The Capacity Region of the Class of Three-Receiver Gaussian MIMO Multilevel Broadcast Channels With Two-Degraded Message Sets

42 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 1, JANUARY 2014

The Capacity Region of the Class of Three-ReceiverGaussian MIMO Multilevel Broadcast Channels

With Two-Degraded Message SetsHon-Fah Chong, Member, IEEE, and Ying-Chang Liang, Fellow, IEEE

Abstract—Nair and El Gamal established the capacity regionof the three-receiver multilevel broadcast channel (MBC) withtwo-degraded message sets. For the three-receiver MBC withtwo-degraded message sets, the output at receiver 2 is a degradedversion of the output at receiver 1. However, no order of degrad-edness is imposed on the output at receiver 3. The transmittersends a common message to all three receivers and a private mes-sage to receiver 1. By considering a specific discrete-memorylessexample, Nair and El Gamal showed that a direct extension of theKörner–Marton region (for the general two-receiver broadcastchannel with degraded message sets) is strictly suboptimal. Theyalso considered a three-receiver Gaussian product MBC andshowed that, restricted to Gaussian inputs, the direct extension ofthe Körner–Marton region is again strictly suboptimal. However,whether Gaussian inputs are optimal remained unresolved. Inthis paper, we show that Gaussian inputs, along with time-sharingbetween rate points obtained with Gaussian inputs, achieve thecapacity region of the three-receiver Gaussian multiple-inputmultiple-output MBC (this includes the three-receiver Gaussianproduct MBC considered by Nair and El Gamal) with two-de-graded message sets. Our proof relies on the channel enhancementtechnique introduced by Weingarten et al. as well as the perturba-tion approach employed by Liu and Viswanath.

Index Terms—Broadcast channel, capacity region, degradedmessage sets, extremal entropy inequality, Gaussian channel,Gaussian perturbation, MIMO.

I. INTRODUCTION

I N this paper, we establish the capacity region of a three-receiver Gaussian multiple-input multiple-output (MIMO)

multilevel broadcast channel (MBC) with two-degraded mes-sage sets. The general three-receiver MBC with two-degradedmessage sets is shown in Fig. 1.In this channel, there is one transmitter with three receivers.

The output at receiver 2 is a degraded version of the output

Manuscript received October 13, 2011; accepted March 08, 2013. Date ofpublication October 09, 2013; date of current version December 20, 2013. Thispaper was presented in part at the 48th Annual Allerton Conference on Com-munication, Control, and Computing, Urbana, IL, USA, 2010.The authors are with the Modulation and Coding Department, Institute for

Infocomm Research, Singapore 138632 (e-mail: [email protected];[email protected]).Communicated by M. Motani, Associate Editor for Communication Net-

works.Digital Object Identifier 10.1109/TIT.2013.2285126

at receiver 1 . However, no order of degradedness is im-posed on the output at receiver 3 , i.e., the outputs at re-ceiver 1 and receiver 3 are not necessarily degraded versions ofeach other. The transmitter has a commonmessage intendedfor all three receivers and a private message intended onlyfor receiver 1.The general two-receiver broadcast channel (BC) with de-

graded message sets was first considered by Körner and Martonin [1], where they established its capacity region. In this channel,the transmitter sends a common message to both receiversand a private message to receiver 1. They showed that thecapacity region is given by the set of all nonnegative rate pairs

such that

(1)

for some . It was later shown by Weingarten et al. [2]that the boundary of the capacity region of the two-receiverGaussian MIMO BC with degraded message sets can beachieved using Gaussian inputs along with time-sharing.In a recent paper [3], Borade et al. introduced MBCs. In [3,

Th. 4], they derived an achievable rate region (a straightforwardextension of the Körner–Marton region) for the three-level, five-receiver BC and conjectured this to be the capacity region. How-ever, Nair and El Gamal [4] showed that this was, in fact, subop-timal. In particular, they studied the subclass of three-receiverMBC with two-degraded message sets as shown in Fig. 1. Theyshowed that the capacity region was given by the following the-orem [4, Th. 1].Theorem 1: The capacity region of the three-receiver MBC,

shown in Fig. 1, with two-degraded message sets is the set of allnonnegative rate pairs such that

(2)

for some .The auxiliary random variable (r.v.) represents the

common message , while the auxiliary r.v. represents partof the private message . The key idea in [4] was to allowreceiver 3 to determine indirectly by decoding ratherthan . In other words, receiver 3 decoded part of the messageintended for receiver 1.

