superposition signaling in broadcast interference networks · superposition signaling in broadcast...

Superposition Signaling in Broadcast Interference Networks

Duong, Q., Tuan, H. D., Tam, H. H. M., Nguyen, H. H., & Poor, H. V. (2017). Superposition Signaling inBroadcast Interference Networks. IEEE Transactions on Communications, 65(11), 4646-4654.https://doi.org/10.1109/TCOMM.2017.2731315

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Download date:17. Jul. 2021

https://doi.org/10.1109/TCOMM.2017.2731315

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Superposition Signaling in Broadcast InterferenceNetworks

Hoang Duong Tuan, Ho Huu Minh Tam, Ha H. Nguyen, Trung Q. Duong and H. Vincent Poor

Abstract—It is known that superposition signaling in Gaussianinterference networks is capable of improving the achievablerate region. However, the problem of maximizing the rate gainoffered by superposition signaling is numerically prohibitive,even in the simplest case of two-user single-input single-outputinterference networks. This paper examines superposition sig-naling for the general networks of multiple-input multiple-output(MIMO) broadcast Gaussian interference networks. The problemof maximizing either the sum rate or the minimal user’s rateunder superposition signaling and dirty paper coding is solvedby a computationally-efficient path-following procedure, whichrequires only a convex quadratic program for each iteration butensures convergence at least to a locally-optimal solution. Numer-ical results demonstrate the substantial performance advantageof the proposed approach.

Index Terms—Gaussian interference networks, multi-userMIMO, superposition signaling, convex quadratic programming.

I. INTRODUCTION

A wireless multi-cell system can be modeled as an inter-ference network with multiple cells and multiple users (e.g.mobile terminals) in each cell. In the two-user case, it isknown that using Gaussian inputs and treating the residualinterference as noise in Gaussian interference networks (GINs)can achieve the sum rate capacity only for certain scenarios,including the low interference regime (see [1] and referencestherein), or under certain sufficient conditions in terms ofmatrix equations [2], [3]. Superposition signaling refers tosplitting signals intended for users to form various signal com-binations at the transmitters. This facilitates partial interferencedecoding to improve the network’s achievable rate region.

The achievable rate region for two-user single-input single-output (SISO) GINs has been investigated in [4]–[14] and [15].

This work was supported in part by the Australian Research CouncilsDiscovery Projects under Project DP130104617, in part by NSERC DiscoveryGrant # RGPIN-2017-05899, in part by the U.K. Royal Academy of Engi-neering Research Fellowship under Grant RF1415/14/22, in part by the U.K.Engineering and Physical Sciences Research Council (EPSRC) under GrantEP/P019374/1, and in part by the U.S. National Science Foundation underGrants ECCS-1343210 and ECCS-1647198.

H. D. Tuan is with the School of Electrical and Data Engineering, Universityof Technology, Sydney, NSW 2007, Australia; Email: [email protected]

H. H. M. Tam was with the Faculty of Engineering and Informa-tion Technology, University of Technology, Sydney, NSW 2007, Aus-tralia. He is now with Thinxtra Solution Pty., Sydney, Australia; Email:[email protected]

H. H. Nguyen is with the Department of Electrical and Com-puter Engineering, University of Saskatchewan, Saskatoon, Canada; Email:[email protected].

T. Q. Duong is with Queen’s University Belfast, Belfast BT7 1NN, UK;Email: [email protected].

H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544, USA; Email: [email protected].

The Han-Kobayashi (H-K) superposition signalling scheme [4]achieves the best known rate region. With H-K signalling,the signal sent by each transmitter is a superposition of twocomponents: (i) a private message that is decoded by theintended receiver only, and (ii) a common message that isdecoded by both receivers. The optimal signal superposi-tion scheme to realize the advantage of H-K signalling iscomputationally prohibitive in the domain of arbitrary inputdistributions and time sharing. Reference [7] was the firstto develop a simplified H-K signalling scheme, which usesindependent and identically distributed (i.i.d.) Gaussian inputdistributions and does not require time sharing. As such, theachievable rate region is defined explicitly via computationallytractable functions of input powers. Since optimization forthese functions is still computationally difficult, [7] proposeda simple power allocation, which achieves the capacity regionto within one bit. For some special weak interference classes,the optimal power allocations for maximizing the sum ratehave been given in [9], [10] and [14].

Inspired by [7], references [16] and [17] derived the co-variances of private and common Gaussian messages forH-K signalling, which achieve the capacity region of two-user multiple-input multiple-output (MIMO) GINs to withina constant gap. A similar result for the K-user cyclic GINwas obtained in [18]. It was also shown in [19]–[21] thatusing non-Gaussian inputs (or in [22] with Gaussian inputs)and treating interference as noise achieves the capacity regionof some special SISO GINs within a constant gap. The within-constant-gap results have a merit in the high signal-to-noiseratio (SNR) regime only, where the achievable rate region issufficiently large. As analyzed in [23], under practical SNRconditions, such results are not better than what is achievedby treating interference as noise. In fact, it is still not knownwhat rate gain H-K superposition signalling with Gaussianinputs can offer even for two-user SISO GINs. Furthermore,it is still not known what gain using non-Gaussian inputs andtreating interference as noise can offer for GINs.

It has been noted that, in the high SNR regime, interfer-ence alignment [24] may achieve a better achievable region.However, a better achievable rate region does not necessarilyyield a better sum rate or better minimal user rate. This issuehas not been treated in depth in previous work, and thus, ourfocus on H-K signalling scheme for optimization is based onthe premise that it is computable and offers a meaningful rategain in MIMO interference networks.

Reference [25] was the first to apply the H-K signallingin multiple-input single-output (MISO) broadcast GINs. Thecommon and private messages are sequentially decoded at

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the users to improve the users’ minimum rate. Inspired by[25], our previous works [23], [26] also examined sequentialdecoding of common and private messages in H-K signallingto maximize either the sum rate or the minimal user’s rate forMIMO broadcast GINs. Such design problems for the MIMOGINs were recast as optimization of d.c. (difference of twoconcave) functions over convex quadratic constraints, whichwere then solved by the so called d.c. iterations (DCI) of d.c.programming (see e.g. [27]–[31]). However, under the optimaljointly decoding as originally considered in H-K signalling, thenonconvex constraints are unavoidable. As such, the designof covariances of private messages and common messagesto maximize either the sum rate or user’s minimum rate ina broadcast GIN is a very difficult nonconvex constrainedoptimization problem. Popular approaches such as Lagrangianmultiplier or convex relaxation are unable even to locatefeasible solutions.

The contribution of the present paper is twofold:• Developing an efficient convex quadratic-based path-

following computation procedure for maximizing the sumrate and minimal user’s rate by H-K signalling in broad-cast MIMO GINs under practical SNRs, which generatesa sequence of feasible and improved points and ensuresconvergence to at least a locally optimum point. Unlike[25], [26] and [23], dirty paper coding (DPC) [32] isemployed to improve the achievable rate regions.

• Numerically demonstrating the benefit of H-K superpo-sition signalling and DPC in MIMO broadcast GINs.

