Energy-efficient Uplink Training Design For Closed-loop MISO Systems

Transcription

1 213 IEEE Wireless Communications and Networking Conference (WCNC): PHY Energy-efficient Uplink raining Design For Closed-loop MISO Systems Xin Liu, Shengqian Han, Chenyang Yang Beihang University, Beijing, China {liuxinjlu, Chengjun Sun Beijing Samsung elecom R&D Center Abstract When the circuit power consumption and the overhead for channel estimation are taken into account, the system designed for maximizing the spectrum efficiency (SE) does not necessarily yield high energy efficiency (EE). In this paper, we strive to optimize the uplink training length towards maximizing the EE of closed-loop multi-antenna systems under the constraint of the SE. he upper bounds of the system EE and the net downlink SE with channel estimation errors are derived, based on which the optimization problem is proved as convex, and the impacts of signal-to-noise ratio (SNR) and circuit power consumption on the optimal training length are analyzed. Analytical and simulation results show that in general the EEoriented optimization leads to a longer training length than the SE-oriented optimization, and it will reduce to the SE-oriented optimization at high SNR and very low SNR regime, or with very high circuit power consumption. I. INRODUCION Energy efficiency (EE) is becoming an important design goal for future cellular networks [1,2]. When the circuit power consumption and the signaling overhead to support high data rate transmission are taken into account, a system with high spectrum efficiency (SE) does not necessarily provide high EE. Closed-loop systems are widely deployed, where data transmission is adapted to the channel variation. o facilitate downlink closed-loop transmission, training symbols are sent in the uplink of time division duplex (DD) systems in order to provide channel state information (CSI) at the base station (BS). Existing studies on the optimization of uplink training mainly focus on the SE maximization. In [3] and [4], the transmit power, position and the number of training symbols were optimized, respectively aimed at maximizing the capacity lower bound and minimizing the Cramer-Rao bound of the channel estimation errors. In [5], the optimal downlink training length was studied. It was shown that the optimal training length equals to the number of antennas at the transmitter, when the optimal power allocation between the training symbols and data is considered. In [6], the optimal resource allocation strategies towards maximizing SE were investigated for multiuser multi-antenna systems. In general, the SE-oriented training design does not maximize the EE. When the circuit energy consumption is taken into account, the relationship between SE and EE becomes his work was supported in part by the National Natural Science Foundation of China (No ), the National Basic Research Program of China (No. 212CB3163) and Beijing Samsung elecom R&D Center. Fig. 1. Uplink training str tr Downlink data sd Frame structure of the considered DD system complicated, depending on the network architectures and transmission schemes [2]. In [7] and [8], the training signals were optimized to maximize the EE for open-loop systems, either without or with the circuit power consumption. Yet the EE-oriented training design for closed-loop systems has not been addressed in literature. In this paper, we study the EE-oriented training length design under the constraint of the SE requirement in closedloop multi-antenna systems. We start with deriving the upper bounds of the net SE and the EE considering imperfect channel estimation and circuit power consumption, based on which the optimization problem is formulated, and the optimal training length under the two criteria of maximizing EE and SE is compared. he impacts of signal-to-noise ratio (SNR) and circuit power consumption on the optimal training length are investigated. Simulation results validate our theoretical analysis. Notations: I denotes the identity matrix. and represent the magnitude and two-norm, respectively. ( ) H denotes conjugate transpose, E[ ] denotes expectation operation, is the Kronecker product, and N denotes the set of non-negative integers. II. SYSEM MODEL Consider a DD multi-input single-output (MISO) system, where one BS with N t antennas serves one single-antenna user. We assume that the user undergoes block fading channel, which remains constant during each uplink-downlink frame but is independent among different frames. he structure of the frame is shown in Fig. 1. Each frame contains symbols, among which tr symbols are for uplink training. Let s tr denote the uplink training symbols, where s tr s H tr = P u tr and P u denotes the transmit power of training symbols. hen the received training symbols at the BS are y u = hs tr + n u, (1) /13/$ IEEE 2589

