mu-miso interference channels with su detection.pdf

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011 4255

Multiuser MISO Interference Channels WithSingle-User Detection: Optimality of Beamforming

and the Achievable Rate RegionXiaohu Shang, Member, IEEE, Biao Chen, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstract—For a multiuser interference channel with multi-antenna transmitters and single-antenna receivers, by restrictingeach transmitter to a Gaussian input and each receiver to asingle-user detector, computing the largest achievable rate regionamounts to solving a family of nonconvex optimization problems.Recognizing the intrinsic connection between the signal power atthe intended receiver and the interference power at the unintendedreceiver, the original family of nonconvex optimization problemsis converted into a new family of convex optimization problems.It is shown that, for such interference channels with each receiverimplementing single-user detection, transmitter beamforming canachieve all boundary points of the achievable rate region.

Index Terms—Achievable rate region, beamforming, Gaussianinterference channel.

I. INTRODUCTION

T HE interference channel (IC) models a multiuser commu-nication system in which each transmitter communicates

to its intended receiver while generating interference to otherreceivers. Determining the capacity region of an IC remains anopen problem except in the case of ICs with strong interfer-ence [1], [2]. To date, the best achievable rate region was es-tablished by Han and Kobayashi in [1], herein termed the HKregion, which combines rate splitting at transmitters, joint de-coding at receivers, and time sharing among codebooks. The HKregion was simplified by Chong, Motani, Garg, and El Gamal[3] and several computable subregions were also proposed in[4]–[6]. Etkin, Tse, and Wang [7, Th. 1] proved that the HK re-gion is within 1-bit of the capacity region of the Gaussian IC.The results in [6], [8] and [9], whose genie-aided approach islargely motivated by [7], established the sum-rate capacity ofthe two-user Gaussian IC in the low interference regime: when

Manuscript received May 13, 2009; revised January 21, 2011; acceptedJanuary 21, 2011. Date of current version June 22, 2011. This work was sup-ported in part by the National Science Foundation under Grants CCF-05-46491,CCF-09-05320, and CNS-09-05398. The material in this paper was presented inpart at the IEEE Global Communications Conference, Honolulu, HI, December2009.

X. Shang was with the Department of Electrical Engineering, Princeton Uni-versity, Princeton, NJ, 08544 USA. He is now with Bell Labs, Alcatel-Lucent,Holmdel, NJ 07733 USA (e-mail: [email protected]).

B. Chen is with the Department of Electrical Engineering and Computer Sci-ence, Syracuse University, Syracuse, NY 13244 USA (e-mail: [email protected]).

H. V. Poor is with the Department of Electrical Engineering, Princeton Uni-versity, Princeton, NJ, 08544 USA (e-mail: [email protected]).

Communicated by H. Bölcskei, Associate Editor for Detection and Estima-tion.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIT.2011.2145230

the interference power is below a certain threshold (referred toas noisy interference in [8]), the results assert the optimality oftreating interference as noise at both receivers, i.e., each re-ceiver should simply implement singe-user detection (SUD). Inaddition, even if the noisy interference condition is not satisfied,practical constraints often limit the receivers to implementingSUD. For example, the receivers may know only the channelsassociated with their own intended links. Under such scenarios,treating interference as noise at each receiver is more practical.

There have been several studies of multiple-input mul-tiple-output (MIMO) and multiple-input single-output (MISO)ICs [10]–[16]. The MISO IC describes, for example, the down-link communications of cochannel cells in which the basestations have multiple antennas and the mobile stations havesingle antennas. The downlink beamforming problem has beenwell studied in [17]–[20]. Less well understood is the downlinktransmission in the presence of interference, both in termsof its fundamental performance limits (i.e., capacity region),as well as in practically feasible transmission schemes. Theassumption of multiantenna transmitters and single-antennareceivers is motivated by the real world constraints in whichminiaturization of mobile units limits the number of antennas.In addition, the asymmetry in available resources at base andmobile stations favors systems in which transmitters are taskedwith heavy processing in exchange for reduced complexity atmobile units. Toward this end, we assume in the present workthat each receiver implements SUD, i.e., it treats interferenceas channel noise. In a preliminary work [12], we showed thatbeamforming is optimal for the entire SUD rate region for atwo-user real MISO IC. This result was used in [21] to charac-terize the beamforming vectors that achieve the boundary ratepoints of the SUD rate region. Later, the result in [12] was alsoused in [22] to derive the noisy-interference sum-rate capacityof the symmetric real MISO IC. In this paper, we generalize theresult of [12] to complex multiuser MISO ICs. We note that theproof in [12] is applicable only to two-user real MISO ICs.

Throughput optimization in a multiuser system under theassumption that each receiver treats interference as channelnoise was considered in [23]–[28]. However, even for thesimple scalar Gaussian IC, computing the largest achievablerate region with SUD at each receiver is in general an openproblem [29]. Exhaustive search over the transmitter power istypically unavoidable due to the nonconvexity of the problem.The difficulty is much more acute for the MISO IC case as oneneeds to exhaust over all covariance matrices satisfying thepower constraints. The complexity increases with the square

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4256 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

of the number of transmit antennas, which renders the com-putation intractable. In this paper we propose an alternativeway of deriving the optimal signaling for the SUD rate regionfor multiuser complex MISO ICs. Our approach is to converta family of nonconvex optimization problems for the originalformulation to an equivalent family of convex optimizationproblems. What is more significant is that, given that each re-ceiver implements SUD, all boundary points of the rate regioncan be achieved by transmitter beamforming.

