Gokhan M. Guvensen, Member, IEEE, and Ender Ayanoglu, Fellow, IEEE. Abstract

Transcription

1 A Generaized Framework on Beamformer esign 1 and CSI Acquisition for Singe-Carrier Massive MIMO Systems in Miimeter Wave Channes Gokhan M. Guvensen, Member, IEEE, and Ender Ayanogu, Feow, IEEE arxiv: v1 [cs.it] 5 Ju 2016 Abstract Recenty, a two-stage beamforming concept under the name of Joint Spatia ivision and Mutipexing JSM), a kind of divide-and-conquer approach based on statistica user-grouping, has been proposed to enabe simpified system operations in massive MIMO. In this study, we estabish a genera framework on the reduced dimensiona channe state information CSI) estimation and pre-beamformer design for frequency-seective massive MIMO systems empoying singe-carrier SC) moduation in time division dupex T) mode by expoiting the joint ange-deay domain channe sparsity in miimeter mm) wave frequencies which is often characterized with imited scattering and hence correatedness in the spatia domain). The main contribution of this work is threefod. First, by an inspiration from the user-grouping idea in the JSM framework), the reduced rank minimum mean square error RR-MMSE) instantaneous CSI estimator, based on generic subspace projection taking the joint ange-deay power profie into account, is derived for spatiay correated wideband MIMO channes. Second, the statistica prebeamformer design is considered for frequency-seective SC massive MIMO channes. We examine the dimension reduction and subspace beamspace) construction on which the RR-MMSE estimation can be reaized as accuratey as possibe. The generaized eigenvector beamspace GEB) appears to be a neary optima pre-beamformer when the eigenspaces of different resovabe muti-path components are assumed to be neary orthogona. Finay, a spatio-tempora domain correator type reduced rank channe estimator, as an approximation of the RR-MMSE estimate, is obtained by carrying out east square LS) estimation in a proper reduced dimensiona beamspace. It is observed that the proposed techniques show remarkabe robustness to the piot interference or contamination) with a significant reduction in piot overhead thanks to the subspace projection. Index Terms The authors are with the Center for Pervasive Communications and Computing CPCC), ept. of EECS, UC Irvine, CA, USA e-mai: g.m.guvensen@uci.edu and ayanogu@uci.edu.)

2 2 Beamforming, massive MIMO, miimeter wave, channe estimation, dimension reduction, reduced rank Wiener fiter, MMSE estimator, user-grouping, ange-deay channe sparsity, singe-carrier, muti-path channe, spatia correation, AoA support, JSM I. INTROUCTION Massive mutipe-input mutipe-output MIMO) systems, which are equipped with a arge number of antenna eements at the base station BS) to serve a reativey smaer number of user terminas UTs) simutaneousy, are beieved to be one of the key technoogies for next-generation ceuar systems such as the upcoming 5G standard [1], [2]. With its potentia arge gains in spectra and energy efficiency, massive MIMO is especiay promising for outdoor ceuar systems operating at miimeter mm) wave frequencies, where arge antenna arrays can be packed into sma form factors, and extremey arge bandwidths are avaiabe for commercia use e.g., up to 7 GHz in the 60 GHz band) [3], [4]. Thus, it is anticipated that massive MIMO systems in the mm wave range form an important part of 5G systems expected to support much arger, e.g., 1000 times faster data rates than the currenty depoyed standards [2]. Instantaneous channe state information CSI) at BS is essentia for massive MIMO transmission, since muti-user precoding at downink or muti-user decoding at upink necessitates accurate CSI in order to capitaize the aforementioned spatia diversity and mutipexing benefits of the channe [5]. In practice, CSI is typicay obtained with the assistance of the periodicay inserted piot signas [5]. This brings the piot overhead, namey, the amount of transmission resources signaing dimensions per time-frequency channe coherence sot) consumed by the training data to be proportiona to the number of active users in the system for upink training, and the number of BS antennas for downink training respectivey [6]. The acquisition of CSI in massive MIMO transmission has been studied extensivey in the iterature. One of the primary frameworks is the frequency division dupex F) mode, where CSI is typicay obtained through expicit downink training and upink imited feedback) [7]. Since use of the F operation imposes a severe imit on the number of BS antennas due to the piot overhead, aternativey, CSI at the BS can be acquired by means of upink training in time division dupex T) mode, where the upink piots provide the BS with downink as we as upink channe estimates simutaneousy via everaging the channe reciprocity [5], [6]. Athough the T mode of operation eiminates the need for feedback and reduces the piot overhead when compared to the F systems, the processing of the signas with very arge dimensionaity, the piot interference, and the piot overhead sti constitute a botteneck for the performance of massive MIMO

