Determinant Based Multiuser MIMO Schedulingwith Reduced Pilot Overhead

Kyeongjun Ko and Jungwoo LeeSchool of Electrical Engineering and Computer Sciences

Seoul National University, Seoul 151-744, KoreaEmail: smuff@wspl.snu.ac.kr and junglee@snu.ac.kr

AbstractIn a multiuser MIMO cellular system where thereare many candidate users, it is critical to select a user groupwhich maximizes the overall throughput of the system. However,the optimal scheduling strategy is computationally prohibitivewhen the total number of users is large. In this paper, we proposea determinant based user selection algorithm which reduces thesearch complexity without much performance degradation. Inorder to reduce pilot overhead by precoding matrix for multiuserMIMO, we also propose a new pilot scheme with only oneset of pilot, and the new pilot scheme is combined with theproposed scheduling algorithm. Simulation results show that theproposed scheduling algorithm with a new pilot scheme reducescomputational complexity and pilot overhead with negligibleperformance degradation compared to the exhaustive schedulingwith a conventional pilot scheme which uses two sets of pilots.

I. INTRODUCTION

Multiuser multiple-input multiple-output (MU-MIMO) sys-tems have drawn much attention recently because of highercapacity compared to single-user MIMO systems [1]. In mul-tiuser single-input single-output (MU-SISO) systems, it wasshown that allocating entire power to the user who has thebest channel is the optimal strategy [2]. On the other hand, inMU-MIMO systems, it has been known that allocating powerto multiple users simultaneously is better, which is differentfrom MU-SISO systems.

However, MU-MIMO systems have some practical issues. AMU-MIMO system needs complicated user selection processcompared to a single-user MIMO (SU-MIMO) system. Itneeds to find a user group which maximizes the throughputwhile a single-user MIMO system finds only the best user. Bythe way, it is not simple to find the best group of users becausea linear precoding matrix to remove inter-user interferenceneeds to be computed for each candidate group of users. Es-pecially, the optimal user selection scheme is computationallyprohibitive when the total number of users is large in a givensystem (cell). Many suboptimal user scheduling algorithmshave been proposed to reduce the computational complexityof the optimal user selection scheme [3],[4],[9],[10].

Inter-user interference occurs when the resources are al-located to multiple users in MU-MIMO systems. The inter-user interference must be eliminated since it limits the sum-capacity of the system. Dirty paper coding (DPC) is theoptimal non-linear precoding technique to avoid the inter-user interference, but its complexity is prohibitive [5]. Zero-forcing beamforming (ZFBF) and block diagonalization (BD)

were proposed to reduce the complexity as linear precodingtechniques [6][8].

The linear precoding techniques require the channel stateinformation (CSI) at the transmitter. The channel estimationhas to be done at each receiver by embedding pilots in thesignal, and the estimated channel has to be fed back to thetransmitter. However, ZF and BD need two sets of pilots. Oneis common pilot, which is used to estimate the raw channel.The other is dedicated pilot, which is used to estimate theeffective channel which is a combined channel of the rawchannel and the precoding matrix. A single-user MIMO systemand a random beamforming system do not need dedicatedpilots since the beamforming matrix is available at the receiver.But, in the ZF scheme and the BD scheme, the receiver doesnot know the beamforming matrix since the beamformingmatrices depend on the selected users, and each receiver doesnot know the users selected in transmitter. Therefore, ZF andBD need dedicated pilots to estimate the effective channel.Obviously, the data rate will suffer with large pilot overheaddue to the two sets of pilots, which may MU-MIMO lessattractive.

In this paper, we propose two new methods. One is the lowcomplexity MU-MIMO scheduling, and the other is a newpilot scheme which uses only one set of (dedicated) pilots.Simulation results show performance comparison between thecombination of two methods and the existing scheduling andpilot schemes.

II. SYSTEM MODEL

We consider an MU-MIMO downlink channel 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 of all the users. The broadcastchannel of MU-MIMO system is 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 i.i.d. complex Gaussian withzero mean and unit variance, and yk is the received signal

978-1-4244-8331-0/11/$26.00 2011 IEEE

vector at the kth user. The transmitted signal, x, is expressedas

x =K

i=1

PiVisi (2)

where K is the number of the simultaneously selected users,si is the ith symbol vector with E[si2] = 1, Pi is the powerallocated for the ith selected user, and Vi is the precodingmatrix for the ith selected user. Then, (1) can be divided intothe desired signal, the interference signal, and the noise asfollows.

yk =

P kHkVksk +HkK

i=k,i =k

PiVisi + nk (3)

where we assume the user index k is re-indexed properly afteruser selection. In the BD scheme, we can express (3) as

yk =

P kHkVisi + nk. (4)

III. DETERMINANT BASED MULTIUSER MIMOSCHEDULING ALGORITHM

A. Power AllocationAn MU-MIMO system can be divided into K independent

parallel SU-MIMO in the BD scheme as in (4). Water-fillingbased power allocation can be used since BS knows perfectchannel information of all the receivers. When we assume eachreceiver receives N streams, there are total N K streams onwhich we use water-filling. The capacity with water-filling [9],[10] is written as

C(H(S), P ) =n

i=1

log2

(1 +

PiM

i

)(5)

where P is the total transmit power, n is the number of totalstreams, S is the set of the selected users, H(S) is the setof channels in S, i(i = 1, . . . , n) is the singular value ofthe ith channel of H(S). i satisfies i =

( MPi

)+, andn

i=1 i = M . The sum rate of the MU-MIMO system withBD and water-filling among the selected users can then begiven by

R(S) =S

C (H(S)V(S), P ) (6)

where V(S) is set of precoding matrix in S.

