LOW-COMPLEXITY LINEAR PRECODING FOR MULTI-CELL MASSIVE MIMO SYSTEMS

Transcription

1 LOW-COMPLEXITY LINEAR PRECODING FOR MULTI-CELL MASSIVE MIMO SYSTEMS Aba ammoun, Axe Müer, Emi Björnson,3, and Mérouane Debbah ing Abduah University of Science and Technoogy (AUST, Saudi Arabia Acate-Lucent Chair, Supéec, France, and 3 T Roya Institute of Technoogy, Sweden ABSTRACT Massive MIMO (mutipe-input mutipe-output has been recognized as an efficient soution to improve the spectra efficiency of future communication systems. owever, increasing the number of antennas and users goes hand-in-hand with increasing computationa compexity. In particuar, the precoding design becomes invoved since near-optima precoding, such as reguarized-zero forcing (RZF, requires the inversion of a arge matrix. In our previous wor [] we proposed to sove this issue in the singe-ce case by approximating the matrix inverse by a truncated poynomia expansion (TPE, where the poynomia coefficients are seected for optima system performance. In this paper, we generaize this technique to muti-ce scenarios. Whie the optimization of the RZF precoding has, thus far, not been feasibe in muti-ce systems, we show that the proposed TPE precoding can be optimized to maximize the weighted max-min fairness. Using simuations, we compare the proposed TPE precoding with RZF and show that our scheme can achieve higher throughput using a TPE order of ony 3. Index Terms Massive MIMO, inear precoding, ow compexity, muti-ce systems, random matrix theory.. INTRODUCTION One of the major chaenges in muti-ce systems is deaing with inter-ce and intra-ce interference. Currenty, the genera trend is to depoy mutipe antennas at the base stations (BSs, thereby enabing muti-user MIMO with fexibe spatia interference mitigation []. User separation in the downin is then performed using inear precoding. Unfortunatey, the use of optima precoding designs is far from being incorporated in current wireess standards such as LTE- Advanced [3]. This can be attributed to the fact that very accurate instantaneous channe state information (CSI is required, which can be cumbersome to achieve in practice [4]. One can imagine that this becomes worse as the number of BS antennas, M, and the number of users,, increase. Interestingy, this is not the case in reaity. As M for a This research has been supported by the ERC Starting Grant 3053 MORE (Advanced Mathematica Toos for Compex Networ Engineering and an Internationa Postdoc Grant from The Swedish Research Counci. fixed, simpe inear precoding, ie maximum ratio transmission (MRT, is asymptoticay optima [5] and robust to CSI imperfections [6]. Nevertheess, MRT is not very appeaing at practica vaues of M and, since it does not activey suppress residua inter-user interference [7]. In fact, a precoding design based on both M and growing arge, with a fixed ratio, yieds better massive MIMO performance [7]. In this regime, RZF precoding is near-optima from a throughput perspective. owever, it requires cacuation of the inverse of the Gram matrix of the joint channe of a users, which is a non-trivia operation with a compexity scaing of M. Fortunatey, the system performance is predictabe in the arge- (M, regime, where advanced toos from random matrix theory provide deterministic approximations of the achievabe rates [7]. In ight of these resuts, we proposed in [] to sove the precoding compexity issues in singe-ce systems using a new famiy of precoding schemes caed TPE precoding. This famiy was obtained by approximating the matrix inverse in RZF precoding by a matrix poynomia. A simiar approach was independenty proposed in [8]. In this paper, we extend the TPE precoding from [] to the scenario of muti-ce massive MIMO systems. A specia focus is paced on the anaysis of reaistic characteristics, incuding user-specific channe covariance matrices, imperfect CSI and piot contamination. The proofs of our resuts can be found in the extended version of this paper; see [9].. SYSTEM MODEL We consider the downin of a muti-ce system composed of L > ces. Each ce consists of a M-antenna BS serving singe-antenna users. We assume a time-division dupex (TDD protoco where the BS acquires instantaneous CSI in the upin and uses it for the downin transmission, expoiting channe reciprocity. The TDD protocos are synchronized across ces, so that piot signaing and data transmission tae pace simutaneousy. The received compex baseband signa at the mth user termina (UT in the jth ce is L y j,m = h x + n j,m ( = where x C M is the transmit signa from the th BS and h C M is the channe vector from that BS to the

