Optimal Space-Time Power Control for MIMO Multiple Access Channel with Partial Feedback

Vincent K.N. Lau, Youjian Liu, Tai-Ann Chen

knlau@lucent.com, eugeneliu@colorado.edu, tac@lucent.com Lucent Technologies and University of Colorado at Boulder

Abstract - The multi-user multiple-input-multiple-output (MIMO) space-time scheduling problem is the prime focus of the paper. Because optimizing the link level performance of the MIMO system does not always imply achieving scheduling-level optimization, the design optimization across the link layer and the scheduling layer is important to fully exploit the temporal and spatial dimensions of the communication channel. This paper is an extension to the single-input-single-output (SISO) investigation in [1] where the optimal scheduler design is to allocate all the power to at most a single user at any particular instant. We found that in the multi-user MIMO system with K ≥≥≥≥ nR (K is the total number of users), the optimal scheduler should allocate all the power to at most nR users at any particular instant. We considered two cases of partial feedback, scalar feedback and the per-antenna vector feedback. With scalar feedback, the optimal power control for system performance is to focus the power on a single transmit antenna only. With the per-antenna vector feedback, both the scheduling performance and the link-level performance are enhanced with nT. Finally, scheduling performance improves as K increases. Keywords: MIMO, Space Time Scheduling, Multi-user, Multiple Access, Transmit Diversity.

I. Introduction

An approach to increase the communication bit rate without increasing the bandwidth or power budget is to make use of multiple transmit and receive antennas. For quasi-static fading channels, it is shown that the order of diversity is the key to link-level performance. Hence, to optimize the link-level performance, one should try to design space-time codes that can fully utilize the intrinsic diversity order. Various space-time coding techniques [4, 5, 6] are based on the full-rank criteria to achieve the maximum link-level diversity. Therefore, designing a full-rank space-time code is a norm in the physical layer domain.

For data transmissions over wireless channels, the cross optimization between the scheduling layer and the link layer is very important to the overall system performance. It attracted less attention in the previous literature due to the enormous complexity involved. There have been some works [7, 8] proposing to take the advantage of the optimization gap by adopting the cooperative scheduling. In [9, 10], the variable-throughput adaptive physical layer design is shown to greatly enhance the scheduling efficiency in a single-input-single-output (SISO) system. It is found that the selection diversity among different users is the key to the performance gain in scheduling. Scheduling optimization for SISO multi-user system has been thoroughly investigated in [1]. The optimal strategies for uplink SIMO with ideal side information has been investigated in [13]. [14] deals with the uplink MIMO optimal strategies with ideal side information. [16] deals with SIMO optimal strategies with scalar feedback. In this paper, we try to provide a unifying theory for the general MIMO uplink scheduling with partial feedback.

In this paper, we formulate the optimal space-time scheduler for a general multi-user MIMO system based on the information theoretical approach. Reverse link is the focus of the paper. Short-term scheduler is considered, meaning that a data burst is assigned based on the instantaneous channel condition. Instead of feeding back the entire channel matrix, we consider two special cases of partial feedback, namely the scalar power feedback and the vector power feedback from the receiver to the transmitter for power adaptation. Based on these, we propose an information theoretical approach to tackle the cross-layer design optimization. A common channel model, multiple block fading, for low-speed mobile communications is assumed. The number of homogeneous users K (with identical fading distributions) is assumed to be greater than nR (total number of receive antennas at the base station). We found that with scalar feedback, achieving full link-level diversity, which is believed to be the right thing to do in designing the physical layer, will hurt the overall system performance. The contributions of this paper can summarized as below:

Optimal Space-Time Scheduling Strategy in the Reverse Link with Partial Feedback: We shall formulate and solve the optimal multi-user MIMO scheduling problem by a simplified information theoretical approach. Optimal scheduling is achieved by allocating resource to at most nR users at a time.

Trade-offs between Link-Level Diversity and Scheduling Diversity: Based on the analytical model, we shall investigate the trade-off involved between the link-level diversity and scheduling diversity in MIMO systems under scalar or vector power feedback. Specifically, the effect of nR, nT, and K on the scheduling performance is investigated.

