IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 3, MARCH 2012 649

Multiuser MIMO User Selection Based on Chordal DistanceKyeongjun Ko and Jungwoo Lee, Senior Member, IEEE

AbstractMultiuser MIMO (MU-MIMO) systems have ad-vantages over single-user MIMO systems in terms of systemperformance. In MU-MIMO systems, inter-user interferenceneeds to be dealt with especially when linear processing is used.The block diagonalization (BD) method is one of techniques thatare widely used to eliminate the inter-user interference. In acellular system where there are many users, the subset of userswhich maximizes the system performance should be selectedsince the base station cannot support all the users in the cell. Inthis paper, we propose a low complexity MU-MIMO schedulingscheme using BD with chordal distance. For a large numberof users, the optimal scheduling technique needs an exhaustivesearch, which is impractical. One of the key ideas of this paperis to use chordal distance as a measure of orthogonality betweendifferent users. Simulation results show the proposed algorithmhas throughput close to the optimal scheduling scheme with lowercomplexity than existing low complexity scheduling algorithms.

Index TermsMU-MIMO systems, scheduling, chordal dis-tance.

I. INTRODUCTION

MULTIPLE-INPUT multiple-output (MIMO) systemshave been drawing a lot of attention because of spectralefficiency and diversity gain [1], [2]. It is known that select-ing the user who has the best channel is optimal in termsof capacity in multiuser single-input single-output (SISO)systems [3]. However, it is optimal to serve multiple userssimultaneously in multiuser MIMO systems [4]. Inter-userinterference needs to be dealt with in multiuser MIMO (MU-MIMO) systems since data are transmitted to multiple users inthe same frequency simultaneously. Dirty paper coding (DPC)is an optimal technique to remove the inter-user interference[5]. It was shown that the interference which is known apriori to the transmitter does not affect the channel capacity.DPC is a nonlinear scheme that achieves the channel capacityin MIMO broadcast channel. However it is impractical forimplementation since its complexity is prohibitive.

In order to lower the complexity, many precoding algo-rithms for MU-MIMO systems have been proposed. Zero-forcing (ZF) and block diagonalization (BD) have been usedwidely among them [6][8]. The ZF scheme removes inter-user interference among the selected users by using a precod-

Paper approved by D. I. Kim, the Editor for Spread Spectrum Transmissionand Access of the IEEE Communications Society. Manuscript received March10, 2011; revised June 28, 2011 and September 10, 2011.

This research was supported in part by the Basic Science Research Program(KRF-2008-314-D00287, 2010-0013397) and the Mid-Career Researcher Pro-gram (2010-0027155) through the NRF funded by the MEST, Seoul R&BDProgram (JP091007, 0423-20090051), the KETEP grant (2011T100100151),the INMAC, and BK21. This paper was presented in part at the Conferenceon Information Sciences and Systems (CISS), Princeton, NJ, March 2008.

The authors are with the School of Electrical Engineering and Com-puter Sciences, Seoul National University, Seoul 151-744, Korea (e-mail:smuff@wmspl.snu.ac.kr; junglee@snu.ac.kr).

Digital Object Identifier 10.1109/TCOMM.2012.020912.110060

ing matrix, which is the pseudo-inverse of the selected userschannels. It is typically used in multiple-input single-output(MISO) systems. On the other hand, the BD scheme usesnull space which is computed by singular value decomposition(SVD). For a given user, a matrix is constructed by stackingall the users channels except its own channel, and finds thenull space by SVD. However, the number of users who can besimultaneously supported with BD is limited by the numberof transmit antennas and the number of receive antennas sinceeach users precoding matrix must lie in the null space of allthe other users channels. It can be considered as an extensionof ZF, and can be used for MU-MIMO systems. In MU-MIMOsystems where there are many users in a cell, the optimalscheduling (user selection) is computationally prohibitive. Theoptimal strategy is a brute-force approach to find the bestsubset of users exhaustively. Many suboptimal user selectionschemes have been proposed to reduce the computationalcomplexity of the optimal (exhaustive) user selection scheme[9],[10],[12].

