A Generalized Framework on Beamformer Design and CSI Acquisition for Single-Carrier Massive MIMO Systems in Millimeter-Wave Channels

Transcription

1 1 A Generaized Framework on Beamformer esign and CSI Acquisition for Singe-Carrier Massive MIMO Systems in Miimeter-Wave Channes Gokhan M. Guvensen, Member, IEEE and Ender Ayanogu, Feow, IEEE Abstract In this paper, we estabish a genera framework on the reduced dimensiona channe state information (CSI) estimation and pre-beamformer design for frequency-seective massive mutipe-input mutipe-output (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. First, based on a generic subspace projection taking the joint ange-deay power profie and user-grouping into account, the reduced-rank minimum mean square error (RR-MMSE) instantaneous CSI estimator is derived for spatiay correated wideband MIMO channes. Second, the statistica pre-beamformer design is considered for frequency-seective SC massive MIMO channes. We examine the dimension reduction probem and subspace (beamspace) construction on which the RR-MMSE estimation can be reaized as accuratey as possibe. 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. Index Terms 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 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 I. INTROUCTION It is anticipated that massive MIMO systems in the mm wave range, 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) [1], [2], form an important part of the next-generation ceuar systems such as the upcoming 5G standard expected to support much arger (e.g., 1000 times faster) data rates than the currenty depoyed systems [3]. In order to capitaize these spatia diversity and mutipexing benefits with its potentia arge gains in spectra and energy efficiency, instantaneous CSI at the base station (BS) is essentia, since muti-user precoding at downink or muti-user decoding at upink necessitates accurate CSI [4]. In practice, CSI is typicay obtained with the assistance of the periodicay inserted piot signas [4]. This brings the piot overhead 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 [5]. We assume instantaneous 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 channe reciprocity [4], [5]. The processing of the signas with very arge dimensionaity, the piot interference, and the piot overhead are thought to be imiting factors for an accurate channe acquisition and throughput of massive MIMO transmission in mm wave especiay in high mobiity or in appications requiring ow atency and short-packet duration 1. Moreover, 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 for an accurate channe acquisition [7], [8]. Recenty, the two-stage beamforming concept under the name of Joint Spatia ivision and Mutipexing (JSM) [9], [10] has been proposed to reduce the dimension of the MIMO channe effectivey, and to enabe massive MIMO gains and simpified system operations [11]. Even though JSM is suggested as an effective reduced-compexity two-stage downink precoding scheme for fat-fading muti-user MIMO systems in frequency division dupex (F) mode initiay, the idea of two-stage beamforming (in [9] [11]) can be appied to both downink and upink transmission in T. 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 1 These features are desirabe for incorporating machine-type communications in next generation systems [3], [6].

3 3 into two steps via the use of a spatia pre-beamformer, which distinguishes intra-group signas from other groups by suppressing the inter-group interference whie reducing the signaing dimension. The major compexity reduction 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 user termina (UT) is reduced consideraby. These motivate the use of the hybrid beamforming architecture [7], [12] [14], where the statistica pre-beamformer (which depends on sowy varying parameters) can be impemented in the anaog radio frequency (RF) domain, whie the subsequent muti-user MIMO precoding/decoding stage can be impemented by standard baseband processing. In this paper, a genera framework on the reduced-dimensiona CSI estimation and the pre-beamformer design based on the statistica user-grouping idea (used in the JSM scheme) is estabished for frequencyseective massive MIMO systems empoying singe-carrier (SC) moduation in T mode. The statistica pre-beamformer was designed to reduce dimensionaity and piot overhead whie mitigating inter-group interference eading to piot contamination (due to inner or outer ce users) by expoiting the channe sparsity indicated by the joint ange-deay domain power profie. The channe sparsity [15], [16], 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, i.e., ange of arriva (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 [15]. Moreover, ony MPCs, undergoing one or two refections, can have significant power [15], [17]. Thus, the wideband massive MIMO channe is expected to be sparse both in the ange and time (deay) domains. The main contributions of the paper are summarized as foows. First, the reduced-rank minimum mean square error (MMSE) channe estimator, based on a subspace projection and the second order statistics, is presented for the first time when the SC transmission in T mode for wideband muti-user spatiay correated MIMO systems is considered. It can be interpreted as the reduced-rank approximation of the fu-dimensiona spatio-tempora domain Wiener fiter by using two different generic transform basis

4 4 sets; namey, the dimension-reducing subspace projection (pre-beamformer), and the Karhunen-Loéve Transform (KLT) characterizing channe sparsity in the joint ange-deay domain. Second, the dimension reduction probem from three different viewpoints reated to the instantaneous CSI estimation accuracy are examined in order to find a good beamspace (subspace in spatia domain). The adopted criteria of the probem resuts in three equivaent optimization probems yieding the same optima pre-beamformer. Finay, a spatio-tempora domain correator type reduced-rank channe estimator as an 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 beamspace so that the number of unknown parameters is reduced whie capturing the signas of the inter-group users, and switching off the interference subspace. 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 [9] [11] is adopted throughout this paper. It is assumed that 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 [9], [10]. 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} for n=0,..., T 1, where h (g k ) L g 1 =0 h (g k ) x (g k ) n } {{ } Intra-Group Signa K g L g 1 + h (g k ) x (g k ) + n { g k Ω g g g } n n k=1 =0 } {{ } η n (g) :Inter-Group Interference + AWGN 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. Here, { (g x k ) n ; L g + 1 n T 1 } are the training symbos for the k th user in group g 3, L g is the channe 2 Ony the UTs, beonging to same group, are assumed to be synchronized for coherent upink SC transmission. 3 Training sequences are assumed to be non-orthogona for synchronized intra-group users for SC transmission in genera. (1)