3 3 transmission especiay in mm wave frequencies even in T mode due to severa reasons. First, in these systems, the eementary operations on the received signas such as matrix inversions and instantaneous CSI acquisition, muti-user precoding, decoding, equaization, adaptive spatia-tempora signa processing etc. become quicky infeasibe with the increasing dimensions especiay for arge number of UTs. Second, for the conventiona orthogona training scheme in T mode [5], [6], the piot overhead woud be prohibitivey arge for mm wave channes, where the signa-to-noise ratio snr) before beamforming is very sma, and thus directiona precoding/beamforming is inevitabe to support onger outdoor inks and to provide sufficient received signa power [8], [9]. However, the design of a directiona beamformer is usuay based on CSI. Moreover, utiizing orthogona piots among a users in the ce is one of the imiting factors on throughput in massive MIMO for high mobiity scenarios where piots must be transmitted more frequenty, and for appications requiring ow atency and short-packet duration [10], [11]. These features are desirabe for incorporating machine-type communications in next generation systems [2], [12]. On the other hand, aowing piot reuse PR) among the intra-ce UTs or non-orthogona piot assignment across the inter-ce UTs in neighboring ces) eads to the piot interference [10] or piot contamination [6], which undermine the vaue of MIMO systems in ceuar networks. Therefore, in order to be abe to expoit the advantages of massive MIMO communications, whie overcoming the signa processing burden due to arge dimensionaity, piot interference, and overhead botteneck, some effective channe dimensionaity reduction techniques, taking the sowy varying channe properties ong-term parameters) such as anges of arriva AoAs), deays, and average power of the arriving waves) into account, must be empoyed. Recenty, the two-stage beamforming concept under the name of Joint Spatia ivision and Mutipexing JSM) [13], [14] has been proposed to reduce the dimension of the MIMO channe effectivey, and to enabe massive MIMO gains and simpified system operations [15], [16]. Even though JSM is suggested as an effective reduced-compexity two-stage downink precoding scheme for muti-user MIMO systems in F mode initiay, the idea of two-stage beamforming in [13], [14], [16]) can be appied to both downink and upink transmission in T. JSM can be seen as a divide-and-conquer approach considering the fact that the channe between a user and BS is spatiay correated. The key idea ies in user-grouping, i.e., partitioning the user popuation supported by the serving BS into mutipe groups each with approximatey the same channe covariance eigenspaces. Then, one can decompose the MIMO beamformer at the BS into two steps via the use of spatia pre-beamformer, which distinguishes intra-

4 4 group signas from other groups by suppressing the inter-group interference whie reducing the signaing dimension. The major compexity reduction in JSM comes from the approach that the pre-beamformer is propery designed based ony on the ong-term parameters described by using the second-order statistics of the channe) and not on the instantaneous CSI which may vary on a much higher rate). In this case, the subsequent operations such as downink muti-user precoding and upink detection/decoding agorithms can be fufied based on the CSI of the effective channe with significanty reduced dimensions thanks to the pre-beamforming projection. At the same time, the training dimension necessary to earn the effective channes of each UT is reduced consideraby. Aso, the JSM scheme motivates the use of anaog/digita MIMO architectures, specificay the so-caed hybrid beamforming [8], [17] [19], recenty proposed as an aternative for fuy digita precoding/decoding in mm wave, where efficient reconfigurabe radio frequency RF) architectures wi be impemented at competitive cost, size, and energy in the near future. In the hybrid beamforming architecture, the statistica pre-beamformer which depends on sowy varying parameters) may be impemented in the anaog RF domain, whie the muti-user MIMO precoding/decoding stage can be impemented by standard baseband processing. In this paper, we estabish a genera framework on the reduced dimensiona CSI estimation and the pre-beamformer design for frequency-seective massive MIMO systems empoying singe-carrier SC) moduation in T mode by expoiting the channe sparsity indicated by the joint ange-deay domain power profie. The channe sparsity [11], [20] [22], which becomes particuary reevant at mm wave frequencies, is observed in practica ceuar systems, where the channes are often characterized with imited scattering and hence correated in the spatia domain; the BS sees the incoming muti-path components MPCs) under a very constrained anguar range AoA support), and the MPCs occur in custers in the ange-deay pane corresponding to the interaction with physica custers of scatterers in the rea word [23]. Moreover, ony MPCs, undergoing one or two refections, can have significant power [23], [24]. On the other hand, the 5G systems, aimed to provide much higher throughput, wi inevitaby be broadband. Thus, the wideband massive MIMO channe is expected to be sparse both in ange and time deay) domain. Recenty, agorithms based on compressed sensing expoiting channe sparsity gained attraction to reaize channe estimation and reducing training overhead, e.g., [25] [27] and the references therein. Nevertheess, the use of joint ange-deay domain sparsity information is overooked in the context of channe estimation with dimension reduction whie taking the piot interference and piot overhead into account for SC systems in the T mode. Here, the reduced rank channe estimation probem based on

5 5 generic subspace projection is handed by an inspiration from the JSM framework, where the statistica pre-beamformer is designed to reduce dimensionaity and piot overhead whie mitigating inter-group interference eading to piot contamination due to intra- or inter-ce UTs). The main contributions of this work are summarized as foows: By an inspiration from the user-grouping idea in the JSM scheme, the reduced rank minimum mean square error RR-MMSE) instantaneous CSI estimator, ying in the sowy varying second order statistics given by the joint ange-deay domain power profie), is derived for spatiay correated wideband MIMO channes. To the best of the authors knowedge, the derivation of the RR-MMSE estimator, provided here, is presented for the first time when the SC transmission with upink training in T mode is considered. The statistica pre-beamformer design is considered for frequency-seective SC massive MIMO channes. The fundamenta approach here is to find a good subspace, on which the RR-MMSE channe estimation can be reaized as accuratey as possibe, so that a minima performance compromise in the subsequent statistica signa processing operations after pre-beamforming is provided. In this paper, we examine the dimension reduction probem by adopting severa criteria based on the instantaneous CSI estimation accuracy. These criteria resut in an equivaent optimization probem, and generaized eigenvector beamspace GEB) appears to be a neary optima pre-beamformer when the eigenspaces of different resovabe MPCs are assumed to be neary orthogona. Moreover, it is observed that RR-MMSE shows remarkabe robustness to the piot interference, and the significant reduction in piot overhead is attained thanks to the dimension reducing subspace projection, which suppresses the inter-group interfering signas. A spatio-tempora domain correator type reduced rank channe estimator as a high snr approximation of the RR-MMSE estimate is derived where the statistica spatia) pre-beamforming and tempora) correator are appied in a successive manner. The key idea is to reaize east square LS) estimation in a proper reduced dimensiona subspace so that the number of unknown parameters is reduced whie capturing the intended part of the group signa and switching off the interference subspace, eading to piot interference via pre-beamforming. This approximate estimator is shown to be constructed based ony on the pre-beamforming matrix determined by the support of the AoAs and deays of the MPCs) without necessitating the knowedge of the exact covariance matrices of the muti-path channe vector.

