05484087

2216 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 7, JULY 2010

Energy-Aware Utility Regions:Multiple Access Pareto Boundary

Eduard Jorswieck, Holger Boche, and Siddharth Naik

AbstractPower management and energy-aware communica-tions systems have become increasingly important in mobilecomputing as well as mobile communications. In future wirelesscommunication systems, the energy efficiency of terminals andbase stations has to be improved significantly. Therefore, wepropose a new utility function, which is the difference of thecapacity and a weighted power cost term. The generally usedindividual power constraint is removed. Next, the utility regionfor single-antenna and multi-antenna multiple access channelsis characterized. We show using basic principles that the single-input single-output (SISO) multiple-access channel (MAC) utilityregion is convex and provide a closed form expression for itsPareto boundary. We need the Pareto boundary to compute effi-cient operating points. Furthermore, the extension to multiple an-tenna channels is indicated by an iterative algorithm for weightedsum utility maximization in multiple-input single-output (MISO)and multiple-input multiple-output (MIMO) MAC. All discussedresults are illustrated by numerical simulations.

Index TermsMultiple access channel, utility region, convexoptimization, multiple-antenna.

I. INTRODUCTION

POWER management and energy efficient communicationis an important topic in future mobile communicationsand computing systems. Currently 0.14% of the carbon emis-sions are contributed by the mobile telecommunications in-dustry[1]. In order to improve the situation, three differentapproaches can be adopted: 1. development of better energysources, 2. network and site optimization, and 3. improvedalgorithms at physical and multiple access layer. The firstapproach lies outside the scope of the current paper and ina different community. The second approach is undertaken in[2] where a energy efficient network protocol is proposed. In[3] an extension and simplification of the protocol is proposed.Still at the network layer, a framework for computation ofthe capacity of ad-hoc networks under energy constraints isproposed in [4]. Finally, in [5], a Mobile Backbone NetworkPower Saving protocol is developed to reduce energy con-sumption. A lot of further recent work on wireless sensornetworks considers energy-aware routing and flow control (e.g.[6][8]).

Manuscript received March 26, 2009; revised February 3, 2010; acceptedFebruary 26, 2010. The associate editor coordinating the review of this paperand approving it for publication was E. Hossain.

E. Jorswieck is with the Dresden University of Technology, Communi-cations Laboratory, Chair of Communication Theory, Dresden, Germany (e-mail: [email protected]).

H. Boche and S. Naik are with the Fraunhofer Institute for Telecommuni-cations, Berlin, Germany (e-mail: {boche, naik}@hhi.de).

This work is supported in part by the German Federal State of Saxonyin the Excellence Cluster Cool Silicon in the framework of the project CoolCellular under grant number 14056/2367.

Digital Object Identifier 10.1109/TWC.2010.07.090437

One example for the third approach is [9] where theenergy efficiency of point-to-point communication systemsis improved by sophisticated adaptation strategies. A codingtheoretic approach is proposed in [10] where green codesfor energy efficient short-range communications are devel-oped. The multiuser scenario is more difficult because ofthe conflicting interests of competing users and the resultingmulti-criteria objective function. However, fewer results areavailable for energy-aware resource allocation at the physicaland multiple access layer. Therefore, there are recent proposalsto define a utility function which incorporates the cost oftransmission, e.g., the price of spending power is consideredin a binary variable in [11] and as an inverse factor in [12].

The utility region consists of utility vectors that are achiev-able by a certain set of resource allocation and transmit strat-egy parameters. In order to operate the system in an efficientway, the Pareto boundary of the utility region needs to be wellunderstood. In Figure 1, we show a typical utility region - therate region of the SISO MAC with successive interferencecancellation (SIC) at the base station [13] under individualpower constraints. The typical polymatroidal structure can beobserved. The line between the operating points in which bothusers apply maximum transmit power and different decodingorders are applied, is achieved by rate splitting [14]. All pointson this line are Pareto optimal. The points on the vertical andhorizontal lines are achieved by a fixed decoding order andthe user who is decoded last varies its transmit power betweenzero and maximum power. These points are called weaklyPareto optimal.

On the Pareto boundary, different efficient operating pointscan be adjusted depending on the fairness criteria, e.g., themaximum sum utility (from an operators point of view),proportional fair utility (from a long-term fairness perspec-tive), or the max-min fair utility point (from a short-term userperspective).

In the current paper, we propose an energy-centric point ofview and define a utility function which is the difference ofthe individual capacity and an individually weighted powercost term. The weights contain context information of thespecific user and could include battery or quality-of-service(QoS) information. Note that in [15] a similar utility functionis proposed to improve the efficiency of the non-cooperativeNash equilibrium. The difference of the current work is thatwe characterize the Pareto boundary and do a centralized(cooperative) optimization of transmit strategies and powerallocation. Using very basic principles, we provide a completecharacterization of the SISO MAC utility region.