5 5 memory of group g muti-path channes,ω g is the set of a UTs beonging to group g with cardinaity Ω g =K g, and{g k } K g k=1 are UT indices formingω 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 CandE { 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. ascn (0, N 0 I N ), and N 0 is the noise power. 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 (g), 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 E s N 0 4. Here, we assume users come in groups, either by nature or by the appication of proper user grouping agorithms in [10], [15], which are out of scope of this work. A. Fundamenta Assumptions on Signa and Channe Mode Each resovabe MPC of the users, beonging to any group g, is assumed to span some particuar anguar sector in the azimuth-eevation pane, capturing oca scattering around the corresponding UTs ange of arriva (AoA) (with respect to the BS). Then, their corresponding cross-covariance matrices can be expressed in the form of E { ( ) h (g k ) h (g k ) H } =ρ (g) R (g) δ gg δ kk δ, (2) for L g 1 ρ (g) =0 = 1 and Tr { R (g) } = 1 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 [15], [16]; 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 (g) 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 R (g) = U (g) Λ (g) ( ) (g) H, U = 0,..., Lg 1, (3) where U (g) is the N r g, matrix of the eigenvectors corresponding to the r g, non-zero or dominant eigenvaues of R (g), given as the diagona eements of the diagona r g, r g, matrixλ (g) in (3). In (2), R (g) can be considered as the common spatia covariance matrix of group g UTs at th deay. In this mode, 4 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.

6 6 each antenna eement at a BS is assumed to see the incoming MPCs at the same common support on the ange-deay pane (simiar to the one in [15]). Aso, the effective rank of R (g), namey, r g, is expected to be much smaer than the number of array eements, N, due to the channe sparsity pronounced at mm wave [15], [17]. When Rayeigh-correated channe coefficients are assumed such as h (g k ) CN ( 0,ρ (g) R (g) ), mutuay independent across the users for a g k, the Karhunen-Loéve representation [18] of the muti-path channe vector beonging to the k th user in group g is given as the foowing by using (3) h (g k ) = ( ρ (g) ) 1/2 (g) U ( ) (g) 1/2 (g Λ c k ), = 0,..., L g 1, (4) where the entries of c (g k ) C r g, 1 CN ( 0, I rg, ). 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. [15], [19], [20]. Thus, their rate of change is significanty ower than that of the sma-scae fading (instantaneous CSI), and they can be estimated with guaranteed accuracy for a intended groups in practice. 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 (g) 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 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 (g) E s δ nn δ gg δ kk, the foowing is obtained { E η n (g) ( ) (g) H } η n = R (g) η δ nn, where R η (g) E s γ (g ) K g g g L g 1 =0 ρ (g ) R (g ) + N 0 I N, (5) andγ (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 the average 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 R (g) η in (5) consists of a the statistica information of the CSI in the spatia domain (i.e., AoA support) for a inner and outer ce users interfering with group g users.

7 7 B. Spatio-Tempora omain Vector efinitions Before eaborating on the detais of the estimation technique, we give the foowing matrix and vector definitions that wi be usefu in the subsequent sections. First, the training matrix (or the convoution matrix), comprising of the transmitted piots with the precursors for k th user in group g is defined as X (g) [ x (g ] k ) k i j. The extended muti-path channe vector of the k th user, beonging to the intended T L g group g, and its corresponding expansion coefficients after Karhunen-Loéve Transform (KLT) in (4) are given as f (g) [ h (g ] k ) k i 1 and b(g) [ c (g ] k) (Lg 1 ) N L g 1 k i 1 by concatenating a MPCs of the r =0 g, 1 kth user in group g. Then, by using the vectors given in (1) and the previousy defined ones, it wi be usefu to construct the foowing vectors that represent the whoe received vector of signas at the BS (in the spatiotempora domain) during the 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 {[ ] } yvec y 0 y 1 y T 1 (6) N T {[ ] } h (g) vec f (g) f (g) f (g), c (g) vec [ ] 1 2 K g b (g) b (g) b (g) ( N L g K g 1 2 K Lg 1 ). (7) g r =0 g, K g In a simiar way, the inter-group interference matrix with respect to the group g in the spatio-tempora {[ ] } domain can be defined as ξ (g) vec η (g) 0 η (g) 1 η (g) with the covariance matrix T 1 N T { R (g) E ξ (g)( ξ (g)) H } = I ξ T R η (g). (8) 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 (g) [ ] X (g) 1 X (g) 2 X (g) K g T K g L g. (9) The extended muti-path channe vector of group g in (7), carrying the compete CSI of a UTs in g, can be expressed in terms of the KLT coefficients (sma-scae fading) given in (4) as [{ (ρ ) (g) 1/2 ( (g) (g) U Λ h (g) = ( I Kg V ) c (g) where Vbdiag } {{ } Υ (g) U ) 1/2 } Lg ] 1. (10) ( In (10),Υ (g) U I Lg ) 1 K g V is a 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, ande c (g)( c (g)) } H = I ( ) Lg 1. Kg r =0 g, The 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 of y in (7). At =0

