Joint Spatial Division and Multiplexing: Realizing Massive MIMO ...

Joint Spatial Division and Multiplexing: Realizing Massive MIMO Gains with Limited Channel State Information Junyoung Nam∗ , Jae-Young Ahn∗ , Ansuman Adhikary† , and Giuseppe Caire† ∗

Mobile Communications Division Electronics Telecommunications Research Institute (ETRI), Daejeon, Korea, email:{jynam, jyahn}@etri.re.kr † Ming-Hsieh Department of Electrical Engineering University of Southern California (USC), CA, email:{caire, adhikary}@usc.edu

(Invited Paper)

Abstract—We propose Joint Spatial Division and Multiplexing (JSDM), an approach to multiuser MIMO downlink that exploits the structure of the channel vectors correlation in order to allow for a large number of base station antennas while requiring a considerably reduced channel state information (CSI) feedback from the users. Thanks to the reduced CSI requirement, JSDM allows the use of a large number of base station antennas also in FDD systems. This work presents the correlated channel model, the main idea of JSDM precoding, and the design of JSDM precoding, formed by the concatenation of a pre-beamforming matrix based only on the structure of the channel covariance, with suitable multiuser precoding based on the CSI of the reduced dimensional transformed channels. We show that in some cases JSDM incurs no loss of optimality with respect to the MIMO broadcast channel with full CSI.

I. I NTRODUCTION We consider a Multiuser MIMO (MU-MIMO) wireless downlink where the base station (BS) has multiple antennas and serves many user terminals (UTs). A major problem to realize in practice the capacity gains promised by MU-MIMO is the availability of channel state information (CSI) at the BS. Analysis in [1] and references therein have shown that the full CSI capacity can be closely approached at high SNR if the channel estimation error decreases at least as fast as the inverse of SNR. In Frequency Division Duplexing (FDD) systems, this is possible if downlink training and CSI feedback make use of a number of channel time-frequency dimensions per channel coherence block at least as large as the number of BS antennas. For typical coherence block length, it turns out that MU-MIMO in FDD systems cannot afford a large number of BS antennas, otherwise the training and feedback overhead consumes the whole system capacity (as quantified by the analysis in [2]). In contrast, Time Division Duplexing (TDD) systems can exploit channel reciprocity for estimating the downlink channels from uplink training. In this case, the training dimension is determined by the number of total UT antennas, while the number of BS antennas can be made as This work was supported by the IT R&D program of MKE/KEIT in Korea [Development of beyond 4G technologies for smart mobile services].

large as desired. It has been recently noticed (see [3], [4]) that using a very large number of BS antennas (much larger than the number of served UTs) is very attractive for TDD systems, both in terms of achieved throughput and in terms of simplification of the downlink scheduling and the signal processing at the BS. In this work, we propose a Joint Spatial Division and Multiplexing (JSDM) approach to potentially achieve throughput gains and system operations simplification similar to massive MIMO for FDD systems, which are by far the majority of currently deployed cellular networks. The main idea of JSDM is to partition the user population into groups that have similar transmit correlation matrices and induce as small inter-group interference as possible by the user location geometry. Then, MU-MIMO precoding is obtained in two stages: first, a prebeamforming stage is used to reduce the effective channel dimension; then, a multiuser precoding scheme (linear or nonlinear) operates on the lower dimensional transformed channel. Only the second stage requires feedback of the instantaneous (transformed) channel realization, while the pre-beamforming stage makes use of only the slowly-varying covariance information, which can be tracked with low protocol overhead. We show that, under some channel conditions, JSDM incurs no loss of optimality with respect to the full CSI case. When these conditions cannot be met, we examine the design of the pre-beamforming matrix and the performance of regularized zero forcing (linear) precoding for the transformed channel resulting from pre-beamforming. II. S YSTEM D ESCRIPTION A. Channel Model Consider a family of spatially correlated channels obeying the well-known Kronecker correlation model [5], [6] 1

1

H = RR 2 W RT 2

(1)

where the elements of W are i.i.d. ∼ CN (0, 1), and R T and R R denote the transmit and receive correlation matrices, respectively. In the typical cellular downlink case, the base

