Reduced feedback SDMA based on subspace packings

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009 1329

Reduced Feedback SDMA Based onSubspace Packings

Patrick Svedman, Member, IEEE, Eduard A. Jorswieck, Senior Member, IEEE, and Björn Ottersten, Fellow, IEEE

Abstract—Herein, we treat Space Division Multiple Access(SDMA) based on partial channel state information and limitedfeedback. We propose a novel framework utilizing subspacepackings, where beamforming, feedback, and scheduling areintegrated. Advantages of the proposed framework are that thefed back supportable rates are based on the post-schedulingSINR and that the feedback implicitly contains informationabout the spatial compatibility of the users. The feasibilityregion of packings of different dimensions is indicated by theallocation outage probability which is derived. Grassmanniansubspace packings, DFT-based packings, and non-orthogonalGrassmannian packings are formulated and studied as candi-dates. Numerical simulations show better performance for theproposed scheme compared to conventional channel quantizationat the receiver with zero-forcing transmission in i.i.d. Rayleighfading. Finally, we propose and evaluate a beam-graph methodto further reduce the feedback load, that can be used in thecontext of tracking quantized beamformers.

Index Terms—Multi-user MIMO, SDMA, feedback.

I. INTRODUCTION

MULTIPLE-INPUT, multiple-output (MIMO) communi-cation systems promise high spectral efficiency and

reliability for mobile systems. Whereas the theoretical as-pects are relatively well understood [1], there is still alarge performance gap between the theoretical capacity andimplementable designs, especially for multiuser systems. Toachieve high spectral efficiency in fading channels, the multi-ple antennas can be used to exploit the spatial dimension andadapt the transmitted and received signals so that interferencefrom unwanted signals is suppressed. To do this efficiently,the transmitters in general require some knowledge of thecommunication channel between themselves and the receivers.Since users in mobile systems typically move, their channelstates change. Therefore, the multi-antenna transmission and

Manuscript received March 26, 2007; revised November 24, 2007 andSeptember 9, 2008; accepted December 5, 2008. The associate editor coordi-nating the review of this paper and approving it for publication was A. Yener.

P. Svedman was with the School of Electrical Engineering, Signal Pro-cessing Lab, Royal Institute of Technology (KTH), Stockholm, Sweden. Heis now with Ericsson Research, Ericsson AB, Stockholm, Sweden (e-mail:[email protected]).

E. A. Jorswieck was with the School of Electrical Engineering, SignalProcessing Lab, Royal Institute of Technology (KTH), Stockholm, Sweden.He is now with the Dresden University of Technology (TUD), Chair ofCommunication Theory, Dresden, Germany.

B. Ottersten is with the ACCESS Linnaeus Center, Signal Processing Lab,Royal Institute of Technology (KTH), Stockholm, Sweden. B. Ottersten isalso with SecurityandTrust.lu, University of Luxembourg.

This work has been performed in the framework of the IST project IST-4-027756 WINNER II, which is partly funded by the European Union. Theauthors would like to acknowledge the contributions of their colleagues. Theideas presented in this paper were conceived during work with the spatial-temporal processing solution of the WINNER II air interface [23].

Digital Object Identifier 10.1109/TWC.2009.071245

reception needs to continuously adapt to the changing condi-tions. In frequency division duplex (FDD) systems and in non-reciprocal time division duplex (TDD) systems, informationabout the channel states needs to be passed between receiversand transmitters. This feedback overhead is intractable, espe-cially in systems with many users and high mobility.

In this paper, we consider the downlink of a flat fading mul-tiuser FDD communication system with multiple antennas atthe basestation and a single antenna at the users. The scenariowith partial channel state information (CSI) at the basestationis studied. In opportunistic beamforming, multiuser diversity isexploited by transmitting a random beam through the multipleantennas [2]. It is likely that the beam suits at least one userand the resulting supportable rate of each user can be fedback as partial CSI to the basestation. This overcomes theneed for explicit CSI at the transmitter. Note that feedback ofsupportable rate implies the use of adaptive modulation andcoding (AMC).

In [3], opportunistic beamforming was extended to spacedivision multiple access (SDMA), where the basestation trans-mits to several users simultaneously. Instead of transmittingonly one beam, several random but orthogonal beams aretransmitted. The users now report the index of their best beamas well as the corresponding supportable rate. One user perbeam can then be scheduled by the basestation. Note that thereported supportable rates can take the inter-beam interferenceinto account, since the users must be able to measure the signalstrength from all beams before feedback. This scheme wasextended to support more beams than transmit antennas in [4],by using Grassmannian line packings. In [5], opportunisticSDMA based on controlled sweeping of the multiple beamsover the cell is treated.

One approach to reduce the feedback load is to let theusers vector quantize their channel states, or equivalentlyoptimal beamformers, using e.g. the generalized Lloyd’s al-gorithm [6]–[8] or random vector quantization [9], [10], andbased on the quantized CSI design the transmitted beamsin the basestation. Another way to generate the quantiza-tion codebook is to use Grassmannian line packings [11].A Grassmannian packing is a collection of subspaces in ahigher dimensional space. An optimal packing is such thatit maximizes some distance measure between the two closestsubspaces in the packing. For the intuitive case where thechannel vector is real and three-dimensional, the optimal linepacking is a collection of one-dimensional subspaces, i.e.lines, in R

3 that pass through the origin and maximize theangle between the two closest lines. Clearly, Grassmannianline packings are promising candidates for the quantization of

1536-1276/09$25.00 c© 2009 IEEE

1330 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009

the i.i.d. Rayleigh fading vector channel [11], [12]. In the caseof spatially correlated Rayleigh fading channels, the packingcan be prewhitened as in [13]. A major disadvantage withGrassmannian packings is the lack of analytical results formost cases, but numerical methods to generate good packingsexist [14], [15].

The step from Grassmannian single-beam beamforming,i.e. line packings, to Grassmannian spatial multiplexing wastaken in [16]. In MIMO channels, several data streams canbe simultaneously transmitted to a user by precoding with aunitary matrix (instead of a vector in the single-stream case) atthe transmitter. The optimal unitary matrix is quantized usinga Grassmannian subspace packing.

Herein, the Grassmannian subspace packing approach isapplied to the multiuser scenario. As in [16], unitary matricesfrom Grassmannian subspace packings are used for transmitterprecoding. The major difference is that we consider SDMA,where one user per beam (or column in the unitary beamform-ing matrix) can be allocated. There is no cooperation or jointsignal processing between the receive antennas (users), andonly a subset of the receiver antennas (users) are supported ata time. We show that, contrary to single-user Grassmannianbeamforming or spatial multiplexing, a denser packing oftendoes not yield better downlink performance due to alloca-tion outages. Additionally, DFT-based and non-orthogonalGrassmannian subspace packings are considered. Furthermore,a graph representation for large packings or codebooks isproposed and evaluated. The beam-graph technique can beused to further reduce the feedback rate when tracking channelvariations, it is also well suited for the previously investigatedGrassmannian beamforming. The contributions of this papercan be summarized as follows:

• A feedback and SDMA scheme based on subspace pack-ings is proposed in Section III. Two benefits of thescheme are:

1) The feedback implicitly contains information aboutthe spatial compatibility of the users.