This paper is organized as follows. In Section II, we outline the multi-user MIMO system model and the channel model. In section III, the partial feedback model and the optimal space-time scheduling problem are

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formulated for the MIMO system based on the information theoretical approach. Optimal power allocation with water-filling strategies in the temporal and spatial domain is derived in Section IV, where scalar feedback is assumed. In Section V, the analysis is extended to accommodate nT-dimension vector power feedback. Numerical results is presented in Section VI. We conclude with Section VII.

II. Multi-user MIMO System Model

We consider the reverse link of a communication system with K mobile users having nT transmit antennas each and a base station with nR receive antennas as illustrated in Figure 1. A widely accepted channel model, namely the block fading channel model, is assumed. We assume a channel code word span over infinite many fading blocks. For each channel use, the k th transmitter, send a complex vector 1Tn

k×∈X � . After matched

filtering, the received vector Y at base station is given by:

1

K

k kk=

= + = +�Y H X Z HX Z� � (1)

where R Tn nk

×∈H � is the random channel matrix (linking user k and the base station) whose elements are unit-

variance circularly symmetric complex Gaussian random variables with identical and independent distribution (i.i.d.), [ ]1 2 K=H H H H� � , †† † †

1 2 K� �= � �X X X X� � , and 1Rn ×∈Z � is the white Gaussian noise vector

with one sided spectral density N0. Perfect channel estimation is assumed at the receiver. In this paper, a symbol X denotes a random matrix or a vector. Symbol X denotes a scalar. Symbols x

and x denote a realization of them. Symbol X� denotes a concatenation of Xk. For any vector x and a set of

integers S, x(S) =� ∈Si ix . †X is the Hermitian of X.

Each of the users has to compete for resource before uplink transmission of payload. The resource allocation is centralized at the base station through the scheduling layer. The system bandwidth is shared among different users in a slot-based manner. A slot consists of a number of tiny contention slots followed by a payload slot as illustrated in figure 2. Users have to send tiny request packets to the base station to compete for the payload slot once every fading block. Base station will broadcast the ownership of the payload slot for the current frame based on the output of the scheduler. In general, the payload slot can be assigned to more than one user and the signals between users can be separated through successive interference cancellation [11]. Since the focus of the paper is on the scheduling performance, we shall assume that the system is designed in such a way that the request contention is not the bottleneck of the throughput and the overhead of contention is hence negligible.

III. Problem Formulation and MIMO Multiple Access Capacity Region

We are interested in maximizing the total reverse link throughput of the system by a proper design of the scheduling algorithm according to available feedback to the transmitters. For each fading block, we assume there is a partial feedback , ( )fb kg h� from the base station to user k, because full feedback requires large

bandwidth. We design the scheduling algorithm for two cases. In the first case, the feedback , ( )fb kg h� is a

scalar. In the second case, it is a vector of length Tn . The design of the scheduling algorithm leads to the quest of the multiple access capacity region as follows.

The capacity region of the considered multiple access channel is the convex hull of the rate vector

1 ... KR R= � �� �R satisfying [11] { }{ }( )( ) ; , {1,.., }ci S i S

S E I S K∈ ∈� �≤ = ∀ ⊂� �HR Y X X H h�

� � (2)

for some input distribution ( ) ( )1 ,1 ,( ) ,..., ( )fb K fb Kg gα αX h X h� � . In addition, the transmitted signal kX satisfies an

average power constraint that ( )†Tr k k kE � � ≤� �X X � . Users can transmit to base station simultaneously and

reliably at rate kR for kth user if the rate vector R belongs to the capacity region.

