Reduced Feedback SDMA Based on

Subspace Packings

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PATRICK SVEDMAN, EDUARD JORSWIECK, AND

BJORN OTTERSTEN

Stockholm 2007

Signal Processing

School of Electrical Engineering

Kungliga Tekniska Hgskolan

IR-EE-SB 2007:015

1

Reduced Feedback SDMA Based on

Subspace Packings

Patrick Svedman, Eduard Jorswieck and Björn Ottersten

KTH School of Electrical Engineering

SE-100 44 Stockholm, SWEDEN.

Email: patrick.svedman@ee.kth.se

Abstract

Herein, we treat Space Division Multiple Access (SDMA) based on partial channel state informa-

tion and limited feedback. We propose a framework utilizingsubspace packings, where beamforming,

feedback, and scheduling are integrated. Advantages of theproposed framework are that the fed back

supportable rates are based on the post-scheduling SINR andthat the feedback implicitly contains

information about the spatial compatibility of the users. The feasibility region of packings of different

dimensions is indicated by the allocation outage probability which is derived. Grassmannian subspace

packings, DFT-based packings, and non-orthogonal Grassmannian packings are formulated and studied

as candidates. Numerical simulations show slightly betterperformance for the Grassmannian subspace

packings in i.i.d. Rayleigh fading. Finally, we propose andevaluate a beam-graph method to further

reduce the feedback load, that can be also used in most quantized beamforming contexts.

I. I NTRODUCTION

Multiple-input, multiple-output (MIMO) communication systems promise high spectral effi-

ciency and reliability for future mobile systems. Whereas the theoretical aspects are relatively

well understood [1], there is still a large performance gap between the theoretical capacity

and implementable designs, especially for multiuser systems. To achieve high spectral effi-

ciency in fading channels, the multiple antennas can exploit the spatial dimension and adapt

the transmitted and received signals so that interference from unwanted signals is suppressed.

To do this efficiently, the transmitters in general require some knowledge of the communication

channel between themselves and the receivers [2]. Since users in mobile systems typically move,

March 27, 2007 DRAFT

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their channel states change. Therefore, the multi-antennatransmission and reception needs to

continuously adapt to the changing conditions. In frequency division duplex (FDD) systems

and in non-reciprocal time division duplex (TDD) systems, information about the channel states

needs to be passed between receivers and transmitters. Thisfeedback overhead may become

intractable, especially in systems with many users and highmobility.

In this paper, we consider the downlink of a flat fading multiuser FDD communication system

with multiple antennas at the basestation and a single antenna at the users. There are many open-

loop schemes where the transmitted signal is designed without knowledge of the channel states

of the users, e.g. space-time codes [3]. However, with some channel knowledge at the transmitter,

the performance can be significantly improved. The most suitable form of this partial channel

state information is an open question, and depends on many scenario-dependent factors, like the

number of active users, the type of traffic, the channel characteristics etc..

One angle to approach the problem is to exploit the diversityof the channel states of the

users. In opportunistic beamforming, multiuser diversityis used by transmitting a random beam

through the multiple antennas [4]. It is likely that the beamsuits at least one user and the

resulting supportable rate of each user can be fed back as partial channel state information to

the basestation. This overcomes the need for explicit channel state information at the transmitter.

Note that feedback of supportable rate implies the use of adaptive modulation and coding (AMC).

For wideband systems, opportunistic beamforming can be combined with OFDMA [5].

In [6], opportunistic beamforming was extended to space division multiple access (SDMA),

where the basestation transmits to several users simultaneously. Instead of transmitting only one

beam, several random but orthogonal beams are transmitted.The users now report the index of

their best beam as well as the corresponding supportable rate. One user per beam can then be

scheduled by the basestation. Note that the reported supportable rates can take the inter-beam

interference into account, since the users must be able to measure the signal strength from all

beams before feedback. This scheme was extended to support more beams than transmit antennas

in [7], by using Grassmannian line packings.

One approach to reduce the feedback load is to let the users vector quantize their channel

states, or equivalently optimal beamformers, using e.g. the generalized Lloyd’s algorithm [8]

or random vector quantization [9]. Another way to generate the quantization codebook is to

use Grassmannian line packings [10]. A Grassmannian packing is a collection of subspaces

March 27, 2007 DRAFT

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in a higher dimensional space. An optimal packing is such that it maximizes some distance

measure between the two closest subspaces in the packing. For the intuitive case where the

channel vector is real and three-dimensional, the optimal line packing is a collection of one-

dimensional subspaces, i.e. lines, inR3 that pass through the origin and maximize the angle

between the two closest lines. Clearly, Grassmannian line packings are promising candidates

for the quantization of the i.i.d. Rayleigh fading vector channel [10], [11]. In the case of

spatially correlated Rayleigh fading channels, the packingcan be prewhitened as in [12]. A

major disadvantage with Grassmannian packings is the lack of analytical results for most cases,

but numerical methods to generate good packings exist [13],[14].

The step from Grassmannian single-beam beamforming, i.e. line packings, to Grassmannian

spatial multiplexing was taken in [15]. In MIMO channels, several data streams can be si-

multaneously transmitted to a user by precoding with a unitary matrix (instead of a vector

in the single-stream case) at the transmitter. The optimal unitary matrix is quantized using a

Grassmannian subspace packing.

Herein, the Grassmannian subspace packing approach is applied to the multiuser scenario.

