Adaptive Downlink Multi-User MIMO Wireless Syfstems for Correlated Channels With Imperfect CSI

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 9, SEPTEMBER 2006 2435

Adaptive Downlink Multi-UserMIMO Wireless Systems for

Correlated Channels with Imperfect CSICheng Wang, Student Member, IEEE, and Ross D. Murch, Senior Member, IEEE

Abstract Multiple-Input Multiple-Output (MIMO) wirelessantenna systems provide increases in capacity without the needfor additional spectrum or power. However the capacity increaseis limited when the number of antennas at the receiver is fixedor restricted (due to mobile size constraints for example). Toovercome this limitation multi-user MIMO can be used, whichallows several users to be served simultaneously in frequency andtime. A disadvantage of these multi-user MIMO systems, when

used in the downlink however, is that they need accurate channelstate information (CSI) at the transmitter and also uncorrelatedchannels among users. In this paper we investigate methods toaddress the problems of multi-user MIMO systems in spatiallycorrelated channels. We adopt the concept of angle betweensubspaces to characterize the inter-user spatial correlation andadapt the algorithm to those conditions. We also investigatethe impact of the accuracy of CSI at the transmitter (CSIT)and whether more limited CSI such as channel correlationinformation alone can be used to provide good multi-user MIMOperformance. Results are presented as various simulated capacitymeasures and we use them to make comparisons between thevarious multi-user MIMO configurations.

Index Terms Multi user, MIMO, multi-user MIMO.

I. INTRODUCTION

MULTIPLE-INPUT Multiple-Output (MIMO) wirelesssystems provide increases in capacity without the needfor additional spectrum or power. It has been shown [1],

[2] that capacity grows linearly with the number of transmit

antennas provided that the number of receive antennas equals

or exceeds the number of transmit antennas in uncorrelated

Rayleigh fading channels. However, capacity grows much

slower as the number of transmit antennas increases when

the number of receive antennas is fixed or limited (due to

mobile size constraints for example), and in fact is bounded

[2]. Motivated by this limitation researchers have investigated

the possibility of multi-user MIMO (MU-MIMO) which can

serve several users simultaneously in frequency and time with

a particular focus on the downlink. It has been shown [9] that

by employing MU-MIMO techniques overall system capacity

can be increased significantly even when the number of receive

Manuscript received March 30, 2004; revised November 30, 2004 andAugust 4, 2005; accepted August 22, 2005. The editor coordinating the reviewof this paper and approving it for publication is M. Uysal. This work was

supported by the Hong Kong RGC grant HKUST 6149/03E.The authors are with the Department of Electrical and Electronic Engineer-ing, The Hong Kong University of Science and Technology, Clear Water Bay,Hong Kong (email: [email protected], [email protected]).

Digital Object Identifier 10.1109/TWC.2006.04202.

antennas at the individual mobile is limited to 1 or 2 in uncor-

related Rayleigh fading channels. However it is unclear what

effect fading correlation has on the spatial separability among

users [13] and consequently its effect on the performance of

MU-MIMO schemes. In addition MU-MIMO schemes need

channel state information (CSI) at the transmitter so that

appropriate signal processing can be performed to separate

multiple users in space. It is therefore important to consider

what types of CSI need to be available at the transmitter and

how accurate it needs to be.

In this paper we investigate methods to address the problems

of MU-MIMO systems in spatially correlated channels. In

particular, we adopt the concept of angle between subspaces

[3] to characterize the inter-user spatial correlation or users

spatial separability and use it to adapt the MU-MIMO schemes

to improve performance. We also investigate the impact of the

accuracy of CSI at the transmitter (CSIT). Two types of CSIT

are considered: instantaneous MIMO channel gain matrix and

also that consisting of the transmit correlation matrix only. Wefind that MU-MIMO algorithms exhibit different sensitivity to

CSIT accuracy in channels with different degrees of fading

correlation. For fairness consideration we use the minimum

capacity among simultaneous users as our performance mea-

sure. Simulation results demonstrate the gain achieved by our

adaptive algorithms.

Previous work on downlink multi-user MIMO systems has

suggested dirty paper coding (DPC) solutions [4], [5], [6], [7]

and it was recently shown in [8] that DPC in fact achieves

the full capacity region of the downlink multi-user MIMO

channel (MIMO BC). However due to the high complexity

of DPC implementation, low complexity sub-optimal schemes

with moderate performance degradation are desirable. Several

sub-optimal schemes have been formulated in [10], [11],

[12], [27], all take the similar approach of forcing the inter-

user interference to zero. Much of the emphasis in these

formulations has been on determining the potential advantages

of these approaches and they have therefore concentrated on

systems that assume perfect CSI is available at the transmitter,

and the channels are uncorrelated. In practice however strong

fading correlation between pairs of transmit antennas often

exists [14], [15], [16], and as we will see later that it will cause

the above formulations to fail to guarantee the instantaneousQoS provisioning, since in certain time interval some users

data rate might be unacceptably low. Also it is unclear what

CSIT and accuracy will be available at the transmitter. Our

1536-1276/06$20.00 c 2006 IEEE


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2436 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 9, SEPTEMBER 2006

work is different in that we focus on the impact of the channel

characteristics on the performance of MU-MIMO formulations

and suggest adaptive approaches to improve performance.

The remainder of the paper is organized as follows. In

section II, preliminaries are introduced, while in section III

a brief summary of several downlink MU-MIMO schemes

is provided. In section IV, inter-user spatial correlation is

analyzed and two adaptive algorithms are proposed. Section V

provides the numerical results. Finally, conclusions are drawn

in Section VI.

I I . PRELIMINARIES

A. Downlink Multi-user MIMO Transmitter Structure

We consider the downlink of a narrowband multi-user

MIMO system with M transmit antennas at the base station(BS) and Nk receive antennas at the k

th user. We denote

such a system as a M (N1, N2, . . . , N K) system, whereK is the number of users that the BS is communicating

with simultaneously in frequency and time. Let the Lk 1vector bk be the transmit data symbol vector for user k,where Lk is the number of parallel data streams transmittedsimultaneously to user k, k = 1, . . . , K . All these data symbolvectors are passed through certain transmit pre-processing

matrix Tk, k = 1, . . . , K , before they are launched into thedownlink channel. The received signal of user k can be writtenas

yk = Hk

Ki=1

Tibi + nk, (1)

where nk

N(0, I) is the additive white Gaussian noise

vector and I is the identity matrix.

B. Channel Model

The downlink channel for each user is modeled as a semi-

correlated NLOS Rayleigh flat fading channel, where it is

assumed that fading is correlated at the transmitter side but

uncorrelated at the receiver side [16], [17]. In this case, the

channel could be modeled as

Hk = GkAk, (2)

where Gk is a Nk Dk matrix with zero-mean unit-variancei.i.d complex Gaussain entries. Ak is the steering matrix ofsize Dk M containing Dk steering vectors of the transmitantenna array corresponding to Dk directions of departure(DOD). Consider a uniform linear array at the BS, the steering

matrix Ak is given by

Ak =1Dk

aT(k1), . . . , aT(kDk)T, (3)with

a() =

1, ej2d sin

, . . . , ej2(M1)d sin

,

where T represents the transpose operation, d is the equidistant

antenna spacing and is the carrier wavelength. With thismodel, angle spread can be modeled by a large number of

discrete DODs, and different degrees of transmit correlation

are adjusted by varying the angle spread. Throughout the paper

we denote a semi-correlated channel with D DODs randomly

and independently distributed in an angle spread as a (D, )channel.