The design of the scheduling algorithm is to choose feedback function , ( )fb kg h� and input distribution

( ), ( )k fb kgα X h� to maximize the capacity region. For Gaussian channel and finite power input, Gaussian

distributed inputs maximize the mutual information in (2) simultaneously for all possible h� [3]. The remaining

148

problem is to choose the conditional variance matrix †,( ) | ( )k k k fb kE g� �= � �p h X X h� � and feedback function

, ( )fb kg h� to maximize total throughput

,

* †

,0

1max E log

Rfb k k

ngR

N

� �= +� �

� �Hp

I HpH�� ��

, (3)

where RnI is a R Rn n× identity matrix and 1 2diag{ , ,..., }K=p p p p� , under the power constraint that

[ ]( )Tr E , {1,2,..., }k k k K≤ ∀ ∈H p� � . Without loss of generality for Gaussian channel and for our particular

interest of the total system throughput, we have let the user set S to include all users. Some insight about the choice of ,fb kg and kp can be obtained by inspecting the full feedback case

and non-feedback case. If full feedback is available to the transmitter (client user), the optimal transmission strategy would be a cascade of space and temporal power water filling matrix and beam-forming matrix [3], i.e., the input covariance matrix is given by

†

,1

,2

,

0 00

0 0T

k

kk k k

k n

ρρ

ρ

� � �= � � ��

p V V

�

� �

� � � �

�

(4)

where V is the unitary beam-forming matrix and ,k tρ is the transmitted power on the tth eigen-channel. Both

kV and ,k tρ are functions of the channel matrix �h . If no feedback is available, the covariance matrix is equal

to a power matrix /Tk T nn I� .

According to the above observation, a reasonable choice for the partial feedback would be to control the power only. (Obviously, it may not be the best choice, which is treated in our two other papers.) For the case

of scalar feedback, we assume the feedback , ( )fb kg h� be kρ and the transmission strategy is given by an

adaptive power control matrix k k kρ=p C , where kC is a constant Hermitian matrix with eigen-

decomposition †k k kU D U . Because Hk is i.i.d. Gaussian and Uk is a constant unitary matrix, Hk Uk has the same

distribution as Hk [3], without loss of generality, we can consider diagonal power control matrix

,1 ,2 ,diag{ , ,..., }Tk k k k k k k nρ ρ ε ε ε= =p D , where ,k tε ’s are positive constants satisfying ,

1

1Tn

k tt

ε=

=� . For

the case of vector feedback, we assume , ( )fb kg h� be ,{ , 1 }k t Tt nρ ≤ ≤ , such that the power control matrix is

(4). Because the beam-forming matrix is a constant matrix, due to similar argument, it is sufficient to consider diagonal power control matrix ,1 ,2 ,diag{ , ,..., }

Tk k k k nρ ρ ρ=p .

IV. Optimal Space Time Power Control - Scalar Feedback

In this section, we shall derive the optimal power control algorithm for multi-user MIMO systems with

scalar feedback. We first fix transmit-antenna power configuration ,1 ,, ...,Tk k k ndiag ε ε� �= � �D and find the

optimal power allocation kρ . Then we look for the best kD . To simplify the optimisation problem (3), we will use an asymptotically tight upper bound for the

determinant. Define 1 2kk T k kn′ =H H D , the mutual information in (3) can be written as

2† †,

1 10 0 0

1 1log log log 1 ( )

T T

T Kt

n n k k tt kT T T

RN n N n N n

γ ρ= =

� ′ ′ ′ ′+ = + ≤ + = ��

� �I p H H I �V p V V H� � �� � (5)

where 1 22[ ... ]K′ ′ ′ ′=H H H H� , 1diag[ ... ]R Rn K nρ ρ=p I I� , †V�V is the eigen-decomposition of †′ ′H H� � , which has T

nonzero eigenvalues kγ . The T eigenvectors corresponding to nonzero eigenvalues are partitioned as

. . .

K K

� � � ��

1,1 1,T

,1 ,T

V V

V V

�

�

. We have used the fact that det( ) det( )+ = +I AB I BA and det( ) iii

≤ ∏A A for non-negative

149

definite matrix. The upper bound is asymptotically tight for large Tn and Rn because †H H� � is dominated by

its diagonal elements. In addition, for the MISO case ( 1Rn = ), the equality holds. Hence, the simplified optimisation problem of MIMO system is to find the power control policy,

{ }Kρρ ...,,1 , so that the Lagrange function J, is optimized.

1 1 1,..., ,..., ,...,

1 1

max max E ( ) E max ( )K K K

K K

k k k kk k

J R Rρ ρ ρ ρ ρ ρ

λ ρ λ ρ= =

� �� � � = − = −� � �� �� � � � �

� �H HH H� �� � (6)

where Lagrange multiplier λk is chosen to satisfy the average power constraint, E [ ]k kρ =H� � .