2 where h = [h 1,, h Nt ] C 1 Nt denotes the complex Gaussian channel vector with zero mean and correlation matrix σh 2I, S tr = I s tr C Nt trnt, and n u is the additive white Gaussian noise (AWGN) at the BS with zero mean and variance σu. 2 With the minimum mean-square error (MMSE) criterion, the uplink channel can be estimated at the BS as ĥ = y u (S H trs tr + σ2 u σh 2 I) 1 S H tr, (2) and the relationship between the estimated channel and the true value of the channel satisfies [9] h = ĥ + e, (3) where e is the channel estimation error with zero mean and correlation matrix σei, 2 the channel estimate ĥ is with zero mean and correlation matrix σ 2 I, and ĥ and e are uncorrelated. ĥ Because S tr S H tr = P u tr I, it is easy to show that σ 2 e = σ 2 ĥ = σ 2 uσ 2 h σ 2 u + σ 2 h P u tr, (4) σ4 h P u tr σ 2 u + σ 2 h P u tr. (5) By exploiting the channel reciprocity, the uplink channel estimate is used as downlink channel for precoding. With CSI at the BS, maximum ratio transmission (MR) [1] can be employed for downlink transmission. hen, the precoding vector is w = ĥ P d (6) ĥ, where P d is the downlink transmit power. he received signal at the user can be expressed as y d = hw H s d + n d = P d hĥh ĥ s d + n d, (7) where s d is the data symbol, and n d is the AWGN at the user with zero mean and variance σ 2 d. A. Net Spectrum Efficiency III. PROBLEM FORMULAION o focus on analyzing the impact of uplink training, we assume that the user has perfect knowledge of downlink channels for coherent detection. From (7), the receive SNR at the user can be expressed as where ϕ = hĥh ĥ SNR = P d σ 2 d = (ĥ+e)ĥh ĥ hĥh ĥ 2 P d σd 2 ϕ 2, (8) ĥ + α, and α = eĥh ĥ. It is easy to show that E[α] =, E[α 2 ] = σ 2 e, and E[ ĥ αh ] =. hen, we can obtain E[ ϕ 2 ] = N t σ 2 ĥ + σ2 e = σ2 h σ2 u + N t σ 4 h P u tr σ 2 u + σ 2 h P u tr. (9) he downlink ergodic capacity using the imperfect CSI with unit bandwidth is [ C = E[log 2 (1 + SNR)] = E log 2 (1 + P ] d σd 2 ϕ 2 ), (1) whose closed-form expression is hard to derive. o facilitate the optimization, we alternatively consider an upper bound of the ergodic capacity using Jensen s inequality. Further considering (9), we have where γ u = Pu σ 2 u C up = log 2 ( 1 + P d σ 2 d σh 2σ2 u + N t σh 4P ) u tr σu 2 + σh 2P u ) tr, log 2 ( 1 + γ d + N t γ d γ u tr 1 + γ u tr (11) σh 2 and γ d = P d σ 2 σd 2 h are the average uplink and downlink receive SNR, respectively. he net spectrum efficiency is defined as SE( tr ) = C( tr), (12) and the upper bound of the net SE is SE up ( tr ) = (1 tr )log 2(1 + γ d + N t γ d γ u tr 1 + γ u tr ). (13) B. Energy Efficiency Considering that the power consumption at the user side is far less than that at the BS, 1 we define the EE of the system as the ratio of the net spectrum efficiency to the total power consumed at the BS during an uplink-downlink frame. he power consumption of the BS during the uplink training phase does not include the transmit power, i.e., P bu = P RXc, (14) where P RXc is the circuit power when the BS operates in receive mode, which includes the power consumed by the RF links, the baseband signal processing (i.e., synchronization and channel estimation), and the power supply and cooling. he power consumption of the BS during the downlink transmission phase includes both the transmit power and the circuit power. Based on the linear power consumption model in [11], the power consumed during the downlink phase is P bd = P d η + P Xc, (15) where η is the efficiency of power amplifier, and P Xc is the circuit power when the BS operates in transmit mode, which includes the power consumed by the RF links, the baseband signal processing (e.g., synchronization, precoding, modulation and coding), and the power supply and cooling. From (14) and (15), the total power consumption during an uplink-downlink frame can be obtained as P tot = P bu tr + P bd ( tr ). (16) 1 Note that the energy efficient design for the user is also important, which is however beyond the scope of this paper. 259