The transmit correlation matrix is then given by

RT xk = E

HHk Hk

= NkAHk Ak, (4)

where H represents the complex conjugate transpose and E[

]

denotes expectation operation.

C. Channel State Information Accuracy

A time division duplex (TDD) system is assumed in this

paper. Two kinds of CSIT will be used, namely H-CSIT and

R-CSIT. By H-CSIT we refer to a transmitter that knows

the instantaneous channel matrix H. While by R-CSIT we

refer to a transmitter that only has the correlation matrix RT x

available.

The transmit correlation matrix is determined by the DODs,

which correspond to the scatterers angular position withrespect to the transmitter. The DODs do not alter rapidly for

small movements of the user [18] and as a consequence so

does the transmit correlation matrix. In most cases the BS

should be able to acquire the transmit correlation matrix [16],

[17], and throughout this paper we will assume if R-CSIT is

known to the BS, then it is accurate.

In contrast to RT x, the components of G describe the

complex path gains, whose phases and amplitudes change

much more rapidly even when the user moves slightly [18].

Since we assume a TDD system, the H-CSIT of the downlink

is estimated from the uplink. Therefore we assume the effect

of Doppler spread, which causes the uplink to be an inaccuratedownlink channel estimate due to the time separation between

them, dominates the error. Taking into account the fact that

DODs change much slower than the complex path gains, we

modify the model in [19] and quantify the problem as

H =

t G +1 2t GwA = t GA +1 2t GwA,(5)

where A is the steering matrix, H = GA is the estimatedchannel matrix,

1 2t GwA is the estimation error that

is uncorrelated with

H, t is the correlation coefficient

between the actual channel gain and its estimate, whichis assumed to be the same for all gains and is given by

t = E[hijhij ]E[|hij|2] E[|hij|2], where hij ,hij re-present the (i, j)th element of H and H respectively and denotes the complex conjugate operation. The entries of Gand Gw are all i.i.d zero-mean complex Gaussian with unity

variance. Thus 2h = 2h

= 1, and the variance of estimation

error is 1 2t . Users are assumed to be in a rich scatteringenvironment, which leads to t = J0(2fd), where J0() isthe zeroth-order Bessel function of the first kind.

III. MULTI-USER MIMO SYSTEMS

Various methods have been proposed for performing multi-

user MIMO communications. A summary of four multi-user

MIMO algorithms is provided here.


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WANG and MURCH: ADAPTIVE DOWNLINK MULTI-USER MIMO WIRELESS SYSTEMS FOR CORRELATED CHANNELS WITH IMPERFECT CSI 2437

A. MU-MIMO ZF Decomposition with Equal Power Alloca-

tion

In this scheme [10], Tk, k = 1, . . . , K is chosen such thateach user receives no interference from the other users, which

can be written in the form

Tk = VkWk, (6)

where Vk is an orthonormal basis of the joint null-spaceKi=1,i=kN(Hi) and can be computed by singular value

decomposition as

Hk =

HT1 , . . . , HTk1, H

Tk+1, . . . , H

TK

T=Uk Uk 00 0

VHkV

H

k

. (7)

Substitute (6) into (1), we get

yk = HkVkWkbk + nk. (8)We can see the multi-user MIMO system denoted by (1)

has been decomposed into K parallel single-user MIMOsystems. By thinking of the equivalent single-user MIMO

channel of user k as Hk = HkVk, the equivalent transmitpre-processing matrix should be chosen according to spatial

waterfilling [2] to maximize the mutual information subject

to the power constraint trace

WHk Wk

= PT/K, wherePT is the total transmit power. In this scheme the constraintM > max

krank

Hk

should be satisfied in order to ensure

the existence ofVk. A detailed description of this scheme can

be found in [10].

B. Max Average Transmit SINR Beamforming

In this scheme [18], transmit beamforming vector tk, k =1, . . . , K is computed by maximizing the average transmitSINR for each user, which is defined as the ratio of the average

desired signal power received at user k and the sum of theaverage interference and noise power introduced to the others

as follows

k =E

Hktkbk 2

Ki=1,i=k E Hitkbk 2 +Ki=1,i=k Ni , (9)where denots the Euclidean vector norm. We can see thatin this scheme each user tries to suppress its interference to

the other users and at the same time tries to transmit efficiently

to itself. By denoting tk =

Pkuk, where uk 2= 1, Pk =PT/K, the average transmit SINR can be further expressedas

k =uHk R

T xk uk

uHk

Ki=1,i=k R

T xi +

1Pk

Ki=1,i=k NiI

uk

, (10)

where we can see that only R-CSIT, i.e. RT xi , i = 1, . . . , K is need of.

To maximize k [18], it can be shown that uk should bechosen as the dominant generalized eigenvector of RT xk andK

i=1,i=k RT xi +

1Pk

Ki=1,i=k NiI

.

C. Max-Min Mutual Information MU-MIMO Scheme with

DPC

In MIMO BC [4], the dirty paper result suggests that

the transmitter encode the users data sequentially such that

through appropriate coding each user sees no interference

from the previously encoded users with full knowledge of

the signals to be transmitted to all the previously encodedusers in such a way that transmit power is not increased. As

a consequence, the mutual information for the two-user case

is given by

I1 = log2 det

I + TH1 HH1

I + H1T2T

H2 H

H1

1H1T1

I2 = log2 det

I + TH2 H

H2 H2T2

(11)

Since the encoding order is arbitrary, one can also achieve

mutual information

I2 = log2 detI + TH2 H

H2 I + H2T1T

H1 H

H2

1

H2T2I1 = log2 detI + TH1 HH1 H1T1 (12)The objective of this scheme is to select the non-zero pre-

processing matrices

T1, T2

, such that the minimum mutual

information IDP between the 2 users is maximized,T1, T2

= arg

(T1,T2)max IDP s.t.

2k=1

trace

TkTHk

= PT,

(13)

where I = min(I1, I2),

I = min(

I1,

I2), IDP = max(I,

I).

We discuss the numerical solution for this nonlinear optimiza-

tion problem in section V.

D. TDMA-MIMO

In this scheme, different users are separated in time, so there

is no inter-user interference. The mutual information of user

k is then given by

Ik =1

Klog2 det

I + THk H

Hk HkTk

, (14)

where the factor 1/K occurs because each of the K users onlyhas 1/K of the time for transmission due to time division.Tk is then chosen as the spatial waterfilling solution with a

total transmit power constraint, i.e. traceTHk Tk = PT tomaximize the mutual information according to [2]. If only R-

CSIT is available [16], it is optimal to transmit along the eigen-

vectors of RT x and a complicated numerical optimization is

needed to find the optimal power allocation. Since the solution

resembles waterfilling in the sense that stronger channel modes

get allocated more power, we use waterfilling to find a sub-

optimal power allocation for simplicity.