2) The fed back supportable rates are based on thepost-scheduling SINR.

• The important concept of allocation outage is discussedand the allocation outage probability is derived as afunction of the system parameters the number of users(K), the number of subspaces (N ) and the number ofsimultaneously transmitted beams (B).

• The beam-graph method for tracking channel variations,which further reduces the feedback rates and the receivercomplexity, is proposed in Section V.

• Numerical system simulation results compare variousflavors of the proposed method with linear zero-forcingprecoding with and without channel quantization andalso opportunistic SDMA. For the studied scenarios,quantized zero-forcing with the same amount of feedbackis outperformed. Opportunistic SDMA is outperformed atthe cost of slightly more feedback.

Here, we would like to stress one important difference ofthe proposed scheme to receiver-quantization of the channeltogether with multiuser beamforming at the transmitter [17].When the channel is quantized to only a few bits, there

is significant quantization distortion. This is acceptable in asingle-user beamforming scenario where the user can feedback a supportable rate, given the fed back beam. In a mul-tiuser beamforming scenario with quantized channel feedback,however, the user can not predict the inter-beam interferenceand can therefore not feed back an accurate supportable rate,leaving the decision to the transmitter. At the transmitter, how-ever, only heavily quantized channels are available, makingaccurate modulation and code-rate assignments in an SDMAscenario difficult.

In the proposed scheme, the users can feed back accuratesupportable rates, since the hypothetical inter-beam interfer-ence is apriori known, due to the special subspace packingstructure. This significant advantage is explained in moredetail in the following sections. The advantage is also clearfrom the numerical results.

In Section II, the system model is described. The ideaof reduced-feedback SDMA using subspace packings is de-scribed in Section III. In Section IV, subspace packings andtheir application to the problem are described. The design ofbeam-graphs and their use is treated in Section V. Numericalresults are given and discussed in Section VI and the paper isconcluded in Section VII.

II. SYSTEM MODEL

We consider the downlink of a cellular multiuser FDDsystem. The basestation with M antennas communicates withK single-antenna users. The wireless channels are assumedto be flat fading and quasi-stationary, i.e. constant duringeach block. One block is also the resource allocation period.The received signal for user k at symbol t, yk(t) ∈ C, isyk(t) = hT

k (t)x(t) + zk(t), where hk(t) ∈ CM×1 is theMISO channel of the kth user, x(t) ∈ CM×1 is the transmittedsignal from the basestation, zk(t) ∈ C is AWGN with varianceσ2

z and (·)T denotes matrix transpose. The quasi-stationaryassumption means that hk(t) is constant for all t within ablock. In the following, the time-index t is omitted for brevity.

The transmitted signal vector, x, is assumed to be

x =1√B

Ws =1√B

B∑i=1

wisi,

where W = [w1 · · ·wB] ∈ CM×B is the beamformingmatrix and s = [s1 · · · sB]T ∈ CB×1 is the vector of i.i.d.symbols. The variable B denotes the number of simultane-ously transmitted symbols as well as the number of usedbeams. The ith data-symbol si ∈ C is transmitted usingthe ith beamforming vector wi ∈ CM×1. The number ofsimultaneous streams B is henceforth called the SDMA factor.The case B = 1 corresponds to single-user beamforming(TDMA). The average transmit power is constrained by settingE

[|si|2]

= 1 and ‖wi‖2 = 1. This means that E[‖x‖2

2

]= 1.

We assume that each user k accurately estimates the channelvector hk through the use of common pilots.

Now, assume that symbol m is dedicated to user k. Thereceived SINR is

γk =|hT

k wm|2Bσ2

z +∑

n�=m |hTk wn|2 .

SVEDMAN et al.: REDUCED FEEDBACK SDMA BASED ON SUBSPACE PACKINGS 1331

The user SINR is maximized if hk is co-linear with wm andhk is orthogonal to wn for all n �= m. Then, the SINR isγ̃k = ‖hk‖2

2Bσ2

z. This means that users are perfectly spatially

compatible if their channel vectors are orthogonal. In practice,this event occurs with zero probability. The beamformingvector that maximizes the received signal power for the kth

user, w̃k = h∗k, is henceforth called the channel-matched

beamforming vector, where (·)∗ denotes complex conjugate.Note that even though the channel-matched beamformingvector is favorable for one user, it may be unfavorable forthe other users, creating significant inter-beam interference.The trade-off between using channel-matched beamformingvectors to each scheduled user, and the inter-beam interferencethey create is a very important aspect of SDMA and one of themain themes of this paper. Also note that vector quantizationof the channel is equivalent to vector quantization of thechannel-matched beamforming vector.

Furthermore, it is assumed that the system supports a finiteset of adaptive modulation and coding (AMC) levels. Giventhe error rate constraint of user k, the supportable AMClevel is uniquely determined by the mapping function Γk(γk),which is assumed known.

III. REDUCED FEEDBACK SDMA BASED ON SUBSPACE

PACKINGS

Assume that a set of beamforming matrices, W ={W1, . . . ,WN}, is a priori known at both the basestationand at the users. W is called a subspace packing. Eachbeamforming matrix consists of B columns (beams/vectors),Wi =

[w1

i · · ·wBi

]. Note that B is not restricted to B ≤ M .

The proposed beamforming and scheduling schemes work ona block basis. Before each block:

1) Each user k finds the overall best beam index1,

(ik, jk) = arg max1≤i≤N,1≤j≤B

γk (i, j)

= arg max1≤i≤N,1≤j≤B

|hTk wj

i |2Bσ2

z +∑

n�=j |hTk wn

i |2.

(1)

This means that, when selecting the best beam, the userassumes that all other beams in Wi than wj

i will be usedfor transmission to other users. The SINR on the bestbeam is γk(ik, jk) and the corresponding supportableAMC level is Γk (γk (ik, jk)), henceforth abbreviated asΓk (ik, jk).

2) Each user feeds back the beam index (ik, jk) and thecorresponding supportable AMC level Γk (ik, jk).

3) According to a scheduling metric, the basestationchooses a subspace i∗ to be used during the next block,i.e. the basestation will use Wi∗ as beamforming matrix.For each beam in Wi∗ , a user is scheduled if possible,with the fed back AMC level. For example, if thebasestation uses maximum throughput scheduling, the

1If B = M and the columns of Wi are orthogonal, (1) is equivalent toarg max1≤i≤N,1≤j≤B |hT

k wji |. In Theorem 2 in Appendix A, we upper

bound the relative loss in SINR from choosing the (i, j) that maximizes|hT

k wji | instead of (1).

subspace is chosen according to

i∗ = arg max1≤i≤N

B∑j=1

max1≤k≤K

Γ̄k (i, j) , (2)

where Γ̄k (i, j) = Γk (γk (ik, jk)) δ(i−ik, j−jk) definesthe channel state information of user k at the bases-tation. The two-dimensional Kronecker delta function,δ(i − ik, j − jk) = 1 for (i, j) = (ik, jk) and zerootherwise. Hence, the inner maximization chooses theuser k with highest known rate for beam (i, j). The outermaximization finds the subspace i with the highest sum-rate over the B beams. Note that (2) is an example andthat other schedulers, for instance proportional fair ormax-min scheduling are compatible with the proposedfeedback and SDMA structure.

