Multicast communications in cognitive radio networks using ...users.wpi.edu/~xhuang/pubs/2015_guo_wcmc.pdf · 1 Active Broadband Networks, 20 Speen St., Framingham, MA 01701, U.S.A

WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob. Comput. 2015; 15:260–275Published online 28 December 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2335

RESEARCH ARTICLE

Multicast communications in cognitive radionetworks using directional antennasWenxuan Guo1* and Xinming Huang2

1 Active Broadband Networks, 20 Speen St., Framingham, MA 01701, U.S.A.2 Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.

ABSTRACT

This paper presents a study on multicast communications in cognitive radio networks (CRNs)using directional antennas.The objective is to maximize the throughput of the CRN. The spectrum is divided into multiple channels and licensed tothe primary network. While the CRN is accessing the spectrum, the interference power is carefully controlled to avoidimpacting the operation of the primary network. The mathematical model is presented and subsequently formulated as amixed integer non-linear programming (MINLP) problem, which is non-deterministic polynomial-time hard. Therefore, agreedy algorithm is designed to approximate the optimal performance. The MINLP problem is then relaxed and an upperbound is developed. Simulation results are presented to compare the performance of the greedy algorithm and the upperbound, which demonstrates the efficacy of the greedy algorithm as well as the tightness of the upper bound. Copyright ©2012 John Wiley & Sons, Ltd.

KEYWORDS

cognitive radio networks; directional antenna; multicast communication

*Correspondence

Wenxuan Guo, Active Broadband Networks, 20 Speen St., Framingham, MA 01701, U.S.A.E-mail: [email protected]

1. INTRODUCTION

In order to fully utilize the scarce spectrum resources,emerging cognitive radio technology becomes a promis-ing approach to exploit the under-utilized spectrum [1].In a cognitive radio network (CRN), unlicensed wirelessusers (secondary users) are allowed to dynamically accessthe licensed bands, as long as the licensed wireless users(primary users) in those particular bands are not inter-fered. Wireless devices equipped with cognitive radiosare enabled to implement various functionalities, includ-ing frequency agility, transmit power control and accesscoordination, which render more efficient use of availablespectrum.

In this paper, we consider a CRN that consists of multi-ple base stations (BSs) and secondary users (SUs). EachBS supports a set of SUs. The spectrum of interest isdivided into a set of multiple orthogonal channels usingfrequency division multiple access, which are licensed toprimary users (PUs). We assume that the channel usagepattern for PUs is quasi-static so that SUs have ampletime to implement primary-user detection and therebyavoid interfering with PUs’ connections. We consider the

scenario of multicast communication in the CRN. Anexample deployment of a CRN is depicted in Figure 1.

Cognitive radio networks may operate in infrastructure-based systems. As a practical application, IEEE 802.22based regional area networks dynamically allocate TVspectrum to SUs [2] while they keep provisioning serviceto PUs. The TV bands are selected because they featurevery favorable propagation characteristics and are scarcelyused because of popularity of cable and satellite TV ser-vices. Therefore, CRN must avoid interfering with PUs inPRN as the licensed TV customers by carefully controllingits beam configuration and channel selection. Because TVsignals are transmitted from BSs to users, only downlinkscenario is considered in this paper.

Directional antenna in the context of wireless net-works can largely reduce the radio interference, therebyimproving the utilization of wireless medium and con-sequently the network performance. In addition, direc-tional antennas permit energy savings by concentratingtransmission energy where it is needed. Directional anten-nas can be loosely classified into omni-directional anten-nas, modestly directional antennas and highly directionalantennas. Omni-directional antennas have fixed beamwidth

260 Copyright © 2012 John Wiley & Sons, Ltd.

W. Guo and X. Huang Directional antenna for cognitive radio networks

Figure 1. An example deployment of a cognitive radio network. Head portraits represent primary users. Cell phones representsecondary users.

and unsteerable orientation. Modestly directional antennashave fixed beamwidth and steerable orientation. At last,highly directional antennas have variable beamwidth andsteerable orientation. In fact, the omni-directional antennasand modestly directional antennas are special cases of thehighly directional antennas. In this paper, we are mainlydiscussing the highly directional antennas.