IV. ADAPTIVE MULTI-U SE R MIMO SYSTEMS

There are two problems caused by the channel that can

prevent MU-MIMO algorithms from operating well: spatialcorrelation among users and CSIT inaccuracy. Spatial corre-

lation among users prevents the separation of two or more

users in space and causes MU-MIMO algorithms to perform

poorly [13]. Inaccurate CSIT will have much greater impact


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on multi-user MIMO than on single-user MIMO because of

the presence of inter-user interference.

A key to overcome these problems is to realize that the

various MU-MIMO algorithms behave differently under differ-

ent channel conditions. For example when the users are highly

spatially correlated, TDMA-MIMO will probably perform

better than sub-optimal MU-MIMO algorithms. In addition,

the same MU-MIMO scheme may exhibit different sensitivity

to H-CSIT accuracy in channels with different degrees of

fading correlation.

Our approach here is to exploit the subspace structure

of the channel. By exploiting the subspace structure of the

channels of different users, we can identify when the inter-user

correlation is likely to cause MU-MIMO schemes to provide

poor performance and adapt them appropriately. By exploiting

the subspace structure of the channels of the same user, we can

investigate the sensitivity of MU-MIMO schemes to inaccurate

H-CSIT.

A. Spatial Correlation between Users

One key measure of the spatial correlation between users

is the relative orientation of the row-spaces of their channel

matrices. This is an indication of how independently the BS

can transmit to each user without affecting the other users.

Here, we suggest the use of the angle between the row-

spaces of users channel matrices to quantify users spatial

correlation. Through the analysis of the next subsection (IV.B),

we can gain some insight into the impact of this angle on sub-

optimal MU-MIMO schemes.

The concept of angle between row-spaces, or more gener-ally subspaces, can be found in [3] and a brief description is

as follows. Suppose Pand Q are two subspaces ofCn, Pand Q are their orthogonal complements respectively. LetP1 = [p1, . . . , pp], Q1 = [q1, . . . , qq ], P2 = [pp+1, . . . , pn]and Q2 = [qq+1, . . . , qn] be the orthonormal basis of thesefour subspaces respectively. Assume p q and p + q nwithout loss of generality [3], angle between P and Q canthen be computed as follows,

Step1, Calculate the SVD of PH1 Q1 as PH1 Q1 = XY

H,

or alternatively calculate the SVD of PH1 Q2 as

PH

1

Q2 = XYH

, where = diag(1, . . . , p),1 . . . p, = diag(1, . . . , p), 1 . . . p.

Step2, Calculate the largest principal angle = arccos(p)or = arcsin(1) as the angle between the twosubspaces Pand Q.

A small angle indicates that the two subspaces are nearly

linearly dependent. We apply this idea to MU-MIMO when

we have H-CSIT available at the BS and the resulting angle

will be referred to as angle between users or ABU. Forexample, ABU of user k and users {1, . . . , k1, k+1, . . . , K }can be computed by substituting the orthonormal basis of the

row-spaces of Hk, H

k into P1 and Q1. Note that for theABU of user k and the remaining users not to be alwayszero, the same constraint as needed in the MU-MIMO ZF

decomposition scheme, i.e. M > maxk

rank

Hk

, should be

satisfied.

When only the transmit correlation matrix RT x is available

at the BS, it is not possible to find the exact row-space of

the channel matrix H, however we can find the row-space of

RT x instead. In fact, it can be easily seen than the row-space

of H is a subspace of the row-space of RT x. So, when only

R-CSIT is available at the BS, ABU of user k and user jwill be computed by substituting the orthonormal basis of the

row-spaces of RT xk , RT xj into P1 and Q1.

B. Impact of ABU on MU-MIMO Schemes

To gain insight into the relationship between ABU and the

direction of departure path, we study the ABU between twousers when each user has only 1 DOD in space for the sake of

clarity. Since Hk = gkak, k = 1, 2, they can be decomposedas

Hk = kukvHk , (15)

where uk = gk/

||gk

||, vk = a

Hk /

M and k =

M

||gk

||.

The row-space of Hk is a one dimensional subspace spannedby vk, which is solely determined by ak . It can be easily

shown that

sin2 = 1 1

M2||aH1 a2||

2

=

1 1M2

1 cos

2 dM(sin1 sin2)

1 cos

2 d

(sin 1 sin2) 1 = 2

0 1 = 2

(16)

where 1 and 2 is the DOD of user1 and user2 respectively.Assume 1 and 2 are uniformly distributed in [

, +],

let Zmn = 2|m n|d/, by using a well known seriesexpansion cos(Z sin ) = J0(Z) + 2

k=1 J2k(Z) cos(2k),

we obtain

E[sin2 ] = 1 1M2

Mm=1

Mn=1

J20 (Zmn). (17)

We next consider the capacity as a function of ABU for both the MU-MIMO ZF decomposition with equal power

allocation scheme and the max average transmit SINR scheme.

i. MU-MIMO ZF decomposition with equal power allocation

schemeWe first consider the special case where there is only 1 DOD

in space for user k, k = 1, . . . , K , K M. Later we gene-ralize to the case for arbitrary number of DODs. For 1 DOD

case, we know that the row-space of Hk is a one dimensional

subspace determined by ak and the row-space of Hk in (7) is

a at most K1 dimensional subspace spanned by Vk, whichis solely determined by a1, . . . , ak1, ak+1, . . . , aK. Then wecan get the following results

cos2 k = vHk

Vk

VHk vk = ||vHk

Vk||2,

sin2 k = vHk VkVH

k vk = ||vHk Vk||2.(18)

where k is the ABU of user k and the remaining users.The equivalent single-user MIMO channel of user k can be

expressed as

Hk = HkVk = k ukvHk Vk = ukk sin(k)v

Hk , (19)


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where vHk = vHk Vk/ sin k, k =

M||gk||. Then we get

Tk =

Pk/KVkvk and consequently

Ck = log2

1 + 2k sin2(k)PT/K

, k = 1, . . . , K . (20)

From the analysis above we can see that Ck is a monotoni-cally increasing function of k, k [0, /2]. When k = 0,the capacity decreases to 0.For a more general case, i.e. without the 1 DOD constraint,the capacity achieved by MU-MIMO ZF decomposition with

equal power allocation scheme for user k is bounded by

log2

1 +

PTKNk

2kmk sin2 k

Ck

Nk log2

1 +PT

KNk2k1 sin

2 k

, k = 1, . . . , K , (21)

when rank

Hk rank(Hk ) < M, where mk is the rank

of Hk, k1 k2 . . . kmk > 0 are the non-zerosingular values of Hk.

Proof: See Appendix I.It is obvious that both the upper and lower bound of Ck,

k = 1, . . . , K are monotonically increasing functions of k,k [0, /2].

ii. Max average transmit SINR beamforming scheme

We assume k = 2 and each user only has 1 DOD in spacein this case. The exact SIR seen at the receiver is given by

SI R1 = SI R2 =

1 + M Psin2 2

/ cos2 . (22)

where P = PT/2.