8 8 the pre-beamforming stage, a T-dimensiona space-time vector y (g) can be formed based on (1) for a intra-ce groups by a inear transformation through a matrix ( Υ (g) ) H ( S I T [ S (g) ] H ) as ] H ) where S (g) sampes{y n } T 1 n=0 y (g) ( Υ (g) ) H ( S y= X (g) [ S (g) h (g) + ( Υ (g) ) H S ξ (g), g=1,..., G (11) is an N statistica pre-beamforming matrix that projects the N-dimensiona received signa in (1) onto a suitabe -dimensiona subspace in spatia domain. III. COVARIANCE-BASE REUCE-RANK CHANNEL ESTIMATION In this section, based on the mode in (11), 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 [18]. The reduced-rank MMSE (RR-MMSE) estimate of the instantaneous CSI can be expressed in the foowing genera form: ĥ (g) =Υ (g) U ĉ(g) =Υ (g) ( ) (g) H U W mmse, y (g) =Υ (g) ( ) (g) H ( ) (g) H U W mmse, Υ S } {{ } Reduced-Rank Wiener Fiter y (12) where the RR-MMSE estimate of h (g), namey, ĥ (g) is written in terms of the RR-MMSE estimate of c (g), namey, ĉ (g) (sma-scae fading) by the KLT through theυ (g) U matrix given in (10). In (12), first, ĉ(g) is formed by a reduced dimensiona inear Wiener fiter (or MMSE fiter) in the spatio-tempora domain through the K g L g T ( W mmse,) (g) H matrix for group g users after projecting (reducing the dimension in kerne space) fu dimensiona observation y in (6) onto a suitabe subspace represented by ( Υ (g) S (pre-beamforming) in (11). Then, the LMMSE estimate of the fu-dimensiona muti-path channe vector of group g, ĥ (g), is constructed by transforming ĉ (g) back to the origina space by KLT throughυ (g) U. The transform matrices ( Υ (g) ) H (g) S andυ U do not depend on the 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 since N. A. Joint Ange-eay omain RR-MMSE Estimator After pre-beamforming, the reduced dimensiona spatio-tempora domain Wiener fiter ( W (g) mmse,) H of group g in (12), depending on the covariances of the intra-group signa and inter-group interference (reated to the joint ange-deay power profie and sparsity information), can be obtained by soving the ) H

9 9 Wiener-Hopf equation [18], based on (11), as 5 whereψ (g) = ( W (g) mmse, =( R y (g) ) 1Ψ ( (g) = ] H ) X (g) [ S (g) by substituting W (g) mmse, Ψ (g) Υ (g) U, and R(g) y [ Ψ (g) ] H+ ( Υ (g) S ) H ( IT R (g) η ) ) (g) 1Ψ (g) Υ S (13) is defined as the covariance matrix of y (g) in (11). Then, in (13) into the expression in (12), and after foowing some mathematica steps using the identities (A B) 1 = ( A 1 B 1), (A+B) 1 = B 1( I+AB 1) 1 and the Kronecker product rue (A 1 A 2 ) (B 1 B 2 )=(A 1 B 1 ) (A 2 B 2 ) successivey, and noting that VV H = L g 1 ρ =0 E Lg, R (g) from (3) and (10), the reduced-rank LMMSE estimate ĥ (g) in (12) can be written expicity as ĥ (g) = L g 1 =0 L g 1 =0 [ IKg ρ E Lg, ] [ ] X (g) H (g) R S (g) R (g) code () [ SNR (g) ()] H + IT ([ ] (g) H ) 1 (g) S R η S (g) 1 y (g). (14) where E Lg, is an L g L g eementary diagona matrix where a the entries except the (+ 1) th diagona one are zero. The matrices SNR (g) SNR (g) ()ρ(g) () and R(g) code ( [S (g) (), appearing in (14), are defined as ] H ) 1 ( (g) [S R η S (g) ] (g) H (g) R S (g) ), (15) R (g) code () (X (g) [ I Kg E Lg,] [ X (g) ] H ). (16) The positive semi-definite matrices SNR (g) () in (15) for = 0,..., L g 1 in the spatia domain can be regarded as the generaized definition of the beamformer output snr for genera rank signa modes [22] 6, and have usefu properties provided in Appendix I. In the tempora domain, the T T positive semidefinite R (g) () matrices are defined as the deterministic correation matrix obtained from the coumns code of X (g) [ I Kg E Lg,] for = 0,..., Lg 1. Effective Channe: The subsequent downink or upink processing, preceded by the pre-beamformer, can access and utiize ony the foowing effective muti-path channe vector of each group user: h (g k ),e f f [ ] (g) H (g S h k ). Then, from the definition of the extended muti-path channe for group g in (7), the effective extended muti-path channe seen after pre-beamforming can be expressed as ( h (g) e f f I Kg L g [ S (g) ] H ) h (g). (17) 5 The deveopment is avaiabe in more detai in [21]. Aso, the reader shoud review the properties of Kronecker products of matrices in a thorough manner to foow the deveopment throughout the paper. 6 If the dimension of S (g) is one, SNR(g) () is the snr at the beamformer output when the beam is steered towards the AoA of the th MPC of group g through the pre-beamformer S (g) for stochastic signas [23]. The maximum vaue of snr is attained when the Capon Beamformer is utiized if the eigenspace of th MPC is rank 1 [24].