Since the submission of this paper, some recent studies ofbeamforming for MISO ICs in the literature have appeared.The characterization of the SUD rate region of the -userMISO IC was studied in [21], [30]–[32], and the optimality ofbeamforming was independently proved in [30] and [31]. Theboundary of the rate region was characterized usingcomplex parameters in [21]. This result was further improvedin [30] by using real parameters. For general multicell systems,only parameters are used per transmitter-receiver pair forspatial transmission design with additional parameters beingused for power control.

The rest of the paper is organized as follows. In Section II,we use a two-user complex MISO IC as an example to ex-plain the basic idea of our problem reformulation. We showthat beamforming is optimal for the SUD rate region of such achannel. This result is generalized to multiuser complex MISOICs in Section III. We prove that beamforming is also optimalfor -user complex MISO ICs with . The obtained resultallows us to compute the SUD rate region for an -user MISOIC, and we illustrate this using a three-user MISO IC example.Numerical examples are provided in Section IV, and we con-clude in Section V.

Before proceeding, we introduce the following notation.• Upper- and lowercase boldface letters, e.g., and , de-

note matrices and vectors respectively.• and denote respectively the transpose and the

Hermitian (conjugate transpose) of a matrix or a vector.Consequently the Hermitian of a scalar is its conjugate.

• is an identity matrix, is an all-zero vector or matrixdepending on the context, and is a diagonal matrixwith its diagonal entries being the elements of the vector .

• means that is a positive semidefinite Hermitianmatrix.

• and denote the trace and the rank, respec-tively, of the matrix .

• denotes the th entry of vector , denotes the throw and th column entry of matrix , and meansthat is an matrix.

• is the absolute value of a scalar , and is the normof a vector , i.e., .

• denotes the angle between two real vectors and, and . If both and are nonzero, then

. Otherwise we letfor convenience.

• denotes expectation.• is the sign of a real scalar , i.e.,

Fig. 1. The two-user Gaussian MISO IC.

• denotes the real part of its complex argument.

II. TWO-USER MISO IC WITH SINGLE USER DETECTION

The two-user Gaussian MISO IC is illustrated in Fig. 1 andthe received signals are defined as

(1)

where and are transmitted signal vectors of user 1 anduser 2 with dimensions and , respectively; and aretwo scalar received signals; and are unit variance circu-larly symmetric complex Gaussian noises; and arecomplex channel vectors; and and are complexchannel vectors. The power constraints at the transmitters arerespectively and , where

and . We assume that the transmittedsignals and are zero-mean Gaussian vectors, whereas eachtransmitter knows all the channel vectors. Each receiver knowsonly the channel vector from its transmitter: receiver 1 (resp. 2)only knows (resp. ). Each receiver decodes its own signalwhile treating the interference from the other user as noise. Theboundary points of the achievable rate region for this channelare characterized by the following family of optimization prob-lems:

(2)

where and . We define the SUD rateregion of a two-user MISO IC as1

(3)

1The SUD rate region defined in (3) is different from the Pareto rate region

which is defined as ��

� � ��

��

� � �� . Actually, the region in (3) is the convex hull of the

Pareto region. The Pareto region can be obtained by directly solving the opti-mization problems (4) and (5) introduced later. Therefore, all the results in thispaper apply also to the Pareto region.

SHANG et al.: MULTIUSER MISO INTERFERENCE CHANNELS WITH SINGLE-USER DETECTION 4257

Problem (2) is a nonconvex optimization problem. For eachpair, all possible and must be exhausted over

to find the solution of problem (2). To obtain the entire SUDrate region, one has to go through this exhaustive search forall the pairs. This exhaustive search is computationallyprohibitive when and become large.

In the following, we first convert problem (2) into a familyof convex optimization problems, and then obtain their closed-form solutions.

A. Problem Reformulation

We first define the following optimization problems:

(4)

(5)

In order for problems (4) and (5) to be feasible, we require

(6)

(7)

We now establish the equivalence between problem (2) andthe above two optimization problems.

Lemma 1: For any nonnegative scalars and , the optimalsolution and for problem (2) is also an optimal solutionfor problems (4) and (5) with and

.Proof: Problem (2) is equivalent to the following optimiza-

tion problem for the same and :

(8)

The equivalence is established as follows. First, the maximum ofproblem (2) is no smaller than that of problem (8), since problem(8) has extra constraints and .On the other hand, the maximum of problem (2) is no greaterthan that of problem (8), since and are also feasible forproblem (8), which are the optimal solutions for problem (2).Therefore, problems (2) and (8) are equivalent. We now recog-nize that problem (8) is equivalent to problems (4) and (5) with

and , which can be solved individually.