Proof: See Appendix II.Both SI R1 and SI R2 are monotonically increasing func-

tions of , [0, /2]. When = /2, both users dont seeany interference from the other user.

For the capacity achieved in this case, through similar

derivations, we obtain

Ck = log2

1 +

||gk||2(1 + M Psin2 )2||gk||2 cos2 + 1/P M + (MP + 2) sin2

,

k = 1, 2. (23)

Denote the second term in the log operation as X, if wetake the derivative of X with respect to , the numerator ofthe result is

2P Msin cos

2||gk||2 + P M||gk||2 sin2 (1 + cos2 )

+2sin2 + P2M2 sin4 + 2P Msin4 +||gk||2P M

1

.(24)

Its easy to see that X is a monotonically increasingfunction of when (/4, /2). When (0, /4), itdepends on P, M and ||gk||2. Nevertheless, for a large rangechoice ofM and P, X is a monotonically increasing functionof for a much larger range than (/4, /2).

From the analysis it follows that ABU is closely related

to the achievable capacity of these sub-optimal MU-MIMOschemes. Small ABU roughly indicates low capacity. As we

show in section V that the probability of ABU to be small,

say less than 30, is about 10% for both the 1 DOD channeland the (50, 20) channel, we can expect that the capacity

achieved by these two schemes at less than 10% outage willbe poor in these two channels.

By using the ABU measure we propose two algorithms

that can adapt to the channel conditions to achieve improved

performance.

C. Adaptive MU-MIMO+TDMA AlgorithmThe motivation of this scheme is to take advantage of

TDMA when the users are highly spatially correlated while

maintaining the good performance provided by MU-MIMO

when the inter-user correlation is low. We use ABU as the

benchmark to determine whether to use TDMA or MU-

MIMO. Here we consider the two-user case to demonstrate

the approach. Note however that the approach is aimed at the

general situation where there is a large pool of users that are to

be served. In this situation we randomly select two users from

this large pool to perform the adaptive MU-MIMO+TDMA

algorithm. When this pair is served the MU-MIMO+TDMA

scheme will then move on to the next timeslot and randomlyselect another pair to serve until all have been handled.

The basic adaptation relies on calculating ABU and when

it is larger than a certain threshold we say these two users are

compatible and decide to use MU-MIMO. When the ABU is

smaller than the threshold, we decide to use TDMA instead.

Since our objective is to improve the capacity at low outage

probability without compromising the good performance at

high outage probability provided by sub-optimal MU-MIMO

schemes, the optimal threshold should be the one that yields

the maximum ergodic capacity. If the threshold is too small,

then even when the users are highly spatially correlated MU-

MIMO will still be used, which leads to poor performanceat low outage probability and consequently a decrease in the

ergodic capacity. However if the threshold is too large, which

means even when the inter-user spatial correlation is low

enough for MU-MIMO to achieve good performance TDMA

will still be used, clearly this leads to a decrease in ergodic

capacity as well. Thus the optimal threshold should be the one

that results in the maximum ergodic capacity.

When H-CSIT is available then MU-MIMO schemes such

as MU-MIMO ZF decomposition with equal power allocation

and TDMA based on instantaneous channel matrix may be

used. When only RT x is known and it is not of full rank,

then the max average transmit SINR beamforming and TDMAbased on RT x should be used. When RT x is of full rank,

TDMA should be used rather than the max average transmit

SINR scheme as we will see in section V that the latter is not

efficient in a channel with low fading correlation.

Also note that as previously mentioned our approach is

general and can be extended to any number of users by

randomly pairing up the users. We then assign a timeslot to

each user pair and then apply the adaptive MU-MIMO+TDMA

algorithm within each timeslot.

D. Adaptive MU-MIMO Grouping AlgorithmIn the previous subsection we introduce an adaptive algo-

rithm, which takes advantage of TDMA when ABU is too

small for sub-optimal MU-MIMO schemes to work efficiently.

Although performance improvement at low outage probability


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can be achieved, this algorithm in some sense reacts to the

disadvantageous condition passively. While ABU cannot be

changed, we can react more actively by exploiting the multi-

user diversity to perform some adaptive user grouping, or

so-called adaptive timeslot allocation. That is rather than

randomly select users from the large pool of users, we can

select users that are compatible and thereby exploit multi-user

diversity. As a result users in the same timeslot will probably

have low spatial correlation and therefore higher capacity can

be achieved.

For the sake of simplicity, we constrain the number of

users in one group to be two. Assume there are 2L activeusers, our problem is how to divide these 2L users into Lgroups, each of which contains two users and occupies one

of the L timeslots. The problem can also be interpreted as:

there are N = C2L2 C2L22 ... C22L!

possible group arrangements,

which arrangement should be chosen. In total there are

M =

2L(2L1)2 mutual angles and each possible arrangement is

associated with a set of L angles, corresponding to the anglesbetween two users in each of the L groups. Denote the sets ofangles as 1, . . . , N with j =

j1, . . . , jL

. In order

to ensure fairness to all the users to some extent, our aim is

to select the arrangement with its minimum angle larger than

the minimum angle of all the other arrangements. That is

j = arg1jN

max min

j

= arg1jN

max min

j1, j2, . . . , jL

(25)

In case several arrangements have the same minimum angle,

then the one with the largest second minimum angle is chosen.This process will continue and repeat, if necessary, until

theres only one arrangement left. The algorithm is as follows,

Step1, Calculate the M mutual angles and sort them inascending order

a1,b1, . . . , am,bm, . . . , aM ,bM

,

where am and bm are user indices. Load all possibleN group arrangements into GAnew. Set m = 1.Step2, Backup GAnew as GAold. Discard group arrange-

ments which put user am and bm in the sametimeslot. Store the remaining group arrangements in

GAnew, set m = m + 1.Step3, Repeat step 2 until there is only 1 arrangement left

in GAnew. In case GAnew is empty, which meansseveral arrangements have the same minimum angle,

restore GAnew with GAold, set m = m + 1 and goback to step 2.

Note, the more active users there are the better the perfor-

mance, but as the number of users increases the complexity

of the algorithm increases too. By intuition, when the user

number reaches a certain value, the performance will reach a

certain limit, in which further increases in the user number

will not gain much. Say n-user adaptive grouping alreadyachieves satisfactory improvement, if the system has more

than n active users in total, we can randomly divide usersinto several clusters with n users in each, then within eachcluster perform the adaptive grouping according to the above

algorithm.

The group size K is not constrained to 2, however it should

satisfy the constraint M > maxk=1,...,K

rank

Hk

when H-CSIT

is available or M > maxk=1,...,K

rank

RT xk

when only R-

CSIT is available, where RT xk is defined similar to Hk , to

ensure there exists non-zero probability for the K users to becompatible. Say each group contains 3 users and there are 3Lactive users in all. Then there are 3 angles associated witheach group and the best group arrangement can be obtained

using the same 3 steps.