10 10 Based on (14), the RR-MMSE estimate of the effective channe h (g) in (17) is constructed as e f f ( ĥ (g) e f f I Kg L g [ S (g) ] H ) ĥ(g) L g 1 H = ( [ ]) X (g) (g) I Kg E Lg, SNR () R (g) code () [ SNR (g) ()] H + IT y (g). (18) =0 =0 ifferent than the previous ow-rank LMMSE approaches in [20], [25], the RR-MMSE estimator in (18) can be interpreted as the reduced-rank approximation of the fu-dimensiona spatio-tempora Wiener fiter by using two different transform basis sets, namey, the dimension reducing subspace projection (pre-beamformer) and the KLT characterizing the joint ange-deay channe sparsity. The RR-MMSE shows remarkabe robustness to the piot interference or contamination due to the use of non-orthogona piots among inter-ce users or piot reuse among intra-ce users since the intergroup interference is mitigated by the statistica pre-beamformer in the spatia domain. In addition to the considerabe reduction in piot interference, the piot overhead is aso reduced when the piot ength T is kept sma by aowing non-orthogona training sequences among intra-group or inter-group users. B. Ange omain RR-MMSE Estimator One can consider the foowing approximation of (18) by assuming that the MPCs of each group have the same AoA support (common anguar sector) with the foowing covariance matrix: R (g) sum L g 1 L g 1 1 =0 ρ (g) This corresponds to the use of anguar information ony when a the AoA supports of each MPC, beonging to the same group, are unified. In (15), by repacing ρ (g) R (g) foowing approximation for the effective channe estimate in (18): R (g). with R (g) sum, one can get the ĥ (g) e f f,2 =( X (g) SNR tota,(g) ) H ( (g) R code ) 1 SNRtota,(g) + I T y (g) (19) where SNR tota,(g) ([ ] (g) H ) 1 ([ (g) S R η S (g) ] (g) H ) (g) S R sums (g), R (g) code [ X(g) X (g)] H. (20) 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, our goa is to find a good subspace (spanned by the coumns of S (g) matrix) on which the reduced dimensiona instantaneous channe estimation can be reaized as accuratey as possibe, so that a minima performance compromise in the subsequent statistica signa processing operations after

11 11 beamforming is provided. The probem of statistica pre-beamformer design is handed for two-stage beamforming framework using JSM in severa recent studies [9], [11], [13], [26] where the bock diagonaization (B) agorithms were investigated in order to reduce the dimensionaity for simpified system operation. A of these recent studies, reated to the pre-beamformer design, consider fat-fading spatiay-correated massive MIMO channe, whereas in this paper, the pre-beamformer design is tacked for frequency-seective massive MIMO systems empoying SC in T mode. A. Beamformer esign Criteria We examine the dimension reduction probem from three different viewpoints based on the instantaneous CSI estimation accuracy. These criteria resut in an equivaent optimization probem yieding the optima dimension-reducing subspace. 1) Reconstruction Error Minimizing Subspace The reduced-rank Wiener fitering in (12) can be seen as the data reconstruction process from noisy observations after dimension reduction. A egitimate goa is the minimization of the reconstruction error according to a criterion. If we denote the reconstruction error vector with e (g) h (g) ĥ (g) for group g channes, the covariance matrix of error R e mmse E{ (h (g) ĥ (g))( h (g) ĥ (g)) H } can be cacuated as foows: where R (g) y R mmse e = R (g) f u R(g) f u F(g) s = ( F (g) ) H (g) s R f u F(g) s { R (g) f ue h (g)( h (g)) } L H g 1 = + ( Υ (g) ) H (g) S R =0 ( R (g) y ξ Υ(g) S ) 1 ( ) (g) H (g) F s R (21) f u, F(g) s ([ ] ) X (g) H (g) S [ ] IKg E Lg, ρ R (g). (22) The R e mmse expression in (21) was obtained by appying spatio-tempora domain Wiener fiter W (g) mmse, in (13) to y (g) in (11), and using the Kronecker product rue successivey. In (22), the first term, named R (g), is cacuated by using the spatio-tempora KLT definition in (10). Moreover, it is important to note f u that the inverse of R e mmse is actuay the Fisher information matrix [18]. Error Voume: The minimization of the estimation error voume, namey, the determinant of R e mmse in (21), can be regarded as one of the important objectives on which -dimensiona subspace S (g) is

12 12 optimized. If one takes the determinant of both parts in (21), the foowing expression is obtained det ( R (g) ) f u V= ( det I T + ) (23) L g 1 R (g) () SNR(g) =0 code () wherev det (R e mmse ) and R (g) f u defined in (22) can be seen as a priori error voume of h (g) before the training period. The mathematica detais of this derivation can be found in Appendix II. Normaized Mean Square Error: The normaized mean square error (nmse) covariance can be defined as the estimation error covariance matrix of KLT coefficients c (g) in (10) as nmse (g) E{( c (g) ĉ (g))( c (g) ĉ (g)) H }. (24) Then, the trace of nmse (g), as an aternative objective function, can be obtained in the foowing compact form as Tr { nmse (g)} L g 1 1 = Tr R (g) () SNR(g) code ()+ I T + L g 1 g r g, T =0 K =0 (25) after some mathematica manipuations given in Appendix II. We woud ike to note that the scaar version of the reation (25) for T= 1, = 1, and K g = 1, L g = 1 that is nmse= 1/(1+ snr), is utiized in the anaysis of communication systems [27], [28]. 2) Mutua Information Preserving Subspace The probem can aso be stated as the preservation of the mutua information with a inear transformation under a rank constraint: rank { S (g) } =. The mutua information between h (g) and y (g) in (11) can be compacty obtained by assuming that both h (g) and y (g) are jointy Gaussian after using the MIMO channe capacity resuts in [29], the covariance matrix of inter-group interference R (g) ξ in (8), and the Kronecker product rue as foows I ( h (g) ; y (g)) = I ( c (g) ; y (g)) = og det I T + B. Neary Optima Soution: Generaized Eigenvector Space L g 1 =0 R (g) () SNR(g) code (). (26) A three criteria in Section IV-A, namey, the minimization of (23) or (25) and the maximization of (26) can be shown to resut in equivaent optimization probems yieding the same optima pre-beamformer S (g). It is possibe to simpify this optimization probem where the optima pre-beamformer depends on the training pattern X (g) in genera.

