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1 This document is downloaded from DR-NTU Nanyang Technological University Library Singapore. Title Energy and spectral efficiency of leaage-based precoding for large-scale MU-MMO systems Authors Tran Tuong Xuan; Teh Kah Chan Citation Tran T. X. & Teh K. C. 05. Energy and Spectral Efficiency of Leaage-Based Precoding for Large-Scale MU-MMO Systems. EEE Communications Letters Date URL Rights 05 EEE. Personal use of this material is permitted. Permission from EEE must be obtained for all other uses in any current or future media including reprinting/republishing this material for advertising or promotional purposes creating new collective wors for resale or redistribution to servers or lists or reuse of any copyrighted component of this wor in other wors. The published version is available at: [

2 Energy and Spectral Efficiency of Leaage-Based Precoding for Large-Scale MU-MMO Systems Tuong Xuan Tran Student Member EEE and Kah Chan Teh Senior Member EEE Abstract We study the capacity performance of signal-toleaage-and-noise ratio precoding scheme SLNR-PS for largescale multiuser multiple-input multiple-output systems. We derive capacity bounds of SLNR-PS when the number of base station antennas N is large. A trade-off between energy efficiency EE and spectral efficiency SE is defined for the case of equal power allocation. Based on this trade-off the closed-form expressions of optimal SE and optimal number of BS antennas for maximizing EE are derived respectively. n addition we consider the EE optimization problem under both maximum transmit power and quality of service constraints. The energy-efficient power allocation scheme is obtained to solve this optimization problem. Numerical simulations verify the benefits of the proposed scheme. ndex Terms Massive MMO system SLNR-based scheme.. NTRODUCTON LNEAR precoding techniques including zero-forcing ZF minimum mean square error bloc diagonalization and maximizing signal-to-leaage-and-noise ratio SLNR have been considered as practical solutions for multiuser multipleinput multiple-output MU-MMO communication systems []. The SLNR-based scheme does not require any dimension condition on the base station BS antennas. Moreover it also taes into account the influence of noise when determining the beamforming vectors for all users. Recently it has been shown that the large-scale MMO LS-MMO systems can greatly improve system performance and reliability []. Motivated by these advantages we analyze the energy efficiency EE and spectral efficiency SE of the SLNR precoding scheme SLNR- PS for LS-MMO systems. n [3] [5] the energy efficiencies of the uplin and downlin LS-MMO systems have been studied based on simulated results only. Thus the trade-off between SE and EE has not been analyzed. n [3] and [4] the authors adopted a power consumption model PCM that only considers the emitted power consumption. However for LS-MMO systems the emitted power consumption the radio frequency circuit power consumption as well as the power consumption for performing digital signal processing DSP should be considered [5] [6]. n this paper we first derive capacity bounds of the SLNR-PS when the number of BS antennas is large. Applying the realistic PCM we analyze the trade-off between SE and EE based on the upper bound UB on the system capacity at high signal-tonoise ratio SNR region for the case of equal power allocation EPA. Based on this trade-off the optimal value of SE and the Manuscript received April 05; revised August 6 05; accepted August Date of publication September 05; date of current version November The associate editor coordinating the review of this paper and approving it for publication was Q. Du. The authors are with the School of Electrical and Electronic Engineering Nanyang Technological University Singapore xuantuon00@ntu.edu.sg; ecteh@ntu.edu.sg. Digital Object dentifier 0.09/LCOMM optimal number of BS antennas that can maximize the overall system EE are also presented respectively. We obtain energyefficient power allocation scheme to maximize the EE under the constraints of the transmit power TP and quality-of-service QoS. n addition the optimal rates of individual users are defined by solving the rate profile optimization problem under the constraints of the TP QoS and EE target.. SYSTEM MODEL We consider a downlin MU-MMO system which has one BS equipped with N transmit antennas and K single-antenna users. The received signal at user is given by [7] y = g H w s + pi g H w is i + n i=i = where s is the data symbol is the transmit power for user with K = P andn is the noise term with n CN0σ. The channel vector of user is given by g = h β where h is the fast fading vector of user with h CN0 ; β is the large-scale fading coefficient. The beamforming vector w is defined as [] w max.eigenvectora where w = A = Ḡ Ḡ H + σ g g H and Ḡ = [g...g g +...g K ] respectively []. The resulting maximum SLNR value ϕ is the largest eigenvalue λ of A which is obtained by solving the following characteristic equation [] [7] det [G G H + σ λ + λ where G =[g...g K ]. Let us define = ] g g H = 0 3 G G H + σ c = λ+ λ g and d H = g H. Applying the matrix determinant lemma i.e. det + cd H = + d H c det ϕ is defined as ϕ = λ = g H. 4 G G H + σ g According to [7] the power of the interference plus noise can be estimated from the power of the leaage plus noise i.e. K i = p i g H w i + σ K i = p i g H i w + σ.thisestimation is generally tight when the leaage power is small compared with the noise power. Moreover as K < N the leaage power converges to zero at low and high SNR regions [7 Eq. 9]. Hence the signal-to-interference-plus-noise ratio SNR of the th user can be approximated by the corresponding SLNR.