0018-9448 © 2013 IEEE

CHONG AND LIANG: CAPACITY REGION OF THE CLASS OF THREE-RECEIVER GAUSSIAN MIMO MULTILEVEL BROADCAST CHANNELS 43

Fig. 1. Multilevel three-receiver BCs with two-degraded message sets.

By considering a specific discrete-memoryless example,Nair and El Gamal showed that the direct extension of theKörner–Marton region by Borade et al. to the three-receiverMBC was strictly suboptimal. They also considered a three-re-ceiver Gaussian product MBC and showed that, restricted toGaussian inputs, the achievable rate region of Borade et al. wasagain strictly suboptimal. However, whether Gaussian inputswere optimal for the class of three-receiver Gaussian productMBC remained unresolved (see [4, p. 4485]).In this paper, we show that Gaussian inputs, along with

time-sharing between rate points obtained with Gaussian in-puts, are indeed optimal for the class of three-receiver GaussianMIMO MBC, which includes the three-receiver Gaussianproduct MBC, with two-degraded message sets. Our proofcombines two key techniques. The first one is the enhancementtechnique introduced byWeingarten et al. [5] and was crucial inthe proof of the capacity region of the Gaussian MIMO BC withprivate messages. The second one is the perturbation approachadopted by Liu and Viswanath to prove an extremal inequalityin [6, Th. 1] (see [6, Appendix C]). The same perturbationapproach was adopted by Weingarten et al. to prove anotherextremal inequality in [7, Lemma 2] as well as by Liu et al.to prove a vector generalization of the Costa entropy powerinequality (EPI) [8, Th. 1].The paper is organized as follows:1) In Section II, we describe the canonic version of thechannel (also known as the aligned channel) and thegeneral linear channel model.

2) In Section III, we briefly describe the main theorem ofthis paper as well as its generalization to general linearchannels.

3) In Section IV, we provide some mathematical prelimi-naries and adopt the perturbation approach to prove twoextremal inequalities which are needed for the proof of ourmain theorem.

4) In Section V, we combine the two extremal inequalitieswith the channel enhancement technique to prove our maintheorem. We then generalize the result to general linearchannel models.

5) In Section VI, we conclude the paper.

II. CHANNEL MODEL

The following notation is used throughout the paper. Arandom vector is denoted with a bold upper-case letter (e.g.,), its realization is denoted with the corresponding bold,

lower-case letter (e.g., ), and its probability density functionis denoted with . We use to denote theexpectation of and to denote the covariance matrixof .

The degraded message sets consists of two independent mes-sages of rates , where is the commonmessage intended for all three receivers and is the privatemessage intended for receiver 1.We first consider a canonic ver-sion of the three-receiver GaussianMIMOMBC (commonly re-ferred to as the aligned channel) given by

(3)

(4)

(5)

where1) is a real input vector of size . We consider a matrixconstraint on the input , where . Thiswill allow us to extend the result to obtain the capacityregion of this channel under various power constraints suchas the average total input power constraint (see [5, Lemma1 and Corollary 1]).

2) is a real output vector of size received by user ,, 2, 3.

3) is a real Gaussian random vector of size withzero mean and a covariance matrix which is assumed tobe strictly positive-definite, i.e., . In addition, werequire that .

We note that the output at receiver 2 is a stochastically de-graded version of the output at receiver 1. Since there is no co-operation between the receivers, the capacity region is equiva-lent to that of a channel in which the output at receiver 2 is givenby

(6)

where .Next, we consider the three-receiver Gaussian MIMO MBC

with general linear channel matrices such that

(7)

(8)

(9)

where1) is a real output vector of size received by user ,

, 2, 3.2) is a real Gaussian random vector of size withzero mean and an identity covariance matrix, i.e.,

. We note that there is no loss of generalitywhen we assume that the noise covariance matrices areidentity matrices.

3) is a linear channel matrix of size . In addition,we require that is degraded with respect to (see [7,Definition 1]). Hence, there exists amatrix of sizesuch that and such that .

We may again verify that the output at receiver 2 is a stochas-tically degraded version of the output at receiver 1 and the ca-pacity region is equivalent to that of a channel in which theoutput at receiver 2 is given by

(10)

where .