978-1-4673-3140-1/12/$31.00 ©2012 IEEE

station (BS) is elevated and free of local scatterers, and the user terminals (UTs) are placed at ground level and are surrounded by local scatterers. Thus, the channel in the form of (1) may reduce to the one-ring model [5], for which we have R R = I . In this case, we let R ! R T for notational simplicity, define R ) and let r∗ denote the number of dominant (nonr = rank(R negligible) eigenvalues of R . In this paper, we consider the downlink of a single-cell FDD system with a BS with M antennas serving K UTs equipped with a single antenna each. By using the Karhunen-Loeve transform, the channel vector from the BS antenna array to a UT can be expressed as 1

h = UΛ2w

(2)

where w ∈ C ∼ CN (00, I ), Λ is an r × r diagonal matrix whose elements are the non-zero eigenvalues of R , and U ∈ CM ×r is a matrix whose columns are the eigenvectors of R corresponding to the non-zero eigenvalues. For the onering model of Fig. 1, a UT located at azimuth angle θ and distance s is surrounded by a ring of scatterers of radius r such that the angular spread (AS) is ∆ ≈ arctan(r/s). Assuming a uniform distribution of the received power from planar waves impinging on the BS array, we have that the correlation coefficient between BS antennas 1 ≤ m, p ≤ M is given by ! ∆ T 1 R ]m,p = ejkk (α+θ)(uum −uup ) dα, (3) [R 2∆ −∆ r×1

T is the wave vector where k (α) = − 2π λ (cos(α), sin(α)) for a planar wave impinging with AoA α, λ is the carrier 2 wavelength, and u m , u p ∈ R are the vectors indicating the position of BS antennas m, p in the two-dimensional coordinate system (see Fig. 1).

r s ∆ ∆

scattering ring

θ

region containing the BS antennas

Fig. 1. A UT at AoA θ with a scattering ring of radius r generating a two-sided AS ∆ with respect to the BS at the origin.

Let H denote the M × K system channel matrix given by stacking the K users channel vectors by columns. The signal vector received by the UTs is given by y = H †V d + z = H †x + z

(4)

where V is the M × s precoding matrix with s the rank of V dd †V † ] (i,e., the number of the input covariance Σ = E[V all independent data streams sent to the users), d is the sdimensional transmitted data symbol vector, and z ∼ CN (00, I )

denotes the Gaussian noise at the UT receivers. The transmit signal vector is given by x = V d . B. Joint Spatial Division and Multiplexing (JSDM) The biggest hurdle that makes FDD large-scale MIMO systems infeasible in practice is the CSI feedback: when M is large, feeding back M × 1 channel vectors by K UTs with sufficiently fine accuracy to enable effective downlink precoding at the BS consumes prohibitively many uplink resources. The motivation of JSDM is that such hurdle may be avoided by exploiting the fact that some UTs have similar transmit correlation matrices (or similar AoA and AS) and further by leveraging a desirable structure among transmit correlation matrices of some groups. A general model for JSDM is presented in the following. Suppose that K UTs are selected to form G groups based on the similarity of their channel correlation matrices. We define K $ = K/G, M $ = M/G and s$ = s/G as the number of UTs, BS antennas and data streams per group, where K $ , M $ and s$ are integers. Also, we enumerate the UTs such that gk = (g − 1)K $ + k is the index of user k in group g, for g = 1, · · · , G and k = 1, · · · , K $ . We assume for simplicity that UTs gk in group g have the same covariance matrix R g = U g Λg U †g , with rank rg and rg∗ ≤ rg dominant eigenvalues. In practice, this condition is not verified exactly, but we can select groups such that this condition is closely approximated. Also, the notion of “dominant eigenvalues” is intentionally left fuzzy, since rg∗ is a design parameter that depends on how much signal power outside the subspace spanned by the corresponding dominant eigenmodes we can tolerate. For future reference, we denote by U ∗g the M ×rg∗ matrix collecting the dominant eigenvectors, U ∗g , U $g ], with U $g of dimension M × (rg − and let U g = [U ∗ rg ), containing the eigenvectors corresponding to the weakest eigenvalues. 1 The channel vector of user gk #is given by h gk" = U g Λg2 w gk#. " We let H g = h g1 , · · · , h gK ! and H = H 1 , · · · , H G denote the group channel and the overall system channel matrices, respectively, reordered according to the index gk . JSDM is based on two-stage precoding given by V = B P , where B ∈ CM ×b is a pre-beamforming matrix that depends only U g , Λg }, on the channels second-order statistics, i.e., on set {U or on some directional information extracted from the channel covariance matrices, and P ∈ Cb×s is a beamforming matrix that depends on the instantaneous realization of the system † transformed channel H , defined by H ! H †B . $ We choose b such that b = b/G is an integer not smaller than s$ , and let B g be the M × b$ pre-beamforming matrix of group g. The received signal (4) can be rewritten as † y = H Pd +z

where



 † H = 

(5)

H †1B 1 H †1B 2 ··· H †1B G H †2B 1 H †2B 2 ··· H †2B G

.. .