In this paper, we study the joint problem of antennadirectionality and channel assignment (ADCA) to maxi-mize the CRN throughput, which is defined as the sumrate of all links supporting SUs. Because transmitters onlyinterfere with user nodes in their directional coverage area,antenna beamwidth and orientation exert a great role innetwork throughput performance. Given a configuration ofdirectional antennas, the network throughput can be greatlyimproved by intelligent channel assignment in terms ofspatial diversity. Therefore, ADCA are mutually affectedand of great importance to CRN throughput.

To formulate the ADCA problem mathematically, wecharacterize behaviors and constraints for multiple param-eters from a multicast CRN. Special attention is givento modeling of antenna directionality, beam covering,channel assignment and interference modeling. Becausethe formulation of the ADCA problem falls into mixedinteger non-linear programming (MINLP), which is non-deterministic polynomial-time hard (NP-hard) in general,we aim to derive a near-optimal solution. In particular, wepropose a greedy algorithm to iteratively increase the over-all throughput. During each iteration, one BS is chosen

to build multicast links with SUs to produce maximumthroughput increase, which can be formulated as a mixedinteger linear programming (MILP) problem and solvedby the branch and bound algorithm. Because the solutionobtained by the proposed greedy algorithm represents alower bound for the objective, we compare it with the upperbound developed later. Simulations show that the perfor-mance obtained by the greedy algorithm are very close tothe upper bound, thus suggesting the following: (i) that theupper bound is very tight; and (ii) that the solution obtainedby the greedy algorithm is near-optimal.

The rest of this paper is organized as follows. InSection 2, we review the related work about channelassignment and directional antenna. In Section 3, wedescribe the network model. In Section 4, we proposea greedy algorithm to solve the ADCA problem. InSection 5, simulation results are presented to compare thesolutions obtained by the greedy algorithm and the upperbound. Section 8 concludes this paper.

2. RELATED WORK

This paper is related to our previous work [3]. Althoughsharing the same objective, this paper is mainly con-cerned with ADCA, while deferring power control as afuture work.

Adaptive phased-array antennas and specifically direc-tional antennas are being given significant attention in

Wirel. Commun. Mob. Comput. 2015; 15:260–275 © 2012 John Wiley & Sons, Ltd. 261DOI: 10.1002/wcm

Directional antenna for cognitive radio networks W. Guo and X. Huang

recent times, especially in the context of ad hoc networks.In [4], the authors presented a constraint formulation interms of MILP, which can be used for an optimal solu-tion of the minimum-energy multicast problem in wire-less ad hoc networks with directional antennas. In [5],the authors identified several criteria and investigated theimpact on overall system performance in the context ofad hoc wireless networks with directional antennas, that isreducing radio interference and improving the utilizationof wireless medium. In [6], the authors consider a wirelessad hoc network where each node employs a single-beamdirectional antenna and is provisioned with limited energy.An online routing algorithm was proposed for successivemulticast communication requests with the aim of maxi-mizing the network lifetime. There are also papers usingdirectional antennas on wireless mesh networks (MWNs).In [7], an analytical model is derived, which allows theincorporation of various node distribution models, radiochannel models and antenna models. In [8], DMesh, aWMN architecture was proposed that combines spatialseparation from directional antennas with frequency sepa-ration from orthogonal channels to improve the throughputof WMNs. In addition, a distributed, directional channelassignment algorithm was proposed for mesh routers thateffectively exploits the spatial and frequency separationopportunities in a DMesh network.