B. Low Complexity MU-MIMO Scheduling AlgorithmThe optimal MU-MIMO scheduling with BD is performed

byRopt(S) = maxS{1, ,KT },1|S|K R(S) (7)

Therefore, the optimal scheduling is computationally pro-hibitive if KT K since the number of all possible pairsof the selected users is

Ki=1

(KTi

). The most important

consideration in selecting users in MU-MIMO systems withZFBF or BD is the orthogonality among the channels of theselected users. If the orthogonality among the selected users isnot large, the beamforming direction of a user is going to bemisaligned from its own channel since the precoding matrix

Fig. 1. Volume comparison by determinant

becomes the null space of channels of the other selected users.That is, the precoding matrix by ZFBF or BD eliminates notonly the undesired interference signal, but also the intendedsignal, and the reduction of the desired signal is roughlyproportional to the distance between its own channel and theprecoding matrix. For example, if the channels among theselected users are orthogonal to each other, the desired signaldoes not suffer degradation because the precoding matrix isaligned with the channel of the desired signal perfectly. In theother extreme case, if the channels among the selected usersis in the same direction or plane, the desired signal is lostcompletely since the precoding matrix is orthogonal to thechannel of the desired signal.

There exist many low complexity MU-MIMO schedulingalgorithms since the optimal scheduling scheme is too com-putationally complex to be used in the practical systems[3],[4],[9],[10]. These schemes achieve performance close tooptimal scheduling algorithm but have high complexity still.

In this paper, determinant is used to estimate orthogonalityand channel power simultaneously in MU-MIMO user se-lection process with low complexity. The absolute value ofthe determinant of a matrix corresponds to the volume ofthe parallelepiped spanned by those vectors, which is roughlyproportional to orthogonality and channel power. Therefore,the determinant based scheme can be regarded as an extendedscheme of Gram-Schmidt orthogonalization in [3]. For exam-ple, Figure 1 shows the relationship between orthogonality anddeterminant, and computes the volume of an geometric objectspanned by vectors. Note that the volume of the cubic figurein Fig. 1 (a) is larger than that in Fig. 1 (b).

The first step of the proposed algorithm is to find the userwhich maximizes capacity among candidates in T . A nextuser that maximizes determinant with the selected user set S ischosen in a successive fashion without computing a precodingmatrix by SVD. The volume S of a figure composed by ageneral m n complex matrix, B, is given by [11]

S =

det(BHB). (8)The determinant between the channel H(S) of the selectedset and the channel of the nth candidate user Hn when H =[H(S)Hn

]is expressed by

det(HHH

)= det

([H(S)H(S)H H(S)HHnHnH(S)H HnHHn

]). (9)

TABLE IDETERMINANT BASED MULTIUSER MIMO SCHEDULING ALGORITHM

Step I: Initialization

T = {1, 2, . . . ,KT }; S = ;H(S) = ; R = 0; X = IM ;

Step II: Loop

FOR i = 1 to Ksi = argmax

nTdet(HnXH

Hn );

Rtemp = R(Stemp), where Stemp = S + {si};IF (Rtemp < R)

break;ELSEH(S) = [H(S)T ,HTsi ]T ;X = IH(S)H(H(S)H(S)H)1H(S);S = Stemp;R = Rtemp; T = T {si};

END-IFEND-FOR

To lower computational complexity of (9), we use the follow-ing identity.

det([

A BC D

])= det(A) det(DCA1B) (10)

From (9) and (10), we have

det([

H(S)H(S)H H(S)HHnHnH(S)H HnHHn

])=

det(H(S)H(S)H)det(HnHHn HnH(S)H(H(S)H(S)H)1H(S)HHn ) =det(H(S)H(S)H)det(Hn(IH(S)H(H(S)H(S)H)1H(S))HHn ).

(11)Note that det(H(S)H(S)H) in (11) is a common value whichhas nothing to do with Hn which is the channel of a candidateuser in T , and it can be skipped in the user scheduling process.The detailed user scheduling process is described in Table I.It should be noted that fewer than K users can be selected inthe user scheduling process. This may occur when a new userwhich is close to the subspace of existing users reduces thenull subspace in the BD scheme.

IV. COMPUTATIONAL COMPLEXITY ANALYSISThe key contribution of the proposed scheduling algorithm

is to achieve low complexity without sacrificing performance.The complexity is counted as the number of flops, denoted as. A flop is defined as a real floating point operation [12]. Eachof a real addition, a multiplication, and a division is countedas one flop [4]. A complex addition and multiplication havetwo flops and six flops, respectively. Although flop countingcannot show complexity precisely in practical systems, it canindicate a rough order of the computational complexity.

We assume KT K,N M, and K = MN in thissection. For an N M complex valued matrix H, the numberof typical matrix operation is representd in [4].

A. Optimal Scheduling AlgorithmThe order of the optimal scheduling algorithm is given by

OP O((

KTK

)K M3

)(12)

B. Suboptimal Scheduling AlgorithmWe analyze computational complexity of the capacity based

algorithm, the Frobenius norm based algorithm [4], and thechordal distance based algorithm [10]. The complexity of thecapacity based algorithm is expressed by

CA < O(KT K2 M3) (13)The complexity of the Frobenius norm based is given by

F O(KT K2 M3) (14)The complexity of the chordal distance based schedulingalgorithm is

CO O(KT K M3). (15)

C. Determinant based Scheduling AlgorithmAs for the determinant based scheduling algorithm, the

complexity is given by

DE /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice