennas (M) grows, not ony the achievabe rate increases, but the gap between two strategies decreases as we. In the case of a singe-antenna system, Fig. 3 shows that the fat-frequency constraint resuts in an infinite SNR penaty. However, average SNR penaty for M = 2 dramaticay reduces to approximatey 1 db. For M =4and M =6, the gap becomes even smaer, around 0.2 db and 0.1 db, respectivey. Fig. 3 shows that performance degradation due to fat-frequency constraint is negigibe, especiay when more than two antennas are empoyed at both ends. We now investigate how the number of channe paths (L) affects the average SNR penaty. The average SNR requirement of Strategy 2 is independent of L and is equa to the fat-fading case since spatia water-pouring is independenty performed for each channe matrix G n, which is statisticay identica to the fat-fading case. On the other hand, the average SNR requirement of Strategy 1 decreases as L grows. This is simiar to the case when diversity order increases in proportion to L [8]. The more sources of transmitted signas are avaiabe at the receiver, the higher the transmission rate can be. From a frequency-domain perspective, as L grows, the frequency response becomes more variabe so that water-pouring over frequency becomes more advantageous. 0-7803-7954-3/03/$17.00 2003 IEEE. 485
Fig. 4. Average SNR penaty due to the fat-frequency constraint in frequency-seective channes for L {2, 3, 4, 6} channe taps and no spatia correation when M =4and N =64. In order to iustrate the effects by L, Fig. 4 pots the average SNR penaty due to the fat-frequency constraint for L {2, 3, 4, 5, 6} when M =4and N =64. Ceary it can be seen that the penaty increases as L grows. When L =2, it is ess than 0.15 db. But it becomes more than 0.2 db as L =5or L =6. In Fig. 4, the advantage of Strategy 1 is not as impressive as the diversity-order increase since the achievabe rate is primariy determined by the rank of R, which is aready fu with probabiity one even for L =1 in uncorreated fading. Aso the advantage in Fig. 4 seems to saturate as L increases. In practice significant paths are often imited to a sma number, and the penaty due to the fatfrequency constraint can be kept sma as ong as a sufficient number of antennas are empoyed at both ends. B. Spatiay Correated Channes When the receiver is ocated in an open pace and no oca scattering occurs, spatia fading at the receive antennas wi be correated and this correated fading can be described by the correation matrix in (3). In this case the average SNR requirement of (1) is strongy affected by the rank of R since it is not necessariy fu rank. Thus the decrease in the average SNR requirement of Strategy 1 becomes more conspicuous as L grows. This is true either when the transmitter knows the channe [3] or not [7]. In contrast to uncorreated fading, where an increase in L does not ower the average SNR requirement of Strategy 2, the average SNR requirement with the fat-frequency constraint aso shows a significant decrease as L grows in the case of correated fading. For instance, we compare performance in Fig. 5 when fading is either correated or uncorreated, and when custer ange spread is either sma (σ θ =0) or arge (σ θ =0.25) if correated. We assume that there are L = 4 custers, whose average anges {θ } are {0,π/4,π/3,π/2}. Asshown in Fig. 5, when the spread is smaer, performance degradation due to the fat-frequency constraint is ess severe. In this exampe, as aready shown in Fig. 4, uncorreated case suffers a penaty of approximatey 0.2 db. As the fading becomes more correated, the penaty decreases up to 0.17 db for arge Fig. 5. Comparison between spatiay correated and uncorreated fading in terms of the average SNR penaty for M =4, L =4,andN =64. spread and up to 0.14 db for sma spread. As mentioned before, the uncorreated fading is the worst case in terms of average SNR penaty. Generay, when there is spatia correation, the average SNR penaty increases, just ike uncorreated case, as L grows. However there is a tendency that the penaty is smaer when the ange spread is narrower. When σ θ =0and when there are ony L =2or L =3custers, the penaty is neary zero and Strategy 2 suffers itte degradation due to the fat-frequency constraint. We see that performance degradation due to fatfrequency constraint becomes ess severe in the presence of spatia correation of fading. IV. BIT-ALLOCATION STRATEGIES The rate-aocation probem becomes the bit-aocation probem when we impose a granuarity constraint on the rates, so that {r m,n } are restricted to be discrete and finite. With this granuarity constraint, the number of possibe bit aocations is imited. The best bit-aocation strategy woud enumerate a possibe aocations and choose the one that has the minimum average SNR requirement. For MIMO-OFDM, however, this exhaustive-search strategy requires high compexity when the number of parae channes (MN) is arge. We