The rest of the paper is organized as follows. Section IIformulates the optimization problem considered in this paperand discusses the challenges in finding solutions. SectionIII proposes a new solution method. Section IV providessimulation results. Section V concludes the paper.

Notation. Deterministic variables are boldfaced. The no-tation 〈A〉 means the trace of matrix A, while |A| is itsdeterminant. The inner product 〈X,Y 〉 between matrices Xand Y is therefore defined as 〈XHY 〉. The inner productbetween vectors x and y is defined as 〈x, y〉 = xHy. A � B(A � B, resp.) for Hermitian symmetric matrices A and Bmeans that A−B is positive definite (semi-definite, resp.). Fornotational simplicity, [X]2 refers to XXH , which is positivesemi-definite ([X]2 � 0 ∀ X). The following properties areused in the paper.

(P1) A � B � 0 implies |A| ≥ |B| and B−1 �A−1 � 0.(P2) 〈[XXX]2A〉 = 〈XXXHAXXX〉, which is a convexquadratic function in XXX whenever A � 0. Alsodefine ||XXX||2 = 〈[XXX]2〉.(P3)

∑ni=1[Xi]

2 = [X]2 and 〈(∑ni=1[Xi]

2)A〉 =〈XHAX〉 for X = [X1 X2 . . . Xn].

II. PROBLEM FORMULATION AND SOLUTION

Consider a communication network consisting of N trans-mitters as illustrated in Fig. 1. Each transmitter (Tx) isequipped with Nt ≥ 1 antennas to serve its K users, eachof which is equipped with Nr ≥ 1 antennas. Define I :={1, 2, . . . , N} and J := {1, 2, . . . ,K}. User j who is served

1,1,1 1H x

1,1, 1KH x

1, ,1 1NH x

1, , 1N KH x

,1,1N NH x

, ,N N K NH x

, ,1N N NH x

,1,N K NH x

Fig. 1: Illustration of an interference network.

by the ith Tx is referred to as user (i, j). Let Hm,i,j ∈ CNr×Ntbe the channel matrix from Tx m to user (i, j). Accordingly,Hi,i,j and Hm,i,j for m 6= i are the direct and interferingchannels with respect to user (i, j). The complex basebandsignal yi,j ∈ CNr received by user (i, j) is

yi,j =

N∑m=1

Hm,i,jxm + ni,j

= Hi,i,jxi +∑

m∈I\{i}

Hm,i,jxm + ni,j

= Hi,i,j

(K∑k=1

xi,k

)+

∑m∈I\{i}

Hm,i,j

(K∑k=1

xm,k

)+ni,j

=

K∑k=1

Hi,i,jxi,k +∑

m∈I\{i}

K∑k=1

Hm,i,jxm,k

+ni,j , (1)

where

• xm is the signal transmitted from Tx m, which is thesuperposition of signals xm,k ∈ CNt intended for allusers (m, k):

xm =

K∑k=1

xm,k.

• ni,j ∈ CNr and its entries are i.i.d. Gaussian noisesamples with zero-means and variances σ2.

The H-K signalling involves a pairing operator a(i, j) thatdescribes which other user, beside user (i, j), decodes thecommon message of user (i, j). When user (i, j) has nocommon message, then a(i, j) is an empty set. Formally, itis a mapping

a : I × J → (I × J ) ∪ {∅}

3

with the restriction that a(i, j) = (i, j) always has i 6= i anda−1(i, j) = {(i, j) : a(i, j) = (i, j)} has cardinality no morethan one.

With ∅ 6= a(i, j) = (i, j), i 6= i, signal xi,j intended foruser (i, j) is a superposition of private message xpi,j ∈ CNtwith covariance QQQp

i,j and a common message xci,j ∈ CNt withcovariance QQQp

i,j , i.e.,

xi,j = xpi,j + xci,j

The user (i, j)’s common message xci,j is to be decoded byuser (i, j), and also by user (i, j). On the other hand, if (i, j) =a(i, j) for some i 6= i, then users (i, j) and (i, j) decode thecommon message xc

i,jof user (i, j).

For simplicity, the following transmit power constraints areconsidered (although other power constraints can be easilyincorporated):

W = {QQQ := (QQQpi,j QQQc

i,j)(i,j)∈I×J : QQQpi,j � 0,

QQQci,j � 0,

∑j∈J〈QQQp

i,j +QQQci,j〉 ≤ PB , i ∈ I}, (2)

Note that xci,j ≡ 0 in (1) and thus QQQci,j ≡ 0 in (2) whenever

a(i, j) = ∅.With dirty-paper coding (DPC) and decoding [32] in a

broadcast network, user (i, j) views the term∑k≤iHi,i,jxi,k

as known non-causally and thus reduces it from the interfer-ence in (1) [33, Lemma 1]. As such, the Nr ×Nr covariancematrix of the interference plus noise at user (i, j) is given as

Mi,j(QQQ) :=∑

(n,k)∈I×J

Hn,i,j(QQQpn,k +QQQc

n,k)HHn,i,j

+σ2INr −∑k≥j

Hi,i,j(QQQpi,k +QQQc

i,k)HHi,i,j

−Hi,i,jQQQci,jHHi,i,j

. (3)

Under nonorthogonal multiple access (NOMA) [34], [35] amessage intended for a user with a worse channel condition isnot only decoded by itself but also by another user (servedby the same transmitter) with a better channel condition.The latter then cancels that message for the former from theinterference in decoding its own message. In H-K signalling,all three messages xp

i,j , xci,j and xa−1(i,j) are jointly decoded

and the corresponding achievable rates rrrpi,j , rrrci,j and rrrca−1(i,j)

satisfy

fpi,j(QQQpi,j ,Mi,j(QQQ)) :=

ln∣∣INr +Hi,i,jQQQ

pi,jH

Hi,i,j(Mi,j(QQQ))−1

∣∣ ≥ rrrpi,j , (4)

f ci,j(QQQci,j ,Mi,j(QQQ)) :=

ln∣∣INr +Hi,i,jQQQ

ci,jH

Hi,i,j(Mi,j(QQQ))−1

∣∣ ≥ rrrci,j , (5)

f ai,j(QQQci,j,Mi,j(QQQ)) :=

ln∣∣∣INr +Hi,i,jQQQ

ci,jHHi,i,j

(Mi,j(QQQ))−1∣∣∣ ≥ rrrc

i,j, (6)

fpci,j(QQQpi,j ,QQQ

ci,j ,Mi,j(QQQ)) :=

ln∣∣INr + (Hi,i,jQQQ

pi,jH

Hi,i,j

+Hi,i,jQQQci,jH

Hi,i,j)(Mi,j(QQQ))−1

∣∣ ≥ rrrpi,j + rrrci,j , (7)

fpai,j(QQQpi,j ,QQQ

ci,j,Mi,j(QQQ)) :=


pi,jH

Hi,i,j

+Hi,i,jQQQci,jHHi,i,j

)(Mi,j(QQQ))−1∣∣∣ ≥ rrrpi,j + rrrc

i,j, (8)

f cai,j(QQQci,j ,QQQ

ci,j,Mi,j(QQQ)) :=


ci,jH

Hi,i,j


)(Mi,j(QQQ))−1∣∣∣ ≥ rrrci,j + rrrc

i,j, (9)

fpcai,j (QQQpi,j ,QQQ

ci,j ,QQQ

ci,j,Mi,j(QQQ)) :=


pi,jH

Hi,i,j +Hi,i,jQQQ

ci,jH

Hi,i,j


)(Mi,j(QQQ))−1∣∣∣ ≥

rrrpi,j + rrrci,j + rrrci,j. (10)