.. .

..

.

.. .

H †GB 1 H †GB 2 ··· H †GB G



 . 

If the estimation and feedback of the transformed channel H can be afforded, the precoding matrix P is determined as a function of H . We refer to this approach as joint group processing. However, the estimation and feedback of the transformed channel may still be too costly in terms of transmission resource. In this case, we can feed back only the G diagonal blocks H g = B †g H g , and treat each group separately. We refer to this approach as per-group processing. In this case, the precoding matrix takes on the block-diagonal ! ! P 1 , · · · , P G ), where P g ∈ Cb ×s , resulting form P = diag(P in the vector broadcast plus interference Gaussian channel y g = H g B gP gd g + †

*

H g B g! P g! d g! + z g , †

(6)

g ! %=g

for g = 1, . . . , G. With per-group processing, it is interesting to choose G groups and design the pre-beamforming matrix such that H g †B g! ≈ 0, for all g $ '= g.

(7)

U g ) '⊆ Exact Block Diagonalization (BD) is possible if Span(U U g! : g $ '= g}) for all g = 1, . . . , G. In particular, Span({U in order to deliver s$ streams to each group we need that U g ) with the orthogonal complement the intersection of Span(U U g! : g $ '= g}) has dimension not smaller than of Span({U s$ . Approximate BD can be achieved by selecting rg∗ domU ∗g ) '⊆ inant eigenmodes for each group g, such that Span(U ∗ $ U g! : g '= g}) for all g = 1, . . . , G. In this case, in Span({U order to deliver s$ streams to each group, the intersection of U ∗g ) with the orthogonal complement of Span({U U ∗g! : Span(U $ $ g '= g}) must have dimension not smaller than s . With pergroup processing, the pre-beamforming creates virtual sectors, i.e., a generalization of spatial sectorization commonly used in current cellular technology. Each group corresponds to a virtual sector, and it is independently precoded under a total sum power constraint and some residual inter-sector interference. Remark on the channel covariance matrix estimation: It is reasonable to assume that the channel covariance matrix R g for each user group changes extremely slowly with respect to the coherence time of the instantaneous channel matrix H g . In addition, it is well-known that for the wide-sense stationary uncorrelated scattering fading model [7] R g is frequencyinvariant. Therefore, all subcarriers in an OFDM system (e.g., LTE [8]) can be used to estimate R g . The dominant channel eigenmodes U ∗g can be tracked for each UT using some wellknown subspace tracking algorithm [9], by exploiting the downlink common training, and fed back to the BS at a low rate. Furthermore, these estimates can be refined at the BS by exploiting the uplink, even though in an FDD system this takes place at a different carrier frequency (see [10]). In this work, we assume that the channel covariance matrix for each user is known, such that the BS can form groups of users with approximately the same dominant eigenspace, as said above.

III. JSDM WITH E IGEN -B EAMFORMING A. Achieving capacity with reduced CSI For simplicity, we assume that the G groups can be chosen in a symmetric manner, such that rg = r for all g. If G groups U 1 , · · · , U G ] is M × rG tall can be found such that U = [U unitary (i.e., rG ≥ M and U †U = I ), we let b$ = r and B = U to obtain exact BD. In this case, the decoupled MUMIMO channel (8) takes on the form y g = H g †P g d g + z g = W †g Λ1/2 g P gd g + z g

(8)

where W g is a r × K $ i.i.d. matrix with elements ∼ CN (0, 1), for all g. The next results show that, under these assumptions, JSDM achieves the same sum capacity of the MU-MIMO downlink with full CSI: Theorem 1: For U tall unitary, the sum capacity of the original MU-MIMO downlink channel (4) with full CSI is equal to the sum capacity of the set of decoupled channels (8). " H ; P ) denote the sum capacity of (4) Proof: Let C sum (H with sum power constraint P and fixed channel matrix H , perfectly known to transmitter and receivers. By the MACBC duality [11], we have + + G + + * + † 1/2 † + H ; P ) = max log +I + C sum (H U g Λ1/2 g W g S g W g Λg U g + + + g=1

