Optimal Linear Precoding in Multi-User MIMO Systems: a Large System Analysis Luca Sanguinetti, Emil Björnson, Mérouane Debbah, Aris Moustakas

Optimal Linear Precoding in Multi-User MIMO Systems: A Large System Analysis Luca Sanguinetti, Emil Björnson, Mérouane Debbah, Aris Moustakas

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Luca Sanguinetti, Emil Björnson, Mérouane Debbah, Aris Moustakas. Optimal Linear Precoding in Multi-User MIMO Systems: A Large System Analysis. 2014 IEEE Global Communications Conference (Globecom), Dec 2014, Texas, United States. 10.1109/glocom.2014.7037420. hal-01098917

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Luca Sanguinetti∗‡, Emil Björnson‡†, Mérouane Debbah‡ and Aris L. Moustakas‡§ ∗Dipartimento di Ingegneria dell’Informazione, University of Pisa, Pisa, Italy ‡Alcatel-Lucent Chair, Ecole supérieure d’électricité (Supélec), Gif-sur-Yvette, France †Dept. of Signal Processing, KTH, Stockholm, and Dept. of Electrical Engineering, Linköping University, Sweden §Department of Physics, National & Capodistrian University of Athens, Athens, Greece

Abstract—We consider the downlink of a single-cell multi-user ratios (SINRs) [5]–[8]. This problem is receiving renewed MIMO system in which the base station makes use of N antennas interest nowadays due to the emerging research area of green K to communicate with single-antenna user equipments (UEs) cellular networks [9]. In particular, we consider the downlink randomly positioned in the coverage area. In particular, we focus on the problem of designing the optimal linear precoding for of a single-cell multi-user MIMO system in which the BS minimizing the total power consumption while satisfying a set makes use of N antennas to communicate with K single- of target signal-to-interference-plus-noise ratios (SINRs). To gain antenna UEs randomly positioned in the coverage area. Under insights into the structure of the optimal solution and reduce the assumption of perfect channel state information (CSI), the the computational complexity for its evaluation, we analyze the solution to the power minimization problem in this context asymptotic regime where N and K grow large with a given ratio and make use of recent results from large system analysis to was originally computed in [10] and later extended to different compute the asymptotic solution. Then, we concentrate on the scenarios in [6]–[8]. In particular, it turns out that the optimal asymptotically design of heuristic linear precoding techniques. linear precoder depends on some Lagrange multipliers whose Interestingly, it turns out that the regularized zero-forcing (RZF) computation can be performed using convex optimization tools precoder is equivalent to the optimal one when the ratio between or solving a fixed-point problem [4]. Although possible, both the SINR requirement and the average channel attenuation is the same for all UEs. If this condition does not hold true but only the approaches do not provide any insights into the structure of the same SINR constraint is imposed for all UEs, then the RZF can optimal values. Moreover, the computation must be performed be modified to still achieve optimality if statistical information for any new realization of the MIMO channel matrix. of the UE positions is available at the BS. Numerical results are To overcome these issues, we follow the same approach as used to evaluate the performance gap in the finite system regime in [11]–[14] and resort to the asymptotic regime where N and and to make comparisons among the precoding techniques. K grow large with a given ratio c = K/N. Differently from I.INTRODUCTION [11]–[14], the asymptotically optimal values of the Lagrange Multiple-Input Multiple-Output (MIMO) technologies are multipliers are computed using the approach adopted in [15], currently being adopted in many wireless communication which provides us a much simpler means to overcome the standards such as fourth generation (4G) cellular networks technical difficulties arising with the application of standard [1]. The main limiting factor in multi-user MIMO systems random matrix theory tools (see for example [13]). As already is the multiple-access interference (MAI). In uplink trans- pointed out in [11]–[14], in the asymptotic regime the optimal missions, MAI mitigation is typically accomplished at the values can be computed in closed-form through a nice and base