The contribution and the organization of the paper are asfollows:

1) The system model, the proposed energy-aware utility

1536-1276/10$25.00 c 2010 IEEE

JORSWIECK et al.: ENERGY-AWARE UTILITY REGIONS: MULTIPLE ACCESS PARETO BOUNDARY 2217

1

2 Weak Pareto Optimal

Weak Pareto Optimal

Pareto Optimal

Utility Region

Fig. 1. Example utility region - SISO MAC with SIC.

function, and the term Pareto-optimality are explainedin Section II.

2) The main results for the SISO MAC are collected inSection III. First, the Pareto optimal power allocationis characterized. Next an efficient algorithm that max-imizes the weighted sum utility is derived. Finally, weshow that the utility region is convex.

3) The main results and differences for the SIMO MACare derived in Section IV. It is shown for the two usercase that the region is not convex. An algorithm for sumutility maximization is derived and low- and high-signal-to-noise-ratio (SNR) behavior is characterized.

4) The MIMO MAC case is studied in Section V. We pro-pose a simple, however an efficient iterative algorithmto solve the weighed sum utility maximization problem.

The paper is concluded in Section VI.

A. Notation

Vectors are denoted in bold letters . Matrices are writtenin bold capital letters . Transpose is [] , and the conjugatetranspose is [] . The matrix (pseudo) inverse is denoted by[]1. () denotes the trace of the matrix , i.e. () =

=1,. () is the th eigenvalue of the matrix .() is the largest eigenvalue of the matrix . ()denotes the real part of the complex variable . + denotesthe set of positive semidefinite matrices. is the l1-norm,i.e. = =1 . The order for matrices is denoted by and this means that the difference is positivesemi-definite. diag() is the vector with diagonal entries of and Diag() is a matrix with entries of the vector on thediagonal.

II. PRELIMINARIES

A. System Model and Channel Model

We consider the MAC, i.e., the uplink, where a numberof terminals wants to transmit their information to a base

station. In the following, we distinguish between three casesdepending on the number of antennas at the terminals orthe base. In the single-input single-output (SISO) MAC allterminals as well as the base have a single transmit or receiveantenna, respecively. The received signal at the base station isgiven by

=

=1

+ (1)

where is the transmitted signal of user and corre-sponds to the complex channel coefficient of user . We denotethe channel gain of user by = 2. The additive whiteGaussian noise is zero-mean complex Gaussian distributedwith variance 2. If the base station has multiple receive an-tennas, we talk about the single-input multiple-output (SIMO)MAC, the received signal vector is given by

==1

+ (2)

where is the transmitted signal of user and is thevector channel from user to the receive antennas at the basestation. The additive white Gaussian noise vector at the baseis zero-mean complex Gaussian distributed with covariancematrix 2 . In both cases (1) and (2), we denote the transmitpower of user as , i.e., [2] = . Finally, the casein which the users as well as the base have multiple antennasis called multiple-input multiple-output (MIMO) MAC. Thereceived vector at the base station is given by

=

=1

+ (3)

where is the transmitted signal vector of user withtransmit covariance matrix =

[

]and the channel

matrix collects the channel realizations from all transmitantennas of user to the base. The additive white Gaussiannoise vector at the base is zero-mean complex Gaussiandistributed with covariance matrix 2 . The transmit powerof user is = tr().

In all scenarios, we assume that the base station has perfectchannel state information (CSI). Further, we assume that thetransmitters are informed about their transmit strategies (eitherpower allocation or transmit covariance matrix) by the basestation via a control channel.

B. Energy-Aware Utilty Function

Define the utility of user for some 0 as() = () (4)

where () is the achievable transmission rate of user inone of our system environments described above and is thepower allocation vector = [1, ..., ].

Let us illustrate the operational meaning of the utilityfunction in (4) in the single-antenna scenario. We denote thesingle-user rate without interference as () = log(1+ )where is the inverse noise power and is the channelpower. The corresponding utility is () = log(1+).In contrast to the pure rate function, the utility in (4) takesthe cost of transmission into account. If some power is


Fig. 2. Utility function ( ) = log(1 + 10) over in linear scale.

spent for transmission it linearly reduces the utility by afactor . Let us call the power price of user . Thisinterpretation is illustrated by the following scenario: if gets too large compared to the log(1 + ) term, i.e., thecosts of allocating power are larger than the gain by reliabletransmission, the user will switch off, i.e., = 0, until thegain by transmission is larger than the cost. The key designquantities are the rate [bit/s], the transmit power [W],and the price per unit of power [bit/s/W].

The variable occurs also in the optimization of the rateswith sum power constraints as Lagrangian dual variables [16].In this context, does not posses the meaning explainedabove. However, the resulting iterative algorithm could havea similar form. The difference to [16] is that we eithercompletely characterize the Pareto boundary (SISO MAC) orpropose an algorithm for the weighed utility maximization(MIMO MAC) which is a non-trivial generalization of [16].

A similar utility function is proposed in [17] for single-antenna systems and used to characterize the Nash equilibriumfor the non-cooperative power control game. Later in [18] theapproach is extended to multiple antenna channels in a relatednon-cooperative game-theoretic setting.

In Figure 2, the utility function is shown for different valuesof . By choosing the power price carefully, we can forcethe user to operate in the energy efficient power regime. Therate function scales logarithmically with the power. From anenergy efficiency point of view, the power regime in whichthe rate scales linearly with the power is of interest.

