Efficient linear precoding for massive MIMO systems using truncated polynomial expansion

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1 Efficient inear precoding for assive MIMO systes using truncated poynoia expansion Axe Müer, Aba Kaoun, Ei Björnson, Mérouane Debbah To cite this version: Axe Müer, Aba Kaoun, Ei Björnson, Mérouane Debbah. Efficient inear precoding for assive MIMO systes using truncated poynoia expansion. IEEE 8th Sensor Array and Mutichanne Signa Processing Workshop (SAM), Jun 014, Coruna, Spain. Proceedings of the IEEE 8th Sensor Array and Mutichanne Signa Processing Workshop (SAM), pp.73-76, 014, < /SAM >. <ha > HAL Id: ha Subitted on 9 Dec 014 HAL is a uti-discipinary open access archive for the deposit and disseination of scientific research docuents, whether they are pubished or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de docuents scientifiques de niveau recherche, pubiés ou non, éanant des étabisseents d enseigneent et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 Efficient Linear Precoding for Massive MIMO Systes using Truncated Poynoia Expansion Axe Müer, Aba Kaoun, Ei Björnson, and Mérouane Debbah Acate-Lucent Chair on Fexibe Radio, Supéec, France, KAUST, Thuwa, Makkah Province, Saudi Arabia Inte Mobie Counications, Sophia Antipois, France, ACCESS, Dept. of Signa Processing, KTH, Sweden {axe.ueer, ei.bjornson, Abstract Massive utipe-input utipe-output (MIMO) techniques have been proposed as a soution to satisfy any requireents of next generation ceuar systes. One downside of assive MIMO is the increased copexity of coputing the precoding, especiay since the reativey antenna-efficient reguarized zero-forcing (RZF) is preferred to sipe axiu ratio transission. We deveop in this paper a new cass of precoders for singe-ce assive MIMO systes. It is based on truncated poynoia expansion (TPE) and iics the advantages of RZF, whie offering reduced and scaabe coputationa copexity that can be ipeented in a convenient pipeined fashion. Using rando atrix theory we provide a cosed-for expression of the signa-to-interference-and-noise ratio under TPE precoding and copare it to previous works on RZF. Furtherore, the su rate axiizing poynoia coefficients in TPE precoding are cacuated. By siuation, we find that to aintain a fixed peruser rate oss as copared to RZF, the poynoia degree does not need to scae with the syste, but it shoud be increased with the quaity of the channe knowedge and signa-to-noise ratio. I. INTRODUCTION Next generation ceuar systes need to cope with the draaticay growing nuber of user terinas (UTs) and data traffic. A proposed soution to this chaenge is to epoy assive utipe-input utipe-output (MIMO) techniques [1]. Such systes are coony defined as acro base stations (BSs) being equipped with any antennas, e.g., hundreds, whie each BS ony serves reative few UTs, e.g., tens, at the sae tie []. These systes iprove spectra and energy efficiency by dense and adaptive spatia reuse, and are robust to channe uncertainty. However, assive MIMO suffers fro potentiay prohibitivey high coputationa copexity pertaining to the precoding. The first works on assive MIMO have disissed this issue, as the coputationay sipe axiu ration transission (MRT) precoding becoes optia for an infinite nuber of BS antennas [3]. More recent works have shown that ore practicay anageabe nubers of BS antennas, ca for ore invoved inear precoding techniques, such as reguarized zero-forcing (RZF) [4]. The coputationa copexity of RZF cannot be negected in assive MIMO systes, as RZF invoves the inversion of very arge atrices [5]. In this paper, we propose a new cass of truncated poynoia expansion (TPE) precoders that repaces the atrix inversion by a atrix poynoia with J ters. TPE precoding features This research has been supported by the ERC Starting Grant MORE (Advanced