2 mth UT in the jth ce, and n j,m CN (0, σ is the additive circuary-symmetric compex Gaussian noise with variance σ. The channe vectors are modeed as Rayeigh fading with h CN ( 0, R ( where the famiy of covariance matrices (R L,L, =,j=,m= satisfy the foowing conditions: Bounded norm: im sup M R < +,, j, m; Trace scaing: im inf M M tr (R > 0,, j, m; Finite dimensiona matrix space for R : It exists a finite integer S > 0 and a inear independent famiy of matrices F,..., F S such that R = S = α,f. Note that these conditions are ess restrictive than the one used in [0], where R was assumed to beong to a finite set of matrices. It is aso in agreement with severa physica channe modes presented in the iterature; for exampe, the one-ring mode with user groups from []. This channe mode considers a finite number G of groups which share approximatey the same ocation and thus the same covariance matrix. Let θ,j,g and,j,g be, respectivey, the azimuth ange and the azimuth anguar spread between the ce and the users in group g of ce j. Moreover, et d be the distance between two adjacent antennas (see Fig. in []. Then, the (u, vth entry of the covariance matrix R for users in group g is [R ] u,v =,j,g,j,g +θ,j,g,j,g +θ,j,g e jd(u v sin α dα. (3 We aso assume that a BSs use Gaussian codeboos and inear precoding, such that the jth ce transmits the signa x j = g j,m s j,m = G j s j (4 m= where G j = [g j,,..., g j, ] C M is the precoding matrix and s j = [s j,,..., s j, ] CN (0, I is the vector containing the data symbos for UTs in the jth ce. The transmission at BS j is subject to a transmit power constraint tr ( G j G j = Pj (5 where P j is the average transmit power per user in the jth ce. The received signa ( can be thus expressed as y j,m = L = = h g, s, + n j,m. (6 A we nown feature of arge-scae MIMO systems is channe hardening, which impies that the effective usefu channe h j,j,m g j,m converges to its average vaue as M. We decompose the received signa as y j,m = E [ ] ( h j,j,mg j,m sj,m + s j,m h j,j,m g j,m E [ ] h j,j,mg j,m + h g, s, + n j,m (, (j,m and assume, simiar to [6,7], that the receiver ony nows the average channe gain E[h j,j,m g j,m], the average sum interference power, E[ h g, ] and the variance of the noise σ. By treating interference as worst-case Gaussian noise, the ergodic achievabe rate of UT m in ce j is where r j,m = og ( + γ j,m (7 E [h j,j,m g j,m ] γ j,m = σ + [ ], E h g, [ ]. E h j,j,m g j,m.. Mode of Imperfect Channe State information Based on a TDD protoco, the channe is estimated in the upin. In each ce, the UTs transmit mutuay orthogona piot sequences, thereby aowing the BS to acquire CSI. Since the same set of orthogona sequences is reused in each ce, the channe estimation is corrupted by inter-ce interference; this is caed piot contamination [6]. To estimate the channe corresponding to UT in ce j, each BS correates the received signa with the piot sequence of that user. This resuts in the processed received signa (8 yj, tr = h j,j, + h j,, + n tr j, (9 ρtr j where n tr j, CN (0, I M and ρ tr > 0 is the effective piot SNR [7]. The MMSE estimate ĥj,j, of h j,j, is ( L ĥ j,j, =R j,j, S j, yj, tr =R j,j, S j, h j,, + n tr j, ρtr = where S j, = ( ρ tr + L = R,j,. The estimated channe vectors at the jth BS to a UTs in its ce is denoted by Ĥ j,j = [ĥj,j, ĥj,j,],..., C M. (0 We aso define Φ j,, = R j,j, S j, R j,, C M M and note that ĥj,j, CN (0, Φ j,j, is independent from the estimation error ĥj,j, h j,j, since the MMSE estimator is used. 3. MULTI-CELL LINEAR PRECODING 3.. Reguarized zero-forcing precoding For muti-ce systems, the optima inear precoding is unnown under imperfect CSI and requires extensive optimization procedures under perfect CSI [4]. Therefore, ony heuristic precoding schemes are feasibe in muti-ce systems. RZF precoding is the state-of-the-art heuristic scheme in terms of