We remark that the optimization problem (8) cannot be solvedindependently as the constraint parameters and depend onthe unknown optimal covariances. That is, unless the optimal

and of problem (2) are obtained, the equivalent optimiza-tion problems in the form of (4) and (5) cannot be parametrized.However, this problem reformulation becomes especially pow-erful when we need to find the entire achievable rate region (orits boundary points) and to study the optimal signaling struc-ture. Even though one cannot solve any individual optimizationproblem (2) by the corresponding problem (8), Lemma 1 es-tablishes the following crucial fact that enables us to obtain theentire SUD rate region without explicitly solving (8):

(9)

where the left-hand side denotes the collection of all the optimalsolutions of problem (2) found by exhausting over and ,and the right-hand side denotes the collection of all the optimalsolutions of problems (4) and (5) found by exhausting overand . Since the SUD rate region is determined by the left-hand side of (9), Lemma 1 successfully converts a family ofnonconvex optimization problems (2) into a family of equivalentconvex optimization problems (4) and (5).

To be more precise, instead of solving the family of non-convex optimization problems by exhausting over and ,one can instead solve the family of convex optimization prob-lems by exhausting and over the range specified by (6) and(7). We now proceed to obtain closed-form solutions for prob-lems (4) and (5).

B. Optimal Solution

By symmetry, we need to solve only problem (4). Assumethat the singular-value decomposition (SVD) of is

(10)

where . Define

(11)

where and is a vector. We have thefollowing lemma.

Lemma 2: Assuming the optimization problem (4) is feasible,the following ’s are optimal:

• If and are linearly independent (consequently, and ), then

(12)

and the achieved maximum is

(13)


where

(14)

(15)

• If and are linearly dependent and , then

(16)

(17)

• If (hence ), then

(18)

(19)

Moreover, for all above cases, we have

(20)

The proof is given in Appendix A.Lemma 2 shows that, for a fixed interference power , trans-

mitter beamforming maximizes the received signal power.2 Ifand are linearly independent, the quadratic constraint of (4)defines a set of beamforming vectors whose projections onhave equal length . Among all these vectors, the one that hasthe largest length of the projection on is the optimal beam-forming vector.

C. The SUD Rate Region of a Two-User MISO IC

With Lemmas 1 and 2, we obtain the SUD rate region of atwo-user MISO IC.

Theorem 1: The SUD rate region of a two-user MISO IC withcomplex channels is

(21)

2In Appendix A, we show that when�� , there exist some matrices thatare not beamforming matrices but still maximize problem (4). This also happenswhen �� and �� are linearly dependent. However, these does not contradict ourconclusion that beamforming is optimal.

where

Furthermore, the boundary points of the rate region can beachieved by restricting each transmitter to implement beam-forming.

Proof: We first assume that is linearly independent of, and that is linearly independent of . Define

then (13) becomes

Similarly, the maximum of problem (5) is

where

Therefore, the achievable rate region determined by problem (2)is

(22)

On defining and

, the interference power and the useful

signal power caused by transmitter 1 are given respectively by

(23)

(24)

Similarly, the interference power and the useful signal powercaused by transmitter 2 are given respectively by

(25)

(26)


Since varies continuously in as varies in(and similarly for ), the region in (22) is the

same as

(27)

When , the useful signal poweror decreases as

the interference power or in-creases. As such, the rate pairs associated withare interior points of the set (27). Therefore, (27) can besimplified into (21).

In the cases where and are linearly dependent, (21) isstill the SUD rate region. This is due to the fact that (17) can beexpressed in (24) when ; and (19) can also be expressedin (24) when and .

For each choice of and , the corresponding can beobtained from (12), (16) or (18), and similarly for .

The rate region in (21) is characterized by and whichalso determine the interference powers (23) and (25) at the tworeceivers. Compared to the original problem (2) which is charac-terized by the slopes of the boundary points and requiresthe solution of a family of nonconvex optimizations, Theorem 1provides an alternative approach to compute the SUD rate re-gion by problem reformulation. Moreover, Theorem 1 showsthat to achieve the rate pairs on the boundary of the SUD rate re-gion, the transmitter can restrict itself to a simple beamformingstrategy.

In the SUD rate region (21), we point out several special ratepairs.

• . This corresponds to a zero-forcing (ZF)beamforming rate pair as both transmitters generate no in-terference to the unintended receiver. Thus

and arethe maximum ZF beamforming rates. In general, this ratepair is in the interior of the SUD rate region.

• . This case shows that user 1can communicate at a rate no greater than

when user 2 is atthe maximum rate . This corre-sponds to a corner point of the rate region.

• . This is the other corner pointof the rate region. User 2 can communicate at a rateno greater than

when user 1 is at the maximum rate.

When and , and and are respectively linearly in-dependent, both transmitters use all their power to achieve thelargest rates. However, if either of the above two vector pairsis linearly dependent, the transmitters do not necessarily use allthe available power. An example is the scalar Gaussian IC.