6 6 II. SYSTEM MOEL We consider a ceuar system based on massive MIMO transmission operating at mm wave bands in the T mode empoying SC in which a BS, having N antennas, serves K singe-antenna UTs. In order to reduce the overhead whie acquiring the instantaneous CSI associated with massive MIMO, two-stage beamforming under the name of JSM is adopted throughout this study. The main idea of JSM scheme is based on partitioning the user popuation supported by the serving BS into mutipe groups in order to enabe massive MIMO gains and simpified system operations [13], [14]. As in JSM-based transmission, K users are partitioned into G groups, where the K g users in group g have statisticay independent but identicay distributed i.i.d.) channes [13], [14], [16] 1. At the beginning of every coherence interva, a users of the intended group g transmit training sequences with ength T. We assume a inear moduation e.g., PSK or QAM) and a transmission over frequency-seective channe for a UTs with a sow evoution in time reative to the signaing interva symbo duration). Under such conditions, the baseband equivaent received signa sampes, taken at symbo rate W ) after puse matched fitering, are expressed as 2 y n = K g {k=1, g k Ω g} L g 1 =0 h g k) x g k) n } {{ } Intra-Group Signa + { g k Ω g g g} K g k=1 L g 1 =0 h g k ) x g k ) n +n n } {{ } η n :Inter-Group Interference + AWGN 1) for n = 0,...,T 1, where h g k) is N 1 muti-path channe vector, namey, the array impuse response of the serving BS stemming from the th muti-path component MPC) of k th user in group g. It can be regarded as the discrete-time equivaent form of the channe response, and obtained after the symbo rate samping of the impuse response, arising as the sum of the contributions from discrete MPCs, without { } any oss of information [28]. Here, x g k) n ; L g +1 n T 1 are the training symbos for the k th 1 Athough JSM is initiay proposed as an effective reduced compexity two-stage downink precoding scheme for mutiuser massive MIMO systems in F mode [13] [15], our focus here is on the reduced dimensiona instantaneous channe acquisition technique and pre-beamformer design, with neary optima accuracy for upink frequency-seective massive MIMO channes in T mode. This can be reaized by expoiting the user-grouping idea inspired from the JSM framework where a sowy-varying spatia correation among the array eements exists. 2 Ony the UTs, beonging to same group, are assumed to be synchronized for coherent upink SC transmission. That is to say, ony intra-group synchronization is sufficient, and no synchronization and/or coordination is required between different group users inter-group). Note that assuming synchronization between upink piots of users can be regarded as a worst-case scenario from a intra-group or inter-group piot interference point of view, since any ack of synchronization wi tend to statisticay decorreate the piots.

7 7 user in group g 3, L g is the channe memory of group g muti-path channes 4, Ω g is the set of a UTs beonging to group g with cardinaity Ω g = K g, and {g k } Kg k=1 are UT indices forming Ω g. The L g 1 symbos at the start of the preambe, prior to the first observation at n = 0, are the precursors. Training { } symbos are seected from a signa consteation S C and E x g k) n 2 is set to E s for a g k. In 1), n n are the additive white Gaussian noise AWGN) vectors during upink piot segment with spatiay and temporariy i.i.d. as CN 0,N 0 I N ), and N 0 is the noise power 5. The first term of 1) is the transmitted signa of the intended group g, named as the intra-group signa of group g users. The second term, η n, namey the inter-group interference, comprises of a the interfering signas, which stem from a inner or outer ce users beonging to different groups other than g. Finay, the average received signa-to-noise ratio snr) can be defined as snr Es N 0 6. A. Fundamenta Assumptions on Signa and Channe Mode In JSM, where oca scattering mode is assumed, the BS sees the incoming MPCs under a very constrained anguar range, and the MPCs tend to occur in custers on the ange-deay pane, corresponding to the interaction with physica custers of scatterers in the rea word [11], [22], [23]. Another important observation, which becomes particuary reevant at mm wave frequencies, is channe sparsity. In other words, most of the channe power is concentrated in a finite region of anges or deays due to the imited scattering, and the number of significant MPCs is reduced to a much ower vaue than that for a microwave system operating in a simiar environment [20], [23]. This sparsity can be resoved in the ange domain with the use of massive array architecture. As in JSM-based systems, each resovabe MPC of the users, beonging to any group g, is assumed to span some particuar anguar sector in azimuth-eevation pane, capturing oca scattering around the corresponding UT s ange of arriva AoA). Then, their corresponding 3 Training sequences are assumed to be non-orthogona for synchronized intra-group users for SC transmission in genera. However, their tempora cross-correation properties affect the accuracy of the CSI acquisition as wi be apparent in the subsequent chapters. In addition to that, piot reuse PR) among inter-group users is feasibe thanks to the pre-beamforming yieding effective suppression for inter-group interference in spatia domain. This brings significant advantage in terms of piot overhead which woud be prohibitivey arge as the number of UTs become arge, since utiizing orthogona piots among a users is one of the imiting factors on throughput in Massive MIMO [10] especiay for appications requiring ow atency and short-packet duration. These two features are desired for incorporating machine-type communications in next-generation ceuar wireess systems [2]. 4 In genera, the muti-path channe is time unimited, so that there are infinite number of nonzero 1 W spaced channe taps. However, it can be we approximated by finite number of nonzero channe coefficients [28] as in 1). 5 The received signa at BS is first pre-fitered by a brick-wa fiter of proper bandwidth before unity gain puse matched fitering and samping at symbo rate W ) without information oss where the compex Gaussian baseband noise process is assumed to have circuar symmetry and a fat power spectra density N 0 in the band of interest. 6 It shows the maximum achievabe snr after beamforming when the beam is steered towards a point, i.e., anguar ocation by assuming that the channe is normaized so that 1 E s N N 0 can be seen as the average received snr at each antenna eement before beamforming.