3 system throughput [4]. Using the notation of [7], the RZF precoding matrix used by the BS in the jth ce is G rzf j = β j (Ĥj,j Ĥ j,j + ϕ j I M Ĥj,j ( where the scaar parameter β j > 0 is set to satisfy the power constraint in (5 and ϕ j is a positive reguarizing parameter. Prior wors have considered the optimization of the parameter ϕ j in the singe-ce case. This parameter provides a baance between maximization of the channe gain at each intended receiver (ϕ j is arge and the suppression of inter-user interference (when ϕ j is sma. To the authors nowedge, a cosed-form optimization of the reguarization parameter for muti-ce scenarios has thus far not been achieved. ence, previous wors have been restricted to the anaysis of intuitive choices of the reguarization parameter ϕ j. In this context, the wor in [7] considers massive MIMO performance of the RZF precoding in the arge-(m, regime, where M and tend to infinity such that 0 < im inf M im sup M < +. ( 3.. Truncated Poynomia Expansion Precoding The matrix mutipications and inversion in ( gives RZF precoding an unfavorabe compexity scaing of M []. Buiding on the concept of TPE used in our wor in [], we propose a new cass of ow-compexity inear precoding schemes aso for the muti-ce case. The proposed precoding originates from the Cayey-amiton theorem which states that the inverse of a M M matrix can be written as a weighted sum of its first M powers. A simpified precoding can then be obtained by considering ony the first matrix powers. Assume that the jth BS empoys a truncation order J j, then the proposed TPE precoding matrix is given by J j G TPE j = n=0 w n,j (Ĥj,j Ĥ j,j n Ĥj,j where {w n,j, j = 0,..., J j } are the J j scaar coefficients that are empoyed by the jth BS and the normaization by contros the energy of the precoding matrix. Note that whie the RZF precoding has ony a singe reguarization parameter, the proposed TPE precoding scheme offers a arger set of J j design parameters. These coefficients define a parametrized cass of precoding schemes ranging from MRT (J j = to RZF precoding, which is achieved at J j = min (M, by seecting w n,j from the characteristic poynomia of Ĥj,jĤ j,j + ϕ j I M. The coefficients are optimized ater in this paper. The compexity of TPE precoding scaes ony as M []. 4. ASYMPTOTIC PERFORMANCE ANALYSIS We provide in this section an asymptotic performance anaysis of the proposed TPE precoding. In particuar, we show that in the arge (M, -regime, the SINR experienced by the mth UT served by the jth ce can be approximated by a deterministic term which depends ony on the channe statistics. Before stating our main resuts, we cast the SINR expression (8 in a simper form. Let w j = [ T w 0,j,..., w Jj,j] and et a j,m C Jj and B C Jj Jj be given by = h j,j,m V n,j ĥ j,j,m, n [0, J j ] [B ] n,p = h V n+p+, h n, p [0, J ] where V n,j = ( Ĥj,jĤ j,j n. Then, the SINR experienced by the mth UT in the jth ce is [ ] E w j a j,m γ j,m = σ + L = E [w B w ] [ ] E w j a j,m. (3 As a j,m and B are of finite dimensions, it suffices to determine an asymptotic approximation of the expected vaue of each of their eements. For that, we introduce the functionas X j,m (t = h j,j,mσ(t, jĥj,j,m Z (t = h Σ(t, h where Σ(t, j = ( tĥj,jĥ j,j + I M. It is easy to see that = ( n X (n j,m (4 n! [B ] n,p = ( n+p+ (n + p +! Z(n+p+. (5 where X (n j,m and Z(n+p+ represent the derivatives of X j,m (t and Z (t at t = 0. Theorem Let X j,m (t and Z (t be X j,m (t = δ j,m(t + tδ j,m (t Z (t = tr (R T (t t tr Φ T (t + tδ,m (t where for each j =,..., L, m =,...,, δ j,m (t are the unique positive soutions to the foowing system of equations: ( δ j,m (t = tr tφ j,j, Φ j,j,m + tδ j, (t + I M and T j (t = ( = (M, regime we have = tφ j,j, +tδ j, (t + I M. In the arge- E [X j,m (t] X j,m (t M, + 0 E [Z (t] Z (t M, + 0.