Lemma 3: [33, Th. 6] If and, then the maximum SUD sum rate is

(28)

where

Therefore, the maximum SUD sum rate for a scalar IC isachieved by letting both users use all the power, or letting oneuser use all the power while keeping the other user silent. Thecontrast between a scalar IC and a MISO IC with linearly in-dependent channels is largely due to the existence of and theinterplay between the spatial diversity and multiuser diversityof a MISO IC. There is a trade-off between the power of the in-tended signal at its own receiver and the interference power atthe other receiver. For a scalar Gaussian IC, these two signalsoverlap in the same subspace and are proportional to each other,i.e., the channels are always linearly dependent. An increase ofthe intended signal power always results in an increase of theinterference power (see (23) and (24) when is 0). The op-timal trade-off is achieved by choosing the appropriate powerat the transmitters. As shown in (28), the optimal trade-off doesnot necessarily require that both users use all the power. But forthe MISO IC with linearly independent channels, the intendedlink and the interference link are in nonoverlapping subspaces.Therefore, the optimal trade-off is achieved by choosing the op-timal beamforming subspaces while using all the power.

Fig. 2 is an illustration of the beamforming vector of a MISOIC. For simplicity, the channel vectors and are assumedto be real vectors with unit lengths. The angle between

and is . The disc with radius contains all possiblebeamforming vectors that satisfy the power constraint. andare on the circle, the projections of vectors and onboth have length . Then all the vectors on the line segmentsatisfy the power constraint and the interference constraint

. Among those vectors, has the greatest length of projec-tion on . Therefore, is the optimal beamforming vector

given the interference constraint . It can be shown that theangle between and is and the length of the projec-tion of on is .

The reduction of the nonconvex optimization problem (2) tothe equivalent optimization problem (4) is obtained by fixing theinterference power while maximizing the useful signal power.This method is equivalent to fixing the useful signal power whileminimizing the interference power. This requires solving thefollowing optimization problems:

The above two problems can be solved in the same way asproblem (4), and the rate region can be similarly obtained. Forthese two proposed methods, the constraints are imposed eitheron the interference powers or the useful signal powers. One canalso combine these two methods. For example, we can impose


Fig. 2. A geometric explanation of beamforming in the MISO IC.

a constraint on the interference power caused by transmitter 1while maximizing the useful power for receiver 1, and in themeantime, impose a constraint on the signal power on receiver 2while minimizing the interference power caused by transmitter2.

D. Interference-Limited SUD Rate Region

As Theorem 1 is obtained by examining the relationship be-tween the interference power and the useful signal power, wecan easily apply it to MISO ICs under interference power con-straints.

Theorem 2: For the MISO IC defined in (1) withand , and with two additional constraints

and on the interference powers, the SUD rateregion is

(29)

where

The proof is straightforward from Theorem 1.When and ,

(29) is exactly (21). Therefore, when the interference constraintsare larger than the above thresholds, these constraints do notchange the SUD rate region. Another extreme case is that inwhich neither user is allowed to generate interference to theother user. This is also the ZF rate region which is a rectangledetermined by .

III. MULTIUSER MISO IC WITH SINGLE-USER DETECTION

In this section, we generalize our study of the two-user case tothe general multiuser MISO IC. The key is, again, the problemreformulation as illustrated in Lemma 1 for the two-user case.For the general -user MISO IC, we prove the optimality ofbeamforming with an SUD receiver. We then give an explicitdescription of the SUD rate region for a three-user MISO IC,and generalize it to the -user case.

A. Optimality of Beamforming for an -User MISO IC

Define the received signal of the th user, , as

(30)

where is the transmitted signal vector of user isthe complex channel vector from the th transmitter to theth receiver, and is unit variance circularly symmetric com-

plex Gaussian noise. The power constraint for user iswhere . As with the two-user case, the input

signals ’s are all zero-mean Gaussian vectors and each re-ceiver treats interference as noise.

Lemma 1 can be easily extended to multiuser MISO ICs, asfollows.

Lemma 4: For any vector with nonnega-tive components, the optimal solution for the fol-lowing optimization problem:

(31)

is also an optimal solution for the problem

(32)

with .Following the same problem reformulation procedure used

in Section II, to characterize the SUD rate region of an -userMISO IC, the key appears to be the solution of (32) whereis a preselected constant denoting the interference power at theth receiver caused by the th transmitter, and and are the

covariance matrix and power constraint for the th transmitter.Unlike problem (4), the optimal covariance matrix for (32)

with any given ’s need not necessarily be a beamformingmatrix. Here is an example.


Example 1: Consider the channels

The optimal covariance matrix with the constraintsand is

By restricting to beamforming, the optimal covariance matrix is

We have

However, the above example does not mean that beam-forming is not optimal for the SUD rate region of an -userMISO IC. The reason is that the optimization problem (32)requires the interference powers to be exactly . Withoutknowing the values of all , exhausting over all willresult in some rate pairs not on the boundaries of the SUD rateregion. As we intend to establish the optimality of beamformingfor achieving the boundary points of the rate region, we resortinstead to the following more general formulation, i.e., theinterference power is bounded by , namely

(33)

Comparing the modified problem (33) with problem (32), wehave relaxed the equality interference power constraints into theinequality interference power constraints. Consequently, for anyvalues of , the maximum of problem (33) is no smaller thanthat of problem (32). In the following, we need to consider onlyproblem (33). Therefore, if we can prove that beamforming isoptimal for (33), then beamforming must be optimal for problem(31) even if it may not be optimal for (32). Such a strategy hasbeen used in Theorem 1 where we let vary ininstead of the entire interval because only in the specifiedinterval does the useful signal power increase as the interferencepower increases. Based on the modified optimization problem(33), we obtain the following theorem.