To summarize steps 1)-3), each user finds its best overallbeam from all subspaces. Then, the user computes the highestAMC level it could support if it is scheduled on its best beamand other users are scheduled on the other beams in the samesubspace. Finally, the basestation selects a subspace accordingto some scheduling metric, based on the fed back AMC levels.Only users that fed back beam-indices for that subspace areconsidered for scheduling.

Example 1: As an example, consider a system with a 2-antenna basestation (M = 2), SDMA factor 2 (B = 2),two possible beamforming matrices (N = 2) and four users(K = 4). Based on their estimated channels, the users computewhich of the four beams (BN = 4) that maximizes theirSINR, according to (1). Assume that the users feed backindices (1, 1), (1, 1), (1, 2) and (2, 2), respectively. In addition,the users feed back the AMC level that the beam can support.Assume that they feed back the levels 1, 2, 3, 2, respectively,i.e. Γ1(1, 1) = 1, Γ2(1, 1) = 2, Γ3(1, 2) = 3 and Γ4(2, 2) = 2.If the basestation chooses to use W1, it may schedule user 1or user 2 on beam w1

1 with AMC level 1 or 2, respectively,and simultaneously user 3 on beam w2

1 with AMC level 3. Ifthe basestation chooses to use W2, it may schedule user 4 onbeam w2

2 with AMC level 2. Beam w12 must remain unused,

since no user fed it back. Now, suppose that the basestationuses maximum rate scheduling, as in (2), and that the AMClevel is proportional to the rate. The maximum sum AMClevel 5 is achieved if W1 is chosen with users 2 and 3 beingscheduled on the beams w1

1 and w21, respectively.

It is important to note that the fed back AMC level, Γk(ik, jk),includes the post-scheduling inter-beam interference, regard-less of which users are scheduled on the other beams. This isnotable since Γk(ik, jk) is computed before the feedback andthe scheduling take place.

The scheduling is simplified since the feedback implicitlyshows which users are spatially compatible. It is likely thattwo users who fed back two different beam-indices withinthe same subspace have channels that are close to orthogonal,assuming that the beams within one subspace are orthogonal.

There is an important trade-off in the selection of thenumber of subspaces N . The more subspaces and beams tochoose from (high N ), the less channel quantization distortioncan be expected on average. This relation is illustrated inFig. 4 in Section VI-A. However, the feedback increases with


N . More importantly, for a limited number of users, theprobability that the basestation can not allocate users to allB beams in any subspace increases with N . If the basestationcan not allocate B beams, there is a significant performanceloss. This is due to the fact that the allocated users assume intheir AMC level computation that the other beams are used fortransmission to other users, creating inter-beam interference.Theorem 1 presents the probability of this event, which iscalled allocation outage.

Definition 1 (Allocation outage): Consider a system whereK users feed back one beam index each out of NB possible.The NB beams are divided into N disjoint sets of B beamseach. An allocation outage is the event that there exists no setwhere all B beam indices have been fed back by at least oneuser.

Theorem 1: Consider the allocation outage event, as inDefinition 1. Additionally assume that the users choose beamindices independently and with equal probability. Then,

Pao � Pr (allocation outage) = 1 +N∑

n=1

(−1)n

(N

n

)

+1

(NB)K

N∑n=1

nB∑m=1

(−1)n+m

(N

n

)(nB

m

)(NB − m)K (3)

Proof: See Appendix B.The result in Theorem 1 is useful in understanding the tradeoffbetween having many beams in the packing (NB) and ahigh SDMA factor (B) on one hand and the probability ofundesired allocation outages on the other hand. Note that theresult only holds exactly in the ideal case where all beams areselected with equal probability. This depends on the used setof beams as well as on the channel statistics.

Example 2: This example describes an allocation outageevent. Consider the same scenario as in Example 1. In thisexample, assume that the four users feed back the beamindices (1, 2), (1, 2), (2, 2) and (2, 2), respectively. Now, thebasestation can not reliably schedule two simultaneous users,since the fed back rates of user 1 and user 2 are valid onlyif they are scheduled on beam w2

1 and w11 is the interference

beam. The fed back rates of user 3 and user 4 are valid onlyif they are scheduled on beam w2

2 and w12 is the interference

beam.The outage probability (3) for a number of packing dimen-

sions is displayed in Fig. 1. For few users (low K), higherSDMA factor (B) than 2 is not suitable due to the highallocation outage probability. The scheduler would rarely finda subspace where all beams could be used. This would resultin fewer spatial streams than B as well as unnecessarily lowfed back supportable rates for the scheduled users. For 10users, SDMA factor 2 and 6 subspaces (K = 10, B = 2 andN = 6), the scheduler can with more than 90% probabilityfind a subspace where 2 users can be scheduled, whichseems to be a suitable operating point. The assumptions inTheorem 1 are validated in Fig. 2 by comparing the theo-retically computed outage probability with the numericallycomputed outage probability using i.i.d. Rayleigh fading usersand Grassmannian subspace packings (c.f. Section IV-A).

For the scenario with many active users, it is clearly suitableto use a higher SDMA factor, enabling many spatial streams

10 20 30 40 5010

−3

10−2

10−1

100

NB=12 B=2NB=12 B=3NB=12 B=4NB=24 B=2NB=24 B=3NB=24 B=4

Number of users K

Pa

o

Fig. 1. Allocation outage probability Pao as a function of the number of usersK . The curves represent packings with in total 12 and 24 beams (NB = 12and NB = 24). The beams are grouped in subspaces (beamforming matrices)of dimensions B = 2, B = 3 and B = 4.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

NB=12 B=2 TheoreticalNB=12 B=2 SimulatedNB=12 B=3 TheoreticalNB=12 B=3 Simulated

Number of users K

Pa

o

Fig. 2. Allocation outage probability Pao as a function of the number ofusers K . The theoretically computed Pao (Theorem 1) is compared withnumerically evaluated outage probability for i.i.d. Rayleigh fading users andGrassmannian subspace packings. The curves represent packings with in total12 beams (NB = 12). The beams are grouped in subspaces (beamformingmatrices) of dimensions B = 2 and B = 3.

with high probability. Note that B > M is not possible withoutinter-beam interference to some users.

Fig. 1 suggests a scheme that would adapt to the numberof active users, which can be assumed to be a slowly varyingparameter. Both the basestation and the users would have aset of packings with various dimensions N and B. Dependingon the number of users, the basestation would broadcast theN and B of the packing to be used. Alternatively, smallerpackings can be generated by removing some matrices in W ,as for the codebook subset restriction in LTE [18].

In the presentation above, single antenna users have beenassumed. However, the proposed scheme is compatible withmost linear and non-linear multi-antenna receiver structures,since the MIMO channel, the transmit beamformer, and theinterfering beamformers are known at all receiving users.The structure proposed here does not, however, allow for the


transmission of several beams to the same user.