As in [15], unitary matrices from Grassmannian subspace packings are used for transmitter

precoding. The major difference is that we consider SDMA, where one user per beam (or column

in the unitary beamforming matrix) can be allocated. There is no cooperation between the receive

antennas (users), and only a subset of the receiver antennas(users) are used at a time. We show

that, contrary to single-user Grassmannian beamforming orspatial multiplexing, a denser packing

often does not yield better downlink performance due to allocation outages. Additionally, DFT-

based and non-orthogonal Grassmannian subspace packings are considered. Furthermore, a graph

representation for large packings or codebooks is proposedand evaluated, which can be used to

further reduce the feedback rate. The beam-graph feedback reduction method is also well suited

for the previously investigated Grassmannian beamforming. The contributions of this paper can

be summarized as follows:

• A feedback and SDMA scheme based on subspace packings is proposed. Two benefits of

the scheme are:

1) The feedback implicitly contains information about the spatial compatibility of the

users.

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

• The important concept of allocation outage is discussed andthe allocation outage probability

is derived.

• Non-orthogonal Grassmannian subspace packings are introduced.

• The beam-graph method for tracking channel variations, which further reduces the feedback

rates, is proposed and evaluated.

• Numerical system simulation results compare various flavors of the proposed method with

opportunistic SDMA and optimal linear precoding.

In Section II, the system model is described. The idea of reduced-feedback SDMA using

subspace packings is described in Section III. In Section IV, subspace packings and their

application to the problem are described. The design of beam-graphs and their use is treated in

Section V. Numerical results are given and discussed in Section VI and the paper is concluded

in Section VII.

II. SYSTEM MODEL

We consider the downlink of a cellular multiuser FDD system.The basestation withM

antennas communicates withK single-antenna users. The wireless channels are assumed to

be flat i.i.d. Rayleigh fading and quasi-stationary, i.e. constant during each block. One block is

also the resource allocation period. The received signal for userk at symbolt, yk(t) ∈ C, is

yk(t) = hTk (t)x(t) + zk(t),

wherehk(t) ∈ CM×1 is the MISO channel of thekth user,x(t) ∈ C

M×1 is the transmitted signal

from the basestation,zk(t) ∈ C is AWGN with varianceσ2z and (·)T denotes matrix transpose.

The quasi-stationary assumption means thathk(t) is constant for allt within a block. In the

following, the time-indext is omitted for brevity.

The transmitted signal,x, is assumed to be

x =1√B

Ws =1√B

B∑

i=1

wisi,

whereW = [w1 · · ·wB] ∈ CM×B is the beamforming matrix ands = [s1 · · · sB]T ∈ C

B×1 is

the vector of i.i.d. symbols. The variableB denotes the number of simultaneously transmitted

symbols as well as the number of used beams. Theith data-symbolsi ∈ C is transmitted using

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the ith beamforming vectorwi ∈ CM×1. The number of simultaneous streamsB is henceforth

called theSDMA factor. The caseB = 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‖22] = 1. We assume that each userk accurately estimates the channel vectorhk.

Now, assume that symbolm is dedicated to userk. The received SINR is

γk =|hT

k wm|2Bσ2

z +∑

n6=m |hTk wn|2

.

The user SINR is maximized ifhk is co-linear withwm and hk is orthogonal town for all

n 6= m. Then, the SINR isγk =‖hk‖

22

Bσ2z

. This means that users are perfectly spatially compatible

if their channel vectors are orthogonal. In practice, this is rarely the case. Instead, the beams

are typically designed to balance the inter-beam interference to fulfill QoS-constraints of the

scheduled users, see e.g. [16].

Furthermore, it is assumed that the system supports a finite set of adaptive modulation and

coding (AMC) levels. Given the error rate constraint of userk, the supportable AMC level is

uniquely determined by the mapping functionΓk(γk), which is assumed known.

III. R EDUCED FEEDBACK SDMA BASED ON SUBSPACEPACKINGS

Assume that a set of a orthogonal beamforming matrices,W = {W1, . . . ,WN}, is a priori

known at both the basestation and at the users. The columns ofeach matrix inW span aB-

dimensional subspace ofCM . W is called a subspace packing. Each beamforming matrix consists

of B orthogonal columns (beams/vectors),Wi =[

w1i · · ·wB

i

]

.

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

1) Each userk finds the overall best beam index1,

(i, j)k = arg max(i,j)

γk ((i, j)) = arg max(i,j)

|hTk w

ji |2

Bσ2z +

∑

n6=j |hTk wn

i |2. (1)

This means that, when selecting the best beam, the userassumesthat all other beams in

Wi thanwji will be used for transmission to other users.

2) Each user feeds back the beam index(i, j)k and the corresponding supportable AMC level

Γk (γk ((i, j)k)).

1For B = M , (1) is equivalent toarg max(i,j) |hTk w

ji |. In Theorem 2 in Appendix I, we upper bound the relative loss in

SINR from choosing the(i, j) that maximizes|hTk w

ji | instead of (1).

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3) According to a scheduling metric, the basestation chooses a subspacei∗ to be used during

the next block, i.e. the basestation will useWi∗ as beamforming matrix. For each beam

in Wi∗, a user is scheduled with the fed back AMC level. For example,if the basestation

uses maximum throughput scheduling, the subspace is chosenaccording to

i∗ = arg maxi

B∑

j=1

maxk

Γk (γk ((i, j)k)) . (2)

In other words, each user finds its best overall beam from all subspaces. Then, the user

computes the supportable rate it would get if it is scheduledon its best beamand other users are

scheduled on the other beams in the same subspace. Finally, the basestation selects a subspace

according to some scheduling metric. Only users that fed back beam-indices for that subspace

are considered for scheduling.