3 With (16) and (12), the EE can be obtained as EE( tr ) = SE( tr) C( tr ) = P tot P bu tr + P bd ( tr ), (17) and further considering (11), the upper bound of the EE is tr EE up ( tr ) = P bu tr + P bd ( tr ) ( log γ ) d + N t γ d γ u tr. (18) 1 + γ u tr With the upper bounds in (13) and (18), the optimization problem of uplink training length aimed at maximizing the EE under the SE constraint can be formulated as max EE up ( tr ) tr (19a) s.t. SE up ( tr ) SE SE (19b) tr (19c) tr N, (19d) where SE is the minimal SE requirement of the user, and SE is a back off margin to ensure an acceptable outage probability due to the usage of the SE upper bound. We will evaluate the impact of using the upper bounds on the design in Section V. IV. ANALYSIS OF HE OPIMAL RAINING LENGH A. Solution of the Optimal raining Length o solve the optimization problem (19), we first relax (19) by regarding tr as a continuous variable within [, ], and then round the optimal solution to the nearest integer. In the sequel, we first show that the relaxed version of the problem (19) is a convex optimization problem. heorem 1: Both the upper bounds of the EE and the SE are concave functions with respect to tr, and their first-order derivatives are monotonically decreasing functions of tr. Proof: See Appendix A. It follows that the relaxed version of (19) omitting the constraint in (19d) is a convex problem, whose solution can be numerically found with efficient algorithms [12]. hough we cannot find its closed-form solution, we are able to compare the difference in the optimal training lengths towards maximizing the EE and towards maximizing the SE. B. Optimal raining Length Difference Under wo Criteria Define trse and tree as the optimal uplink training length of the SE-oriented optimization and the EE-oriented optimization, respectively, where tree is the solution of the relaxed version of problem (19). Denote tr = tree trse as the difference of the optimal training length under the two criteria. heorem 2: tr always holds. If EE up(), then tr = and tree = trse =. If EE up() >, then tr and the equality holds if and only if SE = SE max, where SE max is the maximum achievable net SE. Proof: See Appendix B. heorem 2 indicates that the difference between the optimal training length of the SE-oriented and the EE-oriented optimization depends on the value of EE up(), and the EEoriented optimization usually leads to a longer training length. We will analyze the impact of SNR on EE up() in next subsection. C. Impact of SNR on the Difference in Optimal raining Length Since the average downlink receive SNR is usually larger than the average uplink receive SNR, we set γ d = βγ u with β > 1. One can find from (B.1) in Appendix B that EE up() is a function of SNR. For notational simplicity, in the following we use G(γ u ) to denote EE up(), and then rewrite (B.1) as G(γ u ) = EE up() = 1 [ β(n t 1) γu 2 P bu P bd ln2 1 + βγ u P bd log 2(1 + βγ u )], (2) whose first-order derivative is G (γ u ) = β(n t 1) βγu 2 + 2γ u P bu P bd ln2 1 + βγ u Pbd 2 ln2 β. (21) 1 + βγ u It is easy to show that G (γ u ) > always holds, therefore G (γ u ) is a monotonically increasing function. We can further show that G () < and G ( ) >, hence G(γ u ) first decreases and then increases. Based on L Hospital s rule, we can derive that γ 2 u β(n t 1) lim /( P bu γ u ln2 1 + βγ u P bd log 2(1 + βγ u )) =. (22) hen from (2) and (22), we know G( ) >. Further considering G() = from (2) and the fact that G(γ u ) first decreases and then increases, there must be a positive value of γu o that meets G(γu) o =. When < γ u γu, o we have G(γ u ), i.e., EE up(). According to heorem 2 and the proof, in this case EE up ( tr ) is monotonically decreasing, tree = trse = and tr =. When γ u > γu, o G(γ u ) is an increasing function and G(γ u ) >, i.e., EE up() >. Now from heorem 2 EE up ( tr ) first increases and then decreases, tree trse and tr. In the extreme case where γ u, we can show that tree = trse = 1. o see this, we consider two cases. When tr 1, we have SE up ( tr ) (1 tr )log 2(1 + N t βγ u ) from (13) and EE up ( tr ) tr P bu tr+p bd ( log tr) 2(1 + N t βγ u ) from (18). he asymptotic results of both EE up ( tr ) and SE up ( tr ) are monotonically decreasing functions of tr. herefore, tree = trse = 1 in this case. When tr =, EE up () P bd log 2 (1 + βγ u ) and SE up () log 2 (1 + βγ u ). It is not hard to show that EE up (1) EE up () and SE up (1) SE up (). herefore, we have tree = trse = 1 when γ u. 2591