III. MAIN RESULTS

Since the capacity region, , of the three-receiverGaussian MIMO MBC with two-degraded message sets isconvex due to time-sharing, it can be characterized by the linestangent to the capacity region, i.e., by the solution of

(11)

for . Therefore, if we can show that (11) can beattained by Theorem 1 with Gaussian inputs, we may concludethat Gaussian inputs, along with time-sharing, achieves the ca-pacity region . The main result of this paper is then givenby the following theorem for the aligned channel.Theorem 2: The capacity region of the aligned channel,

, is given by the convex hull of theclosure of all nonnegative rate pairs such that

(12)

(13)

(14)

(15)

for some

(16)

(17)

The achievability of Theorem 2 follows from a direct evalu-ation of Theorem 1 with Gaussian inputs. The main bulk of theproof deals with the converse, i.e., Gaussian inputs are, in fact,optimal.The extension of Theorem 2 to general linear channels is

given in the following theorem and is proved following the linesof [5, Sec. V.B], [7, Sec. V], and [9, Sec. 4].Theorem 3: The capacity region of the general linear channel,

, is given by the convex hull of the closureof all nonnegative rate pairs such that

(18)

(19)

(20)

(21)

for some

(22)

(23)

IV. MATHEMATICAL PRELIMINARIES

In this section, we obtain some intermediate results (seeLemmas 3 and 4) that we shall use in our proof of Theorem 2in Section V. As we shall rely heavily on Fisher informationmatrices and Fisher information inequalities to prove our inter-mediate results, we first begin with the following definition forthe conditional Fisher information matrix:Definition 1: Let be a pair of jointly distributed

random vectors where, for each , the conditionalprobability density function is differentiable. Theconditional Fisher information matrix of given is thendefined as

(24)

We also require the following standard definition for the con-ditional covariance matrix (also known as the MMSE matrix):Definition 2: Given a pair of jointly distributed random vec-

tors , the conditional covariance matrix of given isdefined as

(25)

(26)

Next, we need the conditional form of the Cramer–Raoinequality:Lemma 1: Let be a pair of jointly distributed random

vectors where, for each , the conditional probability den-sity function is differentiable. Let the conditionalcovariance matrix of given be ; then

(27)

Proof: See proof of [7, Corollary 4] and proof of [10,Lemma 13].We also require the following Fisher information inequality

result:Lemma 2: Let be a pair of independent random vec-

tors, of length , where the probability density functions,and , are differentiable. Then

(28)

for any matrix of size .Proof: See proof of [6, Lemma 3].

We can now state and prove the two main results of thissection.Lemma 3: Let , , andand are positive semidefinite matrices such that

and such that

(29)


where , , and and are positivesemidefinite matrices such that

(30)

Then, for every distribution , where, we have

(31)

Proof: Refer to Appendix I.Lemma 4: Let , and and are positive

semidefinite matrices such that and such that

(32)

where , , and , , and are positivesemidefinite matrices such that

(33)

Then, for every distribution , where, we have

(34)

Proof: Refer to Appendix II.Remark 1: Our proof of the two extremal inequalities in

Lemmas 3 and 4 follows the perturbation approach by Demboet al. [11] to prove the classical EPI. Our goal is to show thatthere is a monotone path from any distributions of the randomvectors to the optimal Gaussian distribution. The perturbationapproach was also adopted by Liu and Viswanath to provean extremal inequality in [6, Th. 1] that can be used to givean alternative proof of the capacity region of the two-userGaussian MIMO BC with private messages. It has also been

adopted by Weingarten et al. to prove a more general extremalinequality in [7, Lemma 2]. The extremal inequality in [7,Lemma 2] has been used to prove the capacity region of thedegraded MIMO compound BC [7] as well as the two-userGaussian MIMO BC with common and confidential messages[9]. Recently, Liu et al. also adopted the perturbation approachto prove the generalized Costa EPI [8, Th. 1] that was usedto prove the secrecy capacity regions of the degraded vectorGaussian BC with layered confidential messages. Our proofof the two extremal inequalities in Lemma 3 and Lemma 4is slightly more involved as the differential entropies may beconditioned on two different auxiliary random vectors and. This is also why previous extremal inequalities cannot be

applied in our case.