It is important to note that the fed back AMC level,Γk, includes thepost-schedulinginter-

beam interference, regardless of which users are scheduledon the other beams. This is notable

sinceΓk is computedbeforethe feedback and the scheduling take place.

The scheduling is simplified since the feedback implicitly shows which users are spatially

compatible. It is likely that two users who fed back two different beam-indices within the same

subspace have nearly orthogonal channels, since all beams within one subspace are orthogonal.

The reduction in feedback is due to the quantization of the optimal beamforming vector. It

is indexed with the(i, j)-pair usinglog2(BN) bits. A couple of additional bits are needed for

the AMC level. To reduce the feedback further, Section V proposes a method to order theBN

beams in a graph.

There is an important trade-off in the selection of the number of subspacesN . The more

subspaces and beams to choose from (highN ), the more likely that the beam a user chooses is

close to optimal. This relation is illustrated in Fig. 3 in Section VI-A. However, the feedback

increases with increasingN . More importantly, for a limited number of users, the probability

that the basestation can not allocate users to allB beams in any subspace increases withN . If

the basestation can not allocateB beams, there is a significant performance loss. This is due to

the fact that the allocated users assume in their AMC level computation that the other beams

are used for transmission to other users, creating inter-beam interference. Theorem 1 presents

the probability of this event, which is called allocation outage.

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Definition 1 (Allocation outage):Consider a system whereK users feed back one beam index

each out ofNB possible. TheNB beams are divided intoN disjoint sets ofB beams each. An

allocation outageis the event that there exists no set where allB beam indices have been fed

back by at least one user.

Theorem 1:Consider the allocation outage event, as in Definition 1. Additionally assume that

the users choose beam indices 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 II.

The result in Theorem 3 is useful in understanding the tradeoff between having many beams in

the packing (NB) and a high SDMA factor (B) on one hand and the probability of undesired

allocation outages on the other hand. The outage probability (3) for a number of packing

dimensions is displayed in Fig. 1. For few users (lowK), higher SDMA factor (B) than 2

is not suitable due to the high allocation outage probability. The scheduler would rarely find a

subspace where all beams could be used. This would result in fewer spatial streams thanB as

well as lower fed back supportable rate for the scheduled users. For 10 users, SDMA factor 2

and 6 subspaces (K = 10, B = 2 andN = 6), the scheduler can with more than 90% probability

find a subspace where 2 users can be scheduled, which seems to be a suitable operating point.

For the scenario with many active users, it is clearly suitable to use a higher SDMA factor,

enabling many spatial streams with high probability. Note that B > M is not possible without

inter-beam interference.

For a givenK, the optimalN and B in terms of sum rate is an open problem. A packing

with high NB may enable the users to find a beam close to the optimal, but on the other hand

increases the allocation outage probability. However, suitable packing dimensions may be found

from simulations.

Fig. 1 suggests a scheme that would adapt to the number of active users, which can be assumed

to be a slowly varying parameter. Both the basestation and theusers would have a set of packings

with various dimensionsN and B. Depending on the number of users, the basestation would

broadcast theN andB of the currently used packing.

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In the presentation above, single antenna users have been assumed. However, the proposed

scheme is compatible with most multi-antenna receiver structures, since the MIMO channel,

the transmit beamformer, and the interfering 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. SUBSPACEPACKINGS

In this section, the design ofW is discussed. In Section III, the proposed subspace packing

based SDMA scheme was described. It relies on users finding a beamforming vector from a

finite predefined set that in some sense isclose tothe optimal beamforming vector. At the same

time, the vectors spanning each subspace, or the columns of each Wi, should be orthogonal,

enabling inter-beam interference free SDMA.

When constructing packings for SDMA, there is a fundamental tradeoff between 1) good vector

quantization properties and 2) the orthogonality of vectors that are to be used simultaneously.

For the i.i.d. Rayleigh fading vector channel, the optimal Grassmannian line packing is optimal

in terms of quantization. However, the line packing in general does not contain any orthogonal

vectors, thereby penalizing SDMA. On the other hand, the optimal subspace packing gives

orthogonal matrices that are maximally separated in chordal distance (see Section IV-A). For the

optimal subspace packing, the ensemble of basis vectors from all subspaces, i.e. the columns of

all unitary matrices, do not in general quantize the vector channel as well as the optimal line

packing. This tradeoff is numerically studied in Section VI-A.

Grassmannian and DFT-based packings, which are known from literature, are presented below.

However, they have not been used in the context of SDMA previously. Additionally, we propose

non-orthogonal Grassmannian subspace packings, which quantize the vector channel optimally,

but introduce some extra inter-beam interference.

A. Grassmannian Subspace Packings

The distance measure between subspaces inCM used here is thechordal distance. Other

measures are also feasible, but the chordal distance is chosen for its simplicity and that it yields

the most symmetrical packings [13], [14]. The chordal distance between the subspaces spanned

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by theB orthonormal columns ofWi andWj is given by

dist(Wi,Wj) =√

B − ‖WHi Wj‖2

F , (4)

where (·)H denotes conjugate transpose. The complex Grassmannian manifold G(

B, CM)

is

the collection of allB-dimensional subspaces inCM . A Grassmannian packing is a finite set

of subspaces from the manifold, spanned by the unitary matrices inW. An optimal packing is

such that the minimum distance between any two subspaces is maximized, i.e. fori 6= j

W = arg maxW

minWi,Wj∈W

dist(Wi,Wj) . (5)

For B = 1, the problem reduces to the packing of lines inCM that pass through the origin,

or equivalently, to the packing of points on the unit sphere in CM . For the i.i.d. Rayleigh

fading channel, the optimal Grassmannian line packing is optimal in terms of, for instance,

outage probability [10]. In general, there is no analytic solution to (5). However, for moderate

problem dimensions, there are numerical methods which can provide near-optimal packings.