An important property of the above optimisation problem is summarized in the following theorem.

Theorem 1. For each channel realization and power configuration ′h� , let A be the set of user indices

with non-zero power allocation. We have the cardinality of the set, | | T≤A (T is the rank of †′ ′h h� � ). The theorem can be proved using similar method as in [1] (Appendix A). The optimisation problem (6) can be solved by some nonlinear programming algorithms. A particular

choice is the following orthogonal gradient method. Algorithm 1. Step 1: Set ρ* = [0, 0, ..., 0], A= φ, B={1,2,...,K}.

Step 2: If (i) * *| ( ) | 0,

jj k

J Jarg max k and iiρ ρρ ρ∈ −

� �∂ ∂= >� �∂ ∂� �� �B A

then { }k= ∪A A and )...,,( **1

*Kρρ=ρ , where

* 0jρ = for j ∉ A and * 0jρ > is given by the solution of the equations below for j ∈ A.

*| 0

j

Jjρρ

∂ = ∀ ∈∂

A (7)

and j

Jρ∂

∂ is given by 2

,

21 0 ,1

|| ||

|| v ||

Tt j t T

jKtj t k t k Tk

nJ

N n

γλ

ρ γ ρ==

� ∂ �= − �∂ +�

��

v . Repeat Step 2.

If there is no such k, procedure ends with A obtained above and the optimal power allocation is given by ρ* in (7).

If A = φ, the optimal power allocation is 0, {1,..., }k k Kρ = ∀ ∈ .

The algorithm works because the maximizing function E ( )RH H�� is a concave function of

{ , 1,..., }k k Kρ = , which is a convex region under the power constraint [20, section 4.4]. The algorithm is guaranteed to converge in T steps because there are at most T users with nonzero power.

Therefore, the power allocation strategy in a multi-user MIMO system is to perform water-filling across the spatial and temporal domain. At any instant, at most T = nR users should be allocated non-zero power when K ≥ nR.

A. Special Case: MISO multi-user system

Consider special case when nR = 1, we have the multi-user MISO system. In this case, the equality of

(5) holds. The rank of the matrix †′ ′H H� � equals 1 (i.e. T = 1). Furthermore, the non-zero eigenvalue, 2

11

|| '||K

kk

γ=

=� H . The corresponding eigenvector, 1,1 2,1 ,1

†† † †...

K� �� �V V V is given by †

,11

1k kkγ

′=V H . Hence, there is

only a single term (t = 1) in the capacity region in (5). Using similar method as in [1], for channel realization

k′h , the optimal resource allocation is given by 2

0* 2

'1, arg max

'

0,

kT

kk k k Tk

N nk

n

otherwise

ρ λ λ

+�� � − = �� �= �� ���

hh

. The maximum ergodic

throughput is given by:

0

2*

0

'log ( ) , max k

N kk T

R f dN nµ µ µ

λ∞ �

= = ��

�H

ξξξξ ξξξξ (8)

150

where the distribution ofξξξξ is ( )2 21 1' / ' /

1 1

( ) ( ) ( ) ( )k T T

K K

k j k kj k n nk k

f f F fµ µ λ λ µ λ µ λ≠

= =

= =� � ∏ h hξ ξξ ξξ ξξ ξ, and

21 ' / Tn

Fh

and 2

1 ' / Tnf

h are

the cdf and the pdf of channel respectively and λk is the Lagrange multiplier chosen to satisfy the average power constraint of user-k, k� .

B. Optimal Power Allocation Across nT Transmit Antenna

In previous sub-sections, we have fixed the power allocation ,k tε to find the optimal power control.

Now we want to find the optimal ,k tε . It turns out that in case of partial feedback, the optimal power allocation

strategy for system level perspective is different from that from the link level perspective. This is summarized in the proposition and conjecture below.