8 8 cross-covariance matrices can be expressed in the form of { E h g k) ) } H h g k ) = ρ δ gg δ kk δ, where L g 1 =0 ρ { = 1, Tr } = 1 2) by using the uncorreated oca scattering mode where a MPCs are assumed to be mutuay independent according to the we-known wide sense stationary uncorreated scattering WSSUS) mode [11], [23], [28], [29], the muti-path channe vectors are uncorreated with respect to, and aso mutuay uncorreated with that of the different users independent of whether in the same group or not). In 2), ρ is the power deay profie pdp) of the group g muti-path channes, showing the average channe strength at each deay, and the auto-covariance of each MPCs in group g is given by = U Λ U ) H, = 0,...,Lg 1, 3) where U is the N r g, matrix of the eigenvectors corresponding to the r g, non-zero or dominant eigenvaues of, given as the diagona eements of the diagona r g, r g, matrix Λ in 3). In 2), can be considered as the common spatia covariance matrix of group g UTs at th deay. Under this mode, covers the predetermined sector with a particuar center and anguar spread AS), where the diffuse radiation is incuded by considering intervas of anges for which the th MPC have a continuum of non-resovabe components, each carrying infinitesima scattered energy [23]. That is to say, the th MPC of group g users stems from a particuar scattering region for a given AoA support with respect to the BS. In 3), the effective rank of, namey, r g, is expected to be much smaer than the number of array eements, N, since the channe sparsity of the impuse response [20] is pronounced at mm wave frequencies, where ony MPCs undergoing one or two refections can have significant power [23], [24]. The channe sparsity is the source of significant correation among the antenna array eements, which makes use of pre-beamforming very appeaing in T or F modes in order to reduce the dimensionaity of the muti-path channe 7. This can be reaized by expoiting the near-orthogonaity of the eigenspaces of the MPCs of different user groups in joint ange-deay domain. When Rayeigh-correated channe coefficients are assumed such as h g k) CN ) 0,ρ, mutuay independent across the users for a g k, the Karhunen-Loeve representation [31] of the muti-path channe 7 Simiar ideas woud be appicabe for downink channe estimation in JSM-based systems for the F mode studied in [7], [30]), provided UT is equipped with mutipe antennas, in which case pre-beamformer woud hep simpify the instantaneous CSI acquisition and system operations at UTs, and reduce overheads significanty by suppressing inter-group interference at the precoding stage.

9 9 vector beonging to the k th user in group g is given as the foowing by using 3) h g k) = ρ ) 1/2U Λ where the entries of c g k) C r g, 1 CN 0,I rg, ). ) 1/2c g k ), = 0,...,L g 1, 4) We consider the foowing reaistic assumptions reated to the frequency-seective massive MIMO channe. The subsequent sections are based on these assumptions: Bock Fading assumption, for which the channe is ocay time-invariant over a packet duration, is adopted. Many existing ceuar network standards based on piot-aided channe estimation and coherent detection impicity assume bock fading [32]. WSSUS mode [23], [28], [29] is adopted for sma-scae fading, namey, the normaized sma-scae coefficients c g k) s in 4) are assumed to be mutuay independent, based on which 2) is formed. The cross-covariance matrix in 2) is normaized so that the arge-scae fading parameters such as path-oss and shadowing are incorporated into the average received signa-to-noise ratio snr). These parameters are assumed to be ocay static, and the average channe strength can be easiy earned over a ong period of time. The channe auto-covariance of each group in 3) is sowy varying in time as the AoA of each user signa evoves depending on the user mobiity, variation rate of the scattering environment characteristics, etc. [10], [22], [23], [33]. Aso, this variation is known to be much sower than the actua Rayeigh fading process, then the WSSUS channe mode is a oca approximation with coherence time much arger than the sma-scae fading coherence time. The second order statistics of the muti-path channe vectors, namey, ρ in 3), varying at a much ower rate compared to the instantaneous CSI, can be estimated with guaranteed accuracy for a intended groups in practice, since there are enough time-frequency resources to be expoited for this purpose 8. Mutua couping, AS due to the diffuse scattering, AoA uncertainties of each UT stemming from the use of practica covariance estimation or tracking agorithms, user mobiity, caibration errors or any other spatia correation mismatches can be taken into account by choosing arger AoA support dimension, i.e., r g, in 4)) for each intended user group initiay to construct in 3)), and then, 8 Agorithms for the covariance estimation or signa subspace tracking [21], [22], [34], [35] coud be utiized here to track the sow variations of the user channe covariance matrix together with the user grouping agorithms [14], [23] that partition users having approximatey common subspace characterizing group) in their MPCs. However, subspace tracking and user grouping agorithms are out of the scope of this work.