Matheatica Toos for Copex Network Engineering) and an Internationa Postdoc Grant fro The Swedish Research Counci. reduced, as we as, scaabe coputationa copexity, whie offering perforance that is coparabe to RZF. The TPE technique enabes the baancing of precoding copexity and syste throughput. Using rando atrix theory [6], we derive so-caed arge-scae approxiations for the achievabe user rates for any order J of TPE precoding. These are cosed-for expressions that are tight when the nuber of BS antennas and the nuber of UTs grow arge with a fixed ratio, but aso provide cose approxiations at sa diensions. These expressions aow for optiization of the poynoia coefficients with respect to any etric; we derive the coefficients that axiize the su throughput. We note that the cobination of TPE and precoding is very new and the ony reated work is [7] by Zarei et a.. Unike this paper, the precoding in [7] is conceived to iniize the su-mse of a users. Athough our approach originates fro the sae TPE technique as in [7], our work is ore coprehensive in that we consider a channe ode which takes into account the transit correation at the base station and we optiize the throughput. An extension of this paper for ore invoved singe-ce systes can be found in [5], which aso proves the resuts presented in this paper. The treatent of the uti-ce case under a genera channe ode is avaiabe in [8]. II. SYSTEM MODEL This section defines the singe-ce downink syste with fat-fading channes, inear precoding, and channe estiation errors. The BS is equipped with M antennas and serves K singe-antenna UTs. Denoting the set of copex nubers by C, the received copex baseband signa y k C at the kth UT is y k = h H kx+n k, k = 1,..., K (1) where x C M 1 is the transit signa and h k C M 1 represents the rando channe vector between the BS and the kth UT. The additive circuary-syetric copex Gaussian noise at the kth UT is denoted by n k CN (0, σ ) for k = 1,..., K, where σ is the receiver noise variance. Each channe vector h k, for k = 1,..., K is odeed as h k CN (0 M 1, 1K ) Φ () where the channe covariance atrix Φ C M M has bounded spectra nor, as M ; thus its trace scaes ineary with

3 M. Here, 0 M 1 is a vector with M zeros. We consider a Rayeigh bock-fading ode where h k has a fixed reaization for a coherence period and then takes a new independent reaization. The covariance scaing 1 K is ony for technica reasons. The BS epoys Gaussian codebooks and inear precoding, where g k C M 1 denotes the precoding vector and s k CN (0, 1) is the data sybo of the kth UT. Based on this assuption, the transit signa in (1) can be expressed as K x = g n s n = Gs. (3) n=1 The atrix notation is obtained by etting G = [g 1... g K ] C M K be the precoding atrix and s = [s 1... s K ] T CN (0 K 1, I K ) be the vector containing a UT data sybos. Consequenty, the received signa (1) can be expressed as K y k = h H kg k s k + h H kg n s n +n k. (4) n=1,n k Let G k be the atrix G with coun g k reoved. Then the signa-to-interference-and-noise ratio (SINR) at the kth UT becoes h H k SINR k = g kgk Hh k h H k G kg H k h k+σ. (5) By assuing that each UT has perfect CSI, the ergodic achievabe data rates at the UTs are r k = E[og (1+SINR k )], k = 1,..., K. The transitter is assued to have iperfect knowedge of the instantaneous channe reaization ĥk of each UT, for k = 1,..., K. We adopt the Gauss-Markov foruation [9]: ĥ k = 1 τ h k +τn k (6) where h k is the true channe and n k CN (0 M 1, 1 K Φ) odes the independent error. The scaar paraeter τ [0, 1] indicates the quaity of the instantaneous CSI; τ = 0 gives perfect instantaneous CSI and τ = 1 corresponds to having ony statistica knowedge. The atrix Ĥ = [ĥ1... ĥk] C M K denotes the joint iperfect knowedge of a user channes. III. LINEAR PRECODING Many heuristic inear precoding schees have been proposed in the