V. PROOF OF MAIN RESULTS

We first consider (11) for the aligned channel given by(3)–(5). For the simple case where , the maximum valueof (11) is (obviously) attained by transmitting at maximumcommon rate.For the case where , we have

(35)

where (a) follows from the fact that we can always transmit at arate of , , to receiver 1. Hence, for

, the maximum value of (11) is attained by transmitting atmaximum rate to receiver 1.We consider the most interesting case where . The

maximum value of (11) can be upper bounded by the maximumvalue of the following optimization problem (P1):

(36)

This follows directly from Theorem 1 with the additional con-straint . We note that since ,

is a more relaxed constraint compared to. Hence, the maximum value of (P1) is an upper

bound to the maximum value of (11).We also note that since mutual information is al-

ways nonnegative and, we may easily verify graphically

(refer to Fig. 2) that the constraint , is unnecessary in(P1) for . Since removing the constraint ,


Fig. 2. Redundancy of nonnegative constraints , : (a) the casewhere

; (b) the case where.

does not affect the optimal solution, we will not include thisconstraint in the rest of our proof.Our main objective is to show that themaximum value of (P1)

is attained by zero-mean Gaussian inputs. The capacity region,, will then by attained by zero-mean Gaussian inputs,

coupled with time-sharing. Due to the length of the proof, webreak it up into several steps.We first restrict the solution space to zero-mean Gaussian in-

puts and determined the necessary KKT conditions on the real-izing matrices of an optimal Gaussian rate vector in Lemma 5.If (in Lemma 5), we (partially) enhance receiver 1 andapply the extremal inequality from Lemma 3. If , we donot enhance receiver 1 and apply the other extremal inequalityfrom Lemma 4. We then make use of both extremal inequali-ties to provide an upper bound to the maximum value of (P1).Finally, we show that the upper bound is, in fact, attained bythe maximum value attained by restricting the solution space tozero-mean Gaussian inputs.

A. Solution of (P1) Restricted to Zero-Mean Gaussian Inputs

We first study the solution of (P1) by restricting our attentionto zero-mean Gaussian inputs.We consider the following choicefor the input distributions: ,

, , and. Hence, we have forms a Markov chain.

For the sake of compactness, we define the followingnotations:

We may then consider the following optimization problem(P1-G):

(37)

Let be the optimal solution of (P1-G). Thenecessary conditions that must satisfy aregiven in the following lemma:Lemma 5: The optimal solution, , to

(P1-G) must satisfy the following:

(38)

for some positive semidefinite matrices , and suchthat

(39)

and for some , , where

(40)

Proof: Refer to Appendix III.

B. Partial Enhancement of Receiver 1

Next, we make use of the enhancement technique introducedby Weingarten et al. to establish the capacity region of theGaussian MIMO BC with private messages [5].


If , we enhance receiver 1 as follows:

(41)

where has a covariance matrix defined as follows:

(42)

The new noise covariance matrix has the properties givenin the following lemma:Lemma 6: If the conditions of Lemma 5 hold, the covariance

matrix satisfying (42) has the following properties:

(43)

(44)

(45)

Proof: Equation (43) follows directly from the fact that for, , if , then .To prove (44), we note the following:

(46)

where (a) follows from Lemma 5 and (b) follows from (42).To prove (45), we note the following:

(47)

where (a) follows from (42) and (b) follows from the conditionthat in Lemma 5.If , we enhance receiver 1 and apply the extremal

inequality fromLemma 3. If , we do not enhance receiver1 and we apply the other extremal inequality from Lemma 4.

C. Upper Bound for (P1)

Next, we determine an upper bound for the optimal value of(P1). We may rewrite (P1) as follows:

(48)

where , , satisfies the conditions of Lemma 5.

If , the optimal value of (P1) may be upper boundedby the following:

(49)

where• (a) follows from (42) and the fact that ((43) inLemma 6). Hence, is a stochastically degraded versionof (partial enhancement of receiver 1),

• (b) follows from the fact that Gaussian inputs maximizedifferential entropy for a fixed covariance matrix,


• (c) follows from the fact that when , the optimalsolution of (P1-G) satisfies the conditions of Lemma 3 (set

, , , and) and hence, (31) can be applied,

• and (d) follows from (45) in Lemma 6.If , we may readily verify that the maximum value of

(P1) is again upper bounded by (49). This follows from the factthat the optimal solution of (P1-G) now satisfies the conditionsof Lemma 4 (set , and ) and hence,(34) can be applied.