13 13 Joint Ange-eay omain: In mm wave massive MIMO channes, channe sparsity impies that eigenspaces of each MPC are neary orthogona [15], [16]. In ight of this near-orthogonaity assumption, the prebeamformer of group g can be constructed as [ S (g) S (g) (0) S(g) (1) S(g) (L g 1) where the N d matrix S (g) () can be seen as the sub-beamformer that aows the th resovabe MPC of group g to pass whie suppressing the inter-group interference in the spatia domain, and L g 1 =0 d =. ue to the apparent near-orthogonaity among the different MPCs (especiay for mm wave frequencies), S (g) () is aso expected to reject each MPC of group g other than the one at th deay. Therefore, if the orthogonaity among different MPCs is preserved after pre-beamforming, the eigenspaces of each SNR (g) () matrix (with rank d ) are mutuay orthogona. In this case, the dominant eigenvaues of the ([ ) SNR (g) () are the same as that of the d d ρ (g) (g) S ()] H 1 ([ ) (g) R η S (g) () (g) S ()] H (g) R S (g) () matrix for = 0,..., L g 1, whereas the other eigenvaues of SNR (g) () are neary zero. The eigendecomposition of SNR (g) () in (15) and R(g) code () in (16) matrices for = 0,..., L g 1 can be expressed as R (g) code ()= {m β m>0} β m φ m ] N (27) [ ] φ H, [ (g) {λ m SNR ()=Γ } ] diag n (Γ n=1 ) 1 (28) whereγ [ ] γ 1 γ showing the nth dominant eigenvector γ n in its nth coumn andλ n is defined as the corresponding eigenvaue of the SNR (g) () matrix. In a simiar fashion, φ m andβ m are defined as the m th dominant eigenvector and eigenvaue of R (g) product rue ( R (g) code () SNR(g) code () respectivey. Then, the foowing Kronecker ) ( ) ( ( ) (g) = R code ()φ (g) m SNR n) ()γ = β m λn)( φ m γ n ())( φ m γ n impies that { φ m } γ n is the set of eigenvectors for R(g) () SNR(g) m,n code () with the corresponding set of eigenvaues { } βm λ n for = 1,..., L m,n g 1. By using the eigenspace properties of the SNR (g) () matrix above foowed by the orthogonaity assumption, and the definitions given in (28), the eigenspace of L g 1 R (g) () SNR(g) =0 code () matrix in (23), (25) and (26) can be written as the orthogona direct sum of the eigenspaces of R (g) () SNR(g) code () matrices, i.e., L g 1 { =0 φ m γn} with the corresponding eigenvaues { β m λ n wheree }. This eads to the foowing approximation of the optimization criterion in (25) m,n L g 1 R d Tr { nmse (g)} 1 β =0 m=1 n=1 mλ +E (29) n+ 1 ( Lg 1 K g r =0 g, ) L g 1 R =0 d, and R is the rank of the R (g) code () matrix with R min { } T, K g m,n

14 14 from (16). For a given dimension of the pre-beamformer S (g) () in (27) with d =, and the training pattern determiningβ m, it can be noted that the minimum vaue of the cost function Tr{ nmse (g)} in (29), under the constraint that S (g) eigenvectors of R (g) is a fu coumn rank matrix is achieved by the first d dominant generaized and R (g) η from Appendix I. The minimum vaue of (29) is attained by choosingλ n as the n th dominant generaized eigenvaue of R (g) and R (g) η for = 0,..., L g 1 as noted in Appendix I. One can aso determine the optima d vaues among the possibe aternatives satisfying d = by using the generaized eigenvauesλ n minimizing (29). In order to mitigate the spatia interference stemming from the ack of orthogonaity among different MPCs, the dominant generaized eigenvectors ) ( and R η (g) + ) L g 1 =0, ρ(g) R (g) of ( ρ (g) R (g) can be chosen as the coumns of S (g) (). The generaized eigenvaue probem can be soved by the Lanczos agorithm in a computationay efficient way where d is much smaer than N. It is observed that the piot overhead is significanty reduced since even for T= K g (independent of N), Tr { nmse (g)} approaches zero when d = r g, and R = K g. Moreover, it is important to note that the piot overhead is consideraby high if channe estimation is fufied before dimension reduction, since the snr eve at each antenna eement is expected to be very sma before beamforming in mm wave channes. R (g) Ange-Ony omain: By unification of different MPCs in anguar domain corresponding to the use of sum in (20) as the common AoA support of a MPCs in group g, the optimization criterion in (25) can {( (g) be simpified as Tr R code ) 1 } SNRtota,(g) + I T from Section III-B. The eigendecomposition of the matrices SNR tota,(g) and R (g) code R (g) code = in (20) can be expressed as {m β m >0} β m φ m φm H, SNRtota,(g) =Γ diag [ ] {λ n } n=1 Γ 1 whereγ [ γ 1 γ ]. Then, the minimum cost m n=1 1 β m λ n +1 dominant generaized eigenvectors of R (g) sum and R (g) η as the coumns of S (g). (30) is achieved by choosing the first V. APPROXIMATE CORRELATOR TYPE ESTIMATORS AFTER BEAMFORMING IN HIGH SNR REGIME In this section, a reduced-rank correator-type estimator in the spatio-tempora domain based ony on the pre-beamforming matrix (designed by using ony ong-term channe statistica properties) is constructed. The key idea is to provide a reduced-rank approximation of the Wiener estimator, which performs the optimum weighting of the decorreated channe coefficients in the basis of the eigenvectors given by the coumns of the KLT matrix (10) defined in the spatio-tempora domain. This can be reaized by appying the maximum ikeihood (ML) estimator after a suitabe subspace projection, which suppresses spatia