IV. SUBSPACE PACKINGS

In this section, the design of suitable W for the schemein Section III is discussed. We specifically consider the caseB ≤ M , i.e. we do not consider transmission of more beamsthan transmit antennas. Note that the packings described inSection IV-C naturally extend to B > M , whereas thepackings in Sections IV-A - IV-B do not.

When constructing packings for SDMA, there is a tradeoffbetween 1) good vector quantization properties and 2) theorthogonality of the vectors that are to be used simultaneously.The vector quantization properties are important to maximizethe received signal power, i.e. the fed back vector should beclose to the channel-matched beamforming vector. Note thatthe vector quantization properties of a packing depend on thedistribution of the channel vector. At the same time, it isadvantageous if the beams used simultaneously are orthogonal.If the fed back vector of a user equals the channel-matchedbeamforming vector and the beamforming matrix is orthogo-nal, the user will receive no inter-beam interference and theSINR will be maximized (see the numerator and denominatorin (1)). Note that orthogonal beamforming matrices do notguarantee inter-beam interference free SDMA.

For the i.i.d. Rayleigh fading vector channel, the Grassman-nian line packing is optimal in terms of quantization. However,the line packing in general does not contain any orthogonalvectors, thereby penalizing SDMA. On the other hand, theoptimal subspace packing gives orthogonal matrices that aremaximally separated in chordal distance (see Section IV-A).For the optimal subspace packing, the ensemble of basisvectors from all subspaces, i.e. the columns of all unitarymatrices, do not in general quantize the vector channel aswell as the optimal line packing. This tradeoff is numericallystudied in Section VI-A. The loss in channel quantizationperformance is due to the imposed orthogonal structure ofthe matrices in W .

Grassmannian and DFT-based packings, which are knownfrom literature, are presented below. However, they have notbeen used in the context of SDMA previously. Additionally,we propose non-orthogonal Grassmannian subspace packings,which quantize the vector channel optimally with respect tochordal distance, but do not consist of orthogonal matrices.

A. Grassmannian Subspace Packings

The distance measure between subspaces in CM used here

is the chordal distance. Other measures are also feasible, butthe chordal distance is chosen for its simplicity and that ityields the most symmetrical packings [14], [15]. The chordaldistance between the subspaces spanned by the B orthonormalcolumns of Wi and Wj is given by

dist (Wi,Wj) =√

B − ‖WHi Wj‖2

F ,

where (·)H denotes conjugate transpose and ‖ · ‖F de-notes Frobenius norm. The complex Grassmannian manifoldG

(B, CM

)is the collection of all B-dimensional subspaces

in CM . A Grassmannian packing is a finite set of subspaces

from the manifold, spanned by the unitary matrices in W . Anoptimal packing is such that the minimum distance betweenany two subspaces is maximized, i.e. for i �= j

W̃ = arg maxW

minWi,Wj∈W

dist (Wi,Wj) . (4)

For B = 1, the problem reduces to the packing of lines in CM

that pass through the origin, or equivalently, to the packing ofpoints on the unit sphere in CM . For the i.i.d. Rayleigh fadingchannel, the optimal Grassmannian line packing is optimal interms of, for instance, outage probability [11]. In general, thereis no analytic solution to (4). However, for moderate problemdimensions, there are numerical methods which can providenear-optimal packings. Grassmannian packing construction isdiscussed in more detail in Section VI-A. Numerically con-structed near-optimal Grassmannian packings are henceforthcalled Grassmannian packings.

B. DFT-based Subspace Packings

In [11], it was shown that the columns of the unitary space-time constellations in [19] worked well as beamformers forthe i.i.d. Rayleigh fading channel. The M first elements ofthe columns of an NB-dimensional DFT matrix are taken asbeamforming vectors.

wl =1√M

⎛⎜⎜⎜⎝

1ej2π(l−1)/(NB)

...ej2π(M−1)(l−1)/(NB)

⎞⎟⎟⎟⎠ for l = 1, . . . , NB.

Fortunately, there are orthogonal vectors in the DFT packing,which naturally span the B-dimensional subspaces in the samefashion as Grassmannian subspace packings. A rearrangementof the DFT vectors can create the necessary unitary matricesfor subspace packings of higher dimension than one.

C. Non-orthogonal Grassmannian Subspace Packings

The trade-off between channel vector quantization distortionand beamforming matrix orthogonality was discussed in thebeginning of this section. On one side, we have the Grassman-nian subspace packings, with orthogonal vectors spanning thesubspaces, but which lack in quantization performance, assum-ing i.i.d. Rayleigh fading channel vectors. On the other side,we have the Grassmannian line packings with optimal quan-tization performance with respect to a distance measure, herechordal distance. A natural alternative to the Grassmanniansubspace packings is to construct packings with optimal quan-tization performance, but with vectors spanning the subspaceswhich are non-orthogonal. We have constructed such packings,based on Grassmannian line packings, where the vectors havebeen greedily grouped to yield near-orthogonality within thesubspaces. Such a packing will on average outperform theGrassmannian subspace packing in terms of channel vectorquantization. The greedy non-orthogonal packing constructionis briefly described in Table I. From a set of NB vectors, itcreates N matrices with B columns each.

In [4], a multiuser transmission scheme based on a non-orthogonal Grassmannian line packing was proposed. It differsin several aspects. Firstly, it is applied in the framework of


TABLE ITHE GREEDY ALGORITHM TO CONSTRUCT NON-ORTHOGONAL PACKINGS.

1) Put all vectors from a Grassmannian line packing in the set Wvec.2) Find the two vectors in Wvec with lowest inner product. Remove

them from Wvec and let them be the first columns in a new matrix.3) If the number of columns in the new matrix equals B, save the matrix

and go to step 5). Otherwise go to step 4)4) Find the vector in Wvec with the lowest average inner product with

the columns in the new matrix. Remove it from Wvec and append itas a column in the new matrix. Go to step 3)

5) If Wvec is not empty, go to step 2).6) Now, N matrices with B columns each have been created.

opportunistic SDMA, as in [3], i.e. only one beamformingmatrix is considered. Here, we generate a packing with Nmatrices. Secondly, no construction algorithm is given. Notethat in [4], it was demonstrated by numerical simulations thatin certain scenarios higher rates can be achieved with morebeams than transmit antennas.

V. BEAM-GRAPHS FOR REDUCED FEEDBACK AND

COMPLEXITY

When the channel changes moderately from block to block,the feedback and receiver complexity can be reduced furtherby exploiting structure in the subspace packing. Below, amethod to arrange the beams in W in a graph is proposed. Thepurpose is to reduce the overhead caused by the feedback ofthe beam-index. Instead the branch index in the graph is fedback. This method is in fact even more suitable for systemsusing quantized beamforming (B = 1), since they enjoy amonotonously increasing performance with increasing numberof beams in the codebook. Furthermore, the beam-graphmethod may reduce the receiver complexity significantly, sincethe users only need to search for the best beam among theneighbors in the graph, instead of among all beams in thepacking.