station (BS) using linear multi-user detectors or non- simple expression, which depends only on the user positions linear techniques based on layered architectures. In downlink and SINR requirements. The above results are then used to transmissions, MAI mitigation can only be accomplished at validate the optimality of different heuristic linear precoding the BS using precoding techniques. As shown in [2], the techniques, which are inspired by the widely used regularized capacity-achieving precoding strategy is dirty paper coding zero-forcing (RZF) concept [16]–[18] and its extensions to (DPC). Although optimal, the implementation of DPC requires include arbitrary user priorities [4]. The optimal regularization a tremendous computational complexity at both BS and user parameter is provided in the asymptotic regime. To the best of equipments (UEs). On the other hand, a practical approach authors’ knowledge, this is the first time that such a result is that has received considerable attention (due to its simplicity) found since most of the related works are focused on sum rate is represented by linear precoding or beamforming [3], [4]. maximization. Comparisons are then made with two heuristic In this work, we focus on the problem of designing the op- techniques. The former is the classical RZF precoder [16] timal linear precoding for minimizing the total transmit power while the latter is referred to as position-aware RZF (PA- while satisfying a set of target signal-to-interference-plus-noise RZF) precoder since it relies on knowledge of the UE positions [17]. Interestingly, it turns out that PA-RZF is equivalent to L. Sanguinetti is funded by the People Programme (Marie Curie Actions) FP7 PIEF-GA-2012-330731 Dense4Green. E. Björnson is funded by an the optimal linear precoder when the same SINR constraint is International Postdoc Grant from the Swedish Research Council. A. L. imposed for all UEs. On the other hand, the commonly used Moustakas is the holder of the DIGITEO “ASAPGONE” Chair. This research RZF precoder becomes optimal only when the ratio between has also been supported by the FP7 NEWCOM# (Grant no. 318306), the ERC Starting MORE (Grant no. 305123), the French pôle de compétitivité the SINR requirement and the average channel attenuation is SYSTEM@TIC within the project 4G in Vitro. the same for all UEs. Numerical results are used to evaluate the performance gap in the finite system regime and to make III. OPTIMAL LINEAR PRECODING comparisons among the different precoding techniques. As originally shown in [10], the non-convex optimization problem in (3) can be put in a convex form by reformulating II. SYSTEM MODEL AND PROBLEM FORMULATION the SINRP constraints as second-order cone constraints. In ⋆ We consider the downlink of a single-cell multi-user MIMO doing so, the optimal V is found to be [6]–[8] system in which the BS makes use of N antennas to commu- K −1 nicate with K single-antenna UEs. The K active UEs change ⋆ ⋆ H ⋆ V = λi hihi + NIN H√P (5) over time and are randomly selected from a large set of UEs i=1 ! within the coverage area. The physical location of UE k is X CN×K ⋆ x R2 where H = [h1, h2,..., hK ] and λ = denoted by k (in meters) and it is computed with ⋆ ⋆ ⋆ T ∈ respect to the BS∈ (assumed to be located in the origin). The [λ1, λ2,...,λK ] is the positive unique fixed point of the function l( ): R2 R describes the large-scale channel following equations [4]–[6]: · → + fading at different user locations; that is, l(xk) is the average 1 ⋆ 1 1+ λk = −1 (6) channel attenuation due to path-loss and shadowing at location γk K H ⋆ H xk. The large-scale fading between a UE and the BS is hk λi hihi + NIN hk assumed to be the same for all BS antennas. This is reasonable i=1 P⋆ ⋆ ⋆ ⋆ since the distances between UEs and BS are much larger than for k =1, 2,...,K. Also, P = diag p1,p2,...,pK is a di- the distance between the BS antennas. Since the forthcoming agonal matrix whose entries are such that{ the SINR constrain} ts analysis does not depend on a particular choice of l( ), we in are all satisfied with equality when V = V⋆. Plugging · P ⋆ ⋆ ⋆ ⋆ T keep it generic. Perfect CSI is assumed to be available at the (2) into (4), the optimal vector p = [p1,p2,...,pK ] is BS for analytic tractability. The imperfect CSI case is left for computed as [4] 1 future work. ⋆ 2 −1 p = σ D 1K (7) The BS shall convey the information symbol sk