Note, that the utility function is unimodal, i.e. monotonicincreasing up to the maximum and then monotonic decreasing,and concave with respect to . The optimal single-user oper-ating point is easily found by computing the first derivative ofthe utility with respect to and solving to obtain

=(

1

1

)+. (5)

The corresponding achievable utility is

() = log

(

)+[1

]+. (6)

Note that () in (6) decreases with and increaseswith . Furthermore, the optimal operating point in (5)

= 2 = 3

0 10 20 300.0

0.1

0.2

0.3

0.4

Fig. 3. Optimal single-user power allocation as a function of .

TABLE ICOMPARISON: CAPACITY REGION AND UTILITY REGION WITH PRICING

Capacity without pricing Utility including pricing

Power con-straints

Explicit transmit powerconstraints

Implicit power constraintby prices

SISO region Decoding order dependsonly on weights,Decoding order depends onthe effective channels in-cluding pricing

MIMOregion andoptimiza-tion

Iterative waterfilling (IWF)for sum capacity maxi-mization

Inner optimization withIWF, outer optimizationfor sum transmit power

Single-user range dependson power constraint andchannels

Single-user range dependson channels and prices

corresponds to an effective power constraint of 1 illustratedin Figure 3 where the power as a function of is shown.From (6) follows that zero power is allocated if the slope ofthe rate at zero is smaller than , i.e., if the power priceis too large compared to the rate gain.

From (5) and Figure 3 follows that for all > 0 animplicit power constraint of 1 is given. Therefore, we willnot impose an additional explicit power constraint in thefollowing derivation.

Remark 1: It is a remarkable fact, that we do not needan explicit individual or sum power constraint in the single-user scenario. Instead by using the linear power price thesystem automatically operates in an efficient regime. As aconvenient byproduct the SNR cannot be defined explicitlyand the dependency of the result on the SNR is expressedindirectly by which is the inverse noise variance. We notethat in interference networks, the relation between the price and the maximum transmit power of user is morecomplicated and explicit power constraints could be applied.

The difference of the utility function in (4) compared tothe rate function with additional transmit power constraintis that the utility function forces the user to operate in thepower efficient regime. In Table I, the relationship betweenthe capacity region without pricing and the proposed utilityfunction and its corresponding region are summarized.


C. Pareto Optimality

Next, we study the efficient operating points in the multiuserutility region. The achievable utility region is defined as

0

{1(), ..., ()} + . (7)

We define the Pareto optimality of an operating point, i.e.,utility tuple, as follows.

Definition 1: A utility tuple (1, ..., ) is Paretooptimal if there is no other tuple (1, ..., ) with(1, ..., ) (1, ..., ) and (1, ..., ) = (1, ..., )(the inequality is component-wise).

As shown in Figure 1, all other boundary points of the utilityregion are weak Pareto optimal. At a weak Pareto optimalpoint it is possible to increase the utility of one user withoutdecreasing the other users. We are not interested in theseinefficient operating points because at those operating pointswe could improve the utility of one user without reducing theutility of another user.

III. UTILITY REGION - SISO MAC

It is well known that the capacity region of the SISOGaussian MAC can be theoretically achieved with SIC at thebase station and Gaussian transmit signals. Since our utilityfunction in (4) is linear in the user rates, we are interested inmaximizing the individual rates. Therefore, we consider the user Gaussian MAC with SIC at the base station.

Assume a decoding order of 1 ... 1. Theachievable rate of user is given by

() = log

(1 +

1 + 1

=1

). (8)

Observe that the individual user rates depend on the SICdecoding order.

A. Pareto Boundary of the SISO MAC

The next result parameterizes the Pareto boundary of theutility in (7). We assume a decoding order of 1... 1 induced by

11

22

...

. (9)

The reason for the optimal ordering in (9) is that we coulddefine a new power allocation = and rewrite theutility function in (4) as

() = log

(1 +

1 + 1

=1

)

with the effective channels(11, 22 , ...,

). Therefore, in

(9), the effective channels are ordered in increasing order. We

define the following set of power allocations

={

(1, 2, ..., ) + :

0 1 (

11 11

)+,

0 2 (

12 12 12 1

)+,

...

0 (

1 1

1=1

)+,

...

=(

1

1 1

=1

)+ }. (10)

Note that is a function of 1, ..., 1.Theorem 1: A power allocation is Pareto optimal if and

only if defined in (10) with SIC decoding order in(9).

Proof: The proof consists of two parts. In the first part,we show that the optimal SIC order is characterized by (9).In the second part, we show that the power allocation in (10)corresponds exactly to all Pareto optimal utility points.

The main idea for the first part of the proof is not tostudy the utility region for different decoding orders but thecorresponding power regions, i.e., the set of feasible powerallocations for the effective channels . It is sufficient to

consider two users and + 1 and define =1

=1.