D. Equality of the Maximum Value of (P1-G) and (P1)

Finally, we wish to show that the upper bound for (P1) givenby (49) is equal to the optimal solution of (P1-G).Wemay verifythis by enumerating all possibilities for , , whetheris strictly positive or zero.Alternatively, we note that the following optimization

problem is linear (P2-G):

(50)

(51)

where are the optimal covariance matrices of (P1-G).We readily see that the optimal solution of (P2-G) is the same asthe optimal solution of (P1-G). Moreover, strong duality holdsfor (P2-G).Since also satisfies the KKT condi-

tions for (P2-G) above, the upper bound for (P1) given by (49),which is also the optimal solution of the Lagrange dual problemfor (P2-G), is equal to the optimal solution of (P2-G). There-fore, the upper bound for (P1) given by (49) is also equal to theoptimal solution of (P1-G).Finally, since the optimal solution of (P1) is attained by the

optimal solution of (P1-G), it follows that the optimal solutionof (P1) can be attained with zero-mean Gaussian inputs. Thisconcludes the proof of Theorem 2.

E. Proof of the Capacity Region With General Linear ChannelMatrices

We now extend the proof of Theorem 2 to the case of generallinear channel matrices to prove Theorem 3. Following [5, Sec.5], we may assume without loss of generality that , , 2,3, are square matrices. Otherwise, we may apply singularvalue decomposition (SVD) on the original channel matrices toobtain an equivalent channel (with the same channel capacity)with square linear channel matrices. The new linear channelmatrices will still observe the degradedness order.Next, we apply SVD to the channel matrices , , 2, 3,

as follows:

(52)

where , are orthogonal matrices and are diagonalmatrices.

We now define a new Gaussian MIMO MBC as follows:

(53)

(54)

(55)

where the channel matrices are given by

(56)

(57)

(58)

and . We denote the capacity region of the newchannel defined by (53)–(55) by

Next, we note that we can write whereand . Hence, is degraded

with respect to . Similarly, and are degraded with re-spect to and , respectively. Therefore, the capacity regionof the new channel contains that of the original channel.Next, we may write

(59)

where . We note thatsince , we can set smallenough so that . Hence, is again degraded withrespect to .Since , , 2, 3, are invertible, the capacity region of

the channel defined by (53)–(55) is equivalent to the capacityregion of the following aligned channel:

(60)

(61)

(62)

Applying Theorem 2, the capacity region of the channel definedby (53)–(55) is given by

Finally, we note that as , ,

and thus, complete the proof.

VI. CONCLUSION

In this paper, we resolved an open problem concerning thecapacity region of the Gaussian MIMO three-receiver MBC


with two-degraded message sets. We established two new ex-tremal inequalities following the perturbation approach of Liuand Viswanath [6]. We then applied the extremal inequalities to-gether with the enhancement technique of Weingarten et al. [5]to establish the capacity region.

APPENDIX IPROOF OF LEMMA 3

We consider the following function:

(63)

and where

(64)

(65)

and , are indepen-dent of . We note that can also be expressed asfollows:

(66)

where .When , we obtain the value given by the l.h.s of (31),

and when , we obtain the value given by the r.h.s of(31). We wish to show that for any distributionsatisfying , we must have , .We divide the proof into two parts. In the first part, we determinean exact expression for . In the second part, we determine

a lower bound for . For the sake of compactness, we define

(67)

1) Determine an exact expression forWe consider the first term in (63). We can write

(68)

where and are independent and have the same distri-bution as that of . We then have

(69)

where is zero-mean Gaussian noise vector with an identitycovariance matrix. From [12, (51)], the vector de Bruijn identityis given by

(70)

Hence, we have

(71)

where (a) follows from the scaling property of the Fisher infor-mation matrix (see [13, (2)]). Next, we wish to show that wemay exchange the order of integral and derivative given by

(72)

We note that

(73)

By setting in Lemma 2, we note that

(74)

Hence, we may bound

(75)

Invoking [14, Lemma 2], we may then exchange the order ofintegral and derivative in (72).Following the same approach for the rest of the terms in (63),

we obtain an exact expression for given by

(76)

2) Determine a lower bound forFor the rest of the steps to be well defined, we require that the

inequalities and are strict.Following the same arguments in the proof of [7, Lemma 2],


we can force it to be so by subtracting and adding (we obtainand ), where is chosen to be arbitrarily

small. For all , the inequalities are strict and the rest of theproof holds. By taking and the fact that the differentialentropy is continuous with respect to the variance of the addedGaussian noise, the proof also holds when the inequalities arenot strict. Hence, for the sake of brevity, we assume that theinequalities are strict in the remainder of the proof.Next, we may write since .