to UE k using linear precoding. The symbol vector s = where the (k,i)th element of D CK×K is T K×1 [s1,s2,...,sK ] C originates from a Gaussian code- ∈ ∈ H 1 H ⋆ 2 book with zero mean and covariance matrix Es[ss ] = I . h a for k = i K D γk k k (8) N×K [ ]k,i = | H ⋆ | Denoting by V = [v1, v2,..., vK ] C the precoding h a 2 for k = i ∈ (−| k i | 6 matrix, the received sample yk C at UE k takes the form ∈ ⋆ ⋆ K ⋆ H with ak being the kth column of A = ( i=1 λi hihi + H −1 yk = hk Vs + nk (1) NIN ) H. V⋆ P⋆ p⋆ 2 As seen, in (5) is parameterized by λ and , where where nk (0, σ ) is the additive noise and the entry hk,n λ⋆ needs to be evaluated by an iterative procedure due to the ∼ CN hH C1×N of the row vector k = [hk,1,hk,2,...,hk,N ] is the fixed-point equations in (6). This is a computationally demand- channel propagation coefficient between the ∈th antenna at n ing task when N and K are large since the matrix inversion the BS and the kth UE. We assume a Rayleigh fading channel operation in (6) must be recomputed at every iteration and model hk = l(xk)wk with wk (0, IN ) accounting 2 ∼ CN its computational complexity scales proportionally to N K. for the small-scale fading channel. The SINR at the kth UE ⋆ p Moreover, computing λ as the fixed point of (6) does not is easily written as [18] provide any insights into the optimal structure of both λ⋆ and ⋆ H 2 p . In addition, the parameter values depend directly on the hk vk SINR = . (2) channel vectors hi and change at the same pace as the small- k K { } H 2 2 scale fading (i.e., at the order of milliseconds). hk vi + σ i=1,i=6 k To overcome the above issues, we assume that N,K P with K/N = c (0, 1] and use some recent tools→ in As mentioned earlier, we consider the power minimization large∞ system analysis∈ to compute the so-called deterministic problem whose mathematical formulation is as follows: equivalents of λ⋆ and p⋆. For later convenience, we call : minimize P = tr(VVH ) (3) K K V 1 γ 1 γ P ξ =1 i and A = i . (9) subject to SINR γ k =1, 2,...,K (4) − N 1+ γi K l(xi) k k i=1 i=1 ≥ X X where γk is the given SINR target of UE k obtained as (under A. Asymptotically Optimal Linear Precoding rk the assumption of Gaussian codebooks) γk =2 1 with rk The following theorem provides the solution to the opti- − being the target user rate in bit/s/Hz. For later convenience, mization problem in (3) in the asymptotic regime. T we call γ = [γ1,γ2,...,γK ] . Theorem 1. If N,K with c (0, 1], then → ∞ ∈ 1We limit to observe that might in principle be included following the same ⋆ a.s. max λk λk 0 (10) approach adopted in [16]. See also [19]. k=1,2,...,K − −→

and equal to some ζ 0, i.e., ⋆ a.s. ≥ max pk pk 0 (11) γk k=1,2,...,K | − | −→ = ζ k =1, 2,...,K, (15) l(xk) ⋆ where λk and pk are the deterministic equivalents of λk and ⋆ then λk in (12) takes the form pk, respectively, and are given by K −1 γk ζ 1 γi λk = (12) λk = = ζ 1 . (16) l(xk)ξ ξ − N 1+ γi i=1 ! X and Corollary 2 ([13]). If the same target SINR is imposed for γ σ2 p = k P + (1+ γ )2 (13) each user, i.e., k l(x )ξ2 l(x ) k k k γ = γ1k, (17) with then λ in (12) reduces to cAσ2 k P = (14) −1 ξ γ γ γ λk = = 1 c (18) l(x )ξ l(x ) − 1+ γ being the deterministic equivalent of the transmit power P . k k Proof: Similar results have previously been derived by and P becomes applying standard random matrix theory tools to the right- γ −1 P = cAσ2 1 c . (19) hand-side of (6). However, the application of these tools to the − 1+ γ problem at hand is not analytically correct since the Lagrange IV. HEURISTIC LINEAR PRECODING multipliers in (6) are a function of the channel vectors hk . To overcome this issue, we make use of the same approach{ } Inspired by the optimal linear precoding in (5), we now con- adopted in [15] whose main steps are sketched in [20]. On sider suboptimal precoding techniques that builds on heuristics the other hand, (13) is proved using standard random matrix [4]. To this end, we let V take the following general form theory results (omitted for space limitations). K −1 H The following remarks elaborate on some of the insights V = αihihi + NρIN H√P (20) that are obtained from Theorem 1. i=1 ! Remark 1. In sharp contrast to (6), the computation of X λk where α = [α , α ,...,α ]T is now a given vector with in (12) only requires knowledge of the user position through 1 2 K positive scalars and ρ is a design parameter to be optimized. x . This