4 Coroary The foowing hods in the arge-(m, regime: [ ] E X (n j,m X (n j,m 0, (6 M, + E [ Z (n ] Z (n M, + 0 (7 where X (n j,m and Z (n are the derivatives of X(t and Z (t, respectivey, at t = 0. The deterministic quantities X (n j,m and Z (n are functions of the pth derivatives δ (p, and T(p of δ, (t and T (t at t = 0, p = 0,..., n. Denote by X (0 j,m = tr(φ j,j,m and Z = tr(r. We can compute the deterministic sequences X (n j,m and Z (n as X (n j,m = n = Z (n,jab,m = tr ( + ( n n ( n =0 X ( j,m δ (n j,m + δ(n j,m R T (n δ (n,m n ( n =0 ( tr R T ( δ (n,m Z( Fig.. Iustration of the three-sector site depoyment with L = 3 ces considered in the simuations. these coefficients need to be scaed to satisfy the transmit power constraints tr ( G,TPE G,TPE = w T C w = P (9 where C is a J J matrix with eements given by where δ (p, and T(p can be computed using the iterative agorithm in [0]. Substituting the deterministic equivaent of Theorem into (4 and (5, we get the foowing resut. Coroary 3 Let a j,m and B be given by max = ( n X (n j,m n!, [ B ]n,p = ( n+p+ (n + p +! Z(n+p+. Then, a j,m and B converge as ( [ E B B ], E [a j,m a j,m ] 0. M, + in the arge-(m, regime and the SINRs converge as where γ j,m = γ j,m γ j,m M, + 0 (8 w j aj,ma j,m wj L = w B w w j aj,ma j,m wj. 5. SYSTEM PERFORMANCE OPTIMIZATION In the previous section, we derived deterministic equivaents of the SINR at each UT in the muti-ce system as a function of the poynomia coefficients {w,j, [, L], j [0, J ]} of the TPE precoding. These coefficients can be seected to maximize any system performance metric. Furthermore, n+m+, n, m [0, J ]. (Ĥ, [C ] n,m = tr Ĥ, (0 We woud ie to pre-optimize the weights offine, thus the weights shoud not depend on the instantaneous vaue of the channe, but ony its statistics. To this end, we substitute (9 by its asymptotic approximation wc T w = P ( where [ ] ( C = ( n+m+ n,m (n+m+! tr T (n+m+. In this paper, the performance metric is weighted maxmin fairness. In other words, we maximize the minima vaue of og (+γj,m ν j,m for some user-specific weights ν j,m. If we woud ie to mimic the performance of RZF, then ν j,m shoud be the rate that user m in ce j achieves asymptoticay with RZF precoding. Using deterministic equivaents, the corresponding optimization probem is max min w,...,w L j [,L] m [,] og (+ w j aj,ma j,m wj L = w B w wj aj,ma j,m wj ν j,m subject to w C w = P, [, L]. ( This probem is non-convex, but very simiar to the muti-cast beamforming probems anayzed in []. In particuar, we can instead sove the foowing tractabe reaxed convex probem:

5 Average rate per UT (bit/s/z RZF TPE, J = TPE, J = TPE, J = 3 TPE, J = 4 TPE, J = Number of BS antennas (M Fig.. Comparison between conventiona RZF precoding and the proposed TPE precoding with different orders J = J j j. max ξ (3 W,...,W L,ξ subject to W 0, tr ( C W = P, [, L] a j,mw j a j,m L = tr B W a j,mw j a j,m νj.mξ, j, m. 6. SIMULATION RESULTS In this section, we consider the three-site sector in Fig.. Each of the three BSs is equipped with an horizonta inear array of M antennas. The radiation pattern of each antenna is ( ( θ A(θ = min θ 3dB, 30 db, θ 3dB = 70. We assume that the UTs are divided into G = groups as described in []. The pathoss between UT m in the group g and ce is seected as in []: PL(d j,m = / ( + ( d j,m δ d 0, where δ = 3.7 is the pathess exponent and d 0 = 30 m is the reference distance. We use the channe covariance mode in (3, but scae it by the pathoss and antenna radiation