Applying Lemma 2 with

and

we have

(77)

Substituting (77) into the first two terms of (76), we obtain

(78)

(79)

where (a) follows from the condition (29) of Lemma 3.Similarly, we may obtain an equivalent expression for the

next two terms of (76). Hence, we obtain

(80)

Let us provide a lower bound for :

(81)

where (a) follows from Lemma 1 and (b) follows from the factthat the conditional covariance matrix is smaller than the uncon-ditional covariance matrix.Substituting into the first term of (80), we obtain for the first

two terms

(82)

where (a) follows from condition (30) of Lemma 3, i.e.,.

Next, we consider the last four terms of (80) which can bewritten as

(83)

For the third term above, we have

(84)

where and (a) follows from condition (30)of Lemma 3, i.e., .Next, let us consider a bound for

(85)where . We make use of Lemma 2

with and

. Hence, we obtain

(86)

Substituting into the second term of (83), we obtain

(87)For the second term of (87), we note that

(88)


where (a) follows from condition (30) of Lemma 3, i.e.,and (b) follows from Lemma 2 with .

Finally, combining (83), (84), (87), and (88), we obtain thefollowing:

(89)

where (a) follows from the data processing inequality. To seethis, let be a Gaussian random vector, with covariance

, independent of . Then, by the dataprocessing inequality, for all , we have

(90)

where (a) follows from the fact thatforms a Markov chain. Letting , we obtain

(91)

Following along the lines of the previous analysis (see(70)–(76)), we then obtain

(92)Finally, we obtain (a) in (89) by making use of the fact that

and by setting

(93)

If is not strictly positive definite, we can force it to be soby considering , where is chosen to be arbitrarilysmall. By taking and relying on the fact that is acontinuous function of the elements of and , we obtain thedesired result also for the case where is not strictly positivedefinite. Since , , this completes theperturbation proof of Theorem 3.

APPENDIX IIPROOF OF LEMMA 4

Since the proof follows along the lines of proof of Lemma 3,we will proceed at a much faster pace. We first note that for any

and , there always exists a such

that and (set ).Hence, we may consider the exact same function as in the proofof Lemma 3:

(94)

and where now, we have , is a zero-meanGaussian random vector of covariance and andare as defined in (64) and (65). When , we obtain thevalue given by the l.h.s of (34) and when , we obtain thevalue given by the r.h.s of (34). We wish to show that for anydistribution satisfying , we musthave , .Making use of the condition of Lemma 4, i.e.,

, and following exactly along the lines ofthe proof of Lemma 3, we obtain

(95)

From the proof of Lemma 3, we note that the first six termsof (95) must be nonnegative. Moreover, from the condition ofLemma 4, we have . Therefore, we obtain

(96)

By setting in Lemma 2, we note that

Substituting into (96), we obtain the desired result ,. This completes the proof of Lemma 4.

APPENDIX IIIPROOF OF LEMMA 5

The conditions of Lemma 5 follow from the enhanced FritzJohn necessary conditions for the optimization problem (P1-G).The optimization problem (P1-G) can be expressed as follows:

(97)


where the vector is created by the concatenation of, , the rows of and the rows of ;

is the objective function; and , , are the rateinequality constraints of (P1-G).The set is the set of vectors over which the optimization is

done and is given by , where

(98)

We use to denote the row concatenation of the matrixand to denote the relative interior of the set . We denoteby the space of symmetric matrices and the coneof positive semidefinite matrices.Necessary conditions for (97) can be expressed in terms of

tangent cones, normal cones, and their polars (see [15, Sec.4.6]). Consider a subset of and a vector . The setof all feasible directions of at is denoted by . Theset of all tangents of at is denoted by and is alsoreferred to as the tangent cone of at . We note that sinceevery feasible direction is a tangent, hence, .The polar cone of any cone is defined by

(99)