can reduce compexity by imposing a fixed spatia aocation [1] on top of Strategy 2 with a rate budget of R/M per tone. Instead of an exhaustive search, this strategy fixes the aocation for a tones and a channe reaizations. Combining the fat-frequency constraint with a fixed spatia aocation per tone eads to a totay nonadaptive bit-aocation strategy. If the fixed aocation is carefuy chosen to match the anticipated statistics of MIMO fading channes, this fixed-frequency fixedspace strategy performs we and its achievabe rate approaches Strategy 1 cosey when there are more than two antennas at each end. This is in part due to the ordered nature and reduced variabiity of the eigenvaues { } of MIMO channes [1]. With M = 2 antennas at each end, the fixed-aocation strategy might incur a arge penaty as the number of spatia channes is not sufficient. In this case, the binary-search strategy in [1] can be used instead of the fixed-aocation strategy to 0-7803-7954-3/03/$17.00 2003 IEEE. 486
the fixed aocation is chosen to match the anticipated statistics of fading channes. These resuts impy that a cosed-oop MIMO system need not perform adaptive moduation in order to approach capacity. Instead, a combination of eigenbeamforming and fixed moduation is sufficient. Fig. 6. Performance of the fat-frequency (FF) bit-aocation uncorreated Rayeigh fading with M {2, 4, 6} when L =4and N =64. improve the performance. The binary-search strategy considers two candidates out of a possibe aocations with a budget of R/N and chooses the one with smaer instantaneous SNR requirement in (7). Though the binary search is the simpest form of adaptation, it guarantees good performance for any MIMO channe. Fig. 6 iustrates the performance of the fixed-aocation and binary-search strategies with the fat-frequency constraint for M {2, 4, 6}, where we assume no spatia correation with L =4and N =64. When restricting the spatia aocation to binary search (marked as squares), the fat-frequency strategy incurs an average SNR penaty of between 0.4 db and 0.9 db for M =2compared to the iterative agorithm of [9] (marked as circes), whereas the penaty is negigibe when M =4and M =6. The fixed-frequency fixed-spatia aocation (marked as trianges) is aso neary optima for M =4and M =6, whie its penaty can be arge for M =2. As shown in Fig. 6, the iterative agorithm of [9] is tighty bounded by Strategy 1 (thin ines) whie the binary-search strategy with fat-frequency constraint is tighty bounded by Strategy 2 (thick ines). Thus infinite-precision water-pouring is a good indicator for the performance of practica bit-aocation strategies. REFERENCES [1] J. H. Sung and J. R. Barry, Bit-aocation strategies for MIMO fading channes with channe knowedge at transmitter, in Proc. IEEE VTC 2003 Spring, Jeju, Korea, 2003, pp. 813 817. [2] T. M. Cover and J. A. Thomas, Eements of Information Theory. John Wiey and Sons, 1991. [3] G. G. Raeigh and J. M. Cioffi, Spatio-tempora coding for wireess communication, IEEE Trans. Commun., vo. 46, no. 3, pp. 357 366, March 1998. [4] E. Bigieri, J. Proakis, and S. Shamai (Shitz), Fading channes: information-theoretic and communication aspects, IEEE Trans. Inform. Theory, vo. 44, no. 6, pp. 2619 2692, October 1998. [5] G. Caire, G. Taricco, and E. Bigieri, Optimum power contro over fading channes, IEEE Trans. Inform. Theory, vo. 45, no. 5, pp. 1468 1489, Juy 1999. [6] E. Bigieri, G. Caire, and G. Taricco, Limiting performance of bockfading channes with mutipe antennas, IEEE Trans. Inform. Theory, vo. 47, no. 4, pp. 1273 1289, May 2001. [7] H. Böcskei, D. Gesbert, and A. J. Pauraj, On the capacity of OFDMbased spatia mutipexing systems, IEEE Trans. Commun., vo. 50, no. 2, pp. 225 234, February 2002. [8] A. F. Naguib, On the matched fiter bound of transmit diversity techniques, in Proc. IEEE Commun. Theory Workshop 2001, La Casa De Zorro Desert Resort, CA, 2001, pp. 596 603. [9] J. Campeo, Practica bit-oading for DMT, in Proc. IEEE ICC 1999, Vancourver, Canada, 1999, pp. 801 805. V. CONCLUSIONS We investigated the rate-aocation probem for a cosedoop MIMO-OFDM system using eigenbeamforming. Particuar focus is on simpe rate-aocation strategies instead of highcompexity water-pouring over space and frequency. First we introduced a fat-frequency constraint, which eads to spatia water-pouring by forcing the same tota rate per tone. We showed that the penaty due to the fat-frequency constraint is sma on spatiay uncorreated MIMO-OFDM channes. For exampe, the penaty reative to water-pouring over both space and frequency is ony 0.2 db on 4 4 Rayeigh fading channes with L =4channe taps. It becomes even smaer when fading is spatiay correated. We further reduce the compexity by imposing a fixed spatia aocation on top of the fat-frequency constraint, which eads to a totay nonadaptive rate aocation. Remarkaby this fixed-aocation strategy performs we when 0-7803-7954-3/03/$17.00 2003 IEEE. 487