According to the scheme described in Section III, eachuser needs to feed back one beam index and one AMC leveleach block. This is already a significant reduction from a fullchannel state feedback. However, in systems with many users,high mobility and many beams in W , a further reductionmay be necessary. In mobile wireless systems, it is reasonableto assume a significant temporal channel correlation betweenconsecutive blocks. The channel should not change too muchduring one block to enable accurate rate adaptation andbeamforming.

For feedback in such temporally correlated systems, it isnatural to apply delta modulation. The AMC level is straight-forward to delta modulate by only feeding back ’up one level’,’down one level’ or ’remain on the current level’ commands.How to do this for the beam index is not as trivial since theunit-norm beams are vectors that lie on an M -dimensionalcomplex sphere. As a simple example, consider a set of unit-norm vectors in R

2, which all lie on the unit circle. For thisset, the neighbor relation is easily defined in the angle of thevectors; the two vectors with closest higher and lower angle,respectively, are considered the neighbors of a vector.

A unit-norm vector w ∈ CM×1 can be represented in spher-ical coordinates by 2M − 1 angles. First, the complex-valued

−1

0

1

−1

0

1−1

0

1

v1v2

v 3

Fig. 3. The neighbor directions of v = [−1 0 0] on the R3-sphere are givenby the four segments.

w is transformed into a real vector, v = [v1 · · · v2M ]T =[Re{wT } Im{wT }]T ∈ R2M×1. Since unit-norm vectors are

considered here, only the angular coordinates are of interest.They can be computed as [20]

θ2M−1 = arctanv2M

v2M−1

...

θ1 = arctan

√∑2Mi=2 v2

i

v1.

Having found the spherical coordinates of all vectors in apacking, the nearest neighbors to each vector has to be found.To illustrate the proposed method, consider unit-norm vectorsin R3. The vectors can be represented by two angles, θ1 andθ2. The nearest neighbors in four directions are considered.One neighbor is found in the segment of the sphere with anglesgreater than θ1 and θ2. One neighbor is found in the segmentwith angles greater than θ1 and less than θ2 and one is foundin the segment with angles smaller than θ1 and greater thanθ2. The final nearest neighbor is found in the segment wherethe angles are less than both θ1 and θ2. The four neighborsegments of v = [−1 0 0] on the R3-sphere is illustrated inFig. 3. Note that this is not the only way to find neighbors ona unit sphere.

For vectors on a sphere in R2M , nearest neighbors in 22M−1

directions can be found, using the same approach. Clearly,even for vectors with relatively low dimensions, one can findan overwhelming number of nearest neighbors. For instance,for four transmit antennas (M = 4), 128 nearest neighborscan be found. For packings with less than 128 beams, thismeans that no feedback reduction is achieved, compared withbeam index feedback. However, it is not necessary to use allneighbor directions. A method to remove neighbor directionsis proposed in Table II.

Each neighbor direction signifies a particular change patternin the angles of a vector. These directions can be enumerated


TABLE IITHE ALGORITHM FOR NEIGHBOR DIRECTION REMOVAL.

1) Choose a chordal distance dn. Vectors within this distance areconsidered neighbors here.

2) For each vector in the packing, find the number of neighbors in eachdirection, within a distance dn.

3) Compute the average number of neighbors for each direction.4) Successively remove directions, starting from the direction with

lowest average number of neighbors. If too many directions have0 neighbors, increase dn and go to 2).

5) After each removal, test the graph for connectedness. A graph isconnected if all nodes can be reached from any node [21]. If thegraph is disconnected undo the removal and move on to the nextdirection.

6) Repeat 4)-5) until the desired number of neighbor directions has beenobtained.

and used as feedback entities, instead of the beam index. Now,a beam-graph can be constructed, connecting nearest neighborvectors in all chosen directions. The graph can be constructedoffline, and has to be known at both the basestation and at theusers. Note that the proposed graph construction algorithm isheuristic. It can not be guaranteed that it will find a connectedgraph for a given number of neighbor directions, even if itexists.

As the channels of the users change, the preferred beamschange. The users notify the basestation by feeding back theindex of the neighbor direction instead of the beam index,thereby traversing through the graph. The beam-graph restrictsthe users to change their preferred beams to neighbors inthe beam-graph. This also means that the users only need tosearch among the neighbors in the graph to find which beam-index to feed back, which may result in significantly reducedcomplexity. At the setup of a link or on a regular basis, theusers have to feedback the whole beam index.

Although this is an ad hoc strategy, significant feedback re-duction can be obtained, in particular for packings with manybeams. In Section VI-B, a numerical performance comparisonbetween full beam index feedback and feedback based on abeam-graph is presented.

VI. NUMERICAL RESULTS

A. Packing Study

The construction of optimal Grassmannian packings is stillan open problem. Several numerical methods to constructpackings have been proposed, e.g. [14], [15]. In this paper, wehave used the alternating projections technique for complex-valued subspace packings in [15]. For the cases tabulatedin [15], we obtained very similar minimum distances. Inthe following results, the number of transmit antennas is 4(M = 4).

A measure for the vector quantization performance ofa packing W is the average distortion [13]. The averagedistortion for a number of packings has been numericallyevaluated. The results are displayed in Fig. 4. It is intuitivethat the distortion decreases with the number of beams inthe packing. For Grassmannian packings with 2-dimensionalsubspaces (B = 2), the distortion is slightly higher than forline packings (B = 1). This is due to the fact that for B = 2,the subspaces (planes) are isotropically spread out, but not the

10 20 30 40 50 600.1

0.15

0.2

0.25

0.3

0.35

0.4

Grass. B=1Grass. B=2DFT B=1DFT B=2

Number of beams (NB) in W

Ave

rage

dist

ortion

d(W

)

Fig. 4. The average distortion for a number of Grassmannian and DFT-basedpackings is displayed.

ensemble of lines (i.e. the vectors) that span the subspaces.For the DFT-based packings for a given NB, the quantizationperformance is identical for B = 1 and B = 2. This is due tothe fact that the packings for B = 1 and B = 2 contain thesame vectors, but for B = 2 arranged in orthogonal pairs (c.f.Section IV-B). To conclude, the Grassmannian packings havea superior quantization performance for i.i.d. Rayleigh fadingchannel vectors, especially for packings with many beams.

B. System Study

A system level simulation study has been conducted in orderto evaluate the performance of packings previously discussed.One cell with one basestation and several single-antennausers has been simulated according to the system modelin Section II. The basestation is equipped with 4 transmitantennas (M = 4). The maximum throughput scheduler isused. For each block, it selects the beam/user combinationthat yields the highest sum-rate (c.f. (2)). The average SNRis 5 dB. The simulated channels are generated using Jake’smodel with block fading.

The Grassmannian (Grass.), DFT and non-orthogonal (Non.orth.) packings, as described in Section IV, are used in reducedfeedback SDMA, as described in Section III.

As a comparison, opportunistic SDMA (Opp. SDMA) asin [3] is used. This technique randomly generates one setof orthogonal beams in each block. It relies on the multi-user diversity to provide the scheduler with users that arecompatible with the generated beams. This technique can beseen as a packing with only one subspace, that is randomlyregenerated each block. The Grassmannian, DFT and non-orthogonal packings with multiple subspaces exploit boththe multi-user diversity and the diversity from having manysubspaces. Another advantage with having packings with non-random predefined beamforming matrices is that the usersmay track only the underlying channel, whereas with randombeams they repeatedly need to estimate the discontinuousaggregate channel (i.e. random matrix times channel).