15 15 interference whie reducing dimensionaity, thus yieds optimum bias-variance tradeoff [20], [30]. The ML (or zero-forcing) estimate, which is unbiased, achieves the Cramer-Rao bound (CRB) [18], [30]. The Wiener estimator converges to the ML estimator as snr approaches infinity after beamforming, i.e., E s N 0. The ML estimator, based on the deterministic signa mode in (11), does not expoit spatia correations in the muti-path channe vector, nameyρ (g) R (g). Particuary, ML variance approaches to infinity for ow snr [30]. Therefore, the principe of reducing the number of parameters to be estimated, without osing the intended part of the group g signa, can be adopted again. Then, the reduced-rank ML estimator as a post-processing stage can be constructed on a suitabe subspace spanned by the coumns of the pre-beamformer matrix S (g). By this way, one can reduce the estimation error variance (or MSE) consideraby at the expense of the introduced bias when compared to the conventiona (fu dimensiona) ML estimator, since the noise and interference subspace are switched off by the pre-beamforming in the spatia domain. The neary optima pre-beamformer, constructed by the generaized eigenvector beamspace (GEB) in Section IV-B, is used here to reduce the dimensionaity before appying the ML estimator. The GEB is an appropriate aternative for subspace projection, since it captures a significant portion of a MPCs in group g, whie rejecting inter-group interference. A. High SNR Approximation for Ange omain Estimator First, the reduced-rank correator-type approximation for the ange domain RR-MMSE in (19) is obtained. As mentioned in Section III-B, it corresponds to the case where a MPCs of group g are unified in the anguar domain, and the rank reduction is performed based on the spatia channe properties captured by R (g) sum in (20). If the GEB is used, the SNR tota,(g) S (g) matrix, obtained after pre-beamforming in (20), is competey diagonaizabe (as shown in Appendix I). In this case, the reduced-rank ML estimate of the effective channe in (17) can be obtained by etting E s N 0 in (19). This can be fufied by keeping N 0 fixed in (20), and aowing the training power E s to approach infinity. After foowing the mathematica steps provided in Appendix III, the RR-MMSE estimate of the effective channe in (19) can be approximated as ĥ (g) e f f,2 {( [X (g) ] H X (g) ) 1 [X ] (g) H [ ] } (g) H S y if T K g L g, { [X ] ( (g) H X [ (g) X (g)] ) H 1 [ ] } (g) H S y where X (g) in (9) is assumed to be fu coumn or row rank. if T< K g L g (31)

16 16 B. High SNR Approximation for Joint Ange-eay omain Estimator The joint ange-deay domain RR-MMSE in (18) can aso be approximated in a simiar fashion by obtaining the ML estimate in reduced dimensiona subspace. After pre-beamforming, it is assumed that the eigenspaces of each MPC of group g are neary orthogona, an effect more strongy observed in mm wave channes especiay for the case of a arge number of antenna eements. In this case, the matrices R (g) () SNR(g) code () for = 0,..., L g 1 in (18) have orthogona eigenspaces as expained in Section IV-B, and SNR (g) () matrices are competey diagonaizabe with the use of GEB (see Appendix I). 1) Rank-1 Approximation By assuming that the rank of each MPC is one, i.e., r g, = 1 for a in (3), which is reasonabe in the case of highy directiona propagation, and using the pre-beamformer structure in (27), the foowing approximation for the RR-MMSE estimate of the effective channe in (18) is obtained after some mathematica steps given in Appendix III L g 1 ĥ (g) e f f pinv { X [ ]} (g) I Kg E Lg, [ S (g) } {{ } E ] H, } {{ } y (32) =0 (tempora) correator pre-beamformer where pinv{} is a generaized operation known as the Moore-Penrose pseudoinverse, used to obtain the inverse of singuar or non-square matrices. In (32), E, is defined as a eementary diagona matrix where a the entries except the (+ 1) th diagona one are zero, and the dimension of the pre-beamformer is set as =L g. Here, S (g) E, is steered towards the AoA of the th MPC of group g whie rejecting other MPCs of groups g and inter-group interference. After the beamspace processing, the tempora processing, in the form of correator, is appied in order to differentiate between the MPCs of a group g users at the th deay, and to combat with other interfering sources (with overapping AoA support) by simpy pacing the tempora finger on the th tempora diversity path for a K g intra-group users. 2) Approximation for Genera-Rank Signa Modes The approximate spatio-tempora correator in (32) can be extended to the more genera case, where the rank of each resovabe MPC covariance is greater than one, and there exists significant overap among some of the MPCs in the anguar domain. In this case, one can simpify the probem by partitioning the MPCs of group g into groups in the anguar domain such that some of the MPCs at a specific deay, having approximatey simiar eigenspaces (common AoA support), are paced into the same group. The key idea is to construct MPC groups (resovabe in anguar domain) whose AoA supports are neary orthogona