It should be noted that the constraint (5) in rrrci,j 6= 0 assignsthe following constraint for user (i, j) = a(i, j):

f ai,j

(QQQci,j ,Mi,j(QQQ)) ≥ rrrci,j . (11)

On the other hand, the constraint (6) in rrrci,j

with a−1(i, j) =

(i, j) 6= ∅ results from the following constraint for user(i, j) = a−1(i, j):

f ci,j

(QQQci,j,Mi,j(QQQ)) ≥ rrrc

i,j. (12)

For rp = [rpi,j ](i,j)∈I×J , rc = [rci,j ](i,j)∈I×J and r =(rp, rc), the sum rare maximization problem is thus

maxQQQ,rrr

∑(i,j)∈I×J

(rrrpi,j + rrrci,j) : (2), (4)− (10). (13)

While constraint (2) in (13) is (convex) semi-definite, otherconstraints (4)-(10) are highly nonconvex. Therefore, problem(13) is maximization of a linear objective function subject tononconvex constraints.

To the authors’ best knowledge there is no available methodto handle nonconvex constraints (4)-(10). To understand thecomplexity of these nonconvex constraints, let us revisit thesimplest case of two-user MIMO interference channels con-sidered in [36]:

y1,1 = H1,1,1(xp1,1 + xc

1,1) +H2,1,1(xp2,1 + xc

2,1)+n1,1

y2,1 = H1,2,1(xp1,1 + xc

1,1) +H2,2,1(xp2,1 + xc

2,1)+n2,1.

(14)The authors of [36] considered a two-stage scheme, whichdecodes the common messages in the first stage and thendecodes the private messages in the second stage. The sum

4

achievable rate maximization problem under this scheme isaddressed by performing the following optimization steps foreach grind point (α1, α2) ∈ (0, 1) × (0, 1) of the powerallocation factors:• Solve the private sum-rate maximization [36, (eq. (7)]:

maxQQQpi,1,i=1,2

ln∣∣INr +H1,1,1QQQ

p1,1H

H1,1,1(σINr

+H2,1,1QQQp2,1H

H2,1,1)−1

∣∣+ ln

∣∣INr +H2,2,1QQQp2,1H

H2,2,1(σINr

+H1,2,1QQQp1,1H

H1,2,1)−1

∣∣ : (15)QQQpi,1 � 0, 〈QQQp

i,1〉 ≤ (1− αi)PB , i = 1, 2. (16)

• Suppose Qpi,1(α1, α2) = (Qp

1,1(α1, α2), Qp2,1(α1, α2)) is

a solution found from solving (15)-(16). Then solve thecommon sum-rate maximization [36, eq. (13)]:

maxQQQci,1,rrr

ci,1,i=1,2

rrrc1,1 + rrrc2,1 : (17)

QQQci,1 � 0, 〈QQQc

i,1〉 ≤ αiPB , i = 1, 2, (18)


c1,1H

H1,1,1

×(M1,1(Qp(α1, α2))−1∣∣ ≥ rrrc1,1, (19)


c1,1H

H1,2,1

×(M2,1(Qp(α1, α2))−1∣∣ ≥ rrrc1,1, (20)


c2,1H

H2,1,1

×(M1,1(Qp(α1, α2))−1∣∣ ≥ rrrc2,1, (21)


c2,1H

H2,2,1

×(M2,1(Qp(α1, α2)))−1∣∣ ≥ rrrc2,1, (22)

ln∣∣INr + (H1,1,1QQQ

c1,1H

H1,1,1 +H2,1,1QQQ

c2,1H

H2,1,1)

×(M1,1(Qp(α1, α2)))−1∣∣ ≥

rrrc1,1 + rrrc2,1, (23)

ln∣∣INr + (H1,2,1QQQ

c1,1H

H1,2,1 +H2,2,1QQQ

c2,1H

H2,2,1)

×(M2,1(Qp(α1, α2)))−1∣∣ ≥

rrrc2,1 + rrrc2,1, (24)

where

M1,1(Qp(α1, α2)) =σINr +H1,1,1Q

p1,1(α1, α2)HH

1,1,1

+H2,1,1Qp2,1(α1, α2)HH

2,1,1,

M2,1(Qp(α1, α2)) =σINr +H1,2,1Q

p1,1(α1, α2)HH

1,2,1

+H2,2,1Qp2,1(α1, α2)HH

2,2,1.

As the private sum-rate maximization (15)-(16) is highlynonconvex in QQQp

i,1, the authors of [36] proposed to solve it byalternating optimization between QQQp

1,1 and QQQp1,1, which is still

a difficult nonconvex problem and computationally prohibitive.Although the sum common rate optimization (17)-(24) is con-vex log-det function optimization, it is also computationallydifficult. Again, the authors in [36] proposed to solve it byalternating optimization between QQQc

1,1 and QQQc1,1, which is still

a convex log-det function optimization and computationally

demanding. In summary, the proposed method in [36] forseparate private sum-rate maximization (15)-(16) and commonsum-rate maximization (17)-(24) is already very computation-ally demanding for each (α1, α2) ∈ (0, 1)× (0, 1).

III. PROPOSED SOLUTION

We return to the optimization problem in (13). To givesome insight into its computational challenge, let us rewriteconstraints (4)-(10) as

ln∣∣Mi,j(QQQ) +Hi,i,jQQQ

pi,jH

Hi,i,j

∣∣− ln |Mi,j(QQQ)| ≥ rrrpi,j , (25)


ci,jH

Hi,i,j

∣∣− ln |Mi,j(QQQ)| ≥ rrrci,j , (26)

ln∣∣∣Mi,j(QQQ) +Hi,i,jQQQ

ci,jHHi,i,j

∣∣∣− ln |Mi,j(QQQ)| ≥ rrrci,j, (27)


pi,jH

Hi,i,j +Hi,i,jQQQ

ci,jH

Hi,i,j

∣∣− ln |Mi,j(QQQ)| ≥ rrrpi,j + rrrci,j , (28)


pi,jH

Hi,i,j +Hi,i,jQQQ

ci,jHHi,i,j

∣∣∣− ln |Mi,j(QQQ)| ≥ rrrpi,j + rrrc

i,j, (29)


ci,jH

Hi,i,j +Hi,i,jQQQ

ci,jHHi,i,j

∣∣∣− ln |Mi,j(QQQ)| ≥ rrrci,j + rrrc

i,j, (30)


pi,jH

Hi,i,j +Hi,i,jQQQ

ci,jH

Hi,i,j


∣∣∣− ln |Mi,j(QQQ)| ≥ rrrpi,j + rrrci,j + rrrci,j. (31)