Finally, two versions of a conventional linear zero-forcingSDMA scheme are used. In this scheme [17], users are


10 20 30 40 502

3

4

5

6

7

8

9

ZFSPZFSGrass.Non. orth.DFTOpp. SDMA

Number of users K

Ave

rage

sum

rate

[bps

/Hz]

Fig. 5. Average sum rate as a function of the number of active users. TheSDMA-factor B = 2 for all schemes. For the packing based schemes, thenumber of subspaces N = 16. For ZFS the channels are quantized using a64-vector Grassmannian line packing.

greedily scheduled based on the CSI at the basestation. Thezero-forcing beamforming matrix for the scheduled users iscomputed at the basestation, also based on the available CSI.In one version of the scheme, zero-forcing with user selectionand perfect CSI (ZFSP), the scheduling and zero-forcingbeamforming is based on the true channels to the users. In theother version, zero-forcing with user selection and quantizedCSI (ZFS), the true channels are quantized at the users througha Grassmannian line packing and fed back to the basestation.In both versions, equal power is allocated to all beams, as inthe other schemes.

It is important to note that the AMC level adaptation forZFS is genie-aided in the simulations. Since the basestationdoes not know the quantization error of the available CSI, it isnot possible to exactly compute the resulting post-schedulinguser SINR and therefore not the supportable AMC level. Inthe simulations, it is assumed that the AMC level for ZFS isperfectly adapted to the computed beamforming matrix andthe true CSI. Hence, in the simulations ZFS is not penalizedby outages or unnecessarily large SINR back-off. This kind ofgenie-aided AMC level adaptation is not required for any ofthe other schemes, since the computed supported AMC levelis based on perfect CSI (at the basestation for ZFSP and atthe users for the other schemes).

The number of vectors in the quantization codebook forZFS is equal to the number of columns in W , i.e. log2(NB)bits need to be fed back. Hence the feedback rate of ZFSis identical to that of the proposed subspace packing basedschemes. The feedback rate of the ZFSP scheme would bevery high since the unquantized channels are available atthe transmitter. For the opportunistic beamforming scheme,each user needs to feed back log2(B) bits per block. Forall schemes, the channel magnitude (ZFS and ZFSP) orsupportable AMC level (Opp. SDMA, Grass., Non. orth. andDFT) is not quantized.

In Fig. 5, the average sum rate as a function of the numberof active users is displayed. The SDMA factor for all schemes,B, is 2. The packing-based methods use packings with 16

10 20 30 40 50

2

4

6

8

10

ZFSPZFSGrass.Non. orth.DFTOpp. SDMA

Number of users K

Ave

rage

sum

rate

[bps

/Hz]

Fig. 6. Average sum rate as a function of the number of active users. TheSDMA-factor B = 3 for all schemes. For the packing based schemes, thenumber of subspaces N = 4. For ZFS the channels are quantized using a12-vector Grassmannian line packing.

subspaces (N = 16). The packing-based methods outper-form opportunistic SDMA at the cost of 4 extra feedbackbits per user and block. The DFT-based packing results inslightly worse performance than the Grassmannian. The non-orthogonal SDMA performs slightly better than orthogonalGrassmannian SDMA, since the loss of orthogonality fromgrouping the 64 vectors 2 and 2 is low and a vector betteraligned with the channel can be found on average (see Fig. 4).

The ZFSP with perfect CSI results in a significantly higheraverage sum rate, whereas the ZFS with quantized CSI at thebasestation performs worse for more than 10 users. For ZFS,the channel is quantized using a 64-vector Grassmannian linepacking. The zero-forcing beamforming matrix is based onthe quantized channels, whereas the resulting post-schedulingSINR is based on the true channels. This mismatch signifi-cantly reduces the achieved SINR of the scheduled users. Eventhough it is assumed that the users reliably communicate atrates corresponding to this lower SINR, the loss compared toZFSP with perfect CSI is large.

In Fig. 6, the SDMA factor, B, is 3, but the number of sub-spaces, N , is only 4. These parameters are less beneficial forthe non-orthogonal packing, since it is not possible to find 4groups of 3 nearly orthogonal beams from a Grassmannian linepacking of these dimensions. Even the opportunistic SDMAoutperforms the non-orthogonal scheme. The Grassmanniansubspace packing outperforms the other packings. In thissetting, the performance of the ZFS with quantized CSI ismuch below the other schemes. This is due to the fewer vectorsin the channel quantization codebook (12) and the increasedloss in SINR from the mismatch between the computed zero-forcing beamforming matrix and the true channel. This losstypically increases as the SDMA-factor increases. However,the ZFSP with perfect CSI outperforms the other schemeswith a large margin.

Fig. 7 illustrates the relation between rate, SDMA factor andnumber of users. The figure shows the average sum rate forGrassmannian subspace packings with subspace dimensions1, 2 and 3. The total number of beams, NB, is 12. For


0 10 20 30 40 502.5

3

3.5

4

4.5

5

5.5

6

B=1B=2B=3

Number of users K

Ave

rage

sum

rate

[bps

/Hz]

Fig. 7. Average sum rate for Grassmannian subspace packings with subspacedimensions 1, 2 and 3 as a function of the number of users. For all threecurves, NB = 12.

3 4 5 6 7

2.95

3

3.05

3.1

3.15

3.2

Fullrd = 5%

rd = 10%

rd = 20%

Feedback bits per block

Ave

rage

sum

rate

[bps

/Hz]

Fig. 8. Average sum rate for a system with 1 user, using a Grassmannianline packing with 128 beams. For the Full curve, the whole beam index isfed back each block (7 bits per block). For the curves with relative Doppler(rd) 5-20%, beam-graph feedback with different rates is used.

SDMA factors greater than 1 (B = 2 and B = 3) toperform well, there needs to be many users in the system.Otherwise, allocation outages will occur frequently. Eventhough an allocation outage does not occur, the SINR penaltyfrom scheduling a user with a channel vector poorly aligned tothe beam is greater for higher SDMA factors than for single-beam transmission (B = 1). This is due to the resulting non-orthogonality to the interfering beams. The optimal switchingpoints for the SDMA factor depend on the SNR. For higherSNR, the switching points are shifted to higher number ofusers (to the right in Fig. 7). A framework for finding theoptimal B, given an SNR and the number of users, foropportunistic TDMA and SDMA is developed in [22].