The concept of cognitive radio was first coined by Mitolain his dissertation work [9] and later in his visionary paper[10]. Haykin provides a thorough overview of cognitiveradio and describes the fundamental cognitive capabili-ties and cognitive tasks [11]. A cognitive radio is definedas an intelligent wireless communications system that iscapable of sensing its surrounding environment, learningfrom experience, and adapting certain operating param-eters (e.g., transmit-power, carrier-frequency, and modu-lation strategy) in real-time [11]. There are two primaryobjectives: highly reliable communications whenever andwherever needed and efficient utilization of the radio spec-trum. Akyildiz et al. give a survey on the dynamic spec-trum access for cognitive radios [1,12]. The basic conceptof CRN is described by Thomas et al. together with acase study to illustrate how such a network might operate[13]. In [14], the concept of cognitive radio is extended tomultihop networks.

A number of channel assignment schemes has been pro-posed in recent years. In [15], a distributed and adap-tive approach is proposed to manage spectrum usage indynamic spectrum access networks. However, this paperdoes not take into account the control of transmit power.In addition, the more practical physical interference modelis not studied. In [16], Chin et al. address the problem ofdynamically assigning channels in ad hoc wireless net-works via power control to satisfy their minimum QoSrequirements. The objective then is to maximize the num-ber of co-channel links subject to some stability condi-tions. In [17], a cluster-based multipath topology controland channel assignment scheme is proposed, which explic-itly creates a separation between the channel assignment

and topology control functions, thus minimizes flow dis-ruptions. In [18], Raniwala et al. propose a greedy load-aware channel assignment scheme when network nodes arewith multiple radios. The goal of channel assignment is tobind each network interface to a radio channel such thatthe available bandwidth on each link is proportional to itsexpected load. In [19], Alicherry et al. mathematically for-mulate the joint channel assignment and routing problem,taking into account the interference constraints, the num-ber of channels in the network and the number of radiosavailable at each mesh router. A centralized algorithm isdeveloped to solve the problem to yield the optimizednetwork throughput. The channel assignment algorithmis used to adjust the flow on the flow graph to keep theincrease of interference for each channel to a minimum. In[20], Ramachandran et al. propose an interference-awarechannel assignment algorithm and protocol for multi-radioWMNs. The proposed solution intelligently assigns chan-nels to radios to minimize interference and thus enhancenetwork performance.

Multicast communication on CRNs has been extensivelyresearched recently. In [21], a cross-layer optimizationapproach is proposed to multicast video in CRNs. Theproblem is to optimize the overall received video qual-ity as well as achieving fairness. Important design factorsare considered, including video coding, rate control, spec-trum sensing and spectrum access. However, the modelemployed only involves one BS. In [22], an optimizationframework for multicast scheduling in CRNs is presented.Eventually, two multicast scheduling algorithms are pro-posed accordingly. In [23], Ren et al. address the multicastproblem in CRNs. A low complexity algorithm is pro-posed to construct the minimum-energy multicast tree andtransform the original problem into a directed Steiner treeproblem. However, few research works have addressed themulticast problem for CRNs involving multiple BSs anddirectional antennas.

3. PROBLEM MODELING

We consider a CRN that consists of multiple BSs andSUs. Each BS supports a set of SUs. The spectrumof interest is divided into a set of multiple orthogonalchannels using frequency division multiple access, whichare licensed to PUs. We assume that the channel usagepattern for PUs remain static so that SUs have timeto implement primary-user detection and thereby avoidinterfering with PUs’ connections. From this observation,the studied CRNs should satisfy both demanding delay-sensitive and loss-sensitive applications. Therefore, bothvideo multicast and data multicast can be applied to ourproposed model.

To explore advantages offered by the use of directionalantennas, we consider the scenario of downlink multicasttraffic in the CRN. Each BS employs exactly one channelto support one or multiple SUs.

Table I lists frequently used notations in this section.

262 Wirel. Commun. Mob. Comput. 2015; 15:260–275 © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/wcm


Table I. Notation.