Proposition 1. For sufficiently large K, large SNR, and 1Rn = , the optimal power allocation across

the nT transmit antenna }...,,{ ,1, Tnkkk εε=ε when we have scalar feedback is given by:

,1 ,2 ,

,1 ,

1... ( 1)

1, 0 1 ( )

Tk k k nT

k k t R

for link level performance Kn

t for system level performance K n

ε ε ε

ε ε

� = = = = =��

= = ∀ > ≥��

The proof uses Jensen’s inequality and is omitted to save space. An intuitive explanation is that ξ in

(8) is the maximum of some random variables, whose mean does not change with ,k tε and whose variance is

minimized by choosing ,1 ,2 ,

1...Tk k k n

Tnε ε ε= = = = and is maximized by choosing

,1 ,1, 0 1k k t tε ε= = ∀ > . For link level performance, minimizing variance increases the throughput. For

system level performance, maximizing variance increases the mean of ξ , which result in better throughput.

Similar argument leads to the conjecture that for 1Rn > , the conclusion of Proposition 1 is true for interested cases. It is supported by simulation results in Section VI.

Therefore, from a link level perspective, transmit diversity (nT) enhances link capacity by transforming the fading statistics to AWGN statistics. On the other hand, from a system level perspective, multi-user selection diversity is of importance and the optimal power allocation strategy is to focus transmit power on a single transmit antenna only.

V. Optimal Space Time Power Control – per Antenna Vector Power Feedback

In the previous section, we observe that the role of nR is crucial to the scheduling gain in MIMO multi-

user system due to the distributed MIMO configuration. On the other hand, the role of transmit antenna, nT, actually hurts the scheduling performance in both the MISO and MIMO systems. In this section, we shall illustrate that both the scheduling performance and link level performance could be enhanced by nT > 1 (transmit diversity) when there is additional feedback information, vector feedback as discussed in Section III.

For vector feedback, we assume we can control every transmit antennas’ power. Remember ,k tρ

(Section III.) is the controlled power of tth antenna of kth user. The power constraint is ,

1

E [ ]Tn

k t kt

ρ=

=�H� �. Forming

the Lagrange function similar to (6), we obtain , ,

,1 1

max E max ( )T

k t k t

nK

k k tk t

J Rρ ρ

λ ρ= =

� �� = −� � �

� � �� �H H�

� , which is equivalent to a

virtual Knt-user SIMO system, ,

, ,1 1

E max ( )T

k t

nK

k t k tk t

Rρ

λ ρ= =

� �� −� � �

� � ���H H�

� with the constraint ,k t kλ λ= , which implies

power constraint ,E [ ] /k t k Tnρ =H� � for the virtual Knt-user SIMO system.

Therefore, the optimal vector power control problem is solved by splitting the average power equally among each user’s transmit antennas and solving optimal power control problem of the equivalent Knt-user SIMO system with scalar feedback discussed in the previous Section.

VI. Numerical Results and Discussions

In this section, we shall compare the performance of the optimal space-time scheduler through

numerical results. It is shown that the gain of the space-time schedule is attributed by (i) selection diversity

151

among users, and (ii) the distributed MIMO configuration. To highlight the first effect, we compare the performance of optimal scheduler versus random scheduler. By random scheduler, we mean that the selection of user for power allocation is random at a particular moment while the power allocation to the chosen user follows the optimal water-filling allocation. Hence, the effect of selection diversity is highlighted by comparing their capacities (throughput). To highlight the second effect, we shall consider the system capacities at various nR. It shall be illustrated that both the optimal scheduler and the random scheduler are enhanced by the distributed MIMO configuration. For convenience, we consider homogeneous users where users have identical average power constraint, i.e.,

0k =� � for all k. The system capacity is plotted with reference to the per-user

signal-to-noise ratio, 0 0/ N� .