3 . ERGODC ACHEVABLE CAPACTY From 4 the ergodic capacity of the th user is defined as R = E log [ ] p G H σ G. 5 + The lower bound LB on the ergodic capacity for the th user can be obtained by applying Jensen s inequality where α = R log + E{/α } 6 [ p σ GH G + ]. Following [8] the probability density function pdf of α is approximated by a Gamma function i.e. α Ɣψ ϑ with shape parameter ψ = N K++K μ N K++K θ and scale parameter ϑ = N K++K θ N K++K μ β respectively. Therefore the LB on the ergodic capacity can be approximated as Ř log + ψ ϑ. 7 The values of parameters μ and θ are defined by solving the following equations μ = θ + K i=i = β i φ K i=i = = K i=i = φ m φ where φ =N β i K N + K N + μ andm = β i μk + respectively. n addition 4 can also be rewritten as [7] [9] 8 ϕ = σ gh ϒ g + g H g 9 where ϒ = Ḡ Ḡ H Ḡ Ḡ H and =Ḡ ḠH Ḡ+ σ Ḡ H Ḡ Ḡ H. Following [9] we have ρ zf = g H σ ϒ g = /σ hence E{ρ [G H G zf }= β N K + due to / ] σ [H H H ] χn K+ where H ={h...h K }. We define ρ ad = g H g ; ρ ad and ρ zf are statistically independent and ρ ad is a nondecreasing function of /σ [9]. We denote q = U H g C K whereu is obtained from the singular value decomposition of Ḡ = U V H. We consider a matrix Q C N K that has the same distribution of Ḡ and it is independent of q. According to [9 Eq. 3] at high SNR region ρ ad has an identical distribution of q [Q H Q ] ii hence E{ρ ad } K E{ q [Q H Q ] ii }= N K+. Therefore the UB on the capacity at high SNR region is obtained as R ˆR = log + p β N K + + v 0 σ where v = N K+ K. n the case of EPA since K and N grow large without bound i.e. N K but N K = x < and x then the UB expression in 0 is converged to ˆR log + Pβ x. This result is similar to the one which is obtained by using ZF-PS in []. Thus the capacity of the SLNR-PS converges to that of the ZF-PS when SNR is high. Moreover it is more convenient to obtain 0 than 7 hence the closed-form UB 0 is used for further analysis due to its low complexity. V. ENERGY EFFCENCY OPTMZATON We now study the optimal SE optimal N as well as optimal power allocation OPA for maximizing EE performance. A. Optimum SE and Optimum N for Maximizing EE We assume that the system has unit bandwidth = P/K and σ =. Thus the system EE is given by [5] [6] ω R s P γ σ feed + P c + P sp where ω = σ DC σ MS σ cool γ is the power amplifier and R s = K R is the SE. The parameters σ feed σ DC σ MS σ cool are the loss factors of antenna feeder direct current - direct current power supply main power supply and active cooling system respectively. P sp is the power consumption for performing DSP. P c is the circuit power consumption i.e. P c = Np dac + p mix + p filt + p syn where p dac p mix p filt p syn denote the power consumptions from a digital-to-analog converter a mixer a filter and a frequency synthesizer respectively. Note that P can be defined by obtaining the inverse function of R s. Based on 0 the solution of f R s can be obtained as P = f R s = KRs/K K + v. 3 β N K + Substituting 3 into the trade-off function between the SE and EE can be rewritten as R s b K R s/k K + v + a 4 where a = P c + P sp γβ σ feed N K + and b = ωγβ σ feed N K +. Since ηr s < 0 the system R s EE in 4 is a concave function. By letting ηr s R s = 0 the optimum value of SE for maximizing the EE is obtained as R opt s = K a/k v + 5 ln e where. is the Lambert function i.e. x = xe x ande is Euler s number. For a MMO system with a specific R s requirement we can obtain the optimal number of BS antennas for maximizing EE. Let us define t = N K + 4 can be rewritten as R s c t t 3 c c 4 + t c 4 c 6 + c 5 t KK 6