In the results above, the whole beam index is fed backeach block. In Fig. 8, the impact of the reduced feedbackscheme based on beam-graphs, as described in Section V,is evaluated. A system with one user has been simulatedusing a Grassmannian packing of 128 lines, i.e. B = 1. The

beamforming vectors have been arranged in different graphsdepending on the number of feedback bits per block. Forinstance, for 3 feedback bits per block, each beam (vector) inthe graph is allowed to have 23 = 8 neighbors. The averagesum rate for the full feedback scheme, i.e. the whole beamindex is fed back each block, serves as an upper bound.The performance using beam-graph feedback is shown forthree values of relative Doppler, rd. The relative Doppleris defined as rd = block duration ∗ fd, where fd is themaximum Doppler frequency. For example, rd = 20% couldcorrespond to a system with 2 GHz carrier frequency, a 0.15ms block duration and user speed of 20 m/s. From Fig. 8it is clear that a significant feedback reduction is possiblewithout sacrificing performance to any great extent. Since thepacking contains 128 beams, the full beam index feedback alsomeans 7 feedback bits per block. The relative performanceloss from reducing the feedback from 7 to 3 bits per userand block for rd = 20% is less than 8%. There is a gapbetween the full feedback performance and the beam-graphperformance with 7 feedback bits. This is due to the factthat, using the beam construction method in Section V, somebeams will have no neighbors in some directions. Finally, notethat the performance gap between full feedback and beam-graph feedback will be reduced when there are more usersin the system. This is due the fact that the rate loss due tobeam quantization is accounted for in the supportable ratecomputation by the users. Therefore, a user with temporarilybad beam quantization will less likely be scheduled than auser with perfect beam quantization.

VII. CONCLUSIONS AND DISCUSSION

We have proposed a scheme for reduced feedback SDMAbased on subspace packings. The scheme is based on acodebook of beams (a subspace packing), common to thebasestation and the users. The scheme allows all users tocompute their post-scheduling SINR already before the CSIfeedback. Only a few bits feedback per user and coherencetime is necessary. The feedback also provides informationabout the spatial compatibility of the users. We highlighteda fundamental property of the scheme through the allocationoutage probability. Furthermore, a beam-graph method toreduce the feedback rate even further for large packings wasproposed for tracking situations. The reduced feedback SDMAscheme based on subspace packings shows potential to coversome of the middle ground between opportunistic SDMAand SDMA based on perfect CSI. Several parameters of themethod should be tuned according to the specific scenario.For instance, a scenario with many users would benefit from apacking with many subspaces and with a high SDMA factor.A scenario with few user on the other hand would benefitfrom a packing with fewer subspaces of lower dimension,i.e. fewer simultaneous spatial streams. In a practical systemwith a variable number of users, this insight can be usedto vary the codebook (packing) dimension, for instance bycodebook restriction as in LTE [18]. The proposed schemewas numerically evaluated using different kinds of packings.It outperformed a system with zero-forcing SDMA at thetransmitter based on quantized and fed back channels.


APPENDIX A

In many cases, it is sufficient for the users to find the (i, j)that maximizes |hT

k wji |2 instead of the (i, j) that maximizes

γk (i, j), as in (1). Unfortunately, this may give a sub-optimalchoice. The processing delay induced by the computationof (1) adds to the delay between channel estimation anddata transmission. This delay should be kept to a minimum,otherwise the feedback information will be outdated. Forlarge packings, a large number of divisions can be savedby considering the inner product instead of the SINR whenselecting a beam index. The maximum loss in SINR by usingthe maximum inner product beam selection criterion insteadof the maximum SINR criterion for B ≤ M is quantified inthe following result.

Theorem 2: Assume that 1 ≤ B ≤ M and that for all i thecolumns of Wi are orthonormal. Let

(ak, bk) = arg max1≤i≤N,1≤j≤B

γk (i, j) (5)

(ck, dk) = arg max1≤i≤N,1≤j≤B

|hTk wj

i |2. (6)

denote the optimal and sub-optimal beam choice for the kth

user, respectively. Then, for 1 < B < M , the relative loss inSINR is bounded by

γk (ak, bk)γk (ck, dk)

≤ 1 +‖hk‖2

2

σ2z

(1 − α)B

, (7)

where ‖hk‖22

σ2z

is the instantaneous SNR of the user and αrepresents the worst-case inner product of the packing W ,

α = minu∈C

1×M

‖u‖2=1

max1≤i≤N1≤j≤B

|uwji |2. (8)

Equality in (7) is obtained when hk

‖hk‖2equals the minimizing

u in (8). For B = 1 and B = M , γk (ak, bk) = γk (ck, dk).Proof: Consider user k with channel hk. The optimal

(maximum SINR from (5)) beam from the set W is denotedwb

a. The beam with maximum inner product (as in (6)) isdenoted wd

c . The case B = 1 is trivial, since there is no inter-beam interference. Hence, wb

a = wdc . Now, consider the case

B > 1. If B = M , the channel vector can always be expressedin the coordinate system spanned by the orthonormal columnsof Wi for all i, i.e. hk =

∑Bj=1 βj

i wji , where βj

i = hTk wj

i .Thus,

γk (ik, jk) =|βj

i |2Bσ2

z +∑

n�=j |βni |2

=|βj

i |2Bσ2

z + ‖hk‖22 − |βj

i |2,

since ‖h‖22 =

∑Bj=1 |βj

i |2 due to the orthonormality of thebasis vectors. It is clear that

(ck, dk) = arg max1≤i≤N1≤j≤B

|βji |2 = arg max

1≤i≤N1≤j≤B

γk (i, j) = (ak, bk),

when B = M .If B < M , one additional unit norm vector vi, orthogonal

to Wi, is in general needed to express hk, with

vi =hk − ∑B

j=1 βji w

ji

‖hk − ∑Bj=1 βj

i wji ‖2

.

The channel vector can then be written as a linear combinationof the basis vectors as hk =

∑Bj=1 βj

i wji + βB+1

i vi, whereβB+1

i = hTk vi. Due to the orthonormality of the basis vectors

for all i,

‖hk‖22 =

B∑j=1

|βji |2 + |βB+1

i |2

= |βji |2 +

∑n�=j

|βni |2 + |βB+1

i |2.

Note that

0 ≤∑n�=j

|βni |2 = ‖hk‖2

2 − |βji |2 − |βB+1

i |2 ≤ ‖hk‖22(1 − α),

(9)where α is defined in (8). The minimizing u in (8) is the worst-case unit vector, i.e. the inner product with the closest vector inthe packing is minimal. Since α only depends on W , it can benumerically found offline. Equality in the upper bound in (9) isobtained when hk is co-linear with the minimizing u in (8) andalso in the subspace spanned by Wi, i.e. βB+1

i = 0. Equalityin the lower bound in (9) is obtained when hk is orthogonal tothe interference vectors {wn

i }n�=j , i.e. hk = βji w

ji +βB+1

i vi.The relative SINR loss is


=|βb

a|2Bσ2

z +∑

n�=b |βna |2

Bσ2z +

∑n�=d |βn

c |2|βd

c |2.

(10)The worst case is characterized by

• |βba|2 = |βd

c |2, since

|βdc |2 = max

1≤i≤N1≤j≤B

|βji |2 ≥ |βb

a|2,

which means that two vectors in the packing had the sameinner product and the wrong one was chosen,

•∑

n�=b |βna |2 = 0, i.e. hk is orthogonal to all interference

vectors {wna}n�=b and

•∑

n�=d |βnc |2 = ‖hk‖2

2(1− α), i.e. hk is spanned by Wc

and |hTk wd

c |2 is minimal.