Theorem 3: For an -user MISO IC, the boundary points ofthe SUD rate region can be achieved by restricting each trans-mitter to implement beamforming.

Theorem 3 is proved in the following steps. We first introduceLemma 5 which allows us to solve problem (40). In the process,we establish the fact that the rank of the entire covariance matrixcan be set to the same as that of its submatrices. The final stepis to show that the optimal submatrix need to have a rank nogreater than 1, established via an extension of Sylvester’s Lawof Inertia.

We first introduce Lemma 5.

Lemma 5: Let and be two complex vectors with dimen-sions and respectively, and be apositive semidefinite Hermitian matrix with . If

(34)

and is a preselected positive semidefinite Hermitianmatrix, then

(35)

and the equality can be achieved by choosing , definedas follows.

1) When and , we have

(36)2) When and , we can set

(37)where

with

being the eigenvalue decomposition of ,and being a strictly positive diagonal matrix.

3) When , we can set

(38)

Moreover, for all three cases, we have

(39)


The proof is given in Appendix B. Here are some examples ofLemma 5:

Example 2: Let and in Lemma 5 be two scalars andbe a 2 2 matrix with .• If and , then from (36), we have a

unique , and .

• If and , then from (37) we have a

unique , and .

• If and , then from (37) we have

, and . However, we can also

choose and we have . Therefore,the optimal in this case is not unique.

• If , and , then from (38), we have

, and . However, we can also

choose and .It is seen from the examples that only when

and , or and , is the optimalunique. Otherwise we can choose other optimal s which maynot satisfy (39).

Lemma 5 is useful for the following optimization problem:

(40)

where and are fixed functions. By Lemma 5, wecan convert the above problem into

(41)

Problems (40) and (41) have the same solution. Once the op-timal for problem (41) is obtained, one can construct theoptimal for problem (40) from (36), (37) and (38). As shownby Example 2, the choices of (37) and (38) may not be unique.One can choose ’s that are different from (37) and (38) andstill achieve the same maximum.

With Lemma 5, we prove Theorem 3 as follows.Proof: By symmetry, it suffices to show that for the th

user, the optimal covariance matrix for the following opti-mization problem satisfies :

(42)

where all the ’s are vectors.We first show that problem (42) can be written as

(43)

where and all the ’s, , are vectors,is an matrix, and is defined as

(44)

Obviously, when , problem (42) is exactlythe same as problem (43) if we chooseand . Thus, we need only show the equivalence ofproblems (42) and (43) when .

We first iteratively transform the original problem (42). In theth transformation, the corresponding channel vectors

, and the covariance matrix are updated as and, respectively. Specifically, we keep the first elements

of and apply the SVD of the remainingelements, and update the optimization problem. We formulatethe th, , iteration as follows:

(45)

(46)

(47)

where denotes the th to the th elements of

, and its SVD is

where .For example, in the first transformation, we have

(48)

(49)

(50)

Substituting (49) and (50) into (42), we obtain

(51)


In the second transformation, we have

(52)

(53)

(54)

Since , we have

On substituting (52), (53), and (54) into (51), we have

(55)

We note that the above transformation does not change the formof the previously modified constraints (see the third lines ofproblems (51) and (55)). Now we continue the above procedureup to . Finally, we convert problem (42) into the fol-lowing form:

(56)

where and are vectors and is avector. Furthermore, . Let

(57)

where is an matrix. The quadratic con-straints in problem (56) are

Therefore, the quadratic constraints in problem (56) are relatedonly to . By Lemma 5, problem (56) is equivalent to problem(43).

Thus, in summary, problem (42) is equivalent to problem (43)with all the vectors in (43) being and being .

By Lemma 5, we can construct in such a way that. Let be optimal for

problem (43). To prove Theorem 3, it is equivalent to prove

(58)

Furthermore, since is optimal for problem (43), alsomaximizes under all the constraints in (43) with an extraconstraint . Therefore, to prove (58), it suf-fices to prove that the rank of the optimal covariance matrix forthe following optimization problem is no greater than 1:

(59)

where

(60)

The equivalence is due to the fact that the optimal forproblem (43) is also optimal for problem (59) and vice versabecause of (60). Moreover, since is also optimal forproblem (59), the inequality constraint is active.

The Lagrangian of problem (59) is

(61)

On setting , we have

(62)

where

(63)

and is an matrix.We then introduce the following lemma which is an extension

of Sylvester’s Law of Inertia.