A. Salar Feedback

Effects of transmit diversity (nT) when nR=1: For SISO system (nR = nT = 1), figure 3(a) shows the

ergodic capacity per user of the optimal scheduler versus the random scheduler. It is shown that significant gain of around 5dB at an average capacity of 0.5 bits per channel user per user could be obtained through the channel-adaptive scheduling. In fact, the gain is due to the scheduling diversity where there is a high probability that a user with good channel condition would be selected for transmission in every frame. The effect of scheduling diversity versus the number of active users K could be observed from figure 3(b) where the distribution of

kfξξξξ in Section IV is skewed towards a high value as K increases, meaning that there is a high

probability that ξ would be large. On the other hand, for MISO system (nT > 1 and nR = 1), figure 4(a) and (b) showed the link level

capacity and system capacity of the optimal scheduler at various nT. From section 4.4, we have illustrated that the optimal power allocation strategy (from a system perspective) is to focus transmit power on a single antenna only. However, in order to illustrate the effect of transmit diversity on the system performance, we consider the system throughput when power allocation is uniform across nT = 2 and nT = 8. Notice that although transmit diversity (nT) is beneficial to the link level performance (by transforming the fading statistics into AWGN statistics), it actually hurts the scheduling diversity. This is shown in figure 4(b) where the ergodic capacity drops as the transmit antenna increases. This is because transmit diversity reduces the statistical variation of ξ and therefore, suppress the scheduling diversity. In other words, a 8 x 1 (with equal power allocation on 8 transmit antenna) system is worse than 2 x 1 system in terms of system capacity.

Effect of transmit diversity (nT) when nR > 1: The optimal resource allocation strategy for the MIMO system (nT ≥ 1 and nR ≥ 1) is very similar to the SIMO system and is determined by the admissible index set, A. Again, as illustrated in Section IV the optimal power allocation strategy for system performance is to focus power on single transmit antenna only. However, in order to highlight the effect of transmit diversity on the system performance, we shall plot the results of MIMO system with power uniformly allocated across nT = 1, 2, 4 transmit antenna in figure 5(a). It is shown that the performance of MIMO system is inferior to the SIMO system because the scheduling diversity in forming the set A is reduced. While this enhances the link level performance (as well as the random scheduler performance), the transmit diversity hurts the optimal scheduling performance. Figure 5(b) shows the effect of K on scheduling diversity. Increasing the number of users K helps the scheduling gain because it is more probable to form the set A with larger channel gains. Note that increasing K has no effect on the random scheduler.

Effect of nR: We shall highlight the benefit of having nR receive antenna in the base station by comparing the results of SIMO system versus the SISO system. Figure 6 shows the scheduling gain of SIMO system (nT = 1, nR = 2, 4) versus the SISO system (nT = nR = 1) at various average SNR. Observe that the scheduling performance enhances with nR. The reason for the significant gain is due to the distributed MIMO configuration formed between the mobile users and the base station.

B. Vector Feedback

In the previous subsection, the role of nT is detrimental to the scheduling performance. In this

subsection, we shall illustrate that with nT power feedbacks available to the mobile transmitter, the scheduling performance could also be enhanced by increasing nT.

Figure 7 shows the scheduling performance of MIMO multi-user system with per-antenna power feedback. Observe that as nT increases, the scheduling performance actually increases. This is because nT antenna with independent feedback is equivalent to KnT virtual SIMO users. Hence, increasing nT has the same effect as increasing the total number of virtual users in the system and aids the selection diversity in the scheduling.

152

VII. Conclusion Based on the information theoretical approach, we have developed optimal power allocation schemes

for systems with partial feedback. We show that the optimal strategy for the general MIMO system is to allocation power to at most nR users (when K ≥ nR). This is a generalization of the results obtained for SISO system [1].

We studied the effect of scheduling gain with respect to nR, nT and K. We found that scheduling gain is attributed to (i) distributed MIMO configuration formed between multiple users and base station and (ii) selection diversity among independent users. The first factor is enhanced with increasing nR. For scalar feedback channel, we found that while transmit diversity (nT) is a good for the link level performance, it is detrimental to the scheduling performance in the MISO system and the MIMO system because it hurts the second dimension – selection diversity. In other words, from a system performance perspective, a 2 x 2 system (with uniform power allocation on the 2 transmit antenna) is worse than a 1 x 2 system. On the other hand, when vector power feedback is available, the optimal strategy is to allocate power on a water-filling basis across the spatial, user and temporal domain and increasing nT enhances both the link level and scheduling level performance because selection diversity is not compromised when we have vector feedback. Finally, the scheduling performance is enhanced with K because it is more probable to find good users to form the admissible set A.

Appendix A: Proof of the optimal MIMO power control.