information can be easily observed and estimated l( k) Note that (20) is basically obtained from (5) by setting λ = accurately at the BS because it changes slowly with time k αk/ρ for all k. As before, the power allocation P is computed (relative to the small-scale fading). The Lagrange multiplier according to (7) and satisfies all the SINR constraints with λk is known to act as a user priority parameter that implicitly equality. determines how much interference the other UEs may cause to Observe that if α is set to 1K , then V in (5) reduces to the UE k [4]. Interestingly, its asymptotic value λk is proportional well-known RZF precoder [16]: to the SINR γk and inversely proportional to l(xk) such that − users with weak channels have larger values. Higher priority is K 1 H thus given to users that require high performance and/or have VRZF = hihi + NρIN H√P. (21) i=1 ! weak propagation conditions [4]. X Remark 2. A known problem with using the asymptotically This particular precoding matrix is also known as the transmit optimal power allocation in Lemma 1 is that the target SINRs Wiener filter and signal-to-leakage-and-noise ratio (SLNR) are not guaranteed to be achieved at finite numbers of antennas maximizing beamforming (see Remark 3.2 in [18] for a (see for example [14]). This is because the approximation historical exposition). errors are translated into fluctuations in the resulting SINR On the other hand, if the BS makes use of knowledge of −1 values. However, these errors rapidly vanish also in the finite the user positions and let α be equal to L 1K with L = regime when N is larger than K, which is the regime diag l(x1),l(x2),...,l(xK ) , then the processing matrix V envisioned for massive MIMO systems [21]. It can also be in (5){ reduces to (see also [17],} [19], [20]) avoided by using only the deterministic equivalents λk of K −1 the Lagrange multipliers and computing the power allocation H VPA−RZF = wiw + NρIN H√P (22) coefficients according to (7). This approach retains most of i i=1 ! the complexity benefits of the asymptotic analysis. X which we refer to as PA-RZF precoder in the sequel. The following corollaries can be easily obtained from The- Differently from the optimal linear precoding that requires orem 1 and will be useful later on. to compute the fixed point of a set of equations, the optimiza- Corollary 1. If the ratio between the SINR requirement and tion of a linear precoder in the form of (20) requires only to the average channel attenuation is the same for all UEs and look for the value of ρ minimizing the transmit power. This can generally not be done in closed-form but requires a numerical with µ⋆ being solution of the following fixed point equation: optimization procedure [4]. To overcome this problem, the −1 K K 2 asymptotic regime is analyzed in the sequel. l(xi)γi (l(xi)) µ⋆= . (30) ⋆ 3 ⋆ 3 A. Asymptotic Analysis of the Heuristic Linear Precoding i=1 (1 + l(xi)µ ) ! i=1 (1 + l(xi)µ ) ! X X We keep α generic and look for the value of ρ that Proof: The proof easily follows from the results of minimizes the total transmit power P in (3) when N,K Theorem 1 setting αi =1 for i =1, 2,...,K. with c (0, 1]. In doing so, the following result is obtained.→ ∞ ∈ Corollary 4. If a PA-RZF precoder is used and N,K Theorem 2. If N,K with c (0, 1], then the parameter with c (0, 1], then → ∞ → ∞ ∈ ∈ ρ minimizing the deterministic equivalent of P with V given 1 c by (5) is ρ⋆ = (31) PA−RZF β − 1+ β K ⋆ 1 1 αil(xi) with β > 0 being the average target SINR given by ρ = ⋆ ⋆ (23) µ − N 1+ αil(xi)µ i=1 K X 1 where ⋆ is the solution of the following fixed point equation: β = γk. (32) µ K k=1 K K −1 X α l(x )γ (α l(x ))2 µ⋆= i i i i i . (24) The deterministic equivalent of the minimum transmit power ⋆ 3 ⋆ 3 i=1 (1 + αil(xi)µ ) ! i=1 (1 + αil(xi)µ ) ! reduces to X X −1 In addition, the deterministic equivalent of pk takes the form 2 β P PA−RZF = cAσ 1 c . (33) γ σ2 − 1+ β k x ⋆ 2 (25) pk = ⋆ 2 P + (1 + αkl( k)µ ) l(xk)(µ ) l(xk) Proof: The result follows directly from Theorem 1 setting where αi =1/l(xi) for i =1, 2,...,K. cAσ2 Interestingly, the above results can be used to prove under P = (26) 1 (µ⋆)2F cB which conditions RZF and PA-RZF are optimal. − 2 − is the deterministic equivalent of transmit power with A given Corollary 5. If condition (15) holds true, then RZF becomes by (9) and the optimal linear precoder in the asymptotic regime.