Lemma 6: [34, Th. 7] Let be an matrix and be anHermitian matrix. Let and denote, respectively,

the numbers of positive and negative eigenvalues of a matrixargument. Then we have

By Lemma 6 and the Karush-Kuhn-Tucker (KKT) conditionsthat require , we have

(64)

Since is an Hermitian matrix, we can write its eigen-value decomposition as

(65)

where , and ’s are the eigenvalues in ascendingorder. From (64), we have

(66)

(67)

Since

is an eigenvalue of .Since the optimal for problem (59) satisfies ,

from the KKT conditions we have

(68)

Using (66)–(68) and noticing that the KKT conditions require, we have

(69)

(70)

For the optimal , from the KKT conditions, we have

Since , using (69) and (70) we have

and can be nonzero if . Since, if one diagonal element is zero then all the

elements on this row and this column must be zero. Thus, wehave

In the Proof of Theorem 3, we first use a sequence of SVDsto convert the original optimization problem (42) to (56), andthen use Lemma 5 to further reduce the problem to (59). Theseprocesses effectively reduce the number of antennas in consid-eration to .

Theorem 3 proves the sufficiency of transmitter beamformingfor achieving the SUD rate region. However, it does not meanthat the SUD rate region can be achieved only by beamforming.As shown in the examples, the optimal is not unique when

or . corresponds to the case in whichis orthogonal to , and

corresponds to the case in which is linearly dependent of.

B. SUD Rate Region of -User MISO ICS

Section III-A establishes the optimality of beamforming forthe general -user MISO IC with complex channels when eachreceiver is restricted to SUD. In this section, we first use a three-user real MISO IC as an example to show how we can obtain theSUD rate region by Theorem 3, and then generalize it to -usercomplex MISO ICs.

For a three-user real MISO IC, the optimization problem (42)can be written as

(71)

We first reformulate this problem into (43). Let the SVD ofbe

where . We then update and as follows:

(72)

(73)

(74)

(75)

where , and is the remaining part of . Letthe SVD of be


and . Then we update and as follows:

where is a 2 1 vector. Define

(76)

where is a 2 2 matrix. Then we obtain the following op-timization problem:

(77)

The solution for problem (77) is complex. In the following weobtain the SUD rate region by using the fact that the rank ofthe optimal is no greater than one without directly solvingproblem (77). Therefore, instead of exhausting over all feasible

and and collect all corresponding , we exhaust over allfeasible . We let

(78)

where and can be any values in . Then we havethat the optimal for this is given in (79) if the numberof antennas , and

(80)

if , and

(81)

if .It can be shown that

where

is the angle between the projections of and in ’s or-thogonal subspace. The signal power and the associated inter-ference powers are determined by , and from (77) we have

(82)

(83)

(84)

(79)


Therefore, we have the following theorem.

Theorem 4: The SUD rate region of a three-user MISO ICis given by (85) Furthermore, the boundary points of the rateregion can be achieved by restricting each transmitter to imple-ment beamforming.

In the following, we discuss some special rate triples of theabove region. For simplicity, we assume ,where , and are integers ranging from 1 to 3.

• ZF beamforming rate triple.When , (82)–(84) become

Therefore, the rate triple in set (85) withis the ZF beamforming rate triple. That is,

the beamforming vector is chosen to be orthogonal to thechannel vectors corresponding to the interference linksassociated with the same transmitter.

• Single-user maximum rate surface.When and , (82)–(84) become

Therefore, achieves the maximum with the con-straint . Then we obtain the following result.Let

then the corresponding rate triples form the surface of thethree-dimensional (3-D) SUD rate region with at itsmaximum. Similarly, the maximum rate surface for users2 and 3 can be obtained.

• Projection of the 3-D rate region to the two-dimensional(2-D) rate region.Suppose, say, user 1 is silent. We can recover the 2-D SUD

rate region formed by users 2 and 3 obtained from The-orem 1. That is, the surface of the 3-D SUD rate regioncorresponding to is precisely the 2-D SUD rateregion of the two-user IC consisting of users 2 and 3.

We can similarly obtain the SUD rate region of an -usercomplex MISO IC with using Theorem 3. The onlydifference is how we generate matrix . We generalize it asfollows:

where , and denotes the imaginaryunit. In the case of a real MISO IC, we can simply choose

. The rate region and the corresponding beamforming vectorscan be obtained in the same way as in Theorem 4, and so thedetermination of their quantities is omitted.

IV. NUMERICAL EXAMPLES

In this section, we present the SUD rate region of two-user orthree-user real MISO ICs. For ease of visualization, we take theconvex hull of all the regions.

The achievable rate regions for the symmetric MISO ICs areshown in Figs. 3 and 4. Here, a symmetric MISO IC refers toone with , and .In Fig. 3, the rate region shrinks as varies from to 0, whichcorresponds to the MISO IC with the interference link and thedirect link varying from being orthogonal to colinear. The SUDrate region becomes smaller than that of frequency division mul-tiplexing (FDM) when the direct link and interference link ex-hibit increasing linear dependence. When , neither of thetransmitters generates interference to the other user. Therefore,the IC reduces to two parallel single-user channels without inter-ference, and the ZF beamforming rates achieve the maximum.When , the two transmitters generate the worst interfer-ence and this MISO IC acts as a single antenna IC. Hence, theZF beamforming rates are 0.