10 10 s can be adaptivey updated at a much ower rate compared to the instantaneous CSI earning. In JSM framework, users come in groups, either by nature or by the appication of user grouping agorithms given in [14]. In urban environments, it is typica to observe common custers that create spatiay correated MPCs for many users. That is to say, when each user group is characterized by mutipe scattering custers, some of the custers may significanty overap in the ange-deay pane. In this case, the user seection agorithms described in [23] provide a set of user groups that can be served simutaneousy in the same transmission resource in 1) 9. Spatio-tempora covariance matrix of the inter-group interference in 1) can be cacuated by taking ong-term expectation over a MPCs h g k ) s other than the ones beonging to group in the spatia domain, and transmitted symbos x g k ) n s in the tempora domain. Considering the mutua independence across muti-path channe vectors due to the uncorreated scattering assumption for sma-scae fading in WSSUS mode) given by 2), and considering that the transmitted symbos of different users are { ) } uncorreated incuding the data transmission period), i.e., E x g k) n x g k ) H n = γ E s δ nn δ gg δ kk, the foowing is obtained { E η n η n ) H } = η δ nn, where η E s g gγ g ) K g L g 1 =0 ρ g ) R g ) +N 0 I N, 5) and γ g ) for g g can be regarded as the reative average received power at BS of inter-group users normaized with that of the group g users. In 5), γ g ) s are accountabe for the near-far effect stemming from the fact that received signa strength of different UTs may differ significanty depending on their distance to the BS. Moreover, it is important to note that the N N covariance matrix of the inter-group interference η in 5) consists of a the statistica information of the CSI in the spatia domain i.e., AoA support) for a inner or outer ce users interfering with group g users. The interference covariance matrix, η can be obtained by using the common spatia covariance of each intended group, namey, in 3) at each deay. The mode in 1) can be appied to any SC-based MIMO setting such that both singe-ce or muti-ce, 9 As proposed under the JSM framework, two user grouping techniques [23] can be used to form 1): 1) Spatia Mutipexing, that orthogonaizes two groups in the spatia domain via the pre-beamforming i.e., suppressing common scatterers and/or inter-group interference), enabes us to serve the two groups on the same transmission resource. 2) Orthogonaization serves the user groups, having common MPCs, in different channe transmission resources time or frequency) by using pre-beamforming that aow a the channe eigenmodes incuding the common scatterers) of each group to pass. First technique, yieding higher mutipexing gain, is shown to be effective at high-snr regime, whereas the atter, providing fu muti-path diversity gain, performs better at ow-snr regime [23].

11 11 where piot contamination [6] persists, can be considered 10. Regarding the muti-ce scenario, if ony statistica CSI coordination among ces is possibe, covariance matrix of each MPC, beonging to user groups in neighboring ces, can be exchanged among the BSs. Then, the spatia covariance matrix of the interference η in 5) of each intended group can be cacuated by taking the statistica CSI of the intra-ce as we as the inter-ce groups into account. Then, the statistica pre-beamforming stage can be reaized to suppress a types of interfering sources accordingy 11. Thus, the user grouping strategy with pre-beamforming inspired from JSM [14] can be seen as an appeaing technique for SC upink transmission in order to mitigate the piot contamination effect consideraby in addition to significanty reduced system compexity). Moreover, in this setting, piot reuse PR) can be aowed among the intergroup users, and thus the piot overhead can be significanty reduced. Before concuding this section, it is better to emphasize one more time that AoAs and path strengths change ony when the arge scae geometry of the propagation between the transmitter and receiver significanty changes, thus their rate of change is significanty ower than that of the sma-scae fading, namey, instantaneous CSI. In practice, AoAs and ASs statistica CSI) of each UT can be determined by empoying suitabe compressed sensing toos [9], [27] where the sparse or ow-rank nature of the MIMO channe at mm wave is taken into account 12. Therefore, ong-term earning of AoA supports can be considered as the initia stage for a fine estimation of sma-scae fading coefficients, namey, c g k) s in 4) instantaneous CSI), varying at a much higher rate than that of AoAs [10], [22]. B. Spatio-Tempora omain Vector efinitions In this paper, our main focus is on the upink CSI acquisition that uses both ange-deay domain sparsity information for spatiay correated MIMO channes described in Section II-A. Before eaborating on the detais of the estimation technique, we give the foowing matrix and vector definitions that wi be usefu in the subsequent chapters. First, the training matrix or convoution matrix [37]), comprising 10 Here, piot contamination can be seen as the piot interference stemming from inner or outer ce users beonging to groups apart from g when they use training sequence non-orthogona to that of the users in g. 11 Coordinated piot aocation or scheduing agorithms for mitigating the intra-ce or inter-ce piot contamination in [10], [36] can be expoited, together with the seected user grouping technique and spatia pre-beamforming, in order to provide additiona gain when there exist interfering groups having significant overapping AoA support with that of the intended group g. This aows piot reuse PR) or non-orthogona piot sequences among intra- or inter-ce UTs apart from the intra-group UTs of the intended groups, where the piot ength T can be reduced substantiay compared to the number of array eements in BS, N. 12 ue to the sparse nature of the mm wave channe, compressed sensing CS) agorithms [9], [22], [27] can be empoyed to extract the statistica CSI, namey, the AoA support so that the covariance matrices of each MPC for a inner and outer ce groups, namey, s in 3) can be constructed. In genera, these agorithms can be utiized as the initia acquisition toos of sowy-varying spatia correation statistics necessary for instantaneous channe earning.

12 12 of the transmitted piots with the precursors for k th user in group g, is defined as 13 X k x g k) 0 x g k) 1 x g k) L g+1 x g k) 1 x g k) 0 x g k) L g x g k) T 1 x g k) T 2 x g k) T L g T L g. 6) The extended muti-path channe vector of the k th user, beonging to the intended group g, and its corresponding expansion coefficients after Karhunen-Loeve Transform KLT) in 4) are given as f k h g k) 0 h g k) 1. h g k) L g 1 NL g 1, b k c g k) 0 c g k) 1. c g k) L g 1 Lg 1 =0 r g, ) 1 7) by concatenating a MPCs of the k th user in group g. Then, by using the vectors given in 1) and 7), it wi be usefu to construct the foowing vectors that represents the whoe received vector of signas at BS in space-time domain) during training phase, the concatenated channe vector and its KLT coefficients that incude the channe parameters of a users in group g to be estimated) respectivey: y vec{ [ y 0 y 1 y T 1 [ ] h vec{ f 1 f 2 f K g ] N T } NL g K g c vec{ [ b 1 b 2 b K g ] Lg 1 =0 r g, ) K g } }. 8) In a simiar way, the inter-group interference matrix with respect to group g in space-time domain can be defined as ξ vec{ [ η 0 η 1 η T 1 ] N T }. 9) 13 If the BS has perfect knowedge of the sparsity pattern in ange-deay domain such that some of the non-dominant MPCs are approximatey zero, this a-priori information can be taken into account by simpy setting ρ in 2) to zero for the corresponding deay with zero energy, or construct X k in 6) by extracting the coumns corresponding to the muti-path channe taps possessing ρ = 0.