17 17 in the anguar domain so that the reduced-rank ML estimator for each group of resovabe MPCs can be reaized separatey. By assuming that the eigenspaces of each SNR (g) () matrix in (18) are mutuay orthogona as expained in Section IV-B, and etting E s N 0 (18) is obtained after carrying out simiar mathematica steps to (32):, the foowing genera approximation for ĥ (g) MPC 1 e f f pinv X (g) H I Kg E Lg,m =0 m L S (g) E,n } {{ } n y (33) } {{ } (tempora) correator pre-beamformer where MPC is the tota number of MPC groups, resovabe in anguar domain, in g having neary non-overapping AoA support,l is the set of non-zero (tempora) deays beonging to the th resovabe muti-path group having MPCs with common AoA support in the anguar domain, and MPC 1 =0 L = L g. In (33), the set is defined as { n Z + 1 m=0 d m< n m=0 d m } for > 0, and { n Z + 0<n d 0 } for = 0 where =d and MPC 1 =0 d =. Here, shows the coumn indices of the pre-beamformer matrix S (g) is constructed as in (27), and S (g) in (27) aowed to pass the th resovabe MPC of group g. In (33), S (g) n E,n, whose non-zero coumns equa to that of S (g) () in (27), can be thought as the N beamformer matrix obtained by repacing a sub-matrices in (27) with zero matrix except S (g) (). The sub-beamformer matrix S(g) () is designated to reject other MPCs of group g in addition to the inter-group interference, and to capture a significant portion of the th resovabe MPC (in a simiar way to the one expained in Section IV-B). In (33), pinv{} operation can be seen as the tempora correator preceded by the pre-beamformer. It performs the task of Least Square (LS) type estimation of reduced dimensiona channes corresponding to the th MPC in group g. The form of (33) appears as the decouped spatio-tempora processing where spatia pre-beamforming and tempora (Rake-type) correator are appied in a successive manner. This further simpifies the RR-MMSE estimator in (18). For L g = 1, =N (no dimension reduction), and S (g) = I N, i.e., the spatia covariance structure of the MIMO channe is not expoited, the approximate estimator in (33) reduces to the conventiona LS type CSI acquisition technique, we-known in the iterature [4], [5], which reies on correating the received signa with the known piot sequence and suffering from piot contamination, whereas with the use of pre-beamformer in (33), the inter-group interfering users eading to piot interference are mitigated in the spatia domain. Moreover, the estimator in (33) does not necessitate the a priori power profie given by KLT in the ange-deay domain.

18 18 VI. NUMERICAL RESULTS AN ISCUSSION In this section, we provide some numerica resuts to evauate the performance of the reduced-rank channe estimators and examine the efficiency of the GEB in Section IV-B for the reduced dimensiona processing. Throughout the demonstrations, we consider a massive MIMO system with upink training in T mode where a BS is equipped with a uniform inear array (ULA) of N= 100 antenna eements aong the y-axis 7, and each of K users has a singe receive antenna. In the studied scenario, K users were custered into eight groups (G=8), and each UT is assumed to be ocated at a specific azimuth angeθ aong the ring centered at the origin in x-y pane. Here, we assume users come in groups, either by nature or by the appication of proper user grouping agorithms in [10], [15], which are out of scope of this work. The channe covariance matrix of each group is specified with the center azimuth angeθ (AoA), and can be cacuated in a simiar way to the ones in [9], [11]. In the simuations, our focus is on the channe estimation accuracy of the intended group g with 3 MPCs, i.e., L g = 3. The first two MPCs of group g stem from a azimuth anguar sector [ 1, 1 ] for deays at = 0, 1, and the anguar sector of the ast MPC at = 2 of g is given as [5, 7 ] in azimuth. We assume two users served simutaneousy for group g, i.e., K g = 2. Each of the other 7 groups (interfering with the intended one) consists of three users, i.e., K g = 3, g g and these users have 3 MPCs whose anguar sectors have same supports of AoA (L g = 3, g g) given by [ 29, 26], [ 21, 19], [ 12, 9], [ 5.5, 3.5], [9.5, 12.5], [15, 17], [24, 27] in azimuth respectivey. The channe vector for each user is independenty generated according to the mode (4). The noise power is set as N 0 = 1 so that a db power vaues are reative to 1. In T mode, inter-group users do not need to be synchronized, and even are aowed to use the same sequences during the upink training mode. Intra-group users (of the intended group) use non-orthogona training waveforms composed of 6 symbos (T = 6), and these are obtained by truncating ength-63 Kasami codes [31] by simpy choosing the first T symbos of ast K g Kasami sequences without any optimization 8. Then, the training matrix in (9) can be constructed to be expoited by the BS during the CSI acquisition period. The trace of the estimation error covariance matrix (for the extended channe vector of group g users in (7) given by R e E{ (h (g) ĥ (g))( h (g) ĥ (g)) H } is evauated to compare the performance of different estimators. Here, the channe estimates in the origina space and in the reduced dimensiona subspace 7 Athough the system mode and the proposed estimators are vaid for an arbitrary array structure in this work, ULA is considered for ease of exposition ony. 8 There are more efficient approaches (other than the truncation of Kasami codes) yieding waveforms with better cross- and auto-correation properties and minimizing (23) or (25), but training optimization is beyond the scope of this work