Grassmannian packing construction is discussed in more detail in Section VI-A. Numerically

constructed near-optimal Grassmannian packings are henceforth called Grassmannian packings.

B. DFT-based Subspace Packings

In [10], it was shown that the columns of the unitary space-time constellations in [17] worked

well as beamformers for the i.i.d. Rayleigh fading channel. TheM first elements of the columns

of an NB-dimensional DFT matrix are taken as beamforming vectors.

wl =1√M

1

ej2π(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 theB-

dimensional subspaces in the same fashion as Grassmannian subspace packings. A rearrangement

of the DFT vectors can create the necessary unitary matricesfor subspace packings of higher

dimension than one.

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C. Non-orthogonal Grassmannian Subspace Packings

The trade-off between channel vector quantization distortion and line orthogonality was dis-

cussed in the beginning of this section. On one side, we have the Grassmannian subspace pack-

ings, with orthogonal vectors spanning the subspaces, but which lack in quantization performance.

On the other side, we have the Grassmannian line packings with optimal quantization perfor-

mance. A natural alternative to the Grassmannian subspace packings is to construct packings

with optimal quantization performance, but with vectors spanning the subspaces which are non-

orthogonal. We have constructed such packings, based on Grassmannianline packings, where

the vectors have been greedily grouped to yield near-orthogonality within the subspaces. Such

a packing will on average outperform the Grassmannian subspace packing in terms of channel

vector quantization, but will not enable SDMA without inter-beam interference. The greedy non-

orthogonal packing construction is briefly described in Table I. From a set ofNB vectors, it

createsN matrices withB columns each.

In [7], a multiuser transmission scheme based on a non-orthogonal Grassmannian line packing

was proposed. It differs in several aspects. Firstly, it is applied in the framework of opportunistic

SDMA, as in [6], i.e. only one beamforming matrix is considered. Here, we generate a packing

with N matrices. Secondly, no construction algorithm is given.

V. BEAM-GRAPHS FORREDUCED FEEDBACK

When the channel changes moderately from block to block, the feedback can be reduced

further by exploiting structure in the subspace packing. Below, a method to arrange the beams

in W in a graph is proposed. The purpose is to reduce the overhead caused by the feedback of the

beam-index. Instead the branch index in the graph is fed back. This method is in fact even more

suitable for systems using quantized beamforming (B = 1), since they enjoy a monotonously

increasing performance with increasing number of beams in the codebook.

According to the scheme described in Section III, each user needs to feed back one beam

index and one AMC level each block. This is already a significant reduction from a full channel

state feedback. However, in systems with many users, high mobility and many beams inW,

a further reduction may be necessary. In mobile wireless systems, it is reasonable to assume

a significant temporal channel correlation between consecutive blocks. The channel should not

change too much during one block to enable accurate rate adaptation and beamforming.

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For feedback in such temporally correlated systems, it is natural to apply delta modulation [18].

The AMC level is straightforward 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 the unit-norm beams are vectors that lie on an M -dimensional complex sphere.

As a simple example, consider a set of unit-norm vectors inR2, which all lie on the unit circle.

For this set, the neighbor relation is easily defined in the angle of the vectors; the two vectors

with closest higher and lower angle, respectively, are considered the neighbors of a vector.

A unit-norm vectorw ∈ CM×1 can be represented in spherical coordinates by2M − 1

angles. First, the complex-valuedw is transformed into a real vector,v = [v1 · · · v2M ]T =[

Re{wT} Im{wT}]T ∈ R

2M×1. Since unit-norm vectors are considered here, only the angular

coordinates are of interest. They can be computed as [19]

θ2M−1 = arctanv2M

v2M−1

...

θ1 = arctan

√

∑2Mi=2 v2

i

v1

.

Having found the spherical coordinates of all vectors in a packing, the nearest neighbors to

each vector has to be found. To illustrate the proposed method, consider unit-norm vectors inR3.

The vectors can be represented by two angles,θ1 andθ2. The nearest neighbors infour directions

are considered. One neighbor is found in the segment of the sphere with angles greater than

θ1 and θ2. One neighbor is found in the segment with angles greater than θ1 and less thanθ2

and one is found in the segment with angles smaller thanθ1 and greater thanθ2. The final

nearest neighbor is found in the segment where the angles areless than bothθ1 and θ2. The

four neighbor segments ofv = [−1 0 0] on theR3-sphere is illustrated in Fig. 2. Note that this

is not the only way to find neighbors on a unit sphere.

For vectors on a sphere inR2M , nearest neighbors in22M−1 directions can be found, using

the same approach. Clearly, even for vectors with relativelylow dimensions, one can find an

overwhelming number of nearest neighbors. For instance, for four transmit antennas (M = 4),

128 nearest neighbors can be found. For packings with less than128 beams, this means that

no feedback reduction is achieved, compared with beam indexfeedback. However, it is not

necessary to use all neighbor directions. A method to removeneighbor directions is proposed

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in Table II.