We want to prove the number of nonzero power users T≤A . Observe that the maximizing function

( )R h� in (6) consists of superposition of T terms. However, the maximization problem could not be split into T sub-problems because of the coupling between the sub-problems through the common arguments, [ρ1, ..., ρK].

We first separate the original problem into T sub-problems by introducing additional variables 2

,,

k tk t k t

T

qn

ρ γ=v .

Then 2 ,

1 1 10

1( ) log 1

T K T

k t tt k t

R q rN= = =

� = + = �

� � � �h� , where rt is the tth capacity sub-region. Hence, the original problem

is split into T independent problem with the additional constraint relating the interaction between the T eigen-channels as

[ ],1 ,2 2

1 ,1 ,

0 2,v v

k T k t T

k t k t

q n q nt T

γ γ− = ∀ ∈ . (9)

This introduces additional K x (T - 1) Lagrange multipliers ( )Tkk ,2, ...,, κκ into the optimization

problem. Finally, for the average power constraint term on ρk, we split ρk into � =

ρ=ρ T

tk

k T1. Hence, the

original optimization problem 1 1, ..., , ...,

1

max ( ) max ( )K K

K

k kk

J Rρ ρ ρ ρ

λ ρ=

= −�h h� � is equivalent to

1, 2, , 1,1 2 ,1 ,1 1, 2, ,

,1 ,1 , ' ,2 2, ,..., , ,..., , , ...,

1 1 '2 2 11 ,1 ,

max max maxvt t K t K t t K t

T K T T Kk T k t Tk k

t k t t k tq q q q q q q q qt k t t kk t k t

q n q nJ r r

T Tλ λκ κ

γ γ= = = = =

� �� � � � �� � �� �� � � �= − − + − +� � � � �� � � �� � � �� � � �� �� � � � � �

v (10)

Considering the tth sub-problem in (10), the problem is of the same form as in [1]. Let's define the index set Bt

for the tth subproblem as the set of user indices with the minimum , ''2

,1 2

1 ,1v

Tk

k tt

k

k

Tλ κ

βγ

=

−=

� , ,

, 2

,v

kk t

k t

t k t

Tλ κ

βγ

+=

for t=2,...,K, i.e.

[ ]{ }, ,: , 1, ...,t k t i tk i k K= β ≤ β ∀ ≠ ∈B .

From the results in [1] the optimal received power vector qt = (q1,t, q2,t, …, qK,t ) for the t-th sub-problem has non-zero entry only when k ∈Bt. Note that the set Bt is a function of the Lagrange multipliers

{ }1,2 2,2 ,2 ,, , ..., , ...,K K Tκ κ κ κ κ= . In other words, 0*, =tkq for tk B∉ . However, from (9), we have the

constraint that , , '2 2

, ' , '

k t T k t T

t k t t k t

q n q n

γ γ=

v v for all t ≠ t'. The solution is consistent iff κ is selected such that B1 = B2 = …

BT = B. This is equivalent to the T(|B| - 1) equations , ',k t k tβ β= for all tkk B∈', and for all t ∈ [1, T]. These

equations have |B|(T - 1) unknowns )( ,tkκ . Note that with probability one, these equations will be inconsistent

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when the number of unknowns is smaller than the number of equations. Hence, for consistent solution of tk ,κ ,

we must have ( 1) ( 1)T T− ≥ −B B , which implies T≤B . Therefore, the number of non-zero optimal qk,t is at

most |B| = T. Going back to the original problem in terms of ρk, this implies that the maximum cardinality of the set of user indices with non-zero ρk , A, is at most T.