A suboptimal user selection scheme by using Gram-Schmidtorthogonalization (GSO) in MU-MISO systems was proposedin [9]. However, it may be difficult to use GSO in MU-MIMO systems because the channel is a matrix while GSOcan only be used for vectors. For MU-MIMO systems,two low complexity user scheduling algorithms, the capacitybased algorithm and the Frobenius norm based algorithm,were proposed in [10]. These algorithms are based on agreedy method, and achieve performance close to the optimalscheduling algorithm. But their computational complexity isrelatively high. The capacity based algorithm needs frequentcomputation of SVD, and the Frobenius norm based algorithmalso needs heavy GSO computation. Therefore, our motivationis to further reduce the complexity of user selection in MU-MIMO systems.

In this paper, we propose a new low complexity schedul-ing algorithm with the BD scheme to maximize the totalthroughput. We introduce chordal distance as a new distancemetric. The scheduling algorithm select a new user whohas the maximum chordal distance with previously selectedusers channels. It has lower computational complexity since itrequires less GSO computation than the Frobenius norm basedalgorithm. The proposed scheduling algorithm is extendedto the case where the SVD-based receiver (Rx) processingis used, which will be shown to improve the sum-capacitysignificantly. The rest of this paper is organized as follows.Section II and Section III introduce the system model and thebackground, respectively. The proposed algorithm is describedin Section IV. Section V presents the computational complex-ity analysis, and Section VI presents the simulation resultsfor the proposed scheduling algorithm as well as the receiver

0090-6778/12$31.00 c 2012 IEEE

650 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 3, MARCH 2012

processing (SVD) algorithm. Finally, conclusions are given inSection VII.

II. SYSTEM MODEL

We consider an MU-MIMO downlink system with a singlebase station (BS) which has M transmit antennas, and KTusers with N receive antennas. We assume that the receiversestimate their channels perfectly, and the BS knows the exactchannel state information (CSI) of all the users. The broadcastchannel of an MU-MIMO system is then given by

yk = Hkx+ nk, k = 1, . . . ,KT (1)

where Hk (k = 1,. . . ,KT ) is the N M channel matrix atthe kth user whose channel entries are independent identicallydistributed (i.i.d.) complex Gaussian with zero mean and unitvariance, nk is the complex white Gaussian noise vector forthe kth user whose elements are also i.i.d. complex Gaussianwith zero mean and unit variance, and yk is the received signalvector at the kth user. The transmitted signal, x, is given by

x =

Ki=1

PiVisi (2)

where K is the number of the simultaneously selected users, siis the ith symbol vector with E[si2] = 1, Pi is the allocatedpower for the ith selected user, and Vi is the Mnk precodingmatrix for the ith selected user.

III. BACKGROUND

A. Block Diagonalization

The received signal for the kth user denoted by (1) can bedivided into the intended signal term, the interference term,and the noise by using (2), which is given by

yk =P kHkVksk +Hk

Kj=1,j =k

PjVjsj + nk. (3)

To simplify the notation, we assume that the users are appro-priately re-indexed after the user selection.

The first term of (3) is the desired signal for the kth user,and the second term is the interference term. BD eliminatesthe interference term by finding the null space of the channelmatrices of the other users. To be more precise, the precodingmatrix Vk is found by removing the interference term as in

H(1)Vk = 0.

.

.

H(k1)Vk = 0H(k+1)Vk = 0

.

.

.

H(K)Vk = 0

H(1)

.

.

.

H(k1)

H(k+1)

.

.

.

H(K)

Vk = 0 (4)

where H(k) is the channel matrix of the kth selected user. Vkshould be in the null space of each H(j)(1 j K, j = k)

to eliminate the interference term. The precoding matrix Vkcan be computed by SVD

H(1)

.

.

.

H(k1)

H(k+1)

.

.

.

H(K)

=[U(k) U(k)

] [ 00 0

][R(k)

H

R(k)H

](5)

The columns of R(k) in (5) spans the null space of all H(j)s(1 j K, j = k), and thus it becomes Vk.