In principle, all these nonconvex constraints can be succes-sively and innerly approximated by convex constraints bylinearizing the nonconvex function ln |Mi,j(QQQ)| in (25)-(31)[31]. As a consequence, the nonconvex program (13) canbe successively solved by a sequence of convex programs.However, these convex programs involve log-det functionconstraints (the first term in (25)-(31)), which are althoughconvex but still cannot be handled by the present convexsolvers.1

Next, we present a technique to equivalently express thesemi-definite constraint (2) by a simple convex quadratic func-tion and to successively approximate nonconvex constraints(4)-(10) by convex quadratic constraints. To this end, factorizeeach QQQsi,j , s ∈ {p, c} as

QQQsi,j = [VVV si,j ]2, VVV si,j ∈ CNt×Nt . (32)

The semi-definite constraint (2) in QQQ becomes the convexquadratic constraint in VVV :

WB ={VVV := [VVV p

i,j VVV ci,j ](i,j)∈I×J :∑

j∈J(||VVV p

i,j ||2 + ||VVV c

i,j ||2) ≤ PB , i ∈ I

, (33)

1For convex programs involving log-det functions in their objectives only,there is still no available solver of polynomial-time.

5

while Mi,j(QQQ) defined by (3), which is a linear map in QQQ,becomes a quadratic map in VVV . For notational simplicity, weuse the same notation Mi,j(VVV ) for

Mi,j(VVV ) :=∑

(n,k)∈I×J

Hn,i,j([VVVpn,k]2 + [VVV c

n,k]2)HHn,i,j

+σ2INr −∑k≥j

Hi,i,j([VVVpi,k]2 + [VVV c

i,k]2)HHi,i,j

−Hi,i,j [VVVci,j

]2HHi,i,j

. (34)

The constraints (4)-(10) in QQQ are equivalently expressed as thefollowing constraints in VVV

F pi,j(VVV

pi,j ,Mi,j(VVV )) :=

ln∣∣INr + [Hi,i,jVVV

pi,j ]

2(Mi,j(VVV ))−1∣∣ ≥ rrrpi,j , (35)

F ci,j(VVV

ci,j ,Mi,j(VVV )) :=

ln∣∣INr + [Hi,i,jVVV

ci,j ]

2(Mi,j(VVV ))−1∣∣ ≥ rrrci,j , (36)

F ai,j(VVV

ci,j,Mi,j(VVV )) :=

ln∣∣∣INr + [Hi,i,jVVV

ci,j

]2(Mi,j(VVV ))−1∣∣∣ ≥ rrrc

i,j, (37)

F pci,j(VVV

pi,j ,VVV

ci,j ,Mi,j(VVV )) :=

ln∣∣INr + ([Hi,i,jVVV

pi,j ]

2 + [Hi,i,jVVVci,j ]

2)(Mi,j(VVV ))−1∣∣ ≥

rrrpi,j + rrrci,j , (38)

F pai,j(VVV

pi,j ,VVV

ci,j,Mi,j(VVV )) :=

ln∣∣∣INr + ([Hi,i,jVVV

pi,j ]

2 + [Hi,i,jVVVci,j

]2)(Mi,j(VVV ))−1∣∣∣ ≥

rrrpi,j + rrrci,j, (39)

F cai,j(VVV

ci,j ,VVV

ci,j,Mi,j(VVV )) :=

ln∣∣∣INr + ([Hi,i,jVVV

ci,j ]

2 + [Hi,i,jVVVci,j

]2)(Mi,j(VVV ))−1∣∣∣ ≥

rrrci,j + rrrci,j, (40)

F pcai,j (VVV p

i,j ,VVVci,j ,VVV

ci,j,Mi,j(VVV )) :=

ln∣∣INr + ([Hi,i,jVVV

pi,j ]


2

+[Hi,i,jVVVci,j

]2)(Mi,j(VVV ))−1∣∣∣ ≥

rrrpi,j + rrrci,j + rrrci,j. (41)

With the above developments, the problem in (13) is equiv-alently reformulated as

maxVVV ,rrr

P(rrr) :=∑

(i,j)∈I×J

(rrrpi,j+rrrci,j) : (33), (35)−(41). (42)

It is pointed out that all functions in (35)-(41) are highlynonlinear, nonconcave in variable VVV . As such it is usefulto find their lower bounds, that are global to guarantee therichness of the feasibility region but also sufficiently local fora tight approximation. Our bounding technique is based on thefollowing result, whose proof is given in Appendix A.

Theorem 1: The following inequality holds true for allmatrices XXXi ∈ CNr×Nt , X(κ)

i ∈ CNr×Nt , i = 1, . . . , L and0 ≺MMM ∈ CNr×Nr , 0 ≺M (κ) ∈ CNr×Nr :

ln |INr + (

L∑i=1

[XXXi]2)MMM−1| ≥

ln |INr + (

L∑i=1

[XXX(κ)i ]2)(M (κ))−1|

−〈(L∑i=1

[X(κ)i ]2)(M (κ))−1〉

+2

L∑i=1

<{〈(X(κ)i )H(M (κ))−1XXXi〉}

+〈(M (κ) +

L∑i=1

[X(κ)]2)−1

−(M (κ))−1,MMM +

L∑i=1

[XXXi]2〉. (43)

Next, as

M (κ) +

L∑i=1

[X(κ)i ]2 �M (κ) � 0,

it follows from (P1) that

(M (κ) +

L∑i=1

[X(κ)i ]2)−1 − (M (κ))−1 � 0.

Thus, it follows from (P2) that the right hand side (RHS) of(43) is concave quadratic in XXX , [XXXi]i=1,...,L. Obviously theRHS of (43) is still concave quadratic in X for

MMM =

L∑i=1

Hi[Xi]2HH

i +A,A � 0,

and accordingly, M (κ) =∑Li=1Hi[X

(κ)i ]2HH

i +A.Now, define the following positive combination of [VVV xi,j ]

2,x ∈ {p, c}:

Mpi,j(VVV ) =Mi,j(VVV ) + [Hi,i,jVVV

pi,j ]

2,

Mci,j(VVV ) =Mi,j(VVV ) + [Hi,i,jVVV

ci,j ]

2,Ma

i,j(VVV ) =Mi,j(VVV ) + [Hi,i,jVVVci,j

]2.