In Fig. 4, the rate region also shrinks as the interference gainincreases. However, even if the rate region will not be

worse than the tetragon . Points are the extremepoints of the rate region on the axes. Point denotes the ZFbeamforming rates, which are determined only by and

. Therefore, as increases, the rate region becomes close to

(85)


Fig. 3. The achievable regions for the symmetric MISO Gaussian IC with �� , and � � � � �. � -�denote the respective ZF beamforming rates for � � to � � �. Also plotted for comparison is the FDM achievable rate region.

the tetragon. Point also shows the advantage of multiple an-tenna systems over single antenna systems, since the achievablerate region for the latter case reduces to the triangle defined by

when . It can be shown that point falls insidethe FDM region only if

(86)

i.e., FDM outperforms SUD when the interference link subspaceis close to the direct link subspace and interference gain is suf-ficiently large. Since

for all MISO ICs with and , SUD achievesa larger rate region than FDM for large power constraints, whilethe reverse is true for small power constraints and sufficientlylarge interference gains .

Fig. 5 shows the SUD rate region of a two-user MISO ICunder an interference power constraint. When such a constraintis small, this corresponding rate region is included in the FDMrate region. Since neither user generates interference in FDM,when the interference power is a concern in the system design,one can choose FDM instead of SUD when the interference isrestricted.

Fig. 6 shows the SUD rate region of a three-user MISO ICwith the power constraints and . Thechannels are

where and. The solid curves are the rate regions for one user

being inactive or at the maximum rate. That is, they are the pro-jection of the 3-D rate region onto a 2-D plane with one ratefixed at a constant value. The ZF beamforming rate triple of thischannel is shown in Fig. 7. The two cutting planes that passthrough this rate point show that the ZF beamforming rate pointis not on the boundary of the SUD rate region.

V. CONCLUSION

We have considered MISO ICs in which each transmitteris limited to a Gaussian input and each receiver is limited tosingle-user detection. By exploiting the relation between thesignal power at the intended receiver and the interference powerat the unintended receiver, we have derived a new method to ob-tain the SUD rate region for the MISO IC. It has been shownthat the original family of nonconvex optimization problemsis readily reduced to an equivalent family of convex optimiza-tion problems. As a consequence of restricting each receiver toimplement single-user detection, transmitter beamforming hasbeen shown to be sufficient to achieve all boundary points ofthe SUD rate region.


Fig. 4. The achievable regions for the symmetric MISO Gaussian IC with �� , and � � � � �. � is theZF beamforming rate point for all the choices of �. Also plotted for comparison is the FDM achievable rate region.

APPENDIX

A. Proof of Lemma 2

Since it is straightforward to show (16)–(19) when andare linearly dependent, we need to prove only (12)–(15) underthe condition that and are linearly independent.

Define

(87)

where is a nonnegative real scalar, is a columnvector, and is a Hermitian matrix. Since , we have

(88)

On Substituting (87) and (11) into (4), we have

(89)

(90)

where since is linearly independent of . There-fore the optimization problem (4) is equivalent to

(91)

(92)

Let the SVD of be

(93)

where . We further define

(94)

(95)

(96)

(97)

On substituting (90), (94) and (95) into (88), we have

(98)

We first assume . Then under the constraints in problem(92), we have

(99)


Fig. 5. The interference-limited SUD rate regions of a two-user MISO IC, where �� , and� � � �� .

where (a) is from (93)–(95); (b) is from (96) and (97); and (c) isfrom the facts that and

(100)

and the equality of (c) holds if and only if

(101)

Since , (100) and (101) imply that

(102)

Inequality (d) is from the fact that

(103)

The equality of (103) as well as the equality of (d) holds if andonly if is real, i.e.,

(104)

Inequality (e) holds because of (98) and (100):

(105)

The equality holds under the condition of (102) and

(106)

where the choice of ensures that (104) is satisfied. Thus, insummary, the equality of (99) holds if and only if (102) and(106) are satisfied.

From (94), (95), (102) and (106), the optimal and are

(107)

(108)

Therefore, the optimal covariance matrix for problem (4) is

(109)


Fig. 6. The SUD rate regions of a three-user MISO IC.

Fig. 7. The ZF beamforming rate triple � � �� and � � ��, and the cutting planes that contain the ZF beamforming rate point andare parallel to � -�-� and � -�-� , respectively.

The maximum of problem (4) is

(110)

When , problem (92) is solved by letting

In this case does not change the value of problem (92). Itonly needs to satisfy (88). In Lemma 2, we choose to satisfy

the equality of (88) to make the optimal covariance matrix berank-1. Consequently, the optimal and the maximum of (4)are respectively (12) and (13) with .

B. Proof of Lemma 5

We first consider the special case of . Obviously, (35)holds and we can choose as (38) so that (39) holds. One canalso choose which still achieves the equality of (35)but violates (39). Therefore, in this case is not unique unless

.Another special case is and . Since

we have . To achieve the equality of (35), we choose. Therefore, and (39) holds.