E. Impact of H-CSIT Accuracy and Fading Correlation on

MU-MIMO Schemes

One key aspect of MU-MIMO algorithms is their sensitivity

to inaccurate H-CSIT. As mentioned in section II.C the

primary reason for the inaccuracy is the delay between the

estimation of the CSI and its use. In this subsection we wish

to understand the impact of fading correlation on MU-MIMO

schemes sensitivity to H-CSIT accuracy. The impact of H-

CSIT accuracy itself is explored in the numerical simulationsection that follows.

To appreciate the impact of fading correlation on MU-

MIMO schemes sensitivity to H-CSIT accuracy we assume

that the BS treats the delayed version ofH, i.e. H, as the accu-rate H-CSIT and use it to calculate the transmit pre-processing

matrix as if the actual channel is H. We first considertwo extremes. When the channel is totally uncorrelated, the

elements of the channel matrix are all i.i.d complex Gaussian

random variables with zero mean and unit variance. Even a

small delay between estimation and its use will cause the

subspace structure of the channel matrix to alter significantly

and as a result the performance of MU-MIMO scheme will besignificantly degraded. However when there is only 1 DOD in

space for each user we can prove that the capacity achieved by

adaptive MU-MIMO+TDMA algorithm, adaptive MU-MIMO

grouping algorithm, both incorporated with MU-MIMO ZF

decomposition with equal power allocation scheme, will not be

affected by delayed H-CSIT as long as the relative geometry of

the propagation path remains unchanged, i.e. the DOD remains

the same.

Proof: From the analysis of section IV.B, we know that

the rank-1 channel matrix

Hk =

gkak, k = 1, . . . , K

can be decomposed as Hk = kukvHk , vk is solely de-termined by ak , Vk and Vk are solely determined bya1, . . . , ak1, ak+1, . . . , aK. As a result k, k = 1, . . . , K isalso uniquely determined by a1, . . . , aK.

For TDMA, since there is only one sub-channel for Hk,all the transmit power will be dedicated to this sub-channel

irrespective of the exact value ofk. Thus Tk = PTvk. Aslong as ak remains unchanged, capacity achieved by TDMA

will not be affected even if t = 0.For MU-MIMO ZF decomposition with equal power allo-

cation scheme, as we have shown in section IV.B that

Tk = PT/KVk VH

k vk/ sin k, k = 1, . . . , K . As long asa1, . . . , aK remains unchanged, the capacity achieved by thisscheme will not be affected even if t = 0.

Thus capacity achieved by adaptive MU-MIMO+TDMA

algorithm, adaptive MU-MIMO grouping algorithm, both in-

corporated with MU-MIMO ZF decomposition with equal


7/12


100 80 60 40 20 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DOD of the second user

sin2(A

BU)

Fig. 1. sin2(ABU) in (16) with user 1 having a DOD of 0.

power allocation scheme, will not be affected as long as

a1, . . . , aK remains unchanged.

From this derivation we understand that a critical parameter

characterizing the impact of fading correlation on MU-MIMO

schemes sensitivity to H-CSIT is the amount of change in

the subspace of the channel matrix with different degrees of

fading correlation. This can be defined as the expected value

of the angle between the row-spaces of H and

H(t), where

H(t) is a delayed version of H with correlation coefficient

t as defined in II.C. We denote this as expected row-spaceangle or ERA. ERA provides a measure of the amount of newinformation contained in H(t). The smaller the angle the lessnew information in H(t), as a consequence the smaller theperformance degradation.

V. NUMERICAL RESULTS

Using numerical simulations we wish to demonstrate the

following:

The impact of fading correlations on users spatial sepa-rability

The performance of our adaptive multi-user algorithms

The impact of CSIT accuracy on MU-MIMO schemes in

channels with different degrees of fading correlation

In the simulation, 8 (2, 2) system is investigated. Uniformlinear array (ULA) is used at the BS with equidistant spacing

between antenna elements to be half a wavelength. Semi-

correlated channels with 50 departure waves and 5, 20, 75

angle spread are investigated. We assume the DODs are

randomly and independently distributed in the angle spread

according to uniform distribution. Users are assumed to berandomly and independently distributed in [, ] accordingto uniform distribution around the BS.

The capacity we refer to in this paper is the minimum

mutual information among the K users, where the mutual

0 10 20 30 40 50 60 70 80 9010

2

101

100

ABU in degrees

CDF

1DOD(50,20)(50,75)uncorrelated channel

Fig. 2. CDF of ABU between two users for different degrees of fadingcorrelation.

information of user k is given by

Ik = log2 det

I+THk HHk

I+Hk

Ki=1i=k

TiTHi H

Hk

1HkTk

,

(26)

The optimization problem in (13) involves two nonlinear

optimizations that need to be solved numerically, each of

which consists of2

k=1 2M Lk real variables since Tk, k =1, 2 is a M Lk complex matrix. A Sequential QuadraticProgramming (SQP) method [22] is used. Modifications aremade to the line search, where an exact merit function [23],

[25] is used together with the merit function proposed by [24]

and [26]. The line search is terminated when neither merit

function shows improvement.

To allow benchmarking we compare our algorithms to those

described in section III in the following figures. The notation

R-TDMA and H-TDMA correspond to TDMA using R-

CSIT and H-CSIT respectively. R-MU MIMO and H-MU

MIMO denote max average transmit SINR beamforming and

MU-MIMO ZF decomposition with equal power allocation

respectively. R-MU MIMO+TDMA is short for adaptive

max average transmit SINR beamforming+TDMA using R-CSIT, and H-MU MIMO+TDMA for adaptive MU-MIMO

ZF decomposition with equal power allocation+TDMA using

H-CSIT correspondingly.

We begin by investigate the characteristics of the inter-user

spatial correlation. The function sin2(ABU) in (16) is plottedin Fig. 1, where the DOD of one user is fixed at 0 and theDOD of the other user varies from 90 to 90. In Fig. 2,the CDFs of ABU between two users for different degrees of

fading correlation are shown. We observe that users spatial

separability pattern is quite different in channels with different

degrees of fading correlation. For the uncorrelated channel,

ABU is always relatively large, while for the 1 DOD case and(50, 20) case, ABU is less than 30 for 10% of the time.

In Figs. 3-5, the performance of the adaptive MU-

MIMO+TDMA algorithm is investigated for different angle

spreads. In Fig. 3 we provide a plot of the threshold versus


8/12


0 10 20 30 40 50 60 70 80 904

4.5

5

5.5

6

6.5

Ergodiccapacityperuserinbits/channeluse

Threshold in degrees

RTDMAHTDMARMU MIMOHMU MIMORMU MIMO+TDMAHMU MIMO+TDMA

Fig. 3. Ergodic capacity versus threshold for 8 (2, 2) system with a(50, 5) channel, PT = 15dB.