Applying Theorem 1 at V (κ) = [Vp,(κ)i,j V

c,(κ)i,j ](i,j)∈I×J

givesF pi,j(VVV

pi,j ,Mi,j(VVV )) ≥ Fp,(κ)

i,j (VVV ),

F ci,j(VVV

ci,j ,Mi,j(VVV )) ≥ Fc(κ)i,j (VVV ),

F ai,j(VVV

ci,j,Ma

i,j(VVV )) ≥ Fa,(κ)i,j (VVV )

(44)

with the following concave quadratic functions in VVV

Fp,(κ)i,j (VVV ) := a

p,(κ)i,j + 2<{〈Bp,(κ)

i,j VVV pi,j〉}

+〈Cp,(κ)i,j ,Mp

i,j(VVV )〉, (45)

Fc(κ)i,j (VVV ) := ac,(κ)i,j + 2<{〈Bc,(κ)

i,j VVV ci,j〉}

+〈Cc,(κ)i,j ,Mc

i,j(VVV )〉, (46)

Fa,(κ)i,j (VVV ) := a

a,(κ)i,j + 2<{〈Ba,(κ)

i,j VVV ci,j〉}

+〈Ca,(κ)i,j ,Ma

i,j(VVV )〉, (47)

6

where

0 > ap,(κ)i,j = F p

i,j(Vp,(κ)i,j ,Mi,j(V

(κ)))

−〈[Hi,i,jVp,(κ)i,j ]2(Mi,j(V

(κ)))−1〉, (48)

0 > ac,(κ)i,j = F c

i,j(Vc,(κ)i,j ,Mi,j(V

(κ)))

−〈[Hi,i,jVc,(κ)i,j ]2(Mi,j(V

(κ)))−1〉, (49)

0 > aa,(κ)i,j = F a

i,j(Vc,(κ)

i,j,Ma

i,j(V(κ)))

−〈[Hi,i,jVc,(κ)

i,j]2(Mi,j(V

(κ)))−1〉, (50)

and

Bp,(κ)i,j = (V

p,(κ)i,j )HHH

i,i,j(Mi,j(V(κ)))−1Hi,i,j , (51)

Bc,(κ)i,j = (V

c,(κ)i,j )HHH

i,i,j(Mi,j(V(κ)))−1Hi,i,j , (52)

Ba(κ)i,j = (V

c,(κ)

i,j)HHH

i,i,j(Mi,j(V

(κ)))−1Hi,i,j , (53)

and

0 � Cp,(κ)i,j =Mp

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1, (54)

0 � Cc,(κ)i,j =Mc

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1, (55)

0 � Ca,(κ)i,j =Ma

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1. (56)

Analogously, under the following definitions of the positivecombinations of [VVV ]2

Mpci,j(VVV ) = Mi,j(VVV ) + [Hi,i,jVVV

pi,j ]


2,

Mpai,j(VVV ) = Mi,j(VVV ) + [Hi,i,jVVV

pi,j ]

2 + [Hi,i,jVVVci,j

]2,

Mcai,j(VVV ) = Mi,j(VVV ) + [Hi,i,jVVV

ci,j

]2 + [Hi,i,jVVVci,j

]2,

Mpcai,j (VVV ) = Mi,j(VVV ) + [Hi,i,jVVV

pi,j ]


2

+[Hi,i,jVVVci,j

]2

and by applying Theorem 1, one has

F pci,j(VVV

pi,j ,VVV

ci,j ,Mi,j(VVV )) ≥ Fpc,(κ)

i,j (VVV ),

F pai,j(VVV

pi,j ,VVV

ci,j,Mi,j(VVV )) ≥ Fpa,(κ)

i,j (VVV ),

F cai,j(VVV

ci,j ,VVV

ci,j,Mi,j(VVV )) ≥ F ca,(κ)

i,j (VVV ),

F pcai,j (VVV p

i,j ,VVVci,j ,VVV

ci,j,Mi,j(VVV )) ≥ Fpca,(κ)

i,j (VVV ).

(57)

The various concave quadratic functions in (57) are given as:

Fpc,(κ)i,j (VVV ) :=a

pc,(κ)i,j + 2<{〈Bp,(κ)

i,j VVV pi,j〉}+ 2<{〈Bc,(κ)

i,j VVV ci,j〉}

+ 〈Cpc,(κ)i,j ,Mpc

i,j(VVV )〉, (58)

Fpa,(κ)i,j (VVV ) :=a

pa,(κ)i,j + 2<{〈Bp,(κ)

i,j VVV pi,j〉}+ 2<{〈Ba,(κ)

i,j VVV ci,j〉}

+ 〈Cpa,(κ)i,j ,Mpa

i,j(VVV )〉, (59)

F ca,(κ)i,j (VVV ) :=a

ca,(κ)i,j + 2<{〈Bc,(κ)

i,j VVV ci,j〉}+ 2<{〈Ba,(κ)

i,j VVV ci,j〉}

+ 〈Cca,(κ)i,j ,Mca

i,j(VVV )〉, (60)

Fpca,(κ)i,j (VVV ) := a

pca,(κ)i,j + 2<{〈Bp,(κ)

i,j VVV pi,j〉}

+2<{〈Bc,(κ)i,j VVV c

i,j〉}+ 2<{〈Ba,(κ)i,j VVV c

i,j〉}

+〈Cpca,(κ)i,j ,Mpca

i,j (VVV )〉, (61)

where

0 > apc,(κ)i,j = F pc

i,j(Vp,(κ)i,j , V

c,(κ)i,j ,Mi,j(V

(κ)))

−〈([Hi,i,jVp,(κ)i,j ]2 + [Hi,i,jV

c,(κ)i,j ]2)

×(Mi,j(V(κ)))−1〉, (62)

0 > apa,(κ)i,j = F pa

i,j(Vp(κ)i,j , V

c(κ)

i,j,Mi,j(V

(κ)))


c,(κ)

i,j]2)

×(Mi,j(V(κ))−1〉, (63)

0 > aca,(κ)i,j = F ca

i,j(Vc,(κ)i,j , V

c,(κ)

i,j,Mi,j(V

(κ)))

−〈([Hi,i,jVc,(κ)i,j ]2 + [Hi,i,jV

c,(κ)

i,j]2)

×(Mi,j(V(κ)))−1〉, (64)

0 > apca,(κ)i,j = F pca

i,j (Vp,(κ)i,j , V

c,(κ)i,j , V

c,(κ)

i,j,Mi,j(V

(κ)))


c,(κ)i,j ]2

+[Hi,i,jVc,(κ)

i,j]2)(Mi,j(V

(κ)))−1〉, (65)

and

0 � Cpc,(κ)i,j =Mpc

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1, (66)

0 � Cpa,(κ)i,j =Mpa

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1, (67)

0 � Cca,(κ)i,j =Mca

i,j(V(κ)))−1 − (Mi,j(V

(κ)))−1, (68)

0 � Cpca,(κ)i,j =Mpca

i,j (V (κ)))−1 − (Mi,j(V(κ)))−1. (69)

We now propose the following path-following procedurebased on convex quadratic programming for solving (13).

• Initialization: Initialize any feasible solution V (0) =

(Vp,(0)i,j V

c,(0)i,j )(i,j)∈I×J to the convex power con-

straint (33). Set Qp,(0)i,j = V

p,(0)i,j (V

p,(0)i,j )H , Qc,(0)

i,j =

Vc,(0)i,j (V

c,(0)i,j )H , Q(0) = [Q

p,(0)i,j Q

c,(0)i,j ](i,j)∈I×J and

solve the linear program

maxrrrp=[rrrpi,j ],rrr

c=[rrrci,j ]

∑(i,j)∈I×J

(rrrpi,j + rrrci,j) :

(4)− (10) for Q = Q(0) (70)

to find the optimal solution r(0).• κ-th iteration is to generate V (κ+1) :=

(Vp,(κ+1)i,j V

c,(κ+1)i,j )(i,j)∈I×J and

r(κ+1) := (rp,(κ+1)i,j r

c,(κ+1)i,j )(i,j)∈I×J from

7

Algorithm 1 QP-based path-following algorithm for solving(13)

1: Initialize κ := 0.2: Initialize any feasible solution V (0) =

(Vp,(0)i,j V

c,(0)i,j )(i,j)∈I×J to the convex power constraint

(33). Solve linear program (70) to find the optimalsolution r(0).