iterature, ainy because finding the optia precoding (e.g., in ters of su rate) is very coputationay deanding and thus unsuitabe for fading systes. Aong the heuristic schees we chose RZF precoding [10], which can provide cose-to-optia perforance in any scenarios. A. Review on RZF Precoding in Massive MIMO Systes Suppose we have a tota transit power constraint 1 K tr (GGH ) = P (7) where tr( ) is the trace function and we note that the scaing factor 1 K neutraizes the corresponding channe variance scaing in (), which is possibe via (8) and taking noraization into account. We stress that the tota power P is fixed, whie we et the nuber of antennas M and UTs K grow arge. Siiar to [9], we define the RZF precoding atrix as G RZF = βĥ (ĤHĤ+ξIK = β (ĤĤ H +ξi M Ĥ (8) where the power noraization paraeter β is set such that G RZF satisfies the power constraint in (7). The scaar reguarization coefficient ξ can be seected in different ways, depending on P, σ, τ, and syste diensions [9], [10]. In [9], the perforance of each UT under RZF precoding is studied in the arge-(m, K) regie, where M and K tend to infinity at the sae speed: 0 < i inf K M i sup K M < +. (9) The achievabe user perforance is characterized by SINR k in (5). Athough the SINR is a rando quantity that depends on the instantaneous vaues of the rando users channes in H and the instantaneous estiate Ĥ, it can be approxiated using deterinistic quantities in the arge-(m, K) regie; see [9]. These are quantities that ony depend on the statistics of the channes and are referred to as deterinistic equivaents, since they are aost surey (a.s.) tight in the asyptotic iit. When the deterinistic equivaents are appied at finite M and K, they are often referred to as arge-scae approxiations. The RZF precoding atrix in (8) is a function of the instantaneous CSI at the transitter. The coputation of the precoding atrix incudes arge atrix utipications and inversions, which gives a coputationa copexity of approxiatey MK operations per coherence tie [5]. This copexity scaing is intractabe for arge vaues of M and K. B. Truncated Poynoia Expansion Precoding Next, we derive a new cass of ow-copexity inear precoders based on TPE. The foowing resut fro [11] provides a otivation behind the use of poynoia expansions. Lea 1. For any positive definite Heritian atrix X, X 1 = α ( I (I αx) = α (I αx) (10) =0 where the second equaity hods if the paraeter α is seected such that 0 < α < ax n λ n(x). This ea shows that the inverse of any Heritian atrix can be expressed as a atrix poynoia. More iportanty, the ow-order ters are the ost infuentia ones, since the eigenvaues of (I αx) go to zero as grows arge. Thus, it akes sense to consider a TPE using ony the first J ters. TPE has been successfuy appied for ow-copexity uti-user detection in [11]. Next, we appy this technique to approxiate RZF precoding by a atrix poynoia. Starting fro G RZF in (8), we note that β (ĤĤ H +ξi M Ĥ ( ( ) n βα )(1 αξ) n ( α) (ĤĤH ) Ĥ. (11) =0 n=

4 Inspecting (11), we have a precoding atrix with the structure ) G TPE = w (ĤĤ H Ĥ (1) =0 where w 0,..., w are scaar coefficients. Athough the bracketed ter in (11) provides a potentia expression for w, these are not the optia vaues for J <. Aso, these coefficients do not satisfy the power constraint in (7). The precoding in (1) is coined TPE precoding and actuay defines a whoe cass of precoding atrices for different J. For J = 1, we obtain G = w 0 Ĥ, which equas MRT. Furtherore, RZF precoding can be obtained by choosing J = in(m, K) = K and coefficients based on the characteristic poynoia of (ĤĤH +ξi M (this foows fro the Cayey-Haiton theore). We refer to J as the TPE order. Let the transit signa with TPE precoding at channe use t be denoted x (t) and observe that it can be expressed as x (t) = G TPE s (t) = w x (t) =0 where s (t) is the vector of data sybos at