Symbol Definitions

B The number of BSsN The number of SUsC The number of channelsQ The number of tunable angle beamwidths� A number of very large value� The minimum signal to interference and noise ratio (SINR)bi The ith BSsj The jth SUpj The jth PU�min The minimum angle of beamwidth

tk;qi Binary variable indicating if channel k of bi has beamwidth q�min

xk;qi;j Binary variable indicating the assignment of bi to sj when tk;q

i =1

yk;qi;j Binary variable indicating bi covers sj when tk;q

i =1

yk;q1i;j ,yk;q2

i;j Two binary variables decomposed from yk;qi;j

'ki The antenna orientation of bi on channel k

�ki The beamwidth of bi on channel k

ˇi;j The counter-clockwise angle of the line from bi to sj

lk;qi;j The link between bi and sj when tk;q

i =1

pk;qi;j The received power of lk;q

i;j

di;j The distance of bi and sj

ck;qi;j The rate of lk;q

i;j

N0 The ambient noise power

� The interference threshold value to protect PUs

kj� The channel on which pj operates

pkj�;qi;j� The power received on pj from bi when tkj�;q

i D 1

ykj�;qi;j� Binary variable indicating bi covers pj when tkj�;q

i D 1

ykj�;q1i;j� , ykj�;q2

i;j� Binary variables decomposed from ykj�;qi;j�

ˇi;j� The counter-clockwise angle of the line from bi to pj

BS, base stations; SU, secondary user; PU, primary user.

3.1. Directional antenna model

We use a directional antenna propagation model as shownin Figure 2, where antenna orientation 'u.0� 'u < 2�/ ofnode u is defined as the angle measured counter-clockwisefrom the horizontal axis to the first side of the beam. Theantenna beamwidth is specified as the angle of �u.�min �

�u � 2�/, where �min denotes the minimum angle ofbeamwidth. To model the discrete version of antennabeamwidth, we introduce an integer parameter Q that rep-resents the total number of angle values to which thebeamwidth can be adjusted, that is, �min; 2�min; : : : ;Q�min

(Q�min � 2�). We introduce binary variable tk;qi indicat-ing if the i th BS bi ’s beam on channel k has beamwidth

q�min when tk;qi D 1. Obviously, it is required thatPQ

qD1 tk;qi � 18i ; k.

Definition: A joint downlink channel assignment, powercontrol and antenna directionality scheme is specified by amatrix X, which is defined as (1).

kiϕ

kiθ

u

Figure 2. Directional antenna propagation model.

XD fxk;qi;j jxk;qi;j 2 f0; 1g; i 2 f1; 2; :::; Bg;

j 2 f1; 2; :::; N g; k 2 f1; 2; :::; C g; q 2 f1; 2; :::;Qg(1)

xk;qi;j is denoted as the binary variable with xk;qi;j D 1

indicating the assignment of the kth channel of bi to thej th SU sj when the beamwidth of the transmitter antenna

is of q�min. Similarly, denote yk;qi;j as the binary variable



with yk;qi;j D 1 indicating bi ’s beam on channel k coverssj when bi ’s antenna beamwidth on channel k is of q�min.

xk;qi;j � y

k;qi;j

xk;qi;j � t

k;qi

(2)

Now, we begin to build coverage relations between biand sj on channel k determined by the antenna orienta-

tion 'ki when beamwidth �ki D q�min. It can be easilydrawn as Figure 3. The corresponding mathematical equa-tions are shown in (3) and (4). Let ˇi ;j denote the anglemeasured counter-clockwise from the horizontal axis to thedirectional line from bi to sj . When q�min � ˇi ;j , (3)is depicted by the thick lines in Figure 3(a). Whenˇi ;j � q�min, (4) is depicted by the thick lines inFigure 3(b).

yk;qi;j D

8̂<:̂0 W 0� 'ki < ˇi ;j � q�minI

1 W ˇi ;j � q�min < 'ki � ˇi ;j I

0 W ˇi ;j � 'ki < 2� I

(3)

yk;qi;j D

8̂<:̂1 W 0� 'ki < ˇij I

0 W ˇi ;j � 'ki < 2� C ˇi ;j � q�minI

1 W 2� C ˇi ;j � q�min � 'ki < 2� I

(4)

The relations between yk;qi;j and 'ki are obviously non-linear. In the following two cases, we shall show that theserelations can be linearized [4].