3: repeat4: Solve quadratic program (71) for (V (κ+1), r(κ+1)).5: Set κ := κ+ 1.6: until convergence of the objective in (42), i.e.,

(P(r(κ+1))−P(r(κ)))/P(r(κ)) ≤ ε for a given computa-tional tolerance ε.

(V (κ), r(κ)) by the optimal solution of the convexquadratic program

maxVVV ,rrr

P(rrr) :=∑

(i,j)∈I×J

(rrrpi,j + rrrci,j) : (33), (71a)

Fp,(κ)i,j (VVV ) ≥ rrrpi,j ,F

c,(κ)i,j (VVV ) ≥ rrrci,j , (71b)

Fa,(κ)i,j (VVV ) ≥ rrrc

i,j,Fpc,(κ)

i,j (VVV ) ≥ rrrpi,j + rrrci,j , (71c)

Fpa,(κ)i,j (VVV ) ≥ rrrpi,j + rrrc

i,j,F ca,(κ)

i,j (VVV ) ≥ rrrci,j + rrrci,j, (71d)

Fpca,(κ)i,j (VVV ) ≥ rrrpi,j + rrrci,j + rrrc

i,j, (71e)

(i, j) ∈ I × J , (i, j) = a−1(i, j) (71f)

For ∆ , | ∪(i,j)∈I×J a(i, j)|, the number of quadraticconstraints in (71) is bounded by m = 3N + 7∆ +(NK−∆) while the variable number is n = NKM2

t /2+∆M2

t /2+NK+(NK−∆). So the computational com-plexity of (71) is upper bounded by O(n2m2.5 +m3.5).

It is pointed out that (71b)-(71e) are employed when botha(i, j) 6= ∅ and a−1(i, j) 6= ∅. Other three possible cases are• a(i, j) 6= ∅ but a−1(i, j) = ∅: In this case user (i, j)

needs to decode spi,j and sci,j only. Hence replace (71b)-(71e) with


c,(κ)i,j (VVV ) ≥ rrrci,j ,

Fpc,(κ)i,j (VVV ) ≥ rrrpi,j + rrrci,j .

(72)

• a(i, j) = ∅ but a−1(i, j) = (i, j) 6= ∅: In this case user(i, j) needs to decode spi,j and sc

i,jonly. Hence replace

(71b)-(71e) with


a,(κ)i,j (VVV ) ≥ rrrc

i,j,

Fpa,(κ)i,j (VVV ) ≥ rrrpi,j + rrrc

i,j.

(73)

• Both a(i, j) = ∅ and a−1(i, j) = ∅: In this case user(i, j) needs to decode its private message spi,j only. Then,replace (71b)-(71e) with

Fp,(κ)i,j (VVV ) ≥ rrrpi,j . (74)

Algorithm 1 recaps the above a QP-based path-followingprocedure for solving the sum rate maximization problem(13). The convergence property of the proposed algorithm isestablished in the following proposition.

Proposition 1: Algorithm 1 generates a sequence{(V (κ), r(κ))} of feasible and improved solutions of theoriginal nonconvex program (42) in the sense that

P(r(κ+1)) > P(r(κ)) (75)

as far as (Q(κ+1), r(κ+1)) 6= (Q(κ), r(κ)), which converges atleast to a solution satisfying the KKT condition for optimalityof (13).

Proof: By (44) and (57), every feasible solution to (71) isalso feasible to (42). Then (75) is true because (V (κ), r(κ))is also feasible to (71), while (V (κ+1), r(κ+1)) is its optimalsolution. Furthermore, the sequence {(V (κ), r(κ))} is boundedby constraint (33). By Cauchy’s theorem there is a convergentsubsequence {(V (κν), r(κν))} so

limν→+∞

(P(r(κν+1))− P(r(κν))) = 0.

For every κ, there is ν such that κν ≤ κ and κ + 1 ≤ κν .Therefore

0 ≤ limκ→+∞

(P(r(κ+1))− P(r(κ)))

≤ limκ→+∞

(P(r(κν+1))− P(r(κν)))

= 0,

showing that limκ→+∞

(P(r(κ+1))−P(r(κ))) = 0. Each accumu-

lation point {(V , r)} of the sequence {(V (κ), r(κ))} obviouslysatisfies the KKT condition for optimality [37]. �

Remark. The maximin rate optimization problem, formu-lated as

maxQQQ,rrr

min(i,j)∈I×J

(rrrpi,j + rrrci,j) : (2), (4)− (10), (76)

can also be solved by the proposed path-following algo-rithm when replacing the objective in (70) and (71) withmin(i,j)∈I×J (rrrpi,j + rrrci,j).

IV. NUMERICAL RESULTS

In this section, numerical results are presented to show therate performances achieved by different signalling schemes.For ease of discussion, the conventional signalling involvingonly private messages is referred to as “private only”, whilethe proposed H-K signalling is referred to as “H-K”. Thecomputational tolerance in Algorithm 1 is set as ε = 10−5.Each point plotted for the Monte Carlo simulations is basedon 100 random network realizations.

For convenience, set Hm,i,j =√ηm,i,jhm,i,j for m 6= i.

The entries hm,i,j are independent and identically distributedcomplex Gaussian variables with zero mean and unit variance,which represent the small-scaling fading, whereas ηm,i,j cap-tures the path loss and large-scale fading.

Obviously, the effectiveness of the H-K signalling stronglydepends on the pairing operator a. Unfortunately optimizationof the pairing operator is an intractable combinatorial problem.It is pointed out that a heuristic rule for choosing a based onthe performance of “private only” messaging was proposed in[25] and further developed in [23].

8

A. N = 2, K = 1 with Nt = Nr ∈ {1, 2, 4, 6, 8}, as in [36,Fig. 3]

In this study, the direct channel strengths η1,1,1 = η2,2,1 = 0dB and the inferring channel strengths η1,2,1 = η2,1,1 =−4.7712 dB are selected as in [36, p. 4317]. Fig. 2 plotsthe sum rate performance versus the number of antennas,under a per-Tx power budget PB = 30 dB. For comparison,also included is the performance of the two-stage schemein [36], which is extremely computationally demanding. Itsperformance plotted in Fig. 2 based on only 225 sampledpoints already took hours of computer simulation to obtain.

1 2 3 4 5 6 7 810

20

30

40

50

60

70

80

90

100

Number of antennas (Nt=N

r)

Sum

rat

e (b

ps/H

z)

Two−stage schemeProposed H−K schemePrivate only

Fig. 2: Plots of the sum rate versus the number of antennas.