channe use t and {Ĥs x (t) (t), = 0, = Ĥ(ĤH x (t) 1 ), 1 J 1. This is an iterative way of coputing the J ters in TPE precoding, which reoves precoding precoputation deays and enabes paraeization over utipe processing cores (e.g., appication-specific integrated circuits) and pipeining [11]. The tota coputationa copexity is approxiatey 4JM K per channe use [5], but each ter x (t) is obtained by a sipe atrix-vector utipication which is easiy ipeented in hardware. This eans that the factor J in the copexity expression is not causing any extra deays since it can be handed by adding ore processing cores that operate in parae. Thus, we can seect the TPE order J as arge as needed to obtain a certain precoding accuracy and sti enabe rea-tie coputation of the transit signa x. IV. ANALYSIS AND OPTIMIZATION OF TPE PRECODING Reca the SINR in (5) and observe that g k = Ge k and h H k G kg H k h k = h H k GGH h k h H k g kgk Hh k, where e k is the kth coun of the identity atrix I K. By substituting the TPE precoding atrix (1) into (5), the SINR writes as SINR k = w H A k w w H B k w w H A k w+σ (13) where w = [w 0... w ] T and the atrices A k, B k C J J have entries (nubered fro 0 for convenience) given by [A k ], = h H k (ĤĤ H ) Ĥek e H kĥh (ĤĤ H ) hk (14) [B k ], = h H k (ĤĤ H ) ++1 hk (15) for = 0,..., J 1 and = 0,..., J 1. In the previousy defined arge-(m, K) regie (see (9)), it can be seen that a SINRs in (13) converge to the sae asyptotic equivaent. This is foraized in our ain theore: Theore. In the arge-(m, K) regie, SINR k converges aost surey to with w H Aw SINR k w H (B A)w+σ a.s. M,K + 0 [ A ], = ( 1)+ z z and [ B ]!!, = ( 1)++1 (++1)! x ++1. The foruations for x and z are given in Coroary 3 and for the proof we refer to [5]. Coroary 3. Denote by x 0 and z 0 the deterinistic quantities x 0 1 K tr(φ) and z 0 1 τ 1 K tr(φ). One can iterativey copute the deterinistic sequences x k, and z k, unti a fixed order p as ( ) x = µ +τ µ 1 µ 1 =0 =1 ( ) x ( )µ 1, =0 1 p z = 1 ( ) 1 τ µ z ( )µ 1, 1 p. where µ = 1 K tr(φt ), T = d T(t) dt t=0 and T(t) = (I M + tφ 1+tδ(t) ) 1, with δ(t) > 0 being the unique soution for every t > 0 of the fixed point equation: δ(t) = 1 tr (ΦT(t)). K The atrix T can be coputed iterativey (see [1]). A. Optiization of the Asyptotic SINRs Next, we optiize the asyptotic SINRs with respect to the poynoia coefficients w = [w 0... w ] T under the power constraints in (7). Since the asyptotic SINRs are a equa, this corresponds to the optiization probe axiize w subject to w H Aw w H (B A)w+σ w H Cw = P where A and B are aready given in Theore and [ C ], = ( 1)++1 1 (++1)! K tr (T ++1). (16) Theore 4. Let a be an eigenvector corresponding to the axiu eigenvaue λ ax of ) 1 ) (B A+ σ 1 P C A (B A+ σ P C. (17)

5 Average per UT Rate [bit/sec/hz] 6 4 RZF τ = 0.1 RZF τ = 0.4 RZF τ = 0.7 TPE τ = 0.1 TPE τ = 0.4 TPE τ = Transit Power to Noise Ratio [db] Fig. 1. Average per UT rate vs. transit power to noise ratio for varying CSI errors at the BS (J = 3, M = 18, K = 3). Average per UT Rate [bit/sec/hz] 6 4 RZF TPE J = 4 TPE J = 3 TPE J = Transit Power to Noise Ratio [db] Fig.. Average UT rate vs. transit power to noise ratio for different orders J in the TPE precoding (M = 51, K = 18, τ = 0.1). Then λ ax is the optiu of Probe (16) and attained by P ) w opt = C 1 ( (B A+ σ 1 B A+ σ P C) 1 a P C a. This theore shows that the J poynoia coefficients that axiize the asyptotic SINR can be coputed beforehand, using ony the channe statistics. At finite M and K, there are other poynoia coefficients that provide higher achievabe rates, however, these depend on the current channe estiate Ĥ and thus ust be recoputed in each coherence period. V. SIMULATION RESULTS In this section, we copare the RZF precoding fro [10], utiizing the asyptoticay optia reguarization paraeter fro [9], with the proposed TPE precoding