From (3), we have

yk =P kHkVksk + nk. (6)

An MU-MIMO system is decomposed into K independentSU-MIMO systems by BD. The dimension of Vk is M nkwhere nk = M

Ki=1,i=k N . The number of the transmit

antennas should be larger than the sum of the number ofreceive antennas of any K 1 users for the existence of thenull space of all H(j)s (1 j K, j = k). The condition tobe satisfied is then given by

M > (K 1)N. (7)

From (7), the maximum of K is MN where a is theminimum integer number not smaller than a.

B. Chordal Distance

Let us introduce a distance metric between subspaces.Grassmannian space G(m,n) is the set of all the n-dimensional subspaces of Euclidean m-dimensional space[11]. We need to define the distance between two elementsof G(m,n). We associate principal angles i [0, /2], andprincipal vectors ui P and vi Q for i = 1, . . . , n with twon-planes (n-dimensional hyperplanes) P and Q as follows.Choose u1 P and v1 Q having length 1, and such thatu1 v1 is maximal. Inductively, define ui P and vi Qhaving length 1 and such that ui vi is maximal, subject tothe conditions ui uj = 0 and vi vj = 0 for all 1 j < i[13]. Then set i = arccos(ui vi). The chordal distance isgiven by

dc(P,Q) =

sin2 1 + + sin2 n (8)

A generator matrix for an n-plane P G(m,n) is anm n matrix whose columns span P . The orthogonal groupO(m) acts on G(m,n) by right multiplication of a generatormatrix. Applying a suitable element of O(m) and choosingappropriate basis vectors for the planes, we can reduce thegenerator matrices of any given pair of n-planes P,Q with

KO and LEE: MULTIUSER MIMO USER SELECTION BASED ON CHORDAL DISTANCE 651

n m/2 [14] to the forms of

1 0 00 1 0.

.

.

.

.

.

.

.

.

.

.

.

0 0 10 0 00 0 0.

.

.

.

.

.

.

.

.

.

.

.

0 0 0

,

cos 1 0 00 cos 2 0.

.

.

.

.

.

.

.

.

.

.

.

0 0 cos nsin 1 0 00 sin 2 0.

.

.

.

.

.

.

.

.

.

.

.

0 0 sin n0 0 0.

.

.

.

.

.

.

.

.

.

.

.

0 0 0

. (9)

If A is a generator matrix for P whose columns are orthogonalunit vectors, the projection is represented by the matrixPp = AA

H. From (9), we can see that trace(Pp) is n. For

P,Q G(m,n), with orthonormal generator matrices A andB, and principal angles 1, . . . , n, a simple calculation using(9) shows that

d2c(P,Q) = n(cos2 1 + + cos2 n

)= n trace(AAHBBH)=

1

2Pp Qp2F

(10)

where Pp and Qp are the corresponding projection matrices.Note that the chordal distance definition can be extended to acase where P and Q have different dimensions. For example,P G(m,n) can be an n-plane, and Q G(m,n) can be ann-plane (n n).

IV. LOW COMPLEXITY SCHEDULING ALGORITHM

A. Power Allocation

An MU-MIMO system can be divided into K independentparallel SU-MIMO systems by BD. We can use water-fillingscheme since BS knows perfect channel information aboutall receivers. When we assume each receiver gets N streams,there are total NK streams, and we use water-filling withthese streams. The capacity with water-filling can be writtenby [12]

C(H(S), P ) =n

i=1

log2

(1 +

PiM

i

)(11)

where P is the total transmit power, n is the number of totalstreams, S is set of the selected users, H(S) is the combinedchannel of S, and i (i = 1, . . . , n) is the positive singularvalue of the ith channel of H(S). Note that i satisfies i =( MPi

)+and

ni=1 i = M . The sum rate of the MU-

MIMO system with BD and water-filling among the selectedusers is then given by

R(S) =S

C (H(S)V(S), P ) (12)

where V(S) is the precoding matrix of S.