K 2 1 γi Proof: From (15), it follows that l(xk)γk = (l(xk)) ζ. B = (27) ⋆ K ⋆ 2 Plugging this result into (30) yields µ = ζ from which we i=1 (1 + αil(xi)µ ) X get K x 2 1 (αil( i)) K F2 = . (28) N x ⋆ 2 ⋆ 1 1 γi i=1 (1 + αil( i)µ ) ρRZF = 1 (34) ζ − N 1+ γi X i=1 ! Proof: The proof is detailed in Appendix and operates X in two steps. In the first step, we use the results of Theorem by simple manipulations. Plugging this result into (21) we 1 in [16] to compute SINRk and P , i.e., the deterministic obtain equivalents of SINRk and P , respectively. In the second step, K −1 1 1 we set SINRk = γk for k = 1, 2,...,K and compute the V⋆ = w wH + NI H√P (35) RZF ρ⋆ ρ⋆ i i N corresponding powers pk , which are eventually used to RZF RZF i=1 ! { } X obtain P . The latter takes the form in (26) from which taking ⋆ the derivative with respect to ρ we obtain (23) and (24). which is equal to (5) after replacing λk with λk in (16). As mentioned earlier, this is the first time that the optimal Corollary 6. If condition (17) holds true, then PA-RZF be- value of ρ minimizing the power consumption is given in comes the optimal linear precoder in the asymptotic regime explicit form for a generic heuristic precoding matrix V ⋆ defined as in (20). Most of the existing works have only looked Proof: If γ = γ1k, then (31) reduces to ρPA−RZF = 1 c and V⋆ in (22) becomes equivalent to (5) for the value of ρ that maximizes the sum rate of the network γ 1+γ PA−RZF − ⋆ (see for example [16]). after replacing λk with λk given by (18). From the results of Theorem 2, the optimal value of ρ for V. NUMERICAL RESULTS RZF or PA-RZF easily follows. In this section, Monte Carlo simulations are used to validate Corollary 3. If a RZF precoder is used and N,K with the analysis in the asymptotic regime and to make compar- c (0, 1], then → ∞ ∈ isons between optimal linear precoding and different heuristic K precoding techniques. We assume that the UEs are uniformly ⋆ 1 1 l(xi) ρRZF = ⋆ ⋆ (29) distributed in a circular cell with radius D = 250 m and µ − N 1+ l(xi)µ i=1 minimum distance m. Moreover, we consider X Dmin = 15 1 10 ZF RZF PA-RZF A-OLP 0 10 OLP 0 10

−1 10

−2 10 ZF Average transmit power [Watt] Average transmit power [Watt] RZF PA-RZF A-OLP OLP −3 −1 10 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 8 10 12 14 16 18 20 Rate per user r [bit/s/Hz] N

Fig. 1. Average transmit power in Watt vs. the rate per user r in bit/s/Hz Fig. 2. Average transmit power in Watt vs. N when K = 8 and the user when K = 8 and N = 10. rates takes values within the interval [2, 3] bit/s/Hz. a system in which the large-scale fading is dominated by consumption while satisfying a set of target SINRs. The solu- κ tion to this problem is generally given by solving a set of fixed- the path-loss [22]. This is modelled as l(x) = d0/ x for k k point equations, which is cumbersome in large-scale MIMO x Dmin where κ 2 is the path-loss exponent and the k k ≥ ≥ systems. To simplify the analysis and overcome complexity constant d0 > 0 regulates the channel attenuation at distance issues, we have resorted to the asymptotic regime in which Dmin. In all subsequent simulations, we set κ = 3.76 and −3.53 the number of antennas and users grow large with a given d0 = 10 . In addition, the transmission bandwidth is W = 10 MHz and the total noise power W σ2 is 104 dBm. ratio. The asymptotic solutions to the fixed-point equations − We begin by considering a cellular network in which the have been given in closed form, thereby providing insights on same rate r in bit/s/Hz must be guaranteed to each UE. This the optimal precoding structure. In particular, we have used r these results to prove that the conventional RZF precoding amounts to saying that γk = γ =2 1 for k =1, 2,...,K. Fig. 1 illustrates the average transmit− power in Watt with technique is the optimal one in the asymptotic regime when the