4 * ABLE I LIS OF SIMULAION PARAMEERS Simulation parameters Values Number of antennas at the BS, N t 4 otal number of symbols in a frame, 2 ransmit power of uplink training, P u 23 dbm ransmit power of downlink transmission, P d 46 dbm Uplink/downlink SNR difference, β 23 db Power amplifier efficiency, η 31.1% Circuit power of BS in transmit mode, P Xc 364 W Minimal SE requirement, SE 1 bps/hz Optimal raining Length * tree * trse * tr In summary, we show that tree = trse = at low SNR regime when γ u γu, o and tree = trse = 1 at high SNR regime. In general scenarios, tree is greater than trse. V. SIMULAION RESULS In this section, we evaluate the impact of uplink training length on the EE and the SE of the system via simulations. he simulation parameters listed in able I will be used in the following unless otherwise specified. he maximum transmit powers for uplink training and downlink transmission, the circuit power, and the power amplifier efficiency are configured as in [1]. According to the results in [1], the circuit power ranges from 1 W to 5 W depending on the types of the BS, and the circuit power consumed of BS in uplink receive mode approximately equals to that in downlink transmit mode. herefore, we set P RXc = P Xc in the simulations. he average uplink receive SNR of the user at the cell edge, SNR UL edge, is set as -2 db. he average uplink receive SNR of the user from the BS with distance d is computed as SNR UL = SNR UL edge log 1 (r/d), where the cell radius r = 25 m. In the simulations, the back off margin, SE, is set as.25 bps/hz to ensure SE always satisfied. We first evaluate the impact of using the upper bounds on the training length optimization. We simulate the EEs achieved by the optimization based on the upper bound (11) and based on the true value (1) as a function of average uplink SNR, and find that their results almost overlap. For example, when the average uplink SNR is 3 db, the achieved EE is.39 bit/joule/hz, and the gap between the two results is about bit/joule/hz. his suggests that the training length optimization with the upper bounds has little impact on the performance. Due to the lack of space, we do not show the figure. Figure 2 compares the training length optimized towards the SE and the EE. As shown in the figure, when the SNR exceeds 15 db, tree = trse = 1 and tr =. In general cases, the EE-oriented optimization requires more training symbols than the SE-oriented optimization. hese results agree with our analytical analysis. In Fig.3, we show the impact of circuit power consumption on the difference of the optimal training length tr SNR UL (db) Fig. 2. / (%) tr Optimal training length as a function of uplink SNR SNR UL (db) = 1 w = 1 w = 5 w Fig. 3. Optimal training length difference with different circuit power consumption. under the two criteria. It is shown that when the circuit power consumption is small, there is an obvious difference of training length between the EE-oriented and the SE-oriented optimization. As the circuit power consumption increases, the difference reduces. his is because in this case the total power consumed at the BS in receive mode and in transmit mode gradually become identical with the fixed transmit power, i.e., P bd P bu, and then the EE-oriented optimization will reduce to the SE-oriented optimization according to (B.5) in Appendix B. Figure 4 shows the EE gain of the EE-oriented optimization over the SE-oriented optimization. he EE gain is defined as (EE(trEE ) EE( trse ))/EE( trse ), where EE( tree ) and EE(trSE ) are respectively the EEs achieved by the EE-oriented and the SE-oriented optimization, as defined in Section IV. It is shown that the EE gain decreases with the growing of the circuit power. his is because the optimal training length difference under the two criteria decreases with the increase of the circuit power, as shown in Fig. 3. When the SNR is sufficiently high, the EE gain tends to zero, i.e. 2592