0 1 2 3 4 5 6 7 8 910

2

101

100

Capacity per user in bits/channel-use

CDF

RTDMAHTDMARMU MIMOHMU MIMORMU MIMO+TDMA, threshold=20HMU MIMO+TDMA, threshold=45Dirty Paper Coding

Fig. 4. Capacity CDF for 8 (2, 2) system with a (50, 5) channel, PT =15dB.

ergodic capacity for a (50, 5) channel and show that themaximum ergodic capacity can be achieved. In Figs. 4-5 we

provide capacity CDFs for the various algorithms when theoptimal threshold is selected under various channel conditions.

Both the figures show that the threshold chosen according

to the maximum ergodic capacity criterion indeed achieves

the desired performance, i.e. improve the low outage capacity

without compromising the good performance at high outage

probability. For a (50, 75) channel we observe similar re-sults although we havent shown here due to space limita-

tion and the optimal threshold is 60 and 45 for R-MUMIMO+TDMA and H-MU MMIO+TDMA respectively. Also,

from the flat part around the optimal threshold in Fig. 3, we

conclude that the performance of this adaptive algorithm is

not sensitive to the threshold in a range at least [5, 5].Note that the optimal threshold for the H-MU

MIMO+TDMA algorithm in channels with different degrees

of fading correlation is fairly constant. However, for R-MU

MIMO+TDMA algorithm, the optimal threshold increases as

0 1 2 3 4 5 6 7 8 9 10 1110

2

101

100


CDF

RTDMAHTDMARMU MIMOHMU MIMORMU MIMO+TDMA, threshold=35HMU MIMO+TDMA, threshold=45Dirty Paper Coding

Fig. 5. Capacity CDF for 8 (2, 2) system with a (50, 20) channel,PT = 15dB.

0 1 2 3 4 5 6 7 8 910

2

101

100


CDF

RTDMAHTDMA

RMU MIMOHMU MIMOR4 user groupingR8 user groupingH4 user groupingH8 user groupingDirty Paper Coding

Fig. 6. Capacity CDF for 8 (2, 2) system with a (50, 5) channel, PT =15dB.

the channel correlation decrease. A larger threshold indicates

TDMA should be used more often. Also according to our

simulation result, always using R-TDMA is a good choicefor a (50, 75) channel when only RT x is known. Onepossible explanation is: the less correlated the channel the

less efficient the max average transmit SINR scheme. In both

the figures, we also provide the performance of max-min

mutual information MU-MIMO scheme with DPC, denoted

as Dirty Paper Coding. We can see that the performance

gap between adaptive MU-MIMO+TDMA algorithm and

the one using DPC is large. Besides, one should note that

in general TDMA results in a loss of spatial dimension

compared to the other MU-MIMO schemes.

Figs. 6-7 provide performance comparison between H-

MU MIMO, R-MU MIMO before and after adaptive usergrouping for (50, 5) channel and (50, 20) channel. Thenotation R-n user grouping, H-n user grouping refers

to n-user adaptive grouping incorporated with max averagetransmit SINR scheme using R-CSIT and with MU-MIMO


9/12


0 1 2 3 4 5 6 7 8 9 10 1110

2

101

100


CDF

RTDMAHTDMARMU MIMOHMU MIMOR4 user groupingR8 user groupingH4 user groupingH8 user groupingDirty Paper Coding

Fig. 7. Capacity CDF for 8 (2, 2) system with a (50, 20) channel,PT = 15dB.

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

Capacityat10%outageinbits/channeluse

PT

in dB

RMU MIMOHMU MIMORMU MIMO+TDMA, threshold=20HMU MIMO+TDMA, threshold=45R8 user groupingH8 uer grouping

Fig. 8. Capacity at 10% outage versus PT for 8 (2, 2) system with a(50, 5) channel.

ZF decomposition with equal power allocation scheme using

H-CSIT respectively. We observe that the capacity in the area

from 1% outage to 20% outage has been dramatically im-proved. When compared to the adaptive MU-MIMO+TDMA

algorithm, we find that adaptive MU-MIMO grouping algo-

rithm works much better than the previous one, which is

because of the effective exploitation of the multi-user diversity.

So, when there is enough number of active users, adaptive

MU-MIMO grouping algorithm should be used, otherwise,

adaptive MU-MIMO+TDMA algorithm should be used to

avoid the severe inter-user interference in case users are

highly spatially correlated. In addition, we can see that 8-

user grouping already provides enough degrees of freedom

to achieve desired performance and the performance gap

between 8-user adaptive grouping and the one with DPC,whose complexity is much higher, is reduced compared to

Figs. 4-5. For a (50, 75) channel we observe similar resultsalthough we havent shown here due to space limitation. The

performance of 8-user grouping using H-CSIT is better than

00.10.20.30.40.50.60.70.80.9110

0

10

20

30

40

50

60

70

80

90

t

ERAindegrees

1 DOD(50,5)(50,20)(50,75)uncorrelated channel

Fig. 9. ERA versus t for 8 (2) system with channels of different degreesof fading correlation.

0 1 2 3 4 5 6 7 8 910

2

101

100


CDF

HMU MIMOHMU MIMO when rho=0HMU MIMO+TDMA, threshold=45HMU MIMO+TDMA, threshold=45 when rho=0H8 user groupingH8 user grouping when rho=0

Fig. 10. Capacity CDF for 8 (2,2) system with a (50, 5) channel in thepresence of imperfect channel estimate, PT = 15dB.

adaptive MU-MIMO+TDMA algorithm, however even with

8-user grouping, max average transmit SINR still doesnt

work well in a (50, 75

) channel, which means max averagetransmit SINR scheme is not capable of effectively exploiting

the degrees of freedom provided by channels with low spatial

correlation for multi-user separation.

In Fig. 8 we show the performance of various schemes

in terms of capacity at 10% outage versus PT. From thefigure we can see that all the schemes, except the H-MU

MIMO, show similar performance when PT is low. As PTincreases, the performance gains obtained by exploiting the

ABU measure get more and more significant. For example,

at PT = 20dB the capacity at 10% outage achieved by H-MU MIMO increases from 2 to 5 and 7 bits/channel-use

by employing the adaptive MU-MIMO+TDMA algorithm andadaptive MU-MIMO grouping algorithms respectively.

In order to investigate the impact of fading correlation

on MU-MIMO schemes sensitivity to H-CSIT accuracy, we

provide simulations of ERA versus t for channels with


10/12


0 1 2 3 4 5 6 7 8 9 10 1110

2

101

100


CDF

HMU MIMOHMU MIMO when rho=0.5HMU MIMO+TDMA, threshold=45HMU MIMO+TDMA, threshold=45 when rho=0.5H8 user groupingH8 user grouping when rho=0.5

Fig. 11. Capacity CDF for 8 (2, 2) system with a (50, 20) channel inthe presence of imperfect channel estimate, PT = 15dB.

different degrees of fading correlation in Fig. 9. As expected,

we can see that the less correlated the channel the larger the

ERA, which means the faster the channel subspace changes,

and as a consequence larger performance degradation for the

same value of t is expected. Figs. 10-12 confirm what wehave observed in Fig. 9, i.e. the less correlated the channel

the more sensitive is MU-MIMO scheme to inaccurate H-

CSIT. Also we observe that scheme with adaptive grouping

still achieves certain gain compared with the one without

adaptive grouping. One possible explanation is that when

choosing group arrangement, only the relative relationship of

the minimum angles of each possible arrangement matters,

rather than the accurate values of those angles. This relative

relationship is less sensitive to inaccurate H-CSIT, so the

users in the same timeslot will have potentially low spatial

correlation, and therefore inaccurate H-CSIT has less impact.