pattern. For simpicity, we et the effective piot SNR, ρ tr, and a the downin SNRs, ρ d = P σ, be 5 db. We compare the average user throughput, L L j= m= E[og ( + γ j,m ], of the proposed TPE precoding with that of conventiona RZF precoding. The coefficients of TPE precoding are achieved by soving the reaxed probem in (3 offine. Using Monte- Caro simuations, Fig. shows the performance for = 40 users and different number of antennas at each BS: M {80, 60, 40, 30, 400}. Unie the singe-ce case anayzed in [], where TPE precoding is merey an approximation of RZF precoding, we see that TPE precoding achieves higher user rates for a J j 3 in the muti-ce case. This can be attributed to the tractabe offine optimization of the poynomia coefficients which aows for a better coordination of intra-ce and inter-ce interference, a feature which coud not be impemented for the RZF precoding. 7. CONCLUSION In this paper, we generaize the ow-compexity TPE precoding famiy from [] to muti-ce scenarios. In particuar, we derive deterministic equivaents for the asymptotic SINRs in massive MIMO systems where the number of antennas and users grow arge with a fixed ratio. The most interesting feature of these expressions is that they ony depend on the channe statistics and not on the instantaneous channe reaizations. This enabes us to optimize TPE precoding in an offine manner. The performance of the proposed precoding method is iustrated using simuations. We note that contrary to the singe-ce case, TPE precoding outperforms the RZF precoding in terms of both compexity and throughput. REFERENCES [] A. Müer, A. ammoun, E. Björnson, and M. Debbah, Linear precoding based on truncated poynomia expansion Part I: Large-scae singe-ce systems, IEEE J. Se. Topics Signa Process., Submitted, arxiv: [] D. Gesbert, S. any,. uang, S. Shamai, O. Simeone, and W. Yu, Muti-ce MIMO cooperative networs: A new oo at interference, IEEE J. Se. Areas Commun., vo. 8, no. 9, pp , 00. [3]. oma and A. Tosaa, LTE advanced: 3GPP soution for IMT-Advanced, Wiey, st edition, 0. [4] E. Björnson and E. Jorswiec, Optima resource aocation in coordinated muti-ce systems, Foundations and Trends in Communications and Information Theory, 03. [5] F. Ruse, D. Persson, B.. Lau, E.G. Larsson, T.L. Marzetta, O. Edfors, and F. Tufvesson, Scaing up MIMO: Opportunities and Chaenges with Very Large Arrays, IEEE Signa Process. Mag., vo. 30, no., pp , 03. [6] T.L. Marzetta, Noncooperative ceuar wireess with unimited numbers of base station antennas, IEEE Trans. Commun., vo. 9, no., pp , 00. [7] J. oydis, S. ten Brin, and M. Debbah, Massive MIMO in the UL/DL of ceuar networs: ow many antennas do we need?, IEEE J. Se. Areas Commun., vo. 3, no., pp. 60 7, 03. [8] S. Zarei, W. Gerstacer, R. R. Müer, and R. Schober, Lowcompexity inear precoding for downin arge-scae MIMO systems, in Proc. IEEE PIMRC, 03. [9] A. ammoun, A. Müer, E. Björnson, and M. Debbah, Linear Precoding Based on Truncated Poynomia Expansion Part II: Large-Scae Muti-Ce Systems, IEEE J. Se. Topics Signa Process., Submitted, arxiv: [0] J. oydis, M. Debbah, and M. obayashi, Asymptotic moments for interference mitigation in correated fading channes, in IEEE ISIT, 0. [] A. Adhiary, J. Nam, J. Y. Ahn, and G. Caire, Joint spatia division and mutipexing the arge-scae array regime, IEEE Trans. Inf. Theory, vo. 59, no. 0, 03. [] N. Sidiropouos, T. Davidson, and Z.-Q. Luo, Transmit beamforming for physica-ayer muticasting, IEEE Trans. Signa Process., vo. 54, no. 6, pp. 39 5, 006.