Each neighbor direction signifies a particular change pattern in the angles of a vector. These

directions can be enumerated and used as feedback entities,instead of the beam index. Now,

a beam-graph can be constructed, connecting nearest neighbor vectors in all chosen directions.

The graph can be constructed offline, and has to be known at both the basestation and at the

users. Note that the proposed graph construction algorithmis heuristic. It can not be guaranteed

that it will find a connected graph for a given number of neighbor directions, even if it exists.

As the channels of the users change, the preferred beams change. The users notify the

basestation by feeding back the index of the neighbor direction instead of the beam index, thereby

traversing through the graph. The beam-graph restricts theusers to change their preferred beams

to neighbors in the beam-graph. At the setup of a link or on a regular basis, the users have to

feedback the whole beam index.

Although this is an ad hoc strategy, significant feedback reduction can be obtained, in particular

for packings with many beams. In Section VI-B, a numerical performance comparison between

full beam index feedback and feedback based on a beam-graph is presented.

VI. N UMERICAL RESULTS

A. Packing Study

The construction of optimal Grassmannian packings is stillan open problem. Several numerical

methods to construct packings have been proposed, e.g. [13], [14]. In this paper, we have used the

alternating projections technique for complex-valued subspace packings in [14]. For the cases

tabulated in [14], we obtained very similar minimum distances. In the following results, the

number of transmit antennas is4 (M = 4).

A measure for the vector quantization performance of a packing W is the average distor-

tion [12],

d(W) = Eh

[

‖h‖22 − max

1≤i≤N1≤j≤B

|hTwji |2]

, (6)

where h = g

‖g‖is an isotropically distributed unit vector andg ∈ C

M×1 is i.i.d. zero-mean

complex Gaussian.

The average distortion for a number of packings has been numerically evaluated. The results

are displayed in Fig. 3. It is intuitive that the distortion decreases with the number of beams in

March 27, 2007 DRAFT

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the packing. For Grassmannian packings with 2-dimensionalsubspaces (B = 2), the distortion

is slightly higher than for line packings (B = 1). This is due to the fact that forB = 2, the

subspaces (planes) are isotropically spread out, but not the ensemble of lines (i.e. the vectors) that

span the subspaces. For the DFT-based packings for a givenNB, the quantization performance

is identical forB = 1 and B = 2. This is due to the fact that the packings forB = 1 and

B = 2 contain the same vectors, but forB = 2 arranged in orthogonal pairs (c.f. Section IV-

B). To conclude, the Grassmannian packings have a superior quantization performance for i.i.d.

Rayleigh fading channel vectors, especially for packings with many beams.

B. System Study

A system level simulation study has been conducted in order to evaluate the performance of

packings previously discussed. One cell with one basestation and several single-antenna users

has been simulated according to the system model in Section II. The basestation is equipped

with 4 transmit antennas (M = 4). The maximum throughput scheduler is used. Each block, it

selects the beam/user combination that yields the highest sum-rate (c.f. (2)). The average SNR

is 5 dB. The simulated channels are generated using Jake’s model with block fading.

The Grassmannian, DFT and non-orthogonal packings, as described in Section IV, are used

in reduced feedback SDMA, as described in Section III. In this study, the fed back supportable

rate is not quantized.

As an additional comparison, opportunistic SDMA as in [6] isused. This technique randomly

generates one set of orthogonal beams in each block. It relies on the multi-user diversity to

provide the scheduler with users that are compatible with the generated beams. This technique

can be seen as a packing with only one subspace, that is randomly regenerated each block. The

Grassmannian, DFT and non-orthogonal packings with multiple subspaces exploit both the multi-

user diversity and the diversity from having many subspaces. Another advantage with having

packings with non-random predefined beamforming matrices is that the users may track only

the underlying channel, whereas with random beams they needto estimate the discontinuous

aggregate channel (i.e. random beam times channel).

Finally, the optimal linear downlink multi-user beamformer with uniform power allocation is

used as a comparison. The transmit beamformers are given by the dual uplink MMSE receiver

March 27, 2007 DRAFT

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beamformers [1],

wi = (σ2zI +

1√B

B∑

j=1

j 6=i

hjhHj )−1hi.

In each block, the same users are scheduled for the LMMSE scheme as for the Grassmannian

subspace packing based scheme. This guarantees that they are spatially well separable. Hence,

the gap between the LMMSE and the Grassmannian performance is mainly due to the feedback

quantization distortion. Additionally, the allocation outage events (less thanB users scheduled)

do not penalize the LMMSE scheme.

In Fig. 4, the average sum rate as a function of the number of active users is displayed. The

SDMA factor, B, is 2. The packing-based methods use packings with 32 subspaces (N = 32).

The packing-based methods outperform opportunistic SDMA at the cost of 5 extra feedback

bits per user and block. The DFT-based packing results in slightly worse performance than

the Grassmannian. The non-orthogonal SDMA performs as wellas orthogonal Grassmannian

SDMA, since the loss of orthogonality from grouping 64 the vectors 2 and 2 is low.

In Fig. 5, the SDMA factor,B, is 3, but the number of subspaces,N , is only 4. These

parameters are less beneficial for the non-orthogonal packing, since it is not possible to find 4

groups of 3 nearly orthogonal beams from a Grassmannian linepacking of these dimensions.

Even the opportunistic SDMA outperforms the non-orthogonal scheme. Again, the Grassmannian

subspace packing outperforms the other packings. However,the performance gap to the LMMSE

scheme based on perfect CSI is significant.