13 13 Finay, the compete training matrix that consists of the training data of a users in group g during the signaing interva T is given by [ X ] X 1 X 2 X K g T K gl g. 10) The extended muti-path channe vector of group g in 8), carrying the compete CSI of a UTs in g, can be expressed in terms of KLT coefficients sma-scae fading) given in 4) as h = I Kg V ) c where 11) }{{} Υ U V bdiag [ { ρ ) 1/2U Λ ) } ] 1/2 Lg 1. 12) Here, 11) can be regarded as the generaized Karhunen-Loeve expansion in the spatio-tempora domain or ange-deay domain) by using the corresponding eigenbasis given in 12) with a-priori known channe { } Lg 1 { } Lg 1 power profie ρ Λ in the ange-deay domain. Eigenbeams of group g, namey, U =0 with dimension r g, ) do not need to be orthogona to each other in genera, i.e., overapping can be observed between the MPCs at different deays. In 11), Υ U I Lg 1 ) K g V is an NK g L g K g =0 r g, transform matrix in the spatio-tempora domain) constructed by the eigenbasis of group g at each deay. The KLT coefficients of group g users in 11) are spatiay and temporariy i.i.d. with unity variance such =0 =0 that { E c c ) } H = I Lg 1 Kg. 13) =0 r g,) By using the matrices and vectors in spatio-tempora domain defined above, 1) can be expressed in a more compact matrix form K g ) y = X k I N f k +ξ 14) k=1 where the covariance matrix of the spatio-tempora interference ξ with respect to g can be obtained as since η n s are uncorreated in tempora domain due to 5). ) } H ξ E {ξ ξ = I T R η 15)

14 14 C. Reduced imensiona Spatio-Tempora Mode for Sparse SC MIMO Channes Next-generation wireess networks, composed of massive antenna arrays with severa hundreds of receiving eements, utiize arge dimensiona received signa for upink decoding or downink precoding. In these systems, the eementary operations on the received signas such as matrix inversion and instantaneous CSI acquisition with arge piot overheads) become quicky infeasibe with the increasing dimensions especiay for a arge number of UTs. A good way to enabe the processing of arge-dimensiona signas is the adoption of a pre-processing stage that captures the essence of the input at a reduced dimension. Inspired from the JSM or two-stage beamforming) framework [13], [16], a spatia pre-beamformer, which is to be designed based ony on statistica CSI, not on instantaneous CSI, is expoited to reduce the dimension of the signaing space. Thanks to the dimensionaity reduction brought by the statistica pre-beamforming projection, instantaneous muti-path channe estimation short-term) can be attained at consideraby reduced compexity so that the muti-user precoding at downink or muti-user decoding at upink, necessitating instantaneous CSI for proper operation, can be fufied at reduced dimension with significanty reduced compexity. In ight of the discussion above, pre-beamforming is appied in order to distinguish intra-group signa of group g users from other groups by suppressing the inter-group interference whie reducing the signaing dimension in 14). At the pre-beamforming stage, a T -dimensiona space-time vector y is formed for [ ] ) H ) H a intra-ce groups by a inear transformation through I T matrix caed as where Υ S [ ] ) H y I T y, g = 1,...,G, 16) }{{} ) H Υ S is an N statistica pre-beamforming matrix that projects the N-dimensiona received signa sampes {y n } T 1 n=0 in 1) on a suitabe -dimensiona subspace in the spatia domain14. Our goa is to accompish the dimension reduction with a sma oss of instantaneous CSI of group g UT estimation accuracy so that the foowing stages after CSI acquisition) such as downink precoding or upink decoding can be reaized with a cose performance to that of the fu dimensiona case. The pre-beamformer 14 These motivate the use of anaog/digita MIMO architectures recenty proposed as an aternative for fuy digita precoding/decoding in mm wave communication systems [8], [9], since efficient reconfigurabe radio frequency RF) architectures wi be impemented at competitive cost, size, and energy efficiency in near future [23]. The advantage of impementing pre-beamforming in the anaog RF domain is that the number of RF chains and anaog-to-digita converters ACs) can be reduced so that the cost of baseband processing and baseband to RF moduation scaes with the intermediate dimension which is { } g rank, whie the number of antennas N can be very arge.

15 15 design based ony on the channe statistics for the aforementioned JSM-based massive MIMO systems empoying SC is considered in Section IV. The output of the pre-beamformer ) H y = Υ S y K g [ = X k k=1 [ = X [ = X in 16) can be written expicity as ] ) H f k + ] H ) h +ξ [ ] ) H I T ξ } {{ } ξ ] H ) IKg V ) c +ξ = Ψ c +ξ 17) where [ Ψ X = ] H ) IKg V ). 18) In 17), the second ine foows from the Kronecker product rue A 1 A 2 )B 1 B 2 ) = A 1 B 1 ) A 2 B 2 ) after substituting 14) in its position, the third ine foows from the definitions of the muti-path channe vector and the training matrix given in 8) and 10) respectivey, and the fourth ine foows from the generaized KLT defined in spatio-tempora domain in 11). The expression in 17) is the equivaent spatio-tempora received signa representation of 1) after dimension reduction. This expression, which contains a the reevant training information, channe statistics sparsity information, AoA support, etc.) of intra-group users and inter-group interference, wi be frequenty used in the subsequent sections where the reduced dimensiona channe estimator is constructed based on it. III. COVARIANCE-BASE REUCE RANK CHANNEL ESTIMATION In this section, based on the mode in 17), the reduced dimensiona inear minimum mean square error LMMSE) channe estimator is derived whie the side information ying in the second order statistics of the MPCs of each group is utiized. As it is we-known, the LMMSE channe estimator is often referred to as the Wiener fiter 15. The reduced rank MMSE RR-MMSE) estimate of the instantaneous CSI can be 15 It is actuay a Bayesian approach with a quadratic risk function, i.e., a conditiona mean estimator [31] if intra-group muti-path channe coefficients and inter-group interference in 17) are jointy Gaussian distributed. In this case, the Bayesian estimator based on the maximum a posteriori MAP) estimation rue, yieding the most probabe vaue given the observation y after pre-beamforming, aso coincides with the LMMSE estimate.