Define the feasible power region of user , + 1 for thedecoding order + 1 as +1,+1 and for the otherdecoding order + 1 as +1,+1 . They are given by

+1,+1 ={, +1 2+ :

0 (1

+1+1

+1 )+

,

0 +1 (1 +1+1

+1+1

)+ }

(11)

and for the other (optimal) decoding order

+1,+1 ={, +1 2+ :

0 (1

)+

,

0 +1 (1 +1+1

+1+1

+1+1

)+ }. (12)

In order to prove that the decoding order + 1 is betterthan + 1, we show that

+1,+1 +1,+1using the assumption made above that

+1+1

1. (13)

The idea is illustrated in Figure 4.Obviously, the two regions are described by triangles since

depends linearly on +1 and vice versa. Hence, it sufficesto show that the two corner points and of +1,+1are larger than those of +1,+1 . For the right corner , achieves its maximum value for +1 = 0 in +1,+1but in +1,+1 achieves its maximum value already for


+1

+1,+1

+1,+1(1

)+

(1 +1+1

+1+1

)+

(

+1+1

+1+1 +1+1

)+

(1 +1+1

)

Fig. 4. Illustration of power regions +1,+1 and +1,+1 in the proofof Theorem 1.

+1 =(1 +1+1

)> 0 because of (13). For the

left corner , in +1,+1 , is equal to zero if +1 (

+1+1

+1+1 +1+1

)+

, but in +1,+1 , is equal tozero if +1

(1 +1+1

+1+1

)+

. Therefore, decodingorder +1 has larger power region than decoding order + 1. The final argument for an arbitrary permutation = {, 1, 2, ..., 2, 1} is that successive reorderingof two neighbor indices increases the utility region.

With the Pareto optimal decoding order, the Pareto optimalpower allocation can be obtained iteratively starting by thelast decoded user one. With the range of the optimal powerrange of user +1 is obtained from solving +1()+1 = 0. Inorder to show that all power allocations characterized by (10)are Pareto optimal, there are two parts to prove. The first partis to show that all power allocations / are not Paretooptimal. The second part is to show that all power allocations are Pareto optimal.

All / are not Pareto optimal: = We will show thisdirection by contradiction. We choose a power vector =[1, ..., ] / . Then either one or multiple for 1 1 lie outside the interval specified by (10) or/and isnot equal to .

1) Assume =(

1 1

1=1

)but all =

for 1 1. Then we can increase the utility ofuser , without affecting the utilities of the other users,by choosing = because the global optimumfor (1,

2, ...,

1, ) for fixed

1, ...,

1 is

characterized in (10). Hence, cannot be Paretooptimal.

2) At least one > in (10) for 1 1.Then we can improve the utilities of users , +1, ...,by reducing to

(1 1

1=1

). The utility

of user is improved because the global optimum of(

1, ...,

1, , +1, ..., ) for fixed

1, ...,

1 is

. The utility of the subsequent users + 1, ..., isimproved because they observe less interference fromuser . Hence, cannot be Pareto optimal.

All are Pareto optimal: = We show this directionby contradiction, as well. Assume that there are two powervectors , for which is not Pareto optimal, i.e.,() > (), i.e., there is at least one with () > ()

and () () for all 1 . Consider () and(): Reduce the power of user until () = (). Thenthe following inequality holds

+() > +() +()

for all > 0 because user creates less interference for .Especially, we have that

() > () (). (14)

From (14) follows that the constructed / with theproperty that () > () for which is a contradictionto the first part of the proof. Hence, all must be Paretooptimal.

Remark 2: An important special case from above is the sumutility maximization, i.e.,

max0

log

(1 +

=1

)

=1

. (15)

The optimization problem is a convex optimization, i.e., maxi-mization of concave function over convex constraint set plus atleast one feasible point [19]. Introduce Lagrangian multipliers for non-negativeness constraint, and derive the necessaryand sufficient Karush-Kuhn-Tucker optimality conditions

1 +

=1 = , = 0, 0. (16)

The Lagrangian multiplier is zero if > 0, therefore itmust hold for all active users , = { : > 0}that =

. Therefore, the only active user is the best usercorresponding with max1 . This is interesting sincewe do not have individual or sum power constraints.

For users, the Pareto boundary is a 1 dimen-sional surface parameterized by the powers of 1 users[1, ..., 1].

Corollary 1: For the decoding order in (9), the achievableutilities are characterized by

1(1) = log (1 + 11) 112(1, 2) = log (1 + 11 + 22)

log(1 + 11) 22

(1, ..., ) = log

(1 +

=1

)

log(1 + 1=1

)

(1, ..., 1) = log(

) log(1 +

1=1

)

[1

1=1

]+. (17)

The utility region can also be expressed in terms of the


0 1 2 3 4 5 60

1

2

3

4

5

6

Utility u1

Util

ity u

2

SIC order 12SIC order 21Pareto boundary

Fig. 5. Two-user MAC utility region and its Pareto boundary for 1 =2, 2 = 1, 1 = 3, 2 = 1, = 20.

following equalities

=1

() = log

(1 +

=1

)

=1

for all 1 1 and=1

() = log

(

)[1

1=1

]+

1=1

. (18)

Note that utility is a function of the utilities1, ..., 1. Denote the utility region described by (17) or(18) as , i.e.,

=

().