stated in (1). The perforance easure is the average achievabe UT rate r = 1 K K k=1 E[og (1+SINR k )]. In the siuations, we ode the channe covariance atrix as [Φ] i,j = a j i for a = 0.1, which is known as the exponentia correation ode. We first take a ook at Fig. 1 where M = 18 and K = 3. It considers the TPE order J = 3 and three eves of channe knowedge at the BS: τ {0.1, 0.4, 0.7}. We see that RZF and TPE achieve aost the sae perforance when a bad channe estiate is avaiabe (τ = 0.7). Furtherore, TPE and RZF perfor aost equa at ow SNRs, at any τ. Fig. shows ore directy the reationship between the UT rates and TPE order J. We consider τ = 0.1, M = 51 and K = 18, in order to be in a regie where TPE perfors reativey bad (see Fig. 1) and the precoding copexity is an issue. Fro Fig., we see that choosing a arger vaue for J gives a TPE perforance coser to that of RZF. However, doing so aso requires ore hardware. The proposed TPE precoding never surpasses the RZF perforance, which is natura since TPE precoding is an approxiation of RZF. VI. CONCLUSION In this paper, we have proposed a new cass of owcopexity TPE precoders, which approxiates RZF precoding to any accuracy using the TPE concept with an order in the interva 1 J K. In ters of copexity, TPE precoding has severa advantages in assive MIMO systes: 1) There is no need to copute the precoding atrix beforehand; ) the utistage structure enabes paraeization and pipeining; and 3) the paraeter J can be taiored directy to the avaiabe hardware. Athough the poynoia coefficients depend on the instantaneous channes, we have shown that the SINRs converge to a deterinistic vaue in the arge-(m, K) regie. This enabed us to copute the asyptoticay optia coefficients, using ony the statistics of the channes. REFERENCES [1] K. Hosseini, J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO and Sa Ces: How to Densify Heterogeneous Networks, in Proc. IEEE ICC, 013. [] F. Rusek, D. Persson, B.K. Lau, E.G. Larsson, T.L. Marzetta, O. Edfors, and F. Tufvesson, Scaing up MIMO: Opportunities and chaenges with very arge arrays, IEEE Signa Process. Mag., vo. 30, no. 1, pp , Jan [3] T.L. Marzetta, Noncooperative Ceuar Wireess with Uniited Nubers of Base Station Antennas, IEEE Trans. Coun., vo. 9, no. 11, pp , Nov [4] J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO in the UL/DL of Ceuar Networks: How Many Antennas Do We Need?, IEEE J. Se. Areas Coun., vo. 31, no., pp , Feb [5] A. Müer, A. Kaoun, E. Björnson, and M. Debbah, Linear Precoding Based on Poynoia Expansion: Reducing Copexity in Massive MIMO (extended version), Apri 014, arxiv: v3. [6] W. Hache, Ph. Loubaton, and J. Naji, Deterinistic Equivaents for Certain Functionas of Large Rando Matrices, Annas of Appied Probabiity, vo. 17, no. 3, pp , 007. [7] S. Zarei, W. Gerstacker, R. R. Müer, and R. Schober, Low-Copexity Linear Precoding for Downink Large-Scae MIMO Systes, in Proc. IEEE PIMRC, 013. [8] A. Kaoun, A. Müer, E. Björnson, and M. Debbah, Linear Precoding Based on Poynoia Expansion: Large-Scae Muti-Ce MIMO Systes, IEEE J. Se. Topics Signa Process., Sept. 013, In Press, arxiv: [9] S. Wagner, R. Couiet, M. Debbah, and D. T. M. Sock, Large Syste Anaysis of Linear Precoding in MISO Broadcast Channes with Liited Feedback, IEEE Trans. Inf. Theory, vo. 58, no. 7, pp , 01. [10] C. B. Pee, B. M. Hochwad, and A. L. Swindehurst, A Vector- Perturbation Technique for Near-Capacity Mutiantenna Mutiuser Counication, Part I: Channe Inversion and Reguarization, IEEE Trans. Coun., vo. 53, no. 1, pp , Jan [11] G. Sesser and F. Jondra, Low Copexity Poynoia Expansion Mutiuser Detector for CDMA Systes, IEEE Trans. Veh. Techno., vo. 54, no. 4, pp , Juy 005. [1] J. Hoydis and M. Debbah and M. Kobayashi, Asyptotic Moents for Interference Mitigation in Correated Fading Channes, in Proc. Int. Syp. Inf. Theory (ISIT), 011.