16 16 expressed in the foowing genera form: ) H ) H ĥ = Υ U ĉ = Υ U W mmse, y = Υ U W mmse, Υ S y 19) }{{} Reduced Rank Wiener Fiter where the RR-MMSE estimate of h, namey, ĥ is written in terms of the RR-MMSE estimate of c, namey, ĉ sma-scae fading) by the KLT through Υ U ) H matrix given in 11). This operation does not ead to any oss of information reated to sufficient statistics, since the KLT matrix Υ U i.e, fu coumn rank. is one-to-one, In 19), first, ĉ is formed by a reduced dimensiona inear Wiener fiter or MMSE) in the spatiotempora domain through the W H mmse,) matrix for group g users after projecting reducing the ) H dimension) fu dimensiona observation y in 14) onto a suitabe subspace represented by pre-beamforming) in 16). Then, the LMMSE estimate of the fu-dimensiona muti-path channe vector of group g, ĥ, is constructed by transforming ĉ back to the origina space by KLT through Υ U. Thus, a genera framework for the reduced dimensiona channe estimation probem is estabished here for genera rank signa modes) such that the overa reduced rank estimator given as the ) H ) H, ) H data processing chain: Υ U W mmse, Υ S where the transform matrices Υ S dimensionreducing subspace projection in the Kerne space) and Υ U Υ S transforming back to the origina space by KLT) are composed of different basis set in genera. The Wiener fiter in 19) can be seen as the reduced ) H ) H ) H rank approximation of the fu-dimensiona Wiener fiter, i.e., W fu Υ U W mmse, Υ S { } where the rank of the fu-dimensiona fiter rank W fu = min{nk g L g,nt} is reduced to { ) H ) } H { } rank Υ U W mmse, Υ S = rank W mmse, = min{k g r g,,t}. The transform ) H matrices Υ S and Υ U do not depend on training data and instantaneous CSI, and are to be designed based on ony ong-term channe second order statistics, which brings significant compexity reduction especiay when one considers the use of adaptive fitering and tracking agorithms. As it wi be cear in the seque, the proper design of pre-beamformer determining some overap among eigenspaces of different groups in the joint ange-deay domain. is critica since there aways exists A. Joint Ange-eay omain Reduced Rank MMSE Estimator H The reduced rank Wiener fiter W mmse,) of group g, depending on the covariances of intragroup signa and inter-group interference reated to the joint ange-deay domain sparsity information)

17 17 in 19), can be obtained after the foowing mathematica steps by using the Kronecker product rue A 1 A 2 )B 1 B 2 ) = A 1 B 1 ) A 2 B 2 ) successivey such that W ) 1Ψ mmse, = R y [ ] H = Ψ Ψ + { [ = X = [ + I T { Lg 1 =0 Υ S ) H IT η ) Υ S ) 1 Ψ ] H ) IKg VV H) [ X ] H S ] H ) IT η ) I T X [ I Kg ρ E Lg,][ X ] H ) ) ) } 1 Ψ [ ] ) H R +I T [ ] ) } 1 H R η S Ψ. In 20), the first ine foows from the soution of Wiener-Hopf equation [31] based on 17) defined in the spatio-tempora domain. Here, y is defined as the covariance matrix of y in 17). Then, in the second ine, the expression for y is substituted into its pace expicity by using 13), 15), and 17). The third ine foows from 18) by substituting Ψ and Υ S = I T fourth ine foows from the foowing usefu expression obtained from 3) and 12) VV H = L g 1 =0 20) into their paces. Finay, the ρ E Lg, 21) where E Lg, is an L g L g eementary diagona matrix where a the entries except the +1) th diagona one are zero. Then, substituting W mmse, in 20) into the expression in 19) and using 18), and after some straightforward steps noting that A B) 1 = A 1 B 1 ), A+B) 1 = B 1 I+AB 1 ) 1, and successive use of A 1 A 2 )B 1 B 2 ) = A 1 B 1 ) A 2 B 2 ), the LMMSE estimate ĥ in 19) can be written expicity as ĥ = Lg 1 =0 Lg 1 =0 [ IKg ρ E Lg,][ ] X H R [ ] ) ) H 1 R η S code ) [ SN mimo ) ] H +IT ) 1y. 22)