For the two-user case, the parameterization is illustrated inFigure 5 below.

B. Weighted Sum Utility Maximization - SISO MAC

Consider the following weighted sum utility maximizationproblem

max0

=1

() (19)

with 0 for 1 and

=1 = 1. Fromthe discussion around problem (15), we know that the sumutility is maximized if only the best user transmits. Assumethe optimal decoding order induced by 11 22 ... ,it follows that

1 2 ... . (20)For all other weights, the maximum sum utility solution solvesalso (19). In [20] it is shown that users with larger weights should be decoded later. There, the weights correspondto the buffer occupancy. The inequalities in (20) induce thesame decoding order 1 ... 1 induced by11 22 ... .

1) Characterization of the Optimal Power Allocation:In the following result, we characterize the optimal powerallocation which solves the weighted sum utility maximiza-tion. Based on this characterization, an efficient algorithm isdeveloped which solves the problem (19) in steps.

Theorem 2: The optimal power allocation which solves(19) satisfies for all 0 1

=1

=

(

+1+1+1 1

)+

= (), (21)

where = ( +1) with +1 = 0 by definition.Furthermore, the following special cases can be distinguished:

1) All users allocate zero power if and only if(

1

) 0. (22)

2) For all inverse noise variance only user allocates non-zero power, i.e., we operate in the so calledsingle-user optimality range,

=

(11

1

)1 . (23)

3) There does not exists any single-user optimality rangeat all, i.e., = 0, if and only if

111

0. (24)

Remark 3: Interestingly, the all zero power condition in(22) does not depend on the weights .

Proof: The first step is to rewrite (19) as a convexprogramming problem using the Pareto optimal decodingorder and the assumption on the order of the weights in (20)as=1

() =

=1

log

(1 +

=1

)

=1

. (25)

Hence, the non-negative sum of a concave function withrespect to and the sum of linear functions with respect to thepower vector results in a concave function. The constraintset is convex, a feasible point exists and therefore, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary andsufficient for all 1

=

1 +

=1 = , = 0,

0, 0, (26)where is the Lagrangian multiplier for the non-negativenessconstraint. Evaluate the optimality conditions (26) for = to obtain

1 +

=1 = .

Solving for

=1 and note that = yields

=1

=

(

1

)+. (27)


In the following, assume that 1 . Otherwise, allusers would allocate zero power. Furthermore, evaluate theoptimality (26) for = 1 to obtain

111 +

1=1

+1

1 +

=1 = 11. (28)

Insert (27) into (28) to obtain

111 +

1=1

= 11 1

.

Solve for 1

=1 to obtain

1=1

=

(11

11 1 1

)+. (29)

If the expression in bracket on the right-hand-side (RHS) of(29) is negative, zero power is allocated to users 1, ..., 1 and user is the only active. From (29), the single-useroptimality range in (23) follows by solving for .

Now, we consider = for arbitrary 0 1.For this choice of , it follows from (26) that it holds

1 +

=1

++1

1 + +1

=1 + ...

+1

1 + 1

=1 +

1 +

=1 =

. (30)

For = + 1, (26) yields+1

1 + +1

=1 + ...+

11 +

1=1

+

1 +

=1 =

+1+1+1

. (31)

Subtracting (31) from (30) we obtain

1 +

=1

=

+1+1

+1. (32)

Solving (32) for

=1 and taking the Lagrangian mul-tipliers for non-negativeness of 1, ..., into account weobtain

=1

=

(

+1+1+1 1

)+

= (). (33)

The optimal power allocation 1, ..., has to fulfill the

necessary and sufficient optimality conditions and thereby forall 0 1 the equation (33). This completes the proof.

2) Algorithm for Weighted Sum Utility Maximization:Based on the characterization in Theorem 2, the followingalgorithm described in Algorithm 1 solves (19).

The algorithm 1 works as follows. It checks first the all-power off condition from Theorem 2 in (22). Then, power touser is computed as the difference between () and(+1)

. Then, it starts with = 0 and verifies that the + 1-user optimality condition is not satisfied, e.g., for = 0 it is(23) in Theorem 2. After at most steps the algorithm stopsand returns the optimal power allocation in .

Result: Solve optimization problem (19)Input: Channel realizations 1, ..., , power costs

1, ..., , weights 1, ..., init: = 0;if (0) 0 then

allocate zero power to all users - break;endfor = 0... 1 do

=()(+1)

;

if ( + 1) 0 then = ;return;

endendOutput: Optimal power allocation in 1, ...,

Algorithm 1: Optimal power allocation for weighted sumutility maximization of SISO MAC.

C. Convexity of the Utility Region of SISO MAC

The next result shows that the utility region of the SISOMAC is convex.

Theorem 3: The SISO MAC utility region characterized in(17) is convex. For two utility tuples = [1, ..., ] and =[1, ..., ] inside the utility region, the linear combination() = + (1 ) with 0 1 is also inside theutility region, i.e.,

, = () .