Another thing we observe from Fig. 9 is that large change in

ERA mainly occurs from perfect H-CSIT to small degradation

in t, and the part after that is relatively flat. We have tested,although havent shown here due to space limitation, that

for t changes from 0.6 to 0 for a (50, 20) channel, the

performances of all the three algorithms investigated in Fig.11, are nearly unchanged.

The impact of our investigation may be helpful in defining

MU-MIMO systems for different scenarios. For example in

[14], [15], measurement shows that transmit fading correlation

between the BS antenna pairs does exist for both the indoor

and outdoor scenarios. In an outdoor environment, typically

the BS is elevated above its surroundings, and only waves

transmitted within a small angle spread, e.g. 5, 10, canreach the mobile. In this scenario, max average transmit SINR

scheme with adaptive user grouping is preferred because it

only needs RT x, which can be estimated less often. In [14],

it is reported that in an indoor NLOS case, correlation betweentwo transmit antennas can be quite high as the distance

between the transmitter and the receiver increases. Also, since

the channel varies much slower than the outdoor case, high

quality channel estimation could be obtained. In this scenario

0 1 2 3 4 5 6 7 8 9 10 11 1210

2

101

100


CDF

HMU MIMOHMU MIMO when rho=0.85HMU MIMO+TDMA, threshold=45HMU MIMO+TDMA, threshold=45 when rho=0.85H8 user groupingH8 user grouping when rho=0.85

Fig. 12. Capacity CDF for 8 (2, 2) system with a (50, 75) channel inthe presence of imperfect channel estimate, PT = 15dB.

the MU-MIMO ZF decomposition scheme with adaptive user

grouping is preferred.

VI. CONCLUSION

In this paper we investigated the effect of fading correlation

on MU-MIMO schemes. One effect is that fading correlation

heavily influences the spatial correlation pattern between co-

channel users. The more correlated the channel the more likely

users become highly spatially correlated. By adopting the

concept of angle between subspaces, two algorithms namely

adaptive MU-MIMO+TDMA algorithm as well as adaptiveMU-MIMO grouping algorithm are proposed in order to

increase the system capacity and ensure instantaneous QoS to

users. The other effect is MU-MIMO schemes show different

tolerance to inaccurate H-CSIT in channels with different

degrees of fading correlation. The more correlated the channel,

the less sensitive is the performance of MU-MIMO scheme

to inaccurate H-CSIT. Numerical results show that provided

perfect CSIT is available, both algorithms we propose are

capable of significantly improving the system capacity at low

outage by avoiding severe inter-user interference, with the

latter superior to the former because channel diversity in the

user domain is successfully exploited. For inaccurate H-CSIT,

the two algorithms we propose still achieve gains compared

with the original MU-MIMO ZF decomposition with equal

power allocation scheme.

APPENDIX I

Proof: First,we introduce two inequalities we will use in

the proof.

Proposition 1 ( [20]): Let 1(X) 2(X) . . . n(X)be the singular values of X, then

j (BA

) 1(B

)j (A

) and j (AB

) 1(B

)j (A

).(27)Theorem 1 ( [21]): For any positive definite n n matrix

S, det(S)

1n 1

ntrace

S

. (28)


11/12


The capacity achieved by this scheme for user k is

Ck = log2 det

I + HkWkWHk H

H

k

, (29)

where trace

WHk Wk

= PT/K.From Theorem 1 we have

detI + HkWkWHk HHk

trace

I + HkWkWHk H

H

k

Nk

Nk, (30)

therefore

Ck Nk log2

trace

I + HkWkWHk H

H

k

Nk

. (31)

From Proposition 1 its easy to see that

trace

HkWkW

Hk H

H

k max

H

H

k Hktrace

WkW

Hk = PT

KmaxHHk Hk, (32)

then from (31) and (32) we get

Ck Nk log2

1 +PT

KNkmax

H

H

k Hk

. (33)

On the other hand, since the capacity is achieved by water-

filling over the eigenmodes of HH

k Hk , rather than allocating

all the transmit power to the dominant eigenmode, we have

Ck log2 1 + PTK maxHHk Hk. (34)Let rank

Vk

= lk , then rank

Hk

= rank

HkVk

min{mk, lk} Nk. Assume

HHk Hk =

Uk Uk 2k 0

0 0

UHkUHk

= Uk

2kU

Hk ,

(35)

where

2k =

2k1. . .

2kmk

then

HH

k Hk = VH

k HHk HkVk = V

H

k Uk2kU

Hk Vk. (36)

From Proposition 1, we have

max

HH

k Hk

max

V

H

k Uk

max

2k

max

UHk Vk

= 2k1 sin

2 k, (37)

therefore from (33) and (37) we obtain the upper bound

Ck

Nk log2 1 +

PT

KNk

2k1 sin2 k. (38)

From (36), by straight forward mathematical manipulation,

we obtain

trace

HH

k Hk

2kmk sin2 k. (39)

In addition, since

max

HH

k Hk

trace

HH

k Hk

rank

Hk

trace

HH

k Hk

Nk

,

(40)

together with (34) and (39) we get the lower bound

Ck log2

1 +PT

KNk2kmk sin

2 k

. (41)

APPENDIX II

Proof: Equation (10) in this case can be written as

1 =N1N2

uH1 aH1 a1u1

uH1

aH2 a2 +

1PI

u1,

2 =N2

N1

uH2 aH2 a2u2

uH2 aH1 a1 + 1PIu2 ,(42)

where P = PT/2. For this special case we have

u1 = 1

aH2 a2 +1PI1

aH1 ,

u2 = 2

aH1 a1 +1PI1

aH2 ,(43)

maximize 1 and 2 respectively, and 1, 2 are two scalarsused to normalize u1 and u2. By using (18) we can easily

obtain

1 = 2 = 1Mcos2 (M + 1/P)2 + Msin2

(1/P)2 . (44)

For the exact SIR seen at the receiver side, we have

S IR1 =tH1 H

H1 H1t1

tH2 HH1 H1t2

=21a1

aH2 a2 +1PI1

aH1 a1

aH2 a2 +

1PI1

aH1

22a2

aH1 a1 +1PI1

aH1 a1

aH1 a1 +

1PI1

aH2

. (45)

By straightforward mathematical manipulation and (18), we

get

a1

aH2 a2 +

1

PI1

aH1 a1

aH2 a2 +

1

PI1

aH1

=M2

1/P + Msin22

(M + 1/P)2(1/P)2, (46)

a2

aH1 a1 +

1

PI1

aH1 a1

aH1 a1 +

1

PI1

aH2

=M2 cos2

(M + 1/P)2, (47)

substitute (44), (46) and (47) into (45) we obtain

SI R1 = 1 + M Psin2 2 cos2 . (48)Similarly we get SI R2 =

1 + M Psin2

2cos2 .