5 EE gain (%) =1 w =1 w =5 w estimation, both of which were proved as concave functions of the uplink training length. Both theoretical analysis and simulation results showed that a longer training length is necessary to maximize the EE compared to that maximizing the SE in general cases, while the EE-oriented optimization will be equivalent to the SE-oriented optimization if the uplink SNR is high and very low or the the circuit power consumption is high. APPENDIX A PROOF OF HEOREM SNR UL (db) o simplify the notation, we rewrite (18) as EE up ( tr ) = g 1 ( tr ) g 2 ( tr ), (A.1) Fig. 4. EE gain of the EE-oriented optimization over the SE-oriented optimization. EE * (bit/joule/hz) =1 w =1 w =5 w SE (bit/s/hz) Fig. 5. EE vs. SE (SNR UL = 1 db). the EE-oriented optimization will reduce to the SE-oriented optimization, which agrees with our previous analysis. Figure 5 shows the relationship between the EE and the net SE achieved with different values of downlink circuit power consumption. Because the maximal transmit power is given, the maximal achievable SE is bounded. When the circuit power consumption is high, the EE approximatively grows with the SE monotonically, because in this case the value of tr is very small. When the circuit power consumption is low, on the other hand, the EE first increases and then finally reduces with the increase of the SE, which is led by the difference in the optimal training lengths under the two criteria. According to the trade-off, we can configure the training length reasonably. VI. CONCLUSIONS In this paper, we investigated the design of uplink training length aimed at maximizing the EE for DD closed-loop MISO systems under the constraint on the minimal SE. o facilitate the optimization and analysis, we derived the upper bounds of the EE and the SE considering imperfect channel where g 1 ( tr ) = tr P bu tr+p bd ( tr) and g 2 ( tr ) = log 2 (1 + N t γ d (Nt 1)γ d 1+γ u tr ) >. It is easy to show that g 1( P bu tr ) = [P bu tr + P bd ( tr )] 2 <, g 2( tr ) = (N t 1)γ d γ u 1 >, ln2(1 + γ u tr ) 1 + γ d + γ u tr + N t γ d γ u tr g 1 ( tr ) = 2P bu (P bd P bu ) <, and g [P bd (P bd P bu ) tr ] 3 2 ( tr ) <. hen, we can obtain the derivatives of EE up ( tr ) as EE up( tr ) =g 1( tr )g 2 ( tr ) + g 1 ( tr )g 2( tr ), (A.2) EE up( tr ) =g 1 ( tr )g 2 ( tr ) + 2g 1( tr )g 2( tr ) + g 1 ( tr )g 2 ( tr ). (A.3) We can see that EE up( tr ) < always holds. herefore, EE up ( tr ) is a concave function of tr, and EE up( tr ) is a monotonically decreasing function of tr. Similarly, we rewrite (13) as SE up ( tr ) = f 1 ( tr ) f 2 ( tr ), (A.4) where f 1 ( tr ) = 1 tr and f 2( tr ) = log 2 (1 + N t γ d (N t 1)γ d 1+γ u tr ) >, whose derivatives are as follows. f 1( tr ) = 1 <, tr f 2( tr ) = (N t 1)γ d γ u 1 >, ln2(1 + γ u tr ) 1 + γ d + γ u tr + N t γ d γ u tr f 1 ( tr ) =, and f 2 ( tr ) = g 2 ( tr ) <. hen we have SE up( tr ) =f 1( tr )f 2 ( tr ) + f 1 ( tr )f 2( tr ), (A.5) SE up( tr ) =f 1 ( tr )f 2 ( tr ) + 2f 1( tr )f 2( tr ) + f 1 ( tr )f 2 ( tr ). (A.6) Since SE up( tr ) < always holds, SE up ( tr ) is a concave function of tr, and SE up( tr ) is monotonically decreasing. 2593

6 APPENDIX B PROOF OF HEOREM 2 We begin with providing the expressions of EE up( tr ) and SE up( tr ) at two extreme cases of the training length, i.e., tr = and tr =. From (A.2) and (A.5), we have EE up() = 1 [ (N t 1)γ d γ u P bd ln2(1 + γ d ) P bu 1 P bd log 2(1 + γ d )], (B.1) EE up( ) = 1 g 2 ( ), (B.2) P bu SE up() = (N t 1)γ d γ u ln2(1 + γ d ) 1 log 2(1 + γ d ), (B.3) SE up( ) = 1 f 2 ( ). (B.4) In the following, we consider two cases. 1) EE up() : Because the power consumption of the BS in uplink receive mode is less than that in downlink transmit mode, i.e., P bu P bd < 1, we know from (B.1) and (B.3) that SE up() always holds when EE up(). Further considering heorem 1, which shows that EE up( tr ) and SE up( tr ) are monotonically decreasing functions of tr, we have EE up( tr ) and SE up( tr ) for any tr [, ]. his means that both EE up ( tr ) and SE up ( tr ) are monotonically decreasing functions of tr. As a result, trse = is optimal to maximize the SE up, and tree = is the optimal solution of the relaxed version of problem (19) since it maximizes EE up and satisfies the constraint in (19b). he result implies that uplink training is unnecessary for both the SE-oriented and the EE-oriented optimization in this case, and tr =. 