(9)

where the maximization is over the set of,diagonal K $ × K $ S g ) ≤ P . For input covariance matrices S g , subject to g tr(S any fixed set of matrices S g , define for notation simplicity † 1/2 A g = Λ1/2 g W g S g W g Λg . Notice that A g has dimension r×r and is invertible with probability 1 over the random channel realization. The determinant identity + + G G + + + + * + + + †+ (10) U g A g U †g + = +I + +II + U g A g U g + + + g=1

g=1

can be shown by induction, noticing the following step: for any 1 ≤ g $ ≤ G, + + + + G * + + †+ +I + U gA gU g + + + + g=g ! + + + G + + ++ * + + + † + † −1 †+ = +II + U g! A g! U g! + +I + (II + U g! A g! U g! ) U gA gU g + + + g=g ! +1 + + + + = +II + U g! A g! U †g! + · + + + + G * + + † −1 †+ −1 + Ag! + I ) U g! ) · +I + (II − U g! (A U gA gU g + (11) + + g=g ! +1 + + + G + + ++ * + + + (12) = +II + U g! A g! U †g! + ++I + U g A g U †g ++ , + + g=g ! +1

where (11) follows form the matrix inversion lemma and (12) follows from the the fact that, by assumption, U †g! U g = 0 for all g $ '= g. Using (10) in (9) we obtain H ; P ) = max C sum (H

G * g=1

+ + + † 1/2 + log +II + Λ1/2 g W g S g W g Λg + ,

(13)

which is clearly the same as the dual MAC for the set of decoupled MU-MIMO downlink channels (8). With some (minor) extra effort, it is possible to show that under this orthogonality condition JSDM achieves the whole capacity region, and not only the sum capacity. This simple result suggests the important practical implication that the downlink scheduler at the BS should serve simultaneously group of users for which U is close to tall unitary, so that JSDM (with suitable optimal precoding) is near capacityachieving, while requiring reduced CSI feedback. For JSDM with joint group processing, a valid approach for choosing B consists of eigenbeamforming along the directions of the dominant eigenmodes of each group, i.e., letting B g = U ∗g for some desired value r∗ . When the tall unitary condition can not be met with r replaced by r∗ in practice, this approach with per-group processing may incur excessive inter-sector interference. In this case, an alternative approach based on exact or approximated BD is examined next. B. Block diagonalization B 1 , . . . , B G ] is an M ×b matrix consisting Recall that B = [B of G M × b$ blocks, each corresponding to a particular group g. For some target number of streams per group s$ and some b$ ≥ s$ , our goal is to design the blocks B g such that BD is U †g B g ) ≥ achieved, i.e., U †g! B g = 0 for all g $ '= g and rank(U $ s . A necessary condition for exact zero-forcing of the offU g! : g $ '= g}). B g ) ⊆ Span⊥ ({U diagonal blocks is Span(B ⊥ $ U g! : g '= g}) has dimension smaller than s$ , When Span ({U the rank condition on the diagonal blocks cannot be satisfied. In this case, s$ should be reduced or, as an alternative, approximated BD based on the r∗ < r dominant eigenmodes for each group can be implemented. This consists of replacing U g with U g! : g $ '= g}) has U ∗g in the above conditions. When Span({U dimension M , then exact BD cannot be achieved even for s$ = 1, and therefore approximated BD should be considered in any case. Without loss of generality, we formulate the design B g } for approximated BD with some feasible choice of of {B r∗ and s$ . Following the approach of [12], we define # " (14) Ξg = U ∗1 , . . . , U ∗g−1 , U ∗g+1 , . . . , U ∗G , assumed of rank r∗ (G − 1). Let the SVD of Ξg be given by E 1g , E 0g ]S S g F †g Ξg = [E

(15)

where E 0g is M × (M − r∗ (G − 1)) and contains the left eigenvectors corresponding to the zero singular values. E 0g ) is the orthogonal complement of Clearly, Span(E ∗ $ U g! : g '= g}). We obtain B g by concatenating the Span({U

E 0g ) with eigenbeamforming along the projection onto Span(E dominant eigenmodes of the covariance matrix of the proE 0g )†H g . jected channels of group g, i.e., of the columns of (E Recalling the Karhunen-Loeve decomposition (2), we have that the covariance matrix of the projected channel vector E 0g )†U g Λ1/2 hˆ gk = (E g w gk is given by ˆ g Φg U ˆ g = (E ˆ †, E 0g )†U g Λg U †g E 0g = U R g

(16)

ˆ g. where the expression on the right denotes the SVD of R b! ˆ as the submatrix of U ˆ g that contains the domiDefining U g $ nant b eigenvectors, we eventually obtain b!