B. N = 2, K = 1, Nt = 4, Nr = 2

Here, the statistical performance of MIMO interferencenetworks depicted as in Fig. 3a is analyzed. Following [7],[38], the direct channel strengths are fixed at (η1,1,1, η2,2,1) =(10, 20) (in dB), while the interfering channel strengthsη1,2,1 = η2,1,1 are increased from −5 dB to 20 dB. Thesevalues cover a wide range of channels effects, such as pathloss and shadowing, which may be environment-dependent.The simulation scenarios thus vary from weak MIMO GINsto mixed MIMO GINs. The upper and lower bounds on thesum or minimal rates can be obtained by solving the linearinequality [17, (52a)-(52i)] and [17, (11)- (17)], respectively.Fig. 4 show that both of these bounds are quite loose. Theperformance of the conventional scheme degrades significantlyas the interference channel strength η increases. This is in asharp contrast to the improved performance behavior of theH-K signalling.

C. Three-user cyclic GIN with Nt = 4, Nr = 2

Fig. 3b depicts a three-user cyclic GIN. The direct channelstrengths (η1,1,1, η2,2,1, η3,3,1) are fixed at (10, 20, 5) (in dB),while the interfering channel strengths η2,1,1 = η3,2,1 = η1,3,1are increased from −10 dB to 30 dB for testing differentscenarios. Fig. 5 shows a profound performance improvementachieved by using H-K signalling, especially when the inter-ference channel gain η is large. In contrast, the performanceof the conventional scheme is severely deteriorated.

Tx 1

Tx 2

(a) N = 2, K = 1.

Tx 2Tx 3

Tx 1

(b) Cyclic N = 3, K = 1

Tx 1

Tx 2

(c) N = 2, K = 2

Fig. 3: Different interference networks considered in simula-tion.

D. N = 2, K = 2, Nt = 4, Nr = 2

As shown in Fig. 3c, the direct channel strengths η1,1,1 =η2,2,1 and η1,1,2 = η2,2,2 are, respectively, fixed at 10 dB and15 dB, and the interfering channel strengths η2,1,1 and η2,1,2are set to −50 dB (thus these interfering channels are basicallydisabled). The interfering channel strengths η1,2,1 = η1,2,2are increased from −10 dB to 50 dB. There are two usersper cell so the DPC (which results in the covariance of theinterference-plus-noise as in (3)) is expected to be beneficial.To confirm this fact, we also compare the performance ofthe H-K signalling with a signalling scheme that does not

9

−5 5 10 15 20

50

55

60

65

70

Interference channel gain η (dB)

Sum

rat

e (b

ps/H

z)

H−KPrivate onlyUpper boundLower bound

(a) Sum rate.

−5 5 10 15 20

16

18

20

22

24

26

28

30

32

34


Min

rat

e (b

ps/H

z)

H−KPrivate onlyUpper boundLower bound

(b) Minimum rate.

Fig. 4: Rate performance versus interfering channel strengthfor N = 2, K = 1.

implement DPC. For the latter, the interference-plus-noisecovariance is conventionally calculated as

Mi,j(QQQ) :=∑

(n,k)∈I×J

Hn,i,j(QQQpn,k +QQQc

n,k)HHn,i,j + σ2INr

−Hi,i,j(QQQpi,j +QQQc

i,j)HHi,i,j −Hi,i,jQQQ

ci,jHHi,i,j

.

(77)

Fig. 6 shows the superior performance of the H-K sig-nalling with and without DPC. As expected, a more profoundenhancement in the performance of the H-K signalling isobserved when the intercell interference channel gain is high(e.g., η > 15 dB).

V. CONCLUSIONS

In this paper, we have studied the H-K superposition sig-nalling strategy for multi-user MIMO broadcast interferencenetworks. The ability of the H-K signalling to increase theachievable rate region of a multi-user MIMO Gaussian in-terference network has been previously demonstrated, but itsoptimization has never been adequately addressed. The maincontribution of this paper is to show that such an optimizationproblem can be solved by a path-following procedure based onconvex quadratic programming of low computational complex-ity. In the presence of mild-to-strong interference, simulationresults demonstrated significant rate gains obtained by ouroptimized H-K signalling.

−5 0 5 10 15 20 25 3050

55

60

65

70

75

80


Sum

rat

e (b

ps/H

z)

H−KPrivate only

(a) Sum rate.

−5 0 5 10 15 20 25 3012

14

16

18

20

22

24

Interference channel gain η (dB)M

in r

ate

(bps

/Hz)

H−K

Private only

(b) Minimum rate.

Fig. 5: Rate performance versus interfering channel strengthfor a cyclic channel GIN, N = 3, K = 1.

−5 5 15 25 35 4550

55

60

65

70

75

80

85


Sum

rat

e (b

ps/H

z)

15 20 25 3060

65

70

Proposed H−K, with DPCPrivate only, with DPCH−K, without DPCPrivate only, without DPC

(a) Sum rate.

−5 5 15 25 35 450

5

10

15

20


Min

rat

e (b

ps/H

z)

5 15 25 3513

15

17

Proposed H−K, with DPCPrivate only, with DPCH−K, without DPCPrivate only, without DPC

(b) Minimum rate.

Fig. 6: Rate performance versus interfering channel strengthfor N = 2, K = 2, Nt = 4, Nr = 2.

10

APPENDIX A: PROOF OF THEOREM 1

Lemma 1: The following inequality holds for all XXX , X(κ)

and Y � [X]2, Y (κ) � [X(κ)]2 of appropriate sizes:

ln |INr − [X]2Y−1| ≤ln |INr − [X(κ)]2(Y (κ))−1|

+〈[X(κ)]2(Y (κ) − [X(κ)]2)−1〉−2<{〈(X(κ))H(Y (κ) − [X(κ)]2)−1X〉}+〈(Y (κ) − [X(κ)]2)−1 − (Y (κ))−1,Y〉. (78)

Proof: Define the function

g(X,Y) := ln |INr − [X]2Y−1| on {Y � [X]2}

and mapping

h(X,Y) := XHY−1X on {Y � 0}.

By [39, Appendix C], whenever α ≥ 0, β ≥ 0, α + β = 1,the following matrix inequality holds true

h(α(X,Y) +β(X(κ), Y (κ))) � αh(X,Y) +βh(X(κ), Y (κ)).

It then follows that

INr − h(α(X,Y) + β(X(κ), Y (κ))) �INr − αh(X,Y)− βh(X(κ), Y (κ)).

Therefore

g(α(X,Y) + β(X(κ), Y (κ))) =

ln |INr − h(α(X,Y) + β(X(κ), Y (κ)))| ≥ln |INr − αh(X,Y)− βh(X(κ), Y (κ))| ≥

α ln |INr − h(X,Y)|+ β ln |INr − h(X(κ), Y (κ))| =

αg(X,Y) + βg(X(κ), Y (κ)), (79)

showing that g(·) is a concave function. Note that (79) is basedon the fact that function ln |Z| is concave in Z � 0.