Fig. 6 illustrates the relation between rate, SDMA factor and number of users. The figure

shows the average sum rate for Grassmannian subspace packings with subspace dimensions 1,

2 and 3. The total number of beams,NB, is 12. For SDMA factors greater than 1 (B = 2 and

B = 3) to perform well, there needs to be many users in the system. Otherwise, allocation outages

will occur frequently. Even though an allocation outage does not occur, the SINR penalty from

scheduling a user with a channel vector poorly aligned to thebeam 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 switching points for the SDMA factor depends on the SNR.

For higher SNR, the switching points are shifted to higher number of users (to the right in Fig. 6).

A framework for finding the optimalB, given an SNR and the number of users, for opportunistic

TDMA and SDMA is given in [21].

March 27, 2007 DRAFT

15

In the results above, the whole beam index is fed back each block. In Fig. 7, the impact of

the reduced feedback scheme based on beam-graphs, as described in Section V, is evaluated. A

system with one user has been simulated using a Grassmannianpacking of 128 lines, i.e.B = 1.

The beamforming vectors have been arranged in different graphs depending on the number of

feedback bits per block. For instance, for 3 feedback bits per block, each beam (vector) in the

graph is allowed to have23 = 8 neighbors. The average sum rate for the full feedback scheme, i.e.

the whole beam index is fed back each block, serves as an upperbound. The performance using

beam-graph feedback is shown for three values of relative Doppler, rd. The relative Doppler is

defined asrd = block duration∗ fd, wherefd is the maximum Doppler frequency. For example,

rd = 20% could correspond to a system with 2 GHz carrier frequency, a0.15 ms block duration

and user speed of 20 m/s. From Fig. 7 it is clear that a significant feedback reduction is possible

without sacrificing performance to any great extent. Since the packing contains 128 beams, the

full beam index feedback also means 7 feedback bits per block. The relative performance loss

from reducing the feedback from 7 to 3 bits per user and block for rd = 20% is less than8%.

There is a gap between the full feedback performance and the beam-graph performance with 7

feedback bits. This is due to the fact that, using the beam construction method in Section V,

some beams will have no neighbors in some directions. Finally, note that the performance gap

between full feedback and beam-graph feedback will be reduced when there are more users in

the system. This is due the fact that the rate loss due to beam quantization is accounted for

in the supportable rate computation by the users. Therefore, a user with temporarily bad beam

quantization will less likely be scheduled than a user with perfect beam quantization.

VII. C ONCLUSIONS ANDDISCUSSION

We have proposed a scheme for reduced feedback SDMA based on subspace packings. The

scheme is based on a codebook of beams (a subspace packing), common to the basestation and

the users. The scheme allows all users to compute their post-scheduling SINR already before

the CSI feedback. Only a few bits feedback per user and coherence time is necessary. The

feedback also provides information about the spatial compatibility of the users. We highlighted

a fundamental property of the scheme through the allocationoutage probability. Furthermore, a

beam-graph method to reduce the feedback rate even further for large packings was proposed.

The reduced feedback SDMA scheme based on subspace packingsshows potential to cover some

March 27, 2007 DRAFT

16

of the middle ground between opportunistic SDMA and SDMA based on perfect CSI. Several

parameters of the method should be tuned according to the specific scenario. For instance, a

scenario with many users would benefit from a packing with many subspaces and with a high

SDMA factor. A scenario with few user on the other hand would benefit from a packing with

fewer subspaces of lower dimension, i.e. fewer simultaneous spatial streams. The Grassmannian

subspace packings showed best performance of the low-rate feedback schemes in the numerical

comparison for the studied i.i.d. Rayleigh fading scenario.

Grassmannian subspace packings are well suited for the quantization of i.i.d. Rayleigh fading

channels. However, in many practical systems, there is correlation between the transmitting anten-

nas. For Grassmannian beamforming (using line packings), this can be handled by prewhitening

the packing, as in [12]. Grassmannian subspace packings cannot directly be prewhitened in the

same manner, since the prewhitening destroys the orthogonality between the vectors spanning

each subspace. A possible approach left for future researchis to prewhiten a Grassmannian line

packing, and based on that, construct a non-orthogonal subspace packing, as in Section IV-C.

ACKNOWLEDGMENT

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. The authorswould like to acknowledge the

contributions of their colleagues. The ideas presented in this paper were conceived during work

with the spatial-temporal processing solution of the WINNERII air interface [22].

APPENDIX I

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

ji |2 instead

of the (i, j)k that maximizesγk ((i, j)), as in (1). Unfortunately, this may give a sub-optimal

choice. The processing delay induced by the computation of (1) adds to the delay between

channel estimation and data transmission. This delay should be kept to a minimum, otherwise

the feedback information will be outdated. For large packings, a large number of division can be

saved by considering the inner product instead of the SINR when selecting a beam index. The

maximum loss in SINR by using the maximum inner product beam selection criterion instead

of the maximum SINR criterion is quantified in the following result.

March 27, 2007 DRAFT

17

Theorem 2:Let

(a, b)k = arg max(i,j)

γk ((i, j)) (7)

(c, d)k = arg max(i,j)

|hTk w

ji |2. (8)

denote the optimal and sub-optimal beam choice for thekth user, respectively. Then, for1 <

B < M , the relative loss in SINR is bounded by

γk ((a, b)k)

γk ((c, d)k)≤ 1 +

‖hk‖22

σ2z

(1 − α)

B, (9)

where ‖hk‖22

σ2z

is the instantaneous SNR of the user andα represents the worst-case inner product

of the packingW,

α = minu

T ∈C1×M

‖u‖2=1

max(i,j)

|uwji |2. (10)

Equality in (9) is obtained whenhk

‖hk‖2equals the minimizingu in (10). ForB = 1 andB = M ,

γk ((a, b)k) = γk ((c, d)k).