B g = E 0g Uˆ g .

(17)

The pre-beamforming matrix B g can be interpreted as being orthogonal to the dominant r∗ eigenmodes of other groups and matched to the b$ dominant eigenmodes of the covariance matrix of the projected channels of group g, i.e., the columns E 0g )†H g . If rG ≤ M , we can choose b$ = r∗ = of the matrix (E r and obtain exact BD. IV. P ERFORMANCE ANALYSIS WITH REGULARIZED ZF In this section we provide expressions for the performance analysis of JSDM with joint group processing and per-group processing using the techniques of deterministic equivalents [13]. This analysis technique can be applied as long as the users in each group are randomly selected. Hence, we assume here that for each group a subset of s$ out of the possible K $ users is pre-selected and scheduled for transmission over each time-frequency slot (transmission resource block). This simplified non-opportunistic scheduling requires only the instantaneous CSI feedback from the pre-scheduled users and it is in line with the massive MIMO concept, where hardware augmentation at the BS allows significant simplification in the system operations. Under these assumptions, the transformed channel matrix H has dimension b×s, with blocks H g of dimension b$ ×s$ . Also for simplicity we allocate to all users the same fraction of the total transmit power, pgk = Ps . In the following, we present the deterministic equivalent fixed-point equations for the case of JSDM with joint group processing and linear regularized zero forcing precoding. Analogous analysis can be obtained for the case of per-group processing, and for the linear zero forcing precoding. These are omitted because of space limitation (see details in [14]). For fixed pre-beamforming matrix B and joint group processing, the regularized zero forcing precoding matrix is given by HH † + bαII b )−1H . (18) P rzf = ζ(H The covariance matrix of the transformed channel of users in group g is given by   † † † B 1R g B 1 B 1R g B 2 ··· B 1R g B G B †2R g B 2 ··· B †2R g B G

 B †R B ˜ g =  2 .g 1 R  ..

.. .

.. .

B †GR g B 1 B †GR g B 2 ··· B †GR g B G

 . 

(19)

HH† + +bαII b )−1 , the SINR for a user gk given Letting K = (H by h†gk B K B †hgk |2 pgk ζ 2 |h (21) γgk ,jgp = , 2 h † B K B †h |2 + 1 j gk j%=gk pj ζ |h

1 0.9 0.8 0.7

CDF of eigen values

ζ is a normalization factor chosen to satisfy the power constraint and is given by s / (20) ζ2 = . † tr P rzf B †B P rzf

M →∞

0.2

Υog

0 0

5

10

15

20

Eigen Values

(23)

350

and Γ are given o

300

Capacity ZFBF, JGP RZFBF, JGP ZFBF, PGP RZFBF, PGP

250

(24) (25)

Sum Rate

mog ,

P Γo

0.1

Fig. 2. CDF of the non-zero eigenvalues of the channel covariance matrix R g for M = 100, ∆ = 15o .

pgk ζ 2 (mog )2 2 ζ Υog + (1 + mog )2

where ζ = and the quantities by the system of fixed-point equations 1 .˜ / mog = tr Rg T b −1  G $ * ˜ g! R s + αII b  T =  b ! 1 + mog! 2

0.4

(22)

where for all users gk γgok ,jgp =

0.5

0.3

where the subscript “jgp” stands for joint group processing. The deterministic equivalent of the SINR is given by γgk ,jgp − γgok ,jgp −→ 0

0.6

200

150

g =1

Γ

=

Υog

=

o

G ng ! 1P * (26) b G ! (1 + mog! )2 g =1   G $ * ! ng ,g ng,g  1P  s −1 o )2 + $ bG ! (1 + m s (1 + mog )2 ! g ! g =1,g %=g