K = 8 and N = 10 when r spans the interval from 0.1 ratio between the SINR requirement and the average channel to 5 bit/s/Hz. The curves labelled OLP and A-OLP refer to attenuation is the same for all UEs. A position-aware RZF the performance of the optimal and asymptotically optimal (PA-RZF) precoding that exploits statistical knowledge of the linear precoders, respectively. On the other hand, ZF refers to UE positions has been shown to be asymptotically optimal in the classical zero-forcing precoder. From the results of Fig. 1, realistic scenarios where the SINR constraints are the same it follows that OLP and A-OLP have substantially the same but the path-losses are different. performance. As pointed out in Remark 3, PA-RZF provides APPENDIX the same performance of A-OLP. While PA-RZF achieves If V takes the generic heuristic form in (20), then for any T only a marginal gain compared to RZF, a substantial power given α = [α1, α2,...,αK ] and ρ the following lemma can reduction is obtained with respect to ZF for moderate values be proved using the results of Theorem 1 in [16]. of r. The mean-square-error of the effective user rates (not Lemma 1. If N,K with c (0, 1], then reported here for space limitations) is found to be smaller than → ∞ ∈ a.s. 2% meaning that the performance loss is reasonably negligible. P P 0 (36) − −→ Fig. 2 plots the average transmit power in Watt vs. when a.s. N SINRk SINRk 0 (37) K =8 and the user rates rk are randomly taken within the − −→ interval [2, 3] bit/s/Hz. Although{ } different rates are requested where P and SINRk are given by by the UEs, PA-RZF has substantially the same performance ′ K cµ pil(xi) of A-OLP for any value of N. A significant gap is observed P = (38) K 2 i=1 (1 + αil(xi)µ) with respect to ZF for values of N in the order of K, while all X 2 the schemes guarantee basically the same performance when pkl(xk)µ SINRk = 2 (39) N becomes larger. σ x 2 P + l(xk) [1 + αkl( k)µ] VI. CONCLUSIONS where µ is the solution of the following fixed point equation In this work, we have focused on a single-cell multi-user K −1 1 αil(xi) MIMO system and have studied the problem of designing µ = + ρ (40) N 1+ αil(xi)µ linear precoding techniques for minimizing the total power i=1 ! X and µ′ in (38) is its derivative with respect to ρ. [5] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. To proceed further, we set SINRk = γk for k =1, 2,...,K Tech., vol. 53, no. 1, pp. 18–28, Jan. 2004. and compute the corresponding power P . [6] A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 161–176, Jan. 2006. Lemma 2. If SINRk is set equal to γk for k = 1, 2,...,K, [7] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna then P is found to be downlink with per-antenna power constraints,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2646–2660, Jun. 2007. cAσ2 (41) [8] E. Björnson, G. Zheng, M. Bengtsson, and B. Ottersten, “Robust P = 2 1 µ F2 cB monotonic optimization framework for multicell MISO systems,” IEEE − − Trans. Signal Process., vol. 60, no. 5, pp. 2508–2523, May 2012. with A and µ being given by (9) and (40) whereas B and F2 [9] Y. Chen, S. Zhang, S. Xu, and G. Li, “Fundamental trade-offs on green takes the form in (27) and (28). wireless networks,” IEEE Commun. Mag., vol. 49, no. 6, pp. 30–37, June 2011. Proof: Setting SINRk in (39) equal to γk leads to [10] M. Bengtsson and B. Ottersten, “Optimal and suboptimal transmit beamforming,” in Handbook of Antennas in Wireless Communications, 2 γ P + σ [1 + α l(x )µ]2 L. C. Godara, Ed. CRC Press, 2001. γ k l(xk) k k [11] S. Lakshminaryana, J. Hoydis, M. Debbah, and M. Assaad, “Asymptotic p = k = (42) k f (ρ) n l(x )µ2 o analysis of distributed multi-cell beamforming,” in IEEE 21st Interna- k k tional Symposium on Personal Indoor and Mobile Radio Communica- which used in (38) yields tions (PIMRC), Sept 2010, pp. 2105–2110. [12] Y. Huang, C. W. Tan, and