19 19 after pre-beamforming are defined as ĥ (g) ( W (g)) H y and ĥ (g) e f f ( W (g) ) H e f f y respectivey. For these arbitrary inear estimators, the error covariance can be cacuated by using the R e mmse in (21) (achieved by the reduced-rank Wiener fiter (14)): R e = R mmse e + ( W (g) W mmse (g) ) H ( Ry W (g) W mmse (g) ) (34) where ( W mmse (g) ) HΥ ( ) (g) (g) H ( ) (g) H U W mmse, Υ S in (12), and W (g) is an (NT) (NK g L g ) arbitrary fiter. In a simiar manner, the error covariance matrix of the effective channe estimate (18) can be cacuated in the reduced dimensiona subspace. The covariance matrix of the inter-group interference is evauated by (5) when the anguar sector of each group is provided. The GEB is constructed in a simiar way to the one in Section IV-B by taking the number of resovabe MPCs in the anguar domain as 2, namey,mpc = 2 and setting d 0 = d 1 = 2 in (33). In this work, we compare the performance of dimension reduction based on the GEB (shown to be neary optima under some reaistic assumptions) with that of the conventiona subspace composed of the first dominant eigenvectors of R (g) sum in (20). We ca this conventiona beamspace as discrete Fourier transform (FT) beamspace because the eigenvectors of the spatia correation matrix of the ULA channe are we-approximated by the coumns of the N N unitary FT matrix whose indices correspond to the support of the Fourier transform of the spatia correation function (owing to the Szegö s asymptotic theory) [10] depending on the anguar sector of group g UTs. This conventiona beamspace is known to be information preserving for the spatiay white interference case, and thus, is widey used in practica hybrid beamforming appications, where the beamforming in the RF anaog domain can be impemented by simpe phase shifters [13]. In Figure 1, the beam patterns created by the GEB and FT beamspaces are depicted for =6at snr= 30 (db). The GEB is designed based on the AoA support of the intended group g for = 0, 1, 2 whie taking the anguar ocations of the interfering groups into account. The signas of the inter-group users are assumed to have the same power eve with that of the intended group. As can be seen from the figure, the GEB tries to create deep nus at the anguar ocations of the interfering UTs, whereas the conventiona pre-beamformer ony tries to maximize the captured power of the intended group MPCs for a given dimension. It is expected that as the number of BS antennas increases, the eigenspaces of each group are approximatey orthogona. However, the number of transmit antennas is finite in practice, and there aways exists some overap among the virtua anguar sectors of each group which eads into

20 20 a eakage to the intended group signa. Therefore, as it wi be shown ater, the accuracy of the channe estimation reaized on the reduced dimensiona subspace, spanned by the conventiona FT beamspace, is consideraby ost due to the residua inter-group interference after pre-beamforming. On the other hand, the GEB suppresses the inter-group interference whie aowing the MPCs of the intended group to pass with a negigibe distortion so that the subsequent processing in reduced dimensions, here the instantaneous CSI estimation, can be carried out as accuratey as possibe FT Beamspace ( = 6) Optimized Beamspace ( = 6) Anguar Sector of the intended MPCs Normaized Gain (in db) Interfering Groups Azimuth Ange of Arriva (AoA) Fig. 1. Beam pattern of different pre-beamformers. In Figure 2, the average mean square error (MSE) vaues given by Tr{R e }/K g as a function of the dimension of the spatia domain pre-beamformer () are depicted for both joint ange-deay domain and ange domain RR-MMSE estimators given in (14) at snr= 30 db. For joint ange-deay, the exact knowedge of covariance for each MPC is used, whereas for the ange domain, a common anguar region (obtained by the unification of each deay) is assumed for each MPC (of group g) and used instead of R (g) in (14). Aso, the performance of the fu dimensiona Wiener estimator ( = N) is demonstrated when there are no interfering groups. It is cear that ange domain RR-MMSE estimator is inferior to joint ange-deay domain estimator due to the inefficient use of the training and noise enhancement. Moreover, it is seen that there is a remarkabe performance gap between the performances of RR-MMSE estimators based on two different pre-beamformers (the GEB and the conventiona one) especiay at ower dimensions. Aso, it can be concuded that the RR-MMSE estimator based on the GEB achieves a very cose performance to that of the fu dimensiona estimator even for = 7 (for group g), that is roughy

21 21 15-fod dimension reduction. On the other hand, with the conventiona beamspace, in spite of the optima Wiener fitering after dimension reduction, the MSE performance is not satisfactory for dimensions beow FT Beamspace + RR MMSE Est. (Ange Ony) Opt. Beamspace + RR MMSE Est. (Ange Ony) FT Beamspace + RR MMSE Est. (Joint Ange eay) Opt. Beamspace + RR MMSE Est. (Joint Ange eay) No Interfering Groups, Fu Rank MMSE Est. (Joint Ange eay) 10 1 MSE 10 2 = imension of the Beamspace () Fig. 2. MSE vaues of RR-MMSE estimator versus dimension of the pre-beamformer. In Figure 3, by adopting the same settings used to obtain in Figure 2, the MSE performance for different estimators of the effective channe in -dimensiona spatia subspace are shown. The performance of joint ange-deay RR-MMSE estimator based on the GEB in (18) is used as the performance benchmark. The MSE vaues achieved by different effective channe estimators are normaized by this benchmark for each dimension, and these reative MSE vaues are given as a function of the dimension (starting at 7). Aso, the spatio-tempora correator-type estimators in (31) and (33), obtained after the high snr approximation of the reduced-rank Wiener fiter, are depicted. It is seen that reative performance of these approximate correator-type estimators degrades as the dimension increases. This degradation is expected, since LS-type estimation does not expoit spatia correations in the channe coefficients, and does not appy optimum spatia weights when compared to MMSE fitering. In this case, adding extra dimensions beyond 7 eads to noise enhancement, since the noise subspace is not switched off propery with increasing dimensions, and starts to contaminate the effective channe estimates. In addition to that, the approximate estimator is more sensitive to which pre-beamformer is utiized such that there is a remarkabe performance gap between the GEB and the conventiona beamspaces when the approximate estimator is reaized for both joint ange-