(27)

with n = [n1 , n2 , . . . , nG ]T , n g = [n1,g , n2,g , . . . , nG,g ]T given as n = (II G − J )−1v n g = (II G − J )−1v g

(28) (29)

where J , v and v g are given as . / s! ˜ iT R ˜ jT tr R b J ]ij = (30) [J b(1 + moj )2 / . /5T 1 4 .˜ ˜ GT B †B T (31) tr R 1T B †B T , . . . , tr R v = b . /5T 1 4 .˜ ˜ / ˜ GT R ˜ gT tr R 1T R g T , . . . , tr R (32) vg = b V. R ESULTS In this section, we present some numerical examples especially when the tall unitary condition cannot be satisfied by the system geometry. Moreover, we validate the method of deterministic equivalents by comparing with finite-dimensional Monte Carlo simulations. We set the number of BS antennas to M = 100 and consider G = 6 user groups. The user

100

50

0 0

5

10

15

20

25

30

SNR (in dBs)

Fig. 3. Comparison of sum throughput (bit/s/Hz) vs. SNR (dB) for JSDM with their corresponding deterministic equivalents when r∗ = 6. ‘JGP’ denotes JSDM with joint group processing and ‘PGP’ denotes JSDM with per-group processing.

channel correlation is obtained according to (3) for a uniform circular array with isotropic antenna elements with physical coordinates (λD cos(2πm/M ),λD sin(2πm/M ))T , m = 0, . . . , M − 1,

and D = √

0.5 , so that the minimum (1−cos(2π/M ))2 +sin(2π/M )2 distance between two antennas is λ2 . We choose the AS ∆ = 15o and azimuth AoA for the groups θg = −π +∆+(g −1) 2π G

for g = 1, . . . , G. For the system geometry defined above, the transmit covariance matrix has rank r = 21 for all groups. Figure 2 shows the CDF of the non-zero covariance eigenvalues. Notice that approximately half of the non-zero eigenvalues are close to zero, yielding an effective rank r∗ = 11. We fix s$ = 5, so that the total number of users being served is s = s$ G = 30. We set b$ = 10. Figures 3 and 4 show the performance of the JSDM schemes when the pre-beamforming matrix is designed according to

350

300

Capacity ZFBF, JGP RZFBF, JGP ZFBF, PGP RZFBF, PGP

Sum Rate

250

200

150

100

50

0 0

5

10

15

20

25

30

SNR (in dBs)

Fig. 4. Comparison of sum throughput (bit/s/Hz) vs. SNR (dB) for JSDM with their corresponding deterministic equivalents when r∗ = 12.

the approximate block diagonalization method described in Section III-B with r∗ = 6 and r∗ = 12 respectively. The solid ‘squares’ are obtained through simulations and the dotted ‘x’ are obtained using the corresponding deterministic equivalents. s The regularization parameter is fixed to α = bP , which is optimal [13]. The solid ‘circles’ in green denote the sum capacity of the MIMO BC channel with full CSI (see (4)) computed using the iterative waterfiling approach [15]. VI. C ONCLUDING R EMARKS In this paper we presented a Joint Spatial Division and Multiplexing (JSDM) approach that aims at achieving the benefits in throughput and system operation simplification (scheduling) of massive MIMO, with reduced overhead requirements for downlink channel estimation and CSI feedback. The key features exploited by JSDM are that users can be partitioned into groups with similar transmit correlation and that the tall unitary structure among transmit correlation matrices of some groups can be satisfied by the system geometry (user locations). Then, MU-MIMO precoding is obtained by concatenating a pre-beamforming stage, based only on the eigenmodes of transmit correlation matrices, with a precoding matrix (linear or non-linear) for the reduced-dimension transformed channel. If the eigenmodes of the served user groups form a tall unitary matrix, then JSDM incurs no loss of optimality even with per-group processing. Otherwise, we examined the design of the pre-beamforming matrix for joint group processing and per-group processing, and the performance of regularized zero forcing (linear) precoding for the resulting transformed channel. Our results show that the performance of JSDM with approximate BD and per-group processing improves as the number of dominant eigenmodes r∗ included in the pre-beamforming computation increases. On the other hand, increasing r∗ eventually reduces the number s$ of users scheduled in each group and, under practical subspace tracking and CSI feedback schemes, it increases the cost of the CSI feedback overhead. This shows that the JSDM performance tradeoffs for a given system geometry are

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