Proof: We will prove Theorem 3 in two steps. In thefirst step, we show that uniqueness of solution of weightedsum utility maximization implies convexity. In the secondstep, we show that the solution to the weighted sum utilitymaximization problem in (19) is unique. Denote the convexclosure of as . This corresponds to the linear hull of theextreme points with respect to the weighted sum utility.

1) If max

=1 has a unique solution () for

all = [1, . . . , ], with [0, 1], for all , 1 with =1 = 1 then the region defined in(17) or (18) is strict convex.We show this by contradiction: Assume that is notstrict convex. Then there are at least two (1) and (2)

(extreme points) such that some part of () = (1 )(1)+(2) for 0 1 is not in , i.e., () / for some . It follows that there are intervals [1, 2] with1 < 2 for which () does not belong to the boundaryof . Then, there exists a vector > 0 such that

() argmax

=1

leads to

max

=1

= max

=1

=

=1

(1)

==1

(2)

which contradicts the uniqueness of the solution.


0

0.5

1 0 0.51 1.5

2.5

3

3.5

4

4.5

u2

u1

u 3

Fig. 6. Three-user MAC utility region for fixed channel realization and1 = 2 = 3 = 0.01.

2) The solution to (19) is unique.The programming problem in (19) is a strictly concavemaximization problem over convex constraint set (seethe Proof of Theorem 2). Therefore, there must be aunique global optimum. The iterative Algorithm 1 caneasily locate the unique global optimum.

For the three-user case, the parameterization is illustratedin Figure 6 below. The convexity of the utility region provedin Theorem 3 can be clearly observed.

IV. SIMO MAC

If the base station has multiple receive antennas, the optimalSIC order is far more difficult to characterize. Therefore, webegin by investigating the special case when we have twousers. Denote the channels between the mobiles and the BSas 1, ..., .

A. Two User Case

Theorem 4: The optimal power allocation for the two usercase and decoding order 1 2 is given by

0 2 (

1

2 1

22)+

,

1 =

(1

1 1

12 1

)+(34)

where 1 =221 221+222 . The optimal power allocation for

the decoding order 2 1 is given by

0 1 (

1

1 1

12)+

,

2 =

(1

2 1

22 2

)+(35)

where 2 =211 221+112 .

Proof: The proof for the decoding order 1 2 is shown.The proof for the other decoding order is analogue. Denote

the utility of user one for decoding order 1 2 as 1,1. Theutility of user one can be written as

1,1() = log det( + 1

11 1

1

) 11,

with 1 =[ + 22

2

]. By the matrix-inversion lemma

11 =

[ + 22

2

]1= 2221+222 we obtain the

utility

1,1() = log

(1 + 121 12

1 22

1 + 222) 11. (36)

Next, the upper bound for the power 2 is obtained from themaximum of 2,1(2) by

2,1(2)

2=

221 + 222 2 = 0

which leads to

0 2 (

1

2 1

22)+

. (37)

The optimal power 1 is now obtained from (36) as a functionof 2 by setting the derivative with respect to 1 to zero, i.e.,

1,1()

1=

12 11 + 121 11 1 = 0 (38)

where 1 =222 121+222 . Solving (38) for 1, we obtain the

optimal power allocation

1 =

(1

1 1

12 1

)+which completes the proof.

From the result follows the set of achievable utilities de-noted by 1,1, 1,2 for user one and decoding order 1 2and 2 1, respectively, and 2,1, 2,2 for user two and forthe two decoding orders.

Corollary 2: The utilities on the Pareto boundary for fixeddecoding order 1 2 are

1,1(2) = log

(12 1

1

)+[

1 112 1

]+(39)

2,1(2) = log(1 + 222

) 22. (40)For fixed decoding order 2 1 they are

1,2(1) = log(1 + 121

) 11 (41)2,2(1) = log

(22 2

2

)+[

1 222 2

]+. (42)

Remark 4: In the optimal power allocation as well as in theutility expressions the impact of the spatial correlation of thechannels 1 and 2 can be clearly observed

0 1,2 2212

1 + 222where the lower bound is achieved for completely uncorrelatedchannels 1 and 2.

The characterization of the Pareto boundary is illustrated inFigure 7 below.


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

Utility u1

Util

ity u

2

SIC order 21SIC order 12Boundary 21Boundary 12

Fig. 7. Two-user SIMO MAC utility region and its Pareto boundary for fixedchannel realization and SNR 0 dB.

B. Sum Utility Maximization - SIMO MAC

The sum utility of the SIMO MAC with SIC (arbitrarydecoding order) reads

() = log det

( +

=1

)

=1

. (43)

The corresponding programming problem

max0

() (44)

is a convex optimization problem and standard interior pointmethods could be used to find the optimal power allocationvector. In the next section, we will propose an iterative water-filling type of algorithm to solve the sum utility maximizationof the MIMO MAC. This algorithm could be obviously usedto solve (43) using power instead of transmit covariancematrices for inner optimization. However, in the singletransmit antenna case, we can characterize the optimal powerallocation for low SNR. First, we convert the programmingproblem in (44) into a familiar representation. Define

= =

(45)

and define

=1 = . Then the programming problem(44) is equivalent to

max0

max0=1

log det

( +

=1

) (46)

with effective channel =

. Note that the inner maxi-mization is similar to the SIMO MAC sum rate maximationwith sum power constraint [21]. Next, we are interested ina characterization of the optimal power allocation by theparameter range in which only one user is active, i.e., 1 > 0and 2 = ... = = 0.