For such a concave function, it is true [40] that

g(X,Y) ≤g(X(κ), Y (κ)) + 〈∇g(X(κ), Y (κ)), (X,Y)

−(X(κ), Y (κ))〉 =ln |INr − [X(κ)]2(Y (κ))−1| − 2<{〈(X(κ))H(Y (κ)

−[X(κ)]2)−1(X−X(κ))〉}+〈(Y (κ) − [X(κ)]2)−1 − (Y (κ))−1,Y − Y (κ)〉 =

ln |INr − [X(κ)]2(Y (κ))−1| − 2<{〈(X(κ))H(Y (κ)

−[X(κ)]2)−1X〉}+2〈(X(κ))H(Y (κ) − [X(κ)]2)−1X(κ))〉+〈(Y (κ) − [X(κ)]2)−1 − (Y (κ))−1,Y〉

−〈(Y (κ) − [X(κ)]2)−1 − (Y (κ))−1, Y (κ)〉.

The right hand side (RHS) of the last inequality is the RHSof (78) because

2〈(X(κ))H(Y (κ) − [X(κ)]2)−1X(κ))〉−〈(Y (κ) − [X(κ)]2)−1 − (Y (κ))−1, Y (κ)〉 =

〈[X(κ)]2(Y (κ) − [X(κ)]2)−1〉.

This completes the proof of Lemma 1. �

Now, the proof of Theorem 1 is as follows. By defining

XXX = [XXX1 XXX2 . . . XXXL]

and using (P3) to rewrite

ln |INr + (

L∑i=1

[XXXi]2)MMM−1| =

− ln |INr − [XXX]2(MMM + [XXX]2)−1|.

The inequality (43) then follows from (78) by substituting

XXX ←XXX, MMM + [XXX]2 ← Y,X(κ) ← X(κ), M (κ) + [X(κ)]2 ← Y (κ).

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Hoang Duong Tuan received the Diploma (Hons.)and Ph.D. degrees in applied mathematics fromOdessa State University, Ukraine, in 1987 and 1991,respectively. He spent nine academic years in Japanas an Assistant Professor in the Department ofElectronic-Mechanical Engineering, Nagoya Univer-sity, from 1994 to 1999, and then as an AssociateProfessor in the Department of Electrical and Com-puter Engineering, Toyota Technological Institute,Nagoya, from 1999 to 2003. He was a Profes-sor with the School of Electrical Engineering and

Telecommunications, University of New South Wales, from 2003 to 2011.He is currently a Professor with the Faculty of Engineering and InformationTechnology, University of Technology Sydney. He has been involved inresearch with the areas of optimization, control, signal processing, wirelesscommunication, and biomedical engineering for more than 20 years.

Ho Huu Minh Tam was born in Ho Chi Minh City,Vietnam. He received the B.S. degree in electricalengineering and telecommunications from the HoChi Minh City University of Technology, Vietnam,in 2012, the Ph.D. degree in electrical engineeringfrom the University of Technology Sydney, NSW,Australia, in 2016. He is now with Thinxtra Solu-tion Pty., Sydney, Australia His research interest isin optimization techniques in signal processing forwireless communications and internet of things.

H a H. Nguyen (M’01, SM’05) received the B.Eng.degree from the Hanoi University of Technology(HUT), Hanoi, Vietnam, in 1995, the M.Eng. de-gree from the Asian Institute of Technology (AIT),Bangkok, Thailand, in 1997, and the Ph.D. degreefrom the University of Manitoba, Winnipeg, MB,Canada, in 2001, all in electrical engineering. Hejoined the Department of Electrical and ComputerEngineering, University of Saskatchewan, Saska-toon, SK, Canada, in 2001, and became a fullProfessor in 2007. He currently holds the position

of Cisco Systems Research Chair. His research interests fall into broad areasof Communication Theory, Wireless Communications, and Statistical SignalProcessing. Dr. Nguyen was an Associate Editor for the IEEE Transactions onWireless Communications and IEEE Wireless Communications Letters during2007-2011 and 2011-2016, respectively. He currently serves as an AssociateEditor for the IEEE Transactions on Vehicular Technology. He was a Co-chair for the Multiple Antenna Systems and Space-Time Processing Track,IEEE Vehicular Technology Conferences (Fall 2010, Ottawa, ON, Canadaand Fall 2012, Quebec, QC, Canada), Lead Co-chair for the Wireless AccessTrack, IEEE Vehicular Technology Conferences (Fall 2014, Vancouver, BC,Canada), Lead Co-chair for the Multiple Antenna Systems and CooperativeCommunications Track, IEEE Vehicular Technology Conference (Fall 2016,Montreal, QC, Canada), and Technical Program Co-chair for CanadianWorkshop on Information Theory (2015, St. John’s, NL, Canada). He isa coauthor, with Ed Shwedyk, of the textbook ”A First Course in DigitalCommunications” (published by Cambridge University Press). Dr. Nguyenis a Fellow of the Engineering Institute of Canada (EIC) and a RegisteredMember of the Association of Professional Engineers and Geoscientists ofSaskatchewan (APEGS).

12

Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Sweden in2012. Since 2013, he has joined Queen’s Univer-sity Belfast, UK as a Lecturer (Assistant Profes-sor). His current research interests include small-cellnetworks, physical layer security, energy-harvestingcommunications, cognitive relay networks. He isthe author or co-author of more than 260 technicalpapers published in scientific journals (142 articles)and presented at international conferences (121 pa-

pers).Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IET COMMUNICATIONS, and a Senior Editor for IEEE COMMU-NICATIONS LETTERS. He was awarded the Best Paper Award at the IEEEVehicular Technology Conference (VTC-Spring) in 2013, IEEE InternationalConference on Communications (ICC) 2014, and IEEE Global Communi-cations Conference (GLOBECOM) 2016. He is the recipient of prestigiousRoyal Academy of Engineering Research Fellowship (2016-2021).

H. Vincent Poor (S72, M77, SM82, F87) receivedthe Ph.D. degree in EECS from Princeton Universityin 1977. From 1977 until 1990, he was on the facultyof the University of Illinois at Urbana-Champaign.Since 1990 he has been on the faculty at Princeton,where he is currently the Michael Henry StraterUniversity Professor of Electrical Engineering. Dur-ing 2006 to 2016, he served as Dean of PrincetonsSchool of Engineering and Applied Science. Hisresearch interests are in the areas of informationtheory, statistical signal processing and stochastic

analysis, and their applications in wireless networks and related fields such assmart grid and social networks. Among his publications in these areas is thebook Mechanisms and Games for Dynamic Spectrum Allocation (CambridgeUniversity Press, 2014).

Dr. Poor is a member of the National Academy of Engineering, the NationalAcademy of Sciences, and is a foreign member of the Royal Society. He isalso a fellow of the American Academy of Arts and Sciences, the NationalAcademy of Inventors, and other national and international academies. Hereceived the Marconi and Armstrong Awards of the IEEE CommunicationsSociety in 2007 and 2009, respectively. Recent recognition of his workincludes the 2016 John Fritz Medal, the 2017 IEEE Alexander GrahamBell Medal, Honorary Professorships at Peking University and TsinghuaUniversity, both conferred in 2016, and a D.Sc. honoris causa from SyracuseUniversity awarded in 2017.

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