B. Chordal Distance Based MU-MIMO Scheduling AlgorithmThe sum capacity of an MU-MIMO system with the optimal

scheduling scheme can then be written as

Ropt(S) = maxS{1, ,KT },1|S|KR(S) (13)

From (11) to (13), the optimal scheduling algorithm usesbrute-force exhaustive search over all possible user sets. Thenumber of user subsets with the optimal scheduling algorithmisK

i=1

(KTi

). However, when KT is large, the search com-

plexity of the optimal scheduling algorithm is too high.We propose a new low complexity MU-MIMO scheduling

with chordal distance. Orthogonality among channel matricesof the selected users in BD is a critical issue because of thecharacteristics of BD. For example, the Frobenius norm of Hkcan be large but its orthogonality with the other selected userscan be small, which is not desirable. In (6), we should considernot Hk, but HkVk since the effective channel is HkVk. Evenif the Frobenius norm of Hk is large, the Frobenius norm ofHkVk can be small since Vk eliminates not only the inter-user interference, but also the common subspace of Hk andthe channel matrices of the other selected users. If users areclose to orthogonal, the common subspace removed by Vkcan be made small. We use the chordal distance to select theusers to maximize the channel matrix distance between thepreviously selected users and the new user.

The chordal distance between plane A and B is given by

dc(A,B) =12AoAHo BoBHo F (14)

where Ao and Bo are orthonormal bases for subspaces Aand B, respectively. One of the characteristics of chordaldistance is that it can be measured between two differentdimensional objects. For example, it is possible to estimate thedistance between a plane and a vector in a 3-dimensional spacewith chordal distance. The chordal distance between channelmatrices of the previously selected users and the channelmatrix of a candidate user can be estimated accordingly. Thesummary of the proposed algorithm is described in Table I.

First, the user who has the maximum Frobenius norm ofHk(k T ) is selected among all the users. Since the effectivechannel is HkVk(k T ), we must consider not only thechordal distance between Po and Gk,o, but also the Frobeniusnorm of Hk. The Frobenius norm of the effective channelis roughly proportional to the chordal distance between Poand Gk,o, and the Frobenius norm of Hk. Therefore the nextselected user is the user who has the maximum value ofCk dc(Po,Gk,o), where is the weight factor between thechordal distance and the Frobenius norm. When is large, theFrobenius norm is weighted more than the chordal distance,and vice versa. depends on N and K , as will be shown inSection VI.

V. COMPUTATIONAL COMPLEXITY ANALYSISThe complexity is counted as the number of flops, denoted

as . A real addition, a multiplication, and a division arecounted as one flop. Therefore a complex addition needs twoflops, and a complex multiplication needs six flops [10]. Weassume KT K,N M, and K = MN . For an N M

652 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 3, MARCH 2012

complex valued matrix H, the number of typical matrixoperation [10] is summarized as follows.

Frobenius norm H2F : 2MN real multiplication +2MN real addition = 4MN flops.

Gram-Schmidt orthogonalization (GSO) : 4MN22MNreal multiplication + 4MN2 2MN real addition +2MN real divisions = 8MN2 2MN flops.

Water-filling over n eigenmodes : (1/2)(n2 + 3n) realmultiplication + (n2+3n) real additions + (1/2)(n2+3n)real divisions = 2n2 + 6n flops.

Singular value decomposition (SVD) 24MN2 +48M2N + 54M3 flops.

A. Optimal SchedulingBS performs exhaustive search over all

Ki=1

(KTi

)possible

user sets in the optimal scheduling algorithm. Thus the orderof the optimal scheduling algorithm is given by [10]

OP (KTK

)K

[ (48(K 1)2 + 8)N2M+

24(K 1)NM2 + (54(K 1)3 + 2K2 + 126)N3+8KN

] O

((KTK

)K M3

)

O(KKT KK+

12 M3

)(15)

where the Stirlings approximation to a factorial is used in thelast line.

B. Suboptimal Scheduling AlgorithmThe computational complexity of the capacity based al-

gorithm and the Frobenius norm based algorithm in [10] issummarized here. The complexity order of the capacity basedalgorithm is expressed by [10]

CA