� Case 1: q�min � ˇi ;j

From Figure 3(a), we can observe that yk;qi;j can be

linearly derived by two new binary variables yk;q1i;j

and yk;q2i;j , which are defined in (5) and (6).

yk;q1i;j D

(0 W 0� 'ki < ˇi ;j � q�minI

1 W ˇi ;j � q�min � 'ki < 2� I

(5)

yk;q2i;j D

(1 W 0� 'ki < ˇi ;j I

0 W ˇi ;j � 'ki < 2� I

(6)

Note that (5) and (6) are also depicted by thick lines

in Figure 4(a) and (b), respectively. Because yk;qi;j can

be decomposed into yk;q1i;j yk;q2i;j , we need to derive

linear constraints for yk;q1i;j and yk;q2i;j . A close lookwould lead us to the finding that the binary variablescan be constrained by the quadrangles in Figure 4(a)and (b), respectively. Therefore, the decomposition of

yk;qi;j can be shown by (7).

yk;qi;j D y

k;q1i;j C y

k;q2i;j � 1I

yk;q1i;j �

1

ˇi ;j � q�min'ki I

yk;q1i;j �

1

2� � ˇi ;j C q�min

�'ki � ˇi ;j C q�min

�I

Figure 3. Two possible relations between yk;qi;j and 'k

i . (a) q�min � ˇi;j ; (b) ˇi;j � q�min.

Figure 4. The decomposition of yk;qi;j when �min � ˇi;j .



yk;q2i;j �

�1

ˇij

�'ki � ˇi ;j

�I

yk;q2i;j �

�1

2� � ˇi ;j

�'ki � 2�

�I

(7)

� Case 2: q�min � ˇij ;

From Figure 4, similarly, we can observe that yk;qi;jcan be linearly derived by two new binary variables

yk;q1i;j and yk;q2i;j , which are defined in (8) and (9).

yk;q1i;j D

(1 W 0� 'ki < ˇij I

0 W ˇij � q�min � 'ki < 2� I

(8)

yk;q2i;j D

(0 W 0� 'ki < 2� C ˇi ;j � q�minI

1 W 2� C ˇij � q�min � 'ki < 2� I

(9)(8) and (9) are also depicted by thick lines inFigure 5(a) and (b), respectively.

Similarly, when q�min � ˇi ;j , the decomposition

of yk;qi;j can be shown by (10).

yk;qi;j D y

k;q1i;j C y

k;q2i;j I

yk;q1i;j �

�1

ˇij

�'ki � ˇi ;j

�I

yk;q1i;j �

�1

2� � ˇi ;j

�'ki � 2�

�I

yk;q2i;j �

1

2� C ˇij � q�min'ki I

yk;q2i;j �

1

q�min � ˇi ;j

�'ki � 2� � ˇi ;j C q�min

�I

(10)

3.2. Rate calculation

In this paper, we assume all BSs use the same trans-mit power P on all channels. The more complex issuesof power control will be deferred for future research.Let us denote the link on channel k between BS bi andSU sj when the beamwidth of bi ’s antenna is of q�min

as lk;qi;j . Let di ;j denote the physical distance betweenbi and sj . n denotes the path loss index. The received

power of lk;qi;j is denoted as pk;qi;j , which can be calculatedusing (11).

pk;qi;j D

2�P

q�mindni;j

(11)

The rate of link lk;qi;j , denoted as ck;qi;j , can be calculatedas (12) based on (11).N0 denotes the ambient noise power, B denotes the

number of BSs, N denotes the number of SUs. The termPBaD1

PNbD1

PQqD1 t

k;qa y

k;qa;j p

k;qa;j represents the inter-

ference power on channel k at sj .