12/12


REFERENCES

[1] G. J. Foschini and M. J. Gans, On limits of wireless communications ina fading environment when using multiple antennas, Wireless PersonalCommun., vol. 6, pp. 311-335, Mar. 1998.

[2] E. Telatar, Capacity of multi-antenna Gaussian channels, EuropeanTrans. Telecommun., vol. 10, no. 6, pp. 585-595, 1999.

[3] A. Bjorck and G. H. Golub, Numerical methods for computing anglesbetween linear subspaces, Math. Comp., vol. 27, no. 123, pp. 579-594,

July 1973.[4] G. Caire and S. Shamai, On the achievable throughput of a multiantennaGaussian broadcast channel, IEEE Trans. Inf. Theory, vol. 49, no. 7, pp.1691-1706, July 2003.

[5] W. Yu and J. M. Cioffi, Sum capacity of Gaussian vector broadcastchannels, IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 1875-1892, Sept.2004.

[6] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates,and sum-rate capacity of Gaussian MIMO broadcast channels, IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2658-2668, Oct. 2003.

[7] P. Viswanath and D. N. C. Tse, Sum capacity of the vector Gaussianbroadcast channel and uplink-downlink duality, IEEE Trans. Inf. Theory,vol. 49, no. 8, pp. 1912-1921, Aug. 2003.

[8] H. Weingarten, Y. Steinberg, and S. Shamai, The capacity region of theGaussian MIMO broadcast channel, in Proc. IEEE ISIT04, June 2004.

[9] N. Jindal and A. Goldsmith, Dirty paper coding vs. TDMA for MIMO

broadcast channels, in Proc. IEEE ICC04, vol. 2, pp. 682-686.[10] Lai-U Choi and R. D. Murch, A transmit preprocessing technique for

multiuser MIMO systems using a decomposition approach, IEEE Trans.Wireless Commun., vol. 3, no. 1, pp. 20-24, Jan. 2004.

[11] Q. H. Spencer and M. Haardt, Capacity and downlink transmissionalgorithms for a multi-user MIMO channel, in Proc. 36th AsilomarConf. Signals, Systems, and Computers 2002, vol. 2, pp. 1384-1388.

[12] A. Bourdoux and N. Khaled, Joint TX-RX optimisation for MIMO-SDMA based on a null-space constraint, in Proc. IEEE VTC02, vol. 1,pp. 171-174.

[13] Q. H. Spencer, T. Svantesson, and A. L. Swindlehurst, Performanceof MIMO spatial multiplexing algorithms using indoor channel measure-ments and models, Wireless Commun. Mobile Comput., vol. 4, no. 7,pp. 739-754, Nov. 2004.

[14] P. Kyritsi, D. C. Cox, R. A. Valenzuela, and P. W. Wolniansky,Correlation analysis based on MIMO channel measurements in an indoor

environment, IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 713-720,June 2003.

[15] C. -N. Chuah, G. J. Foschini, R. A. Valenzuela, D. Chizhik, J. Ling,and J. M. Kahn, Capacity growth of multi-element arrays in indoor andoutdoor wireless channels, in Proc. IEEE WCNC00, vol. 3, pp. 1340-1344.

[16] S. A. Jafar, S. Vishwanath, and A. J. Glodsmith, Channel capacity andbeamforming for multiple transmit and receive antennas with covariancefeedback, in Proc. IEEE ICC01, vol. 7, pp. 2266-2270.

[17] M. T. Ivrlac, W. Utschick, and J. A. Nossek, Fading correlation inwireless MIMO communication systems, IEEE J. Sel. Areas Commun.,vol. 21. no. 5, pp. 819-828, June 2003.

[18] D. Gerlach and A. Paulraj, Adaptive transmitting antenna methods formultipath environments, in Proc. IEEE GLOBECOM94, vol. 1, pp. 425-429.

[19] R. Lai-U Choi and R. D. Murch, New transmit schemes and simplifiedreceivers for MIMO wireless communication systems, IEEE Trans.Wireless Commun., vol. 2, pp. 1217-1230, Nov. 2003.

[20] R. Bhatia, Perturbation Bounds for Matrix Eigenvalues. LongmanScientific & Technical, 1987.

[21] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley,1991.

[22] R. K. Brayton, S. W. Director, G. D. Hachtel, and L. Vidigal, A newalgorithm for statistical circuit design based on Quasi-Newton methodsand function splitting, IEEE Trans. Circuits Syst., vol. 26, no. 9, pp.784-794, Sept. 1979.

[23] A. C. W. Grace, Computer-aided control system design using optimiza-

tion techniques, Ph.D. thesis, University of Wales, Bangor, Gwynedd,UK, 1989.

[24] S. P. Han, A globally convergent method for nonlinear programming,J. Optimization Theory and Applications, vol. 22, p. 297, 1977.

[25] K. Madsen and H. Schjaer-Jacobsen, Algorithms for worst case tol-erance Optimization, IEEE Trans. Circuits Syst., vol. 26, no. 9, Sept.1979.

[26] M. J. D. Powell, A fast algorithm for nonlineary constrained optimiza-tion calculations, Numerical Analysis, ed. G.A. Watson, Lecture Notesin Mathematics, Springer Verlag, vol. 630, 1978.

[27] H. Viswanathan, S. Venkatesan, and H. Huang, Downlink capacityevaluation of cellular networks with known-interference cancellation,IEEE J. Sel. Areas Commun., vol. 21. no. 5, pp. 802-811, June 2003.

Cheng WANG (S03) received the Bachelors de-gree in Electronic Science and Engineering fromthe NanJing University, NanJing, JiangSu, China,where she graduated in 2002 and was ranked firstin the department. She is currently working towardthe Ph.D. degree in the Department of Electrical andElectronic Engineering, the Hong Kong Universityof Science and Technology, Kowloon, Hong Kong.

Her research interests include multi-user MIMOwireless communication systems, adaptive resourceallocation and cross-layer design and optimization.

Ross D. Murch (S85M87SM98) is a Pro-fessor of Electrical and Electronic Engineering at

the Hong Kong University of Science and Technol-ogy. His current research interests include multipleantenna systems, compact antenna design, MIMO,WLAN, B3G and Ultra-Wide-Band (UWB) systemsfor wireless communications. He has several USpatents related to wireless communication, over 150published papers and acts as a consultant for indus-try and government. In addition he is an editor forthe IEEE Transactions on Wireless Communications

and was the Chair of the Advanced Wireless Communications SystemsSymposium at ICC 2002. He is also the founding Director of the Center forWireless Information Technology at Hong Kong University of Science andTechnology which was begun in August 1997. He is the program Directorfor the MSc in Telecommunications at Hong Kong University of Scienceand Technology. From August-December 1998 he was on sabbatical leave atAllgon Mobile Communications (manufactured 1 million antennas per week),Sweden and AT&T Research Labs, NJ, USA.

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Adaptive Downlink Multi-User MIMO Wireless Syfstems for Correlated Channels With Imperfect CSI