References

[1] D. Tse and S.V. Hanly, “Multiaccess Fading Channels – Part I: Polymatroid Structure, Optimal Resource Allocation and Throughput Capacities,” IEEE Trans. On Information Theory., vol.44, pp. 2797-2815, Nov 1998. [2] G. J. Foschini and M. J.Gans, “On the limits of wireless communications in a fading environment.,” Wireless Personal Communications., pp. 315-335, Nov 1998. [3] E. Telatar, “Capacity of Multi-antenna Gaussian Channels,” European Trans on Communs., vol. 39, pp. 585-595, Nov 1999. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: performance criterion and code construction,” IEEE Trans. on Information Theory., vol. 45, pp.744-765, March 1998. [5] V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. on Information Theory., vol.45, pp. 1456-1467, July 1999. [6] Y. Liu, M. P. Fitz, and O. Takeshita, “Full rate space-time turbo codes,” IEEE JSAC., vol. 36, pp. 969-980, May 2001. [7] A. Jalali, R. Padovani, and R. Pankaj, “Data Throughput of CDMA-HDR a High Efficiency-High Data Rate Personal Communication Wireless System,” Proceedings of VTC 2001, vol. IT-28, pp. 55-67, Jan. 1982. [8] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushyana, and S. Viterbi, “CDMA/HDR: a bandwidth efficient high speed wireless data service for nomadic users,” IEEE Communication Magazine, vol. 7, pp. 70-77, July 2000. [9] K. Lau and Y. Kwok, “Multiple access control protocol for integrated isochronous and bursty data services.” IEE Proceedings on Communications, pp. 311-316, Dec. 2000.

[10] K. Lau and Y. Kwok, “CHARISMA: a novel channel-adaptive TDMA-based multiple access control protocol for integrated wireless voice and data services,” Proceedings of the IEEE Wireless Communications and Networking Conference, 2000, pp. 507-511, Sept. 2000. [11] R. Ahlswede, “Multi-way communications channels,” Proc. 2nd Int. Symp. Information Thoery., pp. 23-52, 1971. [12] G. P. McCornmick, “Nonlinear Programming,” John Wiley and Sons, 1983. [13] P. Viswanath, D. Tse, V. Anantharam, “Asymptotically Optimal Waterfilling in Vector Multiple Access Channels,” IEEE INFO T; V47, 1, JAN, 2001, p241-267. [14] W. Yu, W. Rhee, JM Cioffi, “Optimal Power Control in Multiple Access Fading Channels with Multiple Antennas,” ICC 2001, Helsinki, Finland, June 2001. [15] A. Kogiantis, L. Ozarow, “Distributed Multi-Antenna Scheduling for Downlink and Uplink Packet Data,” to appear in IEEE Transactions on Wireless Communications, and Lucent Internal Technical Memorandum, July 2001. [16] R. Knopp, P. Humblet, “Information Capacity and Power Control in Single-Cell Multiuser Communications,” ICC 1995. [17] A. Kogiantis, N. Joshi, O Sunay, “On Transmit Diversity and Scheduling in Wireless Packet Data,” ICC 2001, Helsinki, Finland, June 2001. [18] H.C. Huang, H. Viswanathan, A. Blanksby, and M. Haleem, “Multiple antenna enhancements for a high rate CDMA packet data system,” to appear in the Journal of VLSI Signal Processing, 2001. [19] H. Viswanathan, -S. Venkatesan, “The Impact of Antenna Diversity in Packet Data Systems with Scheduling,” Lucent Internal TM, 2001. [20] R. Gallager, “Information Theory and Reliable Communication”, John Wiley and Sons, 1968.

Figure 1: System Model of Multi-user MIMO system. Figure 2: Frame format of uplink and downlink.

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Figure 3a: Comparison of ergodic capacity of optimal scheduler versus random scheduler. nT=nR=1, K=10. (Selection diversity is highlighted by comparing optimal scheduler with respect to random scheduler.)

Figure 3b: Variation of the distribution f(ξk) as the number of active users, K, increases. (Illustration of selection diversity.)

Figure 4a: Link capacity versus nT. (nR=1) (scalar feedback)

Figure 4b: System capacity versus nT . (nR=1) Note that for illustration purpose, the power is uniformly allocated across the Transmit antennas. (Scalar Feedback)

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Figure 5a: MIMO vs SIMO scheduling gain (Scalar Feedback). For the sake of illustration, we assume transmit power is uniformly allocated to nT transmit antennas.

Figure 5b: MIMO scheduling gain vs K. (Scalar Feedback)

Figure 6: Scheduling performance of the SIMO vs SISO system with scalar feedback. nT =1, nR =1, 2,4 and K=4.

Figure 7: Scheduling performance with per-antenna vector power feedback.

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