3.3. Interference constraints

For link lk;qi;j to be reliable, we require that (13)

In practice, � can be the minimum signal to interfer-ence and noise ratio (SINR) required to achieve a cer-tain bit error rate performance at each SU. The value of� depends on specific coding, modulation and detectionschemes being employed.

ck;qi;j D log2

0@1C p

k;qi;j

N0CPBaD1

PNbD1;b¤j

PQqD1 t

k;qa y

k;qa;j p

k;qa;j

1A (12)

pk;qi;j

N0CPBaD1;a¤i

PNbD1

PQqD1 t

k;qa y

k;qa;j p

k;qa;j

� � if xk;qi;j D 1 (13)

Figure 5. The decomposition of yk;qi;j when q�min � ˇij .



Note that (13) can be transformed as (14).

xk;qi;j .�N0C �/

C �

BXaD1;a¤i

NXbD1

QXqD1

tk;qa y

k;qa;j p

k;qa;j � p

k;qi;j � �

(14)where � denotes a very large value.

3.4. Protecting primary users

To protect operations of PUs, we require that the interfer-ence power caused by all SUs on each channel is below athreshold value �, shown as follows:

PBiD1

PQqD1 t

kj�q

i ykj�q

i;j� pkj�;q

i;j� � �8j� 2 P (15)

where kj� denotes the index of the channel on which pj is

receiving signals. ykj�;q

i;j� denotes the binary variable with

yk�j ;qi;j� D 1 indicating bi ’s transmission beam on chan-

nel kj� covers the j th PU pj when the beamwidth of

bi ’s antenna is q�min. Let pkj�;q

i;j� denote the interferencepower on channel kj� from bi to pj when the beamwidthof bi ’s antenna is q�min. As (7) and (10), we then can

derive determinant conditions (16) and (17) on ykj�;q

i;j�

using 'kj�i , q�min and ˇi ;j�, which denotes the angle

from the horizontal axis to the directional line from bito pj .

� Case 1: ˇi ;j� � q�min

ykj�;q

i;j�� y

kj�;q1

i;j�C y

kj�;q2

i;j�� 1I

ykj�;q1

i;j��

1

ˇi;j� � q�min'kj�

iI

ykj�;q1

i;j��

1

2� � ˇi;j�C q�min

�'kj�

i� ˇi;j�C q�min

�I

ykj�;q2

i;j��1

ˇi;j�

�'kj�

i� ˇi;j�

�I

ykj�;q2

i;j��

�1

2� � ˇi;j�

�'kj�

i� 2�

�I

(16)� Case 2: ˇij� � q�min

ykj�;q

i;j�� y

kj ;q1

i;j�C y

kj�;q2

i;j�I

ykj�;q1

i;j��1

ˇi;j�

�'kj�

i� ˇi;j�

�I

ykj�;q1

i;j��

�1

2� � ˇi;j�

�'kj�

i� 2�

�I

ykj�;q2

i;j��

1

2� C ˇi;j� � q�min'kj�

iI

ykj�;q2

i;j��

1

q�min � ˇi;j�

�'kj�

i� 2� � ˇi;j�C q�min

�I

(17)

3.5. Objective

Recall that the objective of the ADCA problem is to max-imize the sum rate achieved by all SUs, which can beformulated as (18).

maxPBiD1

PNjD1

PCkD1

PQqD1 x

kqi;j c

k;qi;j

(18)

where C denotes the number of channels.Remark: In this paper, we assume the channel usage

pattern for PUs is fairly static. With the objective of max-imizing the overall throughput, the optimal solution tothe configuration of CRN, in terms of antenna orienta-tion, beamwidth and channel assignment, tends to be static.Therefore, it is not necessary to introduce scheduling in aslotted time system to the formulation.