2) EE up() > : Considering EE up( ) < from (B.2), and the fact that EE up( tr ) is monotonically decreasing from heorem 1, in this case EE up ( tr ) first increases and then decreases. Let tree o denote the training length that only maximizes EE up, i.e., without the constraint on the minimal SE. Since EE up ( tr ) is a concave function, we know that EE up(tree o ) =. In the following, we prove that tree o > trse no matter SE up() or SE up() >. When SE up(), since SE up( tr ) is monotonically decreasing with tr, SE up( tr ) < and SE up ( tr ) is monotonically decreasing. herefore, we have trse =, and tree o > trse. When SE up() >, considering that SE up( ) < from (B.4) and the fact that SE up( tr ) is monotonically decreasing, SE up ( tr ) first increases and then decreases with tr. Since SE up ( tr ) is concave, trse can be found from SE up(trse ) =. From (13) and (18), we can obtain the relationship between the upper bounds of the EE and the SE as SE up ( tr ) EE up ( tr ) = P bu ( + ( P bd P bu 1)( tr )). (B.5) By taking the derivative with respect to tr respectively to the left and right hand sides of (B.5) and considering that EE up(tree o ) =, we obtain SE up( o tree) = SE up (tree o ) +. (B.6) o P bd P 1 tree bu Obviously, SE up(tree o ) <. Since SE up(trse ) = and SE up( tr ) is monotonically decreasing, tree o > trse holds. If tree o satisfies constraint (19b), we know that tree = tree o > trse. Now we see what will happen if tree o does not satisfy (19b). Denote [tr min, tr max ] as the feasible set of tr. It s easy to see that trse min [tr, tr max ] since trse maximizes the SE so that the constraint in (19b) is satisfied. Further considering the fact tree o > trse, if tree o is outside of the feasible set, there must be tree o > tr max. As we have analyzed, EE up ( tr ) first increases and then decreases with tr, and the transition point is at tree o. herefore, we have EE up (tr min ) EE up (trse ) EE up(tr max ) < EE up (tree o ). Consequently, tree = tr max trse in this case. he equality tree = trse holds if and only if tr max = trse, i.e., if SE equals to the maximum achievable net SE, SE max. REFERENCES [1] G. Auer, V. Giannini, C. Desset, I. Godor, P. Skillermark, M. Olsson, M. Imran, D. Sabella, M. Gonzalez, O. Blume, and A. Fehske, How much energy is needed to run a wireless network? IEEE Wireless Commun. Mag., vol. 18, no. 5, pp. 4 49, Oct [2] G. Li, Z. Xu, C. Xiong, C. Yang, S. Zhang, Y. Chen, and S. Xu, Energyefficient wireless communications: tutorial, survey, and open issues, IEEE Wireless Commun. Mag., vol. 18, no. 6, pp , Dec [3] S. Adireddy, L. ong, and H. Viswanathan, Optimal placement of training for frequency-selective block-fading channels, IEEE rans. Inf. heory, vol. 48, no. 8, pp , Aug. 22. [4] M. Dong and L. ong, Optimal design and placement of pilot symbols for channel estimation, IEEE rans. Signal Process., vol. 5, no. 12, pp , Dec. 22. [5] B. Hassibi and B. Hochwald, How much training is needed in multipleantenna wireless links? IEEE rans. Inf. heory, vol. 49, no. 4, pp , Apr. 23. [6] M. Kobayashi, N. Jindal, and G. Caire, raining and feedback optimization for multiuser MIMO downlink, IEEE rans. Commun., vol. 59, no. 8, pp , Aug [7] M. Gursoy, On the capacity and energy efficiency of training-based transmissions over fading channels, IEEE rans. Inf. heory, vol. 55, no. 1, pp , Oct. 29. [8] Z. Xu, C. Yang, G. Li, S. Zhang, Y. Chen, and S. Xu, Energy-efficient power allocation between pilots and data symbols in downlink OFDMA systems, in Proc. IEEE Globecom, Dec [9] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and ime Series Analysis. Addison-Wesley, [1]. Lo, Maximum ratio transmission, IEEE rans. Commun, vol. 47, no. 1, pp , Oct [11] A. Fehske, P. Marsch, and G. Fettweis, Bit per joule efficiency of cooperating base stations in cellular networks, in Proc. IEEE Globecom, Dec. 21. [12] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University Press, Dec