Theorem 5: If the users are ordered according to

121

22

2 ...

2

(47)

then there is a range - the so called single-user range - of0 such that 1 12 and

= 1

( 122 2211222 1 22

+1

12)

(48)

10 5 0 5 100

0.5

1

1.5

2

SNR [1/2] dB

pow

er a

lloca

tion

psu

m u

tility

us(

p)

p2

p1

p3

us(p)

Fig. 8. Three-user SIMO MAC sum utility and corresponding optimal powerallocation. Single-user optimality range = 4.5632 dB as given in (48).

such that only one user is active with the optimal powerallocation

1 =

(1

1 1

12).

Proof: As in [21, Theorem 6], we sort the users accordingto their effective channels

12 22 ... 2 (49)which is the same order as in (47). Then the result says thatfor fixed 0, only the first user is supported, i.e., 1 = and 2 = ... = = 0, if and only if

1

[ + 1

1

]11 2

[ + 1

1

]12. (50)

Using the matrix-inversion lemma again, it follows from (50)

12 22 (1222 1 22

). (51)

On the other hand, in the single-user range, we have theoptimal power allocation

=(

1

1 1

12). (52)

Using the optimal single-user power allocation from (52)in (51) leads to the desired single-user optimality range in(48).

Let us illustrate the single-user optimality range in Figure 8.Three users with power prices = [1.2, 1, 0.8], four receiveantennas and three channel realizations given by

= [1,2,3] =0.73 0.08 0.51 + 0.01 0.31 0.060.54 + 0.09 0.02 + 0.57 0.02 0.721.53 0.56 1.08 + 0.49 0.22 + 0.660.30 0.16 1.20 0.16 0.69 0.80

(53)

are used to compute the optimal power allocation (using theiterative single-user optimization outline in the MIMO MACsection) and the corresponding maximum sum utility. Theoptimal user order for this set of channels and power pricesis 2 1 3 for small SNR.

One interesting observation is that the order of the usersregarding their power allocation changes completely from


small SNR 2 1 3 to high SNR 3 1 2. Forhigh SNR, the optimal power allocation converges to 1 , inthe example above this corresponds to

1 11.2

0.833, 2 1, 3 1.25.The low- and high- SNR characterization corresponds wellwith the behavior of the power allocation observed in Figure8. In the next section, we allow the transmitter to have multipleantennas and study the sum utility maximization of the MIMOMAC.

V. MIMO MAC

In the MIMO MAC, each multiple antenna transmittercan optimize its transmit covariance matrix including itsoverall transmit power tr().

In [22], an iterative waterfilling algorithm is proposed tosolve the sum capacity maximization of the MIMO MAC.The idea is to perform alternating optimization by keepingall users transmit covariance matrices fixed expect user and optimize the utility with respect to user . This is doneiteratively until the algorithm converges. The convergencefollows because the objective is increased in each step andit is concave and bounded. Therefore, the iterative algorithmmust converge. A generalization to a broader class of utilityfunctions is developed in [21]. In [23], an algorithm basedon dual decomposition is developed to solve the sum ratemaximization for the Gaussian broadcast channel. The powerprices correspond exactly to the dual variables.

However, the iterative waterfilling algorithm and the dualdecomposition approach cannot be applied to the weightedsum utility maximization problem. Therefore, we present herean alternative approach. Consider the following programmingproblem

max0

=1

log det

( +

=1

)

log det( +

1=1

)

tr(). (54)Define +1 = 0, the objective function can be reformulatedas

=1

( +1) log det( +

=1

)

=1

tr(). (55)

The objective function in (55) is jointly concave in1, ...,and therefore the algorithm developed in [24] can be adapted.The main difference is that here we do not have a traceconstraint on and thus do not need the normalization step.

The convergence of Algorithm 2 can be shown followingsimilar arguments as in [24]. If in Algorithm 2 is chosenas tr() then it corresponds to the normalization step in[24]. The convergence rate of the fixed point optimizationalgorithm is only linear [25] and any Newton style algorithmhas local super-linear convergence. However, it provides asimple universal approach to weighted sum rate maximization.

Result: Solve weighted sum utility optimization problemInput: Channel realizations 1, ..., , power costs

1, ..., , inverse noise power .init: = 0 for all 1 ;while required accuracy not reached do

for = 1... do

=1

=

( +1)1/2

[ +

=1

]1

1/2

endendOutput: Optimal 1, ...,

Algorithm 2: Weighed sum utility maximization forMIMO MAC.

10 5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Inverse noise power [dB]

power

allo

catin

p1,p

2

p1, = [0.8, 0.2]

p2, = [0.8, 0.2]

p2, = [0.5, 0.5]

p1, = [0.5, 0.5]

Fig. 9. Two-user MIMO ( = 2, = 2) MAC optimal weighted sumutility power allocation.