Hence,


≤ Bσ2z + ‖hk‖2

2(1 − α)Bσ2

z

= 1 +‖hk‖2

2

σ2z

(1 − α)B

.

APPENDIX B

In order to derive the allocation outage probability inTheorem 1, we prove the following Lemma.

Lemma 1: Consider a set A with a elements and a subsetB ⊆ A with b ≤ a elements. Independently, with uniformprobability and with replacement, k elements are taken fromA. The taken elements, with duplicates removed, constitutethe set K ⊆ A, with cardinality |K| ≤ min(k, a). Then,

P̄k(a, b) � Pr (B ⊆ K)

= 1 +1ak

b∑m=1

(−1)m

(b

m

)(a − m)k. (11)


Proof: First, define the b dependent events {Em}m=1,...,b,where Em is the event that element m in B is not in K. Then,

P̄k(a, b) = Pr(B ⊆ K) = 1 − Pr( b⋃

m=1

Em

),

where⋃

denotes the union of events. Poincaré’s formula statesthat [24, page 89]

Pr( b⋃

m=1

Em

)=

b∑m=1

(−1)m−1Sm, (12)

where

Sm =∑

· · ·∑

1≤j1<···<jm≤b

Pr (Ej1 ∩ · · · ∩ Ejm) , (13)

where ∩ denotes AND. But

Pr (Ej1 ∩ · · · ∩ Ejm) =(a − m)k

ak

for all distinct j1, . . . , jm. Therefore,

Sm =(a − m)k

ak

∑· · ·

∑1≤j1<···<jm≤b

1

=(a − m)k

ak

(b

m

),

since the number of ways the m integer indices can be chosenso that 1 ≤ j1 < · · · < jm ≤ b equals

(bm

). Finally,

P̄k(a, b) = 1 +1ak

b∑m=1

(−1)m

(b

m

)(a − m)k.

Equipped with Lemma 1, the allocation outage probabilitycan be determined. First, we define the N dependent events{Dn}n=1,...,N , where Dn is the event that all beam indicesin the nth set have been fed back by at least one user.Then, Pao = 1 − Pr

(⋃Nn=1 Di

), where

⋃denotes the union

of events. As in the proof of Lemma 1, using Poincaré’sformula (12)-(13) and the fact that

Pr (Dj1 ∩ · · · ∩ Djn) = P̄K(NB, nB)

for all distinct j1, . . . , jn and where P̄k(a, b) is defined inLemma 1 gives

Pao = 1 +N∑

n=1

(−1)n

(N

n

)P̄K(NB, nB)

= 1 +N∑

n=1

(−1)n

(N

n

)

+1

(NB)K

N∑n=1

nB∑m=1

(−1)n+m

(N

n

)(nB

m

)(NB − m)K .

REFERENCES

[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge University Press, 2005.

[2] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamformingusing dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, pp. 1277–1294, June 2002.

[3] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channelswith partial side information,” IEEE Trans. Inform. Theory, vol. 51, pp.506–522, Feb. 2005.

[4] N. Zorba and A. I. Pérez-Neira, “A multiple user opportunistic scheme:the Grassmannian approach,” in Proc. Int. Zürich Seminar. on Commun.,pp. 134–137, Feb. 2006.

[5] V. K. N. Lau, “Spatial-multiplexing phase-sweep transmit diversity (SM-PSTD) for multiantenna base stations with mobile-assisted schedulingand incremental rate feedback,” IEEE Trans. Veh. Technol., vol. 55, pp.490–498, Mar. 2006.

[6] A. Narula, M. Lopez, M. Trott, and G. Wornell, “Efficient use of sideinformation in multiple-antenna data transmission over fading channels,”IEEE J. Select. Areas Commun., vol. 16, pp. 1423–1436, Oct. 1998.

[7] V. K. N. Lau, Y. Liu, and T.-A. Chen, “Capacity of memoryless chan-nels and block-fading channels with designable cardinality-constrainedchannel state feedback,” IEEE Trans. Inform. Theory, vol. 50, pp. 2038–2049, Sept. 2004.

[8] P. Xia and G. B. Giannakis, “Design and analysis of transmit-beamforming based on limited-rate feedback,” IEEE Trans. SignalProcessing, vol. 54, pp. 1853–1863, May 2006.

[9] W. Santipach and M. L. Honig, “Signature optimization for CDMA withlimited feedback,” IEEE Trans. Inform. Theory, vol. 51, pp. 3475–3492,Oct. 2005.

[10] C. K. Au-Yeung and D. J. Love, “On the performance of random vectorquantization limited feedback beamforming in a MISO system,” IEEETrans. Wireless Commun., vol. 6, pp. 458–462, Feb. 2007.

[11] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “Onbeamforming with finite rate feedback in multiple-antenna systems,”IEEE Trans. Inform. Theory, vol. 49, pp. 2562–2579, Oct. 2003.

[12] D. J. Love, R. W. Heath, and T. Strohmer, “Grassmannian beamform-ing for multiple-input multiple-output wireless systems,” IEEE Trans.Inform. Theory, vol. 49, pp. 2735–2747, Oct. 2003.

[13] D. J. Love and R. W. Heath, “Limited feedback diversity techniques forcorrelated channels,” IEEE Trans. Veh. Technol., vol. 55, pp. 718–722,Mar. 2006.

[14] J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing lines, planes,etc.: packings in Grassmannian spaces,” Experimental Math., vol. 5, pp.139–159, 1996.

[15] J. A. Tropp, “Topics in sparse approximation,” Ph.D. dissertation, TheUniversity at Texas Austin, 2004.

[16] D. J. Love and R. W. Heath, “Limited feedback unitary precoding forspatial multiplexing systems,” IEEE Trans. Inform. Theory, vol. 51, pp.2967–2976, Aug. 2005.

[17] G. Dimic and N. D. Sidiropoulos, “On downlink beamforming withgreedy user selection: performance analysis and a simple new algo-rithm,” IEEE Trans. Signal Processing, vol. 53, pp. 3857–3868, Oct.2005.

[18] 3GPP. (2008, May) TS 36.211: Physical channels and modulation.[Online]. Available: http://www.3gpp.org/

[19] B. M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens, andR. Urbanke, “Systematic design of unitary space-time constellations,”IEEE Trans. Inform. Theory, vol. 46, pp. 1962–1973, 2000.

[20] S. Hassani, Mathematical Physics. Springer, 1999.[21] W. Mayeda, Graph Theory. John Wiley and Sons, 1992.[22] E. Jorswieck, P. Svedman, and B. Ottersten, “Performance of TDMA

and SDMA based opportunistic beamforming,” IEEE Trans. WirelessCommun., vol. 7, pp. 4058–4063, Nov. 2008.

[23] IST WINNER II. (2006, Nov.) D3.4.1: The WINNER II air interface:refined spatial-temporal processing solutions. [Online]. Available:https://www.ist-winner.org

[24] W. Feller, An Introduction to Probability Theory and its Applications.John Wiley and Sons, 1962, vol. I.

Reduced feedback SDMA based on subspace packings

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