Proof: Consider userk with channelhk. The optimal (maximum SINR from (7)) beam

from the setW is denotedwba. The beam with maximum inner product (as in (8)) is denoted

wdc . The caseB = 1 is trivial, since there is no inter-beam interference. Hence, wb

a = wdc .

Now, consider the caseB > 1. If B = M , the channel vector can always be expressed in the

coordinate system spanned by the orthonormal columns ofWi for all i, i.e. hk =∑B

j=1 βji w

ji ,

whereβji = hT

k wji . Thus,

γk ((i, j)k) =|βj

i |2Bσ2

z +∑

n6=j |βni |2

=|βj

i |2Bσ2

z + ‖hk‖22 − |βj

i |2,

since‖h‖22 =

∑Bj=1 |β

ji |2 due to the orthonormality of the basis vectors. It is clear that (c, d)k =

arg max(i,j) |βji |2 = arg max(i,j) γk ((i, j)) = (a, b)k whenB = M .

If B < M , one additional unit norm vectorvi, orthogonal toWi, is in general needed to

expresshk, with

vi =hk −

∑Bj=1 β

ji w

ji

‖hk −∑B

j=1 βji w

ji‖2

.

The channel vector can then be written as a linear combination of the basis vectors ashk =∑B

j=1 βji w

ji + βB+1

i vi, whereβB+1i = hT

k vi. Due to the orthonormality of the basis vectors for

March 27, 2007 DRAFT

18

all i,

‖hk‖22 =

B∑

j=1

|βji |2 + |βB+1

i |2 = |βji |2 +

∑

n6=j

|βni |2 + |βB+1

i |2.

Note that

0 ≤∑

n6=j

|βni |2 = ‖hk‖2

2 − |βji |2 − |βB+1

i |2 ≤ ‖hk‖22(1 − α), (11)

where α is defined in (10). The minimizingu in (10) is the worst-case unit vector, i.e. the

inner product with the closest vector in the packing is minimal. Sinceα only depends onW,

it can be numerically found offline. Equality in the upper bound in (11) is obtained whenhk is

co-linear with the minimizingu in (10) and also in the subspace spanned byWi, i.e. βB+1i = 0.

Equality in the lower bound in (11) is obtained whenhk is orthogonal to the interference vectors

{wni }n6=j, i.e. hk = β

ji w

ji + βB+1

i vi. The relative SINR loss is

γk ((a, b)k)

γk ((c, d)k)=

|βba|2

Bσ2z +

∑

n6=b |βna |2

Bσ2z +

∑

n6=d |βnc |2

|βdc |2

. (12)

The worst case is characterized by

• |βba|2 = |βd

c |2, since |βdc |2 = max(i,j) |βj

i |2 ≥ |βba|2, which means that two vectors in the

packing had the same inner product and the wrong one was chosen,

•∑

n6=b |βna |2 = 0, i.e. hk is orthogonal to all interference vectors{wn

a}n6=b and

•∑

n6=d |βnc |2 = ‖hk‖2

2(1 − α), i.e. hk is spanned byWc and |hTk wd

c |2 is minimal.

Hence,

γk ((a, b)k)

γk ((c, d)k)≤ Bσ2

z + ‖hk‖22(1 − α)

Bσ2z

= 1 +‖hk‖2

2

σ2z

(1 − α)

B.

APPENDIX II

PROOF OFTHEOREM 1

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

Lemma 1:Consider a setA with a elements and a subsetB ⊆ A with b ≤ a elements.

Independently, with uniform probability and with replacement, k elements are taken fromA.

The taken elements, with duplicates removed, constitute the setK ⊆ A, with cardinality |K| ≤min(k, a). Then,

Pk(a, b) , Pr (B ⊆ K) = 1 +1

ak

b∑

m=1

(−1)m

(

b

m

)

(a − m)k. (13)

March 27, 2007 DRAFT

19

Proof: First, define theb dependentevents{Em}m=1,...,b, whereEm is the event that element

m in B is not in K. Then,

Pk(a, b) = Pr(B ⊆ K) = 1 − Pr

(

b⋃

m=1

Em

)

,

where⋃

denotes the union of events. Poincaré’s formula states that[23, page 89]

Pr

(

b⋃

m=1

Em

)

=b∑

m=1

(−1)m−1Sm, (14)

where

Sm =∑

· · ·∑

1≤j1<···<jm≤b

Pr (Ej1 ∩ · · · ∩ Ejm) , (15)

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 them integer indices can be chosen so that1 ≤ j1 < · · · < jm ≤ b

equals(

bm

)

. Finally,

Pk(a, b) = 1 +1

ak

b∑

m=1

(−1)m

(

b

m

)

(a − m)k.

Equipped with Lemma 1, the allocation outage probability can be determined. First, we define

theN dependentevents{Dn}n=1,...,N , whereDn is the event that all beam indices in thenth 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’s formula (14)-(15) and the fact

that

Pr (Dj1 ∩ · · · ∩ Djn) = PK(NB, nB)

for all distinct j1, . . . , jn and wherePk(a, b) is defined in Lemma 1 gives

Pao = 1 +N∑

n=1

(−1)n

(

N

n

)

PK(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 .

March 27, 2007 DRAFT

20

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Theory, vol. 48, pp. 1277–1294, June 2002.