B. Rao, “Large system analysis of power 2 σ 2 minimization in multiuser MISO downlink with transmit-side channel ′ K γ P + [1 + α l(x )µ] cµ 1 i l(xi) i i correlation,” in International Symposium on Information Theory and its P = . (43) − µ2 K n 2 o Applications (ISITA), Oct. 2012, pp. 240–244. i=1 (1 + αil(xi)µ) [13] R. Zakhour and S. Hanly, “Base station cooperation on the downlink: X Large system analysis,” IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. Solving with respect to P produces 2079–2106, Apr. 2012. [14] A. T. H. Asgharimoghaddam and N. Rajatheva, “Decentralizing ′ 2 cµ Aσ the optimal multi-cell beamforming via large system analysis,” in ′ (44) P = 2 µ . Proceedings of the IEEE International Conference on Communications, − µ 1+ c 2 B µ Sydney, Australia, June 2014. [Online]. Available: http://arxiv.org/abs/ Observing that the derivative of µ in (40) with respect to ρ is 1310.3843 2 [15] R. Couillet and M. McKay, “Large dimensional analysis and opti- ′ µ 2 the result in (41) follows from (44). mization of robust shrinkage covariance matrix estimators,” Journal of µ = 1−µ F2 Taking the derivative of in (41) with respect to yields Multivariate Analysis, vol. 131, no. 0, pp. 99 – 120, 2014. P ρ [16] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large system (the mathematical details are omitted for space limitations) analysis of linear precoding in correlated MISO broadcast channels under limited feedback,” IEEE Transactions on Information Theory, ′ µ P =2c2Aσ2µ′ A−B (45) vol. 58, no. 7, pp. 4509–4537, July 2012. 2 2 [17] R. Muharar, R. Zakhour, and J. Evans, “Optimal power allocation and (1 µ F2 cB) − − user loading for multiuser MISO channels with regularized channel with inversion,” IEEE Trans. Commun., vol. 61, no. 12, pp. 5030–5041, Dec. 2013. K 2 K 1 (αil(xi)) 1 αil(xi)γi [18] E. Björnson and E. Jorswieck, “Optimal resource allocation in coordi- = = . nated multi-cell systems,” Foundations and Trends in Communications A K 3 B K 3 i=1 (1 + αil(xi)µ) i=1 (1 + αil(xi)µ) and Information Theory, vol. 9, no. 2-3, pp. 113–381, 2013. X X [19] L. Sanguinetti, A. L. Moustakas, and M. Debbah, “Interference ⋆ From (45), it turns out that the optimal µ is such that µ = management in 5G reverse TDD HetNets: A large system analysis,” / from which using (40) the optimal ρ is found to be in submitted to IEEE J. Sel. Areas Commun., July 2014. [Online]. B A Available: http://arxiv.org/abs/1407.6481 the form of (23) in the text. [20] L. Sanguinetti, A. L. Moustakas, E. Björnson, and M. Debbah, “Large ACKNOWLEDGMENT system analysis of the energy consumption distribution in multi-user MIMO systems with mobility,” submitted to IEEE Trans. Wireless The authors thank Dr. Romain Couillet for helpful dis- Commun., June 2014. [Online]. Available: http://arxiv.org/abs/1406.5988 cussions on the large system analysis of the optimal linear [21] F. Rusek, D. Persson, B. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with precoding and in particular for the results of Theorem 1. very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40– 60, Jan. 2013. REFERENCES [22] Further advancements for E-UTRA physical layer aspects (Release 9). [1] Q. Li, G. Li, W. Lee, M. il Lee, D. Mazzarese, B. Clerckx, and Z. Li, 3GPP TS 36.814, Mar. 2010. “MIMO techniques in WiMAX and LTE: a feature overview,” IEEE Commun. Mag., vol. 48, no. 5, pp. 86–92, May 2010. [2] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3936–3964, Sept 2006. [3] A. Gershman, N. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson, and B. Ottersten, “Convex optimization-based beamforming,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 62–75, May 2010. [4] E. Björnson, M. Bengtsson, and B. Ottersten, “Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure [lecture notes],” IEEE Signal Processing Magazine, vol. 31, no. 4, pp. 142–148, July 2014.