The normal cone of at , denoted by , is obtainedfrom the polar cone by means of a closure operation.As and are continuously differentiable in an open set con-

taining , the enhanced Fritz John optimality conditions hold(see [15, Proposition 5.2.1] and [16, Proposition 2.1]). Hence,if is a local minimum of (97), there exists , ,satisfying the following:1)2) and not all 03) , with .We may verify that condition (IV) in [15, Proposition 5.2.1]

and [16, Proposition 2.1] is stronger than condition (3) above(which is in the classical Fritz John optimality conditions).As , and are nonempty convex sets such that

is not empty, we can write (see [15,Problem 4.23, p. 267])

(100)

Since , and are convex sets, from [15, Proposition4.6.3], we also have

(101)

We first note that , where , ,and , must be a valid tangent in

since it is a feasible direction in . We also have, where , and , must

be a valid tangent in .

Since is a self-dual cone and , for , ,we have

(102)

where .Similarly, we have

(103)

Next, we note that , where , , and, must be a valid tangent in .

Hence, we have

(104)

It is easy to verify that

(105)

or equivalently,

(106)

where , , and

(107)

(108)

(109)

Moreover, if the matrices , , satisfy(107)–(109), they must also satisfy

(110)

(111)

(112)

Since is regular ( is nonempty and convex), the condi-tions of Lemma 5 hold if we can show that the enhanced FritzJohn conditions hold with . To accomplish this, we wishto show that is pseudonormal [16, Definition 3.2]. If ispseudonormal, the enhanced Fritz John conditions cannot be


satisfied with , so that can be taken to be equal to1.Wemake use of constraint qualification CQ5a in [16] to show

that is pseudonormal. More specifically, we show that thereexists a

(113)

such that

(114)

It is easy to verify that ,where and are strictly positive, is a feasible direction in

and, hence, is a feasible tangent in . Moreover,must hold, . Hence, by [16, Propo-

sition 3.1], is pseudonormal.

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Hon-Fah Chong (M’05) received the Bachelor’s degree, the Masters degreeand the Ph.D. degree in Electrical and Computer Engineering from the NationalUniversity of Singapore in 2000, 2002 and 2008, respectively. Currently, he isworking as a Research Scientist at the Institute for Infocomm Research in Sin-gapore. His main research interests are information theoretical problems relatedto the broadcast channel, the relay channel and the interference channel.

Ying-Chang Liang (F’11) is a Principal Scientist at the Institute for InfocommResearch (I2R), Agency for Science, Technology and Research (A*STAR), Sin-gapore. He was a visiting scholar with the Department of Electrical Engineering,Stanford University, from December 2002 to December 2003, and taught grad-uate courses in National University of Singapore from 2004–2009. His researchinterest includes cognitive radio networks, dynamic spectrum access, reconfig-urable signal processing for broadband communications, information theory andstatistical signal processing.Dr. Liang was elected a Fellow of the IEEE in 2011 for contributions to cogni-

tive radio communications, and has received five Best Paper Awards, includingIEEE ComSoc APB outstanding paper award in 2012, and EURASIP Journal ofWireless Communications and Networking best paper award in 2010. He alsoreceived the Institute of Engineers Singapore (IES)’s Prestigious EngineeringAchievement Award in 2007, and the IEEE Standards Association’s Certificateof Appreciation Award in 2011, for contributions to the development of IEEE802.22, the first worldwide standard based on cognitive radio technology.Dr. Liang currently serves as Editor-in-Chief of the IEEE JOURNAL ON

SELECTED AREAS IN COMMUNICATIONS-Cognitive Radio Series, and is on theeditorial board of the IEEE Signal Processing Magazine. He was an AssociateEditor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, and served as a GuestEditor of five special issues on emerging topics published in IEEE, EURASIPand Elsevier journals in the past years. He is a Distinguished Lecturer of theIEEE Communications Society and IEEE Vehicular Technology Society, and isa member of the Board of Governors of the IEEE Asia-Pacific Wireless Com-munications Symposium. He served as technical program committee (TPC)Co-Chair of 2010 IEEE Symposium on New Frontiers in Dynamic SpectrumAccess Networks (DySPAN’10), General Co-Chair of 2010 IEEE InternationalConference on Communications Systems (ICCS’10), and Symposium Chair of2012 IEEE International Conference on Communications (ICC’12).

Documents

The Capacity Region of the Class of Three-Receiver Gaussian MIMO Multilevel Broadcast Channels With Two-Degraded Message Sets