18 18 The matrices SN mimo ) and R code ), appearing in 22), are defined as SN mimo ) ρ [ ] H R η S ) 1 [ ] ) H R, 23) code ) X [ I Kg E Lg,][ X ] H ). 24) The matrices SN mimo ) in 23) and R code ) in 24) for = 0,...,L g 1 have usefu properties expained in Appendix I. Briefy, the positive semi-definite SN mimo ) matrix in the spatia domain can be regarded as the generaized definition of the beamformer output snr for genera rank { } signa modes [34], [38] 16. As it was shown in our previous work [38], Tr SN mimo ) is actuay the expected vaue of the point signa-to-interference noise ratio sinr) over the eigenspace AoA support) of the th MPC in group g, where the point sinr is defined as the output snr after beamforming when the beam is steered towards a point, i.e., anguar ocation in the AoA support of th MPC. In the tempora domain, the T T positive semi-definite code ) matrix is defined as the deterministic correation matrix obtained from the coumns of X [ ] I Kg E Lg, where the coumns with index { +1)+k 1)Lg } Kg k=1 are the same as the +1) th coumn of the training matrix X k, k = 1,...,K g in 6), and the other eements are set to zero. In 22), the SN mimo ) matrix is responsibe for spatia processing ony, utiizing the eigenspaces of the intended group g at th deay and the inter-group interference after subspace projection onto. On the other hand, R code ) is responsibe for tempora processing ony, utiizing the tempora cross-correation properties of the piot sequences assigned to each UT. In order to harness the spatia mutipexing in each group, one consider the effective muti-path channe vector of each group user h g k),eff, seen after pre-beamforming as h g k),eff [ ] H h g k ) [ h eff I KgL g ] H ) h 25) from the definition of the extended muti-path channe for group g in 7) and 8). The subsequent stages at the transmitter or receiver, preceded by the pre-beamformer, fufi intra-group processing such as mutiuser precoding inner beamformer) at downink or muti-user decoding at upink in reduced dimensiona subspace. These stages can access and utiize ony this reduced dimensiona effective channe in 25). By using 22) and the definition in 25), the RR-MMSE estimate of the effective channe, seen after 16 If the dimension of is one, in this case SNR mimo ) is the snr at the beamformer output when the beam is steered towards the AoA of th MPC of group g through pre-beamformer for stochastic signas. The maximum vaue of snr is attained when the Capon Beamformer is utiized if the eigenspace of th MPC is rank-1. [39]

19 19 pre-beamforming, is constructed as [ ĥ eff I KgL g = Lg 1 =0 ] H ) ĥ [ )H Lg 1 X I Kg E Lg,]) SNR mimo ) =0 code ) [ SN mimo ) ] H +IT ) 1y. In 26), it is observed that the compexity of cacuating the instantaneous CSI estimate, stemming mainy from the matrix inversion, is substantiay reduced thanks to the pre-beamformer which reduces the dimensionaity with suitabe projection subspace whie increasing the snr eve snr before beamforming, 1 E i.e., s N N 0 is typicay very ow especiay at mm wave frequencies). The size of the matrix to be inverted in 22) or 26) is independent of the number of array eements N. Moreover, the form of the RR-MMSE estimate in 26) is suitabe to be used in decision-directed iterative channe estimation such that the decoded data can be expoited to construct code ) in adaptive channe fitering or tracking mode. It appears that the form of muti-path channe estimate in joint ange-deay domain in 26) is couped spatio-tempora processing in genera, meaning that the spatia and tempora processing need to be accompished jointy. As to the effectiveness of the proposed RR-MMSE estimator in terms of the piot contamination effect, it is possibe to attain considerabe reduction in piot interference intra- or inter-ce) together with piot overheads where the piot ength T is kept sma by aowing non-orthogona sequences among intra-group users and piot reuse among inter-group users. This can be achieved with the use of [ ] H optima joint spatio-tempora processing in 26), where the statistica pre-beamformer suppresses the inter-group interfering signas eading to piot interference, and X [ I Kg E Lg,]) H is a kind of tempora Rake-type) correator used to differentiate different MPCs having overapping AoA support in the spatia domain. In 26), X [ I Kg E Lg,]) H simpy seects the th deayed signa, i.e., paces a tempora finger on the th tempora diversity path for a K g intra-group users, whie SN mimo ) is accountabe { } Lg 1 for appying optimum spatia weights given the power profie ρ Λ in the ange-deay domain [ ] =0 H after spatia) beamforming in order to suppress the inter-group interference effectivey. The derivation of the RR-MMSE estimator provided here is presented for the first time when the SC upink transmission in T mode is considered for frequency-seective muti-user spatiay correated MIMO channes with a given ong-term joint ange-deay power profie. ifferent than the previous ow-rank LMMSE approaches in [33], [40], [41], the RR-MMSE estimator here can be interpreted as 26)

20 20 the reduced rank approximation of the optima spatio-tempora Wiener fiter in reduced dimensiona) transformed domain by using two different generic transform basis sets for projection onto a suitabe subspace pre-beamformer) and KLT whie there exists overap between eigenspaces of different groups in joint ange-deay domain in genera. Thus, for the mode here, which provides a genera description for massive MIMO based transmission empoying SC in frequency-seective fading, the proposed covariancebased reduced rank estimator here, confirm, compare, and compement many previous works, where the piot interference due to the use of non-orthogona piots in intra- or inter-ce users persists, by changing severa system and mode parameters. B. Ange omain Reduced Rank MMSE Estimator One can consider the foowing approximation of 26) by assuming that the MPCs of each group have the same AoA support common anguar sector) with the foowing covariance matrix: Lg 1 R sum =0 ρ. 27) This corresponds to the use of anguar information ony when a the AoA supports of each MPC, beonging to the same group, are unified as in 27). In 23), by repacing ρ get the foowing approximation for the effective channe estimate in 26): with sum, one can ĥ eff,2 = X SNR tota, mimo ) H ) 1y code SNRtota, mimo +I T 28) where SNR tota, mimo and code SNR tota, mimo matrices are defined accordingy as [ ] ) H 1 [ ] ) H R η R sum, 29) code X[ X ] H. 30) This estimator is caed the ange domain RR-MMSE estimator, which wi be used in pre-beamformer design and performance comparison in the seque. IV. NEARLY OPTIMAL BEAMFORMER ESIGN In this section, we consider the pre-beamformer design based ony on the second order channe statistica information of user groups in 1). The probem of statistica pre-beamformer design is handed for two-stage beamforming framework using JSM in severa recent studies [13], [15], [16], [18] where