Next, we need only to prove Lemma 5 when and. Let . Then we have

(111)–(112) at the bottom of the next page, where in (a) wedefine

(113)


and is and is . In (b) we define

(114)

The lower part of the matrix on the right-hand side of (114) mustbe the all-zero matrix, since the second matrix in the quadraticform (111) is positive semidefinite. In (c), we let the SVD ofand be respectively

(115)

(116)

Since

(117)

we have

(118)

(111)

(112)


and

(119)

Therefore (d) holds with equality when

(120)

(121)

From (113) and (115) we have

(122)

Therefore, (e) holds. Thus, in summary, the equality of (112)holds when (120) and (121) hold. Therefore, (34) is true and theoptimal satisfies

(123)

In the following we obtain that achieves the equality of (35).1) When , from (120) and (121) we have

(124)

(125)

From (114) and (124), we have

(126)

where the last equality is from (122).2) When is still given by (125). However,

can be any unitary matrix since . Therefore,there are different choices of that achieve the equalityof (35). We choose for convenience. Then, from(114) and (124), we have

(127)

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Xiaohu Shang (S’07–M’09) received the B.S. and M.S. degrees in electronicsand information engineering from Huazhong University of Science and Tech-nology, Wuhan, China, in 1999 and 2002, respectively, and the Ph.D. degree inelectrical engineering from Syracuse University, Syracuse, NY, in 2008.

From 2002 to 2003, he worked at Guoxin Lucent Technologies NetworkTechnologies Co., Ltd., Shanghai, China. In 2007, he worked as a summer in-tern student at Communications and Statistical Sciences Research Departmentof Bell Labs, Alcatel-Lucent, Murray Hill, NJ. From 2008 to 2010, he was withPrinceton University, Princeton, NJ, as a Postdoctoral Research Associate. SinceAugust 2010, he has been with Bell Labs, Alcatel-Lucent, Holmdel, NJ, as aResearch Scientist. His area of interest mainly focuses on multiuser informa-tion theory and MIMO systems.

Dr. Shang received the Graduate School All University Doctoral Prize, andthe Wilbur R. LePage Award from Syracuse University in April 2009.

Biao Chen (S’96–M’99–SM’07) received the B.E. and M.E. degrees in elec-trical engineering from Tsinghua University, Beijing, China, in 1992 and 1994,respectively, and the M.S. degree in statistics and the Ph.D. degree in electricalengineering in 1998 and 1999, respectively, from the University of Connecticut,Storrs.

From 1994 to 1995, he was with AT&T (China) Inc., Beijing. From 1999to 2000, he was with Cornell University, Ithaca, NY, as a Postdoctoral Asso-ciate. Since 2000, he has been with Syracuse University, Syracuse, NY, wherehe is currently a Professor with the Department of Electrical Engineering andComputer Science. His area of interest mainly focuses on signal processing andinformation theory for wireless sensor and ad hoc networks and in multiusermultiple-input multiple-output (MIMO) systems.

Prof. Chen is an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL

PROCESSING and a past Associate Editor for the IEEE COMMUNICATIONS

LETTERS and the EURASIP Journal on Wireless Communications and Net-working (JWCN). He received an NSF CAREER Award in 2006

H. Vincent Poor (S’72–M’77–SM’82–F’87) received the Ph.D. degree in elec-trical engineering and computer science from Princeton University, Princeton,NJ, in 1977.

From 1977 until 1990, he was on the faculty of the University of Illinois atUrbana-Champaign. Since 1990 he has been on the faculty at Princeton, wherehe is the Dean of Engineering and Applied Science, and the Michael HenryStrater University Professor of Electrical Engineering. His research interests arein the areas of stochastic analysis, statistical signal processing and informationtheory, and their applications in wireless networks and related fields. Amonghis publications in these areas are Quickest Detection (Cambridge UniversityPress, 2009), coauthored with Olympia Hadjiliadis, and Information TheoreticSecurity (Now Publishers, 2009), coauthored with Yingbin Liang and ShlomoShamai.

Dr. Poor is a member of the National Academy of Engineering and the Na-tional Academy of Sciences, a Fellow of the American Academy of Arts andSciences, and an International Fellow of the Royal Academy of Engineering(U.K.). He is also a Fellow of the Institute of Mathematical Statistics, the OpticalSociety of America, and other organizations. In 1990, he served as President ofthe IEEE Information Theory Society, in 2004–2007 as the Editor-in-Chief ofthese TRANSACTIONS, and in 2009 as General Co-Chair of the IEEE Interna-tional Symposium on Information Theory, held in Seoul, Korea. He received aGuggenheim Fellowship in 2002 and the IEEE Education Medal in 2005. Recentrecognition of his work includes the 2008 Aaron D. Wyner Distinguished Ser-vice Award of the IEEE Information Theory Society, the 2009 Edwin HowardArmstrong Achievement Award of the IEEE Communications Society, the 2010IET Ambrose Fleming Medal for Achievement in Communications, and the2011 IEEE Eric E. Sumner Award.

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