[5] P. Svedman, S. K. Wilson, L. J. Cimini, and B. Ottersten, “Opportunistic beamforming and scheduling for OFDMA

systems,”IEEE Trans. on Commun., 2006, to appear.

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Inform. Theory, vol. 51, pp. 506–522, Feb. 2005.

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on Commun., pp. 134–137, Feb. 2006.

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on Signal Processing, vol. 54, pp. 1853–1863, 2006.

[9] C. K. Au-Yeung and D. J. Love, “On the performance of randomvector quantization limited feedback beamforming in a

MISO system,”IEEE Trans. on Wireless Commun., 2006, to appear.

[10] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple-antenna

systems,”IEEE Trans. on Inform. Theory, vol. 49, pp. 2562–2579, 2003.

[11] D. J. Love, R. W. Heath, and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless

systems,”IEEE Trans. on Inform. Theory, vol. 49, pp. 2735–2747, 2003.

[12] D. J. Love and R. W. Heath, “Limited feedback diversity techniques for correlated channels,”IEEE Trans. on Veh. Techn.,

vol. 55, pp. 718–722, 2006.

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Experimental Math., vol. 5, pp. 139–159, 1996.

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

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Inform. Theory, vol. 51, pp. 2967–2976, 2005.

[16] M. Schubert and H. Boche, “Iterative multiuser uplink and downlink beamforming under SINR constraints,”IEEE Trans.

on Signal Processing, vol. 53, pp. 2324–2334, 2005.

[17] B. M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens,and R. Urbanke, “Systematic design of unitary space-time

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[18] J. G. Proakis,Digital Communications. McGraw-Hill Book Co., 2001.

[19] S. Hassani,Mathematical Physics. Springer, 1999.

[20] W. Mayeda,Graph Theory. John Wiley and Sons, 1992.

[21] E. Jorswieck, P. Svedman, and B. Ottersten, “On the performance of TDMA and SDMA based opportunistic beamforming,”

IEEE Trans. on Wireless Commun., 2007, submitted.

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[Online]. Available: https://www.ist-winner.org

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

March 27, 2007 DRAFT

21

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 usersK

Pao

Fig. 1. Allocation outage probabilityPao as a function of the number of usersK. The curves represent packings with in total

12 and 24 beams (NB = 12 and NB = 24). The beams are grouped in subspaces (beamforming matrices) of dimensions

B = 2, B = 3 andB = 4.

1) Put all vectors from a Grassmannian line packing in the setWvec.

2) Find the two vectors inWvec with lowest inner product. Remove them fromWvec and let them be the first columns in

a new matrix.

3) If the number of columns in the new matrix equalsB, save the matrix and go to step 5). Otherwise go to step 4)

4) Find the vector inWvec with the lowest average inner product with the columns in the new matrix. Remove it from

Wvec and append it as a column in the new matrix. Go to step 3)

5) If Wvec is not empty, go to step 2).

6) Now, N matrices withB columns each have been created.

TABLE I

THE GREEDY ALGORITHM TO CONSTRUCT NON-ORTHOGONAL PACKINGS.

March 27, 2007 DRAFT

22

−1

0

1

−1

0

1−1

0

1

v1v2

v3

Fig. 2. The neighbor directions ofv = [−1 0 0] on theR3-sphere are given by the four segments.

1) Choose a chordal distancedn. Vectors within this distance are considered neighbors here.

2) For each vector in the packing, find the number of neighbors in each direction, within a distancedn.

3) Compute the average number of neighbors for each direction.

4) Successively remove directions, starting from the direction with lowestaverage number of neighbors. If too many

directions have 0 neighbors, increasedn and go to 2).

5) After each removal, test the graph for connectedness. A graph is connected if all nodes can be reached from any node [20].

If the graph is disconnected undo the removal and move on to the next direction.

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

TABLE II

THE ALGORITHM FOR NEIGHBOR DIRECTION REMOVAL.

March 27, 2007 DRAFT

23

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

ortio

nd(W

)

Fig. 3. The average distortion for a number of Grassmannian and DFT-based packings is displayed.

0 50 100 150 200 2503

4

5

6

7

8

LMMSEGrass.Non. orth.DFTOpp. SDMA

Number of usersK

Ave

rage

sum

rate

[bps

/Hz]

Fig. 4. Average sum rate as a function of the number of active users. The SDMA-factorB = 2 and the number of subspaces

N = 32.

March 27, 2007 DRAFT

24

0 50 100 150 200 2503

4

5

6

7

8

9

LMMSEGrass.Non. orth.DFTOpp. SDMA

Number of usersK

Ave

rage

sum

rate

[bps

/Hz]

Fig. 5. Average sum rate as a function of the number of active users. The SDMA-factorB = 3 and the number of subspaces

N = 4.

0 10 20 30 40 502.5

3

3.5

4

4.5

5

5.5

6

B=1B=2B=3

Number of usersK

Ave

rage

sum

rate

[bps

/Hz]

Fig. 6. Average sum rate for Grassmannian subspace packings with subspace dimensions 1, 2 and 3 as a function of the

number of users. For all three curves,NB = 12.

March 27, 2007 DRAFT

25

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. 7. Average sum rate for a system with 1 user, using a Grassmannianline packing with 128 beams. For the Full curve,

the whole beam index is fed back each block (7 bits per block). For the curves with relative Doppler (rd) 5-20%, beam-graph

feedback with different rates is used.

March 27, 2007 DRAFT