Corollary 3: The Algorithm 2 converges always to a globaloptimal set of transmit covariance matrices 1, ..., .

A. Illustration of MIMO MAC Weighted Sum Utility Maxi-mization

In Figure 9, the optimal power allocation is shown for a two-user MIMO ( = 2, = 2) MAC with fixed but randomchannel matrices and power prices = [1.2, 1]. The weightsare 1 = 0.8, 2 = 0.2. The sum utility optimization 1 =2 = 0.5 is shown for comparison.

Furthermore, the difference to the case with single antennatransmitters can be clearly observed. At different points multiple eigenmodes of the two transmitters are activated.Therefore, the order of the allocated power varies. Note thatfor sum utility maximization asymptotically, the allocatedpower of user converges to . In general, the quantita-tive behavior of the optimal power allocation is difficult tocharacterize for medium SNR values because it depends onthe spatial signatures of the involved users and their weights.

VI. CONCLUSION

We proposed a utility function for energy-aware resourceallocation and transmit optimization in multiuser uplink sys-tems based on the difference of the individual capacity and


a weighted power penalty term. For the SISO MAC, wecompletely characterized the utility region and proposed an ef-ficient algorithm to achieve any point on the Pareto boundary.The region is always convex. However, in the SIMO MAC, theutility region is no longer convex due to the additional spatialdegrees of freedom and due to the different decoding ordersat the base station. In the MIMO MAC case, we extendedthe iterative waterfilling algorithm to solve the maximum sumutility problem. The important question, how to choose theprices in order to optimize system utility functions is left forfuture research. The decentralized implementation of the twoproposed algorithms is another interesting research direction.

ACKNOWLEDGMENT

The authors would like to thank Marcin Wiczanowski andZhijiat Chong for helpful discussions.

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Eduard A. Jorswieck (S01-M05-SM08) re-ceived his Diplom-Ingenieur degree and Doktor-Ingenieur (Ph.D.) degree, both in electrical engi-neering and computer science from the Berlin Uni-versity of Technology (TUB), Germany, in 2000and 2004, respectively. He was with the FraunhoferInstitute for Telecommunications, Heinrich-Hertz-Institute (HHI) Berlin, from 2001 to 2006. In 2006,he joined the Signal Processing Department at theRoyal Institute of Technology (KTH) as a post-docand became a Assistant Professor in 2007. Since

February 2008, he has been the head of the Chair of Communications Theoryand Full Professor at Dresden University of Technology (TUD), Germany.

His research interests are within the areas of applied information theory,signal processing and wireless communications. He is senior member ofIEEE and elected member of the IEEE SPCOM Technical Committee. From2008-2010 he serves as an Associate Editor for IEEE SIGNAL PROCESSINGLETTERS. In 2006, he was co-recipient of the IEEE Signal Processing SocietyBest Paper Award.

Holger Boche (M04 - SM07) received his Dipl.-Ing. and Dr.-Ing. degrees in Electrical Engineeringfrom the Technische Universitaet Dresden, Ger-many, in 1990 and 1994, respectively. In 1992he graduated in Mathematics from the Technis-che Universitaet Dresden, and in 1998 he receivedhis Dr.rer.nat. degree in pure mathematics fromthe Technische Universitaet Berlin. From 1994 to1997, he did postgraduate studies in mathematicsat the Friedrich-Schiller Universitt Jena, Germany.In 1997, he joined the Heinrich-Hertz-Institut (HHI)

fr Nachrichtentechnik Berlin. Since 2002, he has been Full Professor forMobile Communication Networks at the Technische Universitt Berlin at theInstitute for Communications Systems. In 2003, he became Director of theFraunhofer German-Sino Lab for Mobile Communications, Berlin, Germany,and since 2004 he has also been Director of the Fraunhofer Institute forTelecommunications (HHI), Berlin, Germany. He was Visiting Professor at theETH Zurich during winter term 2004 and 2006 and at KTH Stockholm duringsummer term 2005. Prof. Boche received the Research Award TechnischeKommunikation" from the Alcatel SEL Foundation in October 2003, theInnovation Award" from the Vodafone Foundation in June 2006, GottfriedWilhelm Leibniz Prize from the Deutsche Forschungsgemeinschaft (GermanResearch Foundation) in 2008. He was co-recipient of the 2006 IEEE SignalProcessing Society Best Paper Award and recipient of the 2007 IEEE SignalProcessing Society Best Paper Award. He is a member of IEEE SignalProcessing Society SPCOM and SPTM Technical Committee. He was electeda member of the German Academy of Sciences (Leopoldina) in 2008 and ofBerlin Brandenburg Academy of Sciences and Humanities in 2009.

Siddharth Naik received his MS in EE from theRoyal Institute of Technology in 2004. He was aresearch engineer at the SPANN laboratory at IIT,Mumbai in 2004. He has been a research associateat the Heinrich Hertz Institute and has been pursuinghis PhD in Electrical Engineering at the TechnicalUniversity of Berlin, since 2006. His research inter-ests are game theory, mechanism design, probabilitytheory and convex optimization.

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