4 where c =ωγ σ feed β c =p dac +p mix +p filt c 3 =p syn + P sp c 4 = γ σ feed β c 5 = K Rs/K and c 6 = K c + c 3. By letting η t = 0 we have c c 4 t 3 c 5 t + KK = 0. 7 The solutions of 7 are defined as t j = 3c c 4 u j Z + δ 0 u j Z with j { 3} whereu = u = +i 3 u 3 = i 3 and Z = δ 3 + δ 4δ3 0 / with δ 0 = 3c c 4 c 5 and δ = 54c c 4 KK. ft j is real and t j > K the optimal number of BS antennas is obtained as N opt = min{ t j +K }where. denotes the round number. B. Optimal Power Allocation for Maximizing EE The optimization problem for maximizing EE under both transmit power and QoS constraints is formulated as K R max Q ξ K +u s.t. C : R C min C : 8 K P Fig.. Capacity bounds versus SNR for both cases of EPA and OPA. ωγ σ feed where ξ = and u = P c + P sp /ω. The objective function Q is a ratio of two functions of. The result in 8 is a fractional programming FP problem which is nonconvex. By applying the properties of FP the objective function is equivalent to K R η ξ K + u whereη is the EE when is equal to the optimal value p i.e. η = K R p ξ K p +u [0]. Thus 8 can be rewritten as max R η ξ + u s.t. C and C. 9 The optimization in 9 has been transformed into a convex optimization problem. The Lagrange function is obtained as L γτ= K R η ξ K + u + K R C min K τ P. 0 ϱ Based on the Karush-Kuhn-Tucer KKT conditions the { OPA for the th user } is obtained as p = +ϱ max 0 τ+ξη ln β +v N K+ whereϱ and τ are Lagrange multipliers respectively. They are updated by the gradient method which can be defined as ϱ i + = max{0 ϱ i ϱ R C min } and τi+=max{0τi τp K }where ϱ and τ are the positive iteration steps. We consider the rate optimization problem for individual users based on C and C and the EE target η t i.e. C 3 : K R η t ξ K + u. Let us define the achievable rate region R as a set which composes of rate-tuples for K users i.e. R ={r R K + r = R with C C and C 3 } []. We need to obtain the Pareto boundary PB of R at which it is impossible to improve a user s rate without degrading others rate. Any rate-tuple on the PB of R can be achieved as a solution of the rate profile optimization problem i.e. max { }R R s.t. R z R C C and C 3 Fig.. Capacity bounds versus N for both cases of EPA and OPA when SNR = 5dB. where z is the target ratio between the achievable rate of user and the users sum rate []. For a given rate-profile vector z with K z = the optimal solution of R can be defined by applying bisection method hence the corresponding Pareto optimal rate-tuple for user is obtained as z R. V. N UMERCAL RESULTS AND DSCUSSONS The active users are assumed to be uniformly distributed in the cell with radius R = 800 m the large-scale fading is β = u whereu r /r h 3.8 is a log-normal random variable with standard deviation σ sh = 8dBandr h = 00 m []. We set K = 0 users p dac = 5.6 mwp mix = 30.3 mwp filt = 0 mw p syn = 50 mw P sp = 5 mw σ DC = 7.5% σ MS = 9% σ cool = 0% and σ feed = 3dB respectively [5] [6]. Firstly we examine the accuracy of the derived capacity bounds of SLNR-PS for both cases of EPA and OPA. For the case of OPA we use the well-nown water-filling algorithm WFA to allocate power for each user while the QoS requirement is satisfied. We set N = 30 and σ =. n Fig. the SLNR-PS outperforms the ZF-PS. The LB on the capacity is very close to simulated results and the UB on the capacity is very tight to the simulation at high SNR region. This fact can be shown more evidently by observing Fig. where N is increased. Clearly all bounds are very tight when N > 50. These results validate our theoretical analysis. The WFA provides an improvement in capacity at low SNR region but there is no difference in system performance between two cases of EPA and OPA at high SNR region or N is large. n Fig. 3 we investigate the trade-offs between EE and SE and between EE and N based on both the UB and the LB on the capacity via numerical results. The y-axis on the left and

5 Fig. 3. For the case of EPA and β = : a EE versus SE with different N; b EE versus N with different R s requirements. Fig. 4. For both cases of EPA and OPA β = : a EE versus SE with different N;bEEversusN with different SNR. x-axis on the bottom of Fig. 3 i.e. named Fig. 3a shows the EE versus the SE with different N whereas the y-axis on the right and x-axis on the top of Fig. 3 i.e. named Fig. 3b shows the EE versus N with different R s requirements. n this example we set β = then we have optimum values of SE which is obtained from 5 equal to R opt s = and bits/s/hz corresponding to N = and 00 respectively. We observe that the R opt s is increased as N increases from 50 to 00 but the maximum EE at R opt s is decreased because P c is increased. On a different note there are two operating points of SE for a given EE as shown in Fig. 3a. However the operating point corresponding to lower SE should be avoided. n Fig. 3b the optimal numbers of N for maximizing EE are equal to N opt = and 33 corresponding to the given R s = and 60 bits/s/hz respectively. The accuracy of the optimal values has been demonstrated in Fig. 3 respectively where the maximum EEs achieved at the optimum values of SE and N are mared by a circle. Moreover the maximum EEs at R opt s and N opt obtained from the UB on the capacity are close to the corresponding ones which are obtained from the LB on the capacity. Next we consider the EE optimization problem 9 with the effect of β in Fig. 4. Similar to Fig. 3 we have Fig. 4a and 4b which compare the EE of the proposed algorithm with that of EPA. At low SNR region or when N is not large such as N = 50 in Fig. 4a the EE is substantially improved by applying the OPA algorithm. For example as we set C min = bit/s/hz and SNR = 0 db in Fig. 4b the system with OPA has an 3.7% improvement of the EE at N opt as compared to that of the EPA. Moreover the optimum values of R opt s and N opt for the case of OPA are closed to the corresponding values for the case of EPA as N is large or at high SNR region. Observing Fig. 5. Plot of three-user rate region at SNR = 0 db. Figs. 3 and 4 the effect of β significantly decreases the system EE and SE performance. Finally by solving optimization problem in with different rate-profile vector z the complete Pareto boundary of R is illustrated in Fig. 5 for K = 3 users when N = 00 C min = bit/s/hz SNR = 0 db and η t = bit/joule. The boundary loos lie a box with rounded edges and the colour bar show the individual user rate. For example for a given z ={0.63; 0.46; 0.59}wehaveR =.8 bits/s/hz then Pareto optimal per-user rates are R =.94 R =.63 and R 3 = 6.6 bits/s/hz respectively. V. CONCLUSON The capacity bounds of the SLNR-PS and the trade-off between SE and EE have been derived and validated by numerical results when N is large. The closed-form expressions of the optimal SE and optimal N corresponding to maximum EEs have also been derived based on this trade-off for the case of EPA. By applying the energy-efficient power allocation our results have shown that the SLNR-PS with OPA can improve the system EE as compared to that of EPA at low SNR region. REFERENCES [] M. Sade A. Tarighat and A. H. Sayed A leaage-based precoding scheme for downlin multi-user MMO channels EEE Trans. Wireless Commun. vol. 6 no. 5 pp. 7 7 May 007. [] F. Ruse et al. Scaling up MMO: Opportunities and challenges with very large arrays EEE Signal Process. Mag. vol. 30 no. pp Jan. 03. [3] H. Q. Ngo E. G. Larsson and T. L. Marzetta Energy and spectral efficiency of very large multiuser MMO systems EEE Trans. Commun. vol. 6 no. 4 pp Apr. 03. [4] H. Yang and T. L. Marzetta Performance of conjugate and zero forcing beamforming in large-scale antenna system EEE J. Sel. Areas Commun. vol. 3 no. pp Feb. 03. [5] D. Ha K. Lee and J. Kang Energy efficiency analysis with circuit power consumption in massive MMO systems in Proc. EEE PMRC London U.K. Sep. 03 pp [6] G. Auer et al. How much energy is needed to run a wireless networ? EEE Trans. Wireless Commun. vol. 8 no. 5 pp Oct. 0. [7] P. Patcharamaneepaorn S. Armour and A. Doufexi On the equivalence between SLNR and MMSE precoding schemes with single-antenna receivers EEE Commun. Lett. vol. 6 no. 7 pp Jul. 0. [8] P. Li D. Paul R. Narasimhan and J. Cioffi On the distribution of SNR for the MMSE MMO receiver and performance analysis EEE Trans. nf. Theory vol. 5 no. pp Jan [9] Y. Jiang M. K. Varanasi and J. Li Performance analysis of ZF and MMSE equalizers for MMO systems: An in-depth study of the high- SNR regime EEE Trans. nf. Theory vol. 57 no. 4 pp Apr. 0. [0] D. W. K. Ng E. S. Lo and R. Schober Energy-efficient resource allocation for secure OFDMA systems EEE Trans. Veh. Technol. vol. 6 no. 6 pp Jul. 0. [] R. Mochaourab P. Cao and E. Jorswiec Alternating rate profile optimization in single stream MMO interference channels EEE Signal Process. Lett. vol. no. pp. 4 Feb. 04.