4. DESIGN OF A GREEDYALGORITHM

In this section, we present a greedy algorithm. This algo-rithm increases the sum rate of CRN iteratively until it cannot be increased. The main idea is presented in Section 4.2,which includes maximum throughput estimation, BS sort-ing and channel usage implementation. The details of eachmodule is described in Section 4.3.

4.1. Intuition

The main difficulty for the ADCA problem lies in the

objective function as (18), where ck;qi;j is a nonlinear term

with xk;qi;j and yk;qi;j in the denominator contained in a log-arithmic function. It is obviously very hard to approximatethe optimal solution via relaxing the objective function.

Then, we realize the variables xk;qi;j and yk;qi;j in the objec-tive function represents interference power from multipleBSs. If we divide the large optimization task and distributesubtasks to each BS heuristically, the objective functionfor each smaller optimization problem would be mucheasier in the sense that interference from only one BS isconsidered during one iteration.

4.2. Description of the greedy algorithm

Our greedy algorithm increases the overall networkthroughput iteratively and terminates until the overall net-work throughput can not be increased further. We assumethat each BS maintains one database recording positioninformation of all PUs and SUs, current channel assign-ment, antenna orientation and beamwidth, named as Tableof Assignment Links (TAL). At the beginning of an itera-tion, each BS pretends that the existing links that connectitself are annihilated. Then each BS proceeds calculatingthe best possible throughput increase under the interfer-ence constraints imposed by both PUs and existing links



Figure 6. The flow chart showing how our greedy algorithmworks.

within the CRN. After calculating the potentially maxi-mum throughput increase, each BS exchanges its resultwith other BSs, and finally, the globally largest is iden-tified as associated with one BS. Then, the chosen BS

is required to implement its channel assignment, antennaorientation and beamwidth to best increase the overall net-work throughput. At the last step of each iteration, newresults of assigned links should be updated in the TALsof all BSs. The basic diagram of our greedy algorithm isshown in Figure 6.

The notation used in this section is listed in Table II.

4.3. Details of each module

4.3.1. Table of assignment links establishment.

We first present the method of TAL establishment. Foreach BS, the TAL records updated information of the posi-tions of PUs and SUs, channel usage pattern, antennaorientation and associated beamwidth all over CRN. Theinformation in TAL will help each BS avoid failing existinglinks and impacting PUs by generating excessive interfer-ence while establishing new links.

4.3.2. Maximum throughput estimation.

We now present the method for each BS to bring thelargest increase to the sum rate performance. As the BSis trying to update the setting of beams on each channel,

Table II. Notation.

Symbol Definitions

xk;qj Binary variable indicating the assignment of the

current BS to sj on channel k of beamwidth q�min

yk;qj Binary variable indicating the current BS covers sj on

channel k of beamwidth q�min

ykj�;qj� Binary variable indicating the current BS covers pj on

channel kj� of beamwidth q�min

ˇj The counter-clockwise angle of the line from thecurrent BS to sj

'k The counter-clockwise angle of the beam onchannel k of the current BS

tqk Binary variable indicating if the current BS’s beam

on channel k has beamwidth q�min

E The set of existing linkspk;q

j The received power at sj from current BS when tqk =1

pkj�;qj� The received power at pj from current BS when tq

kj�=1

ˇj� Then counter-clockwise angle of the line fromcurrent BS to pj

il The index of BS of the lth existing linkjl The index of SU of the lth existing linkkl The channel of the lth existing linkql�min The beamwidth of the lth existing linkIkj The overall interference power of the assigned links

at sj on the kth channelIj� The overall interference power of the assigned links

at pj on the kj�th channelck;q

j The rate of the link between the current BS and sj

cl The rate of the lth existing linkP The set of all PUsR The set of SUs not connected to BSs

BS, base stations; SU, secondary user; PU, primary user.



it should be aware of increasing interference to existinglinks. Therefore, the achievable rate on other links wouldbe probably deteriorated. Moreover, hypothetically anni-hilating all the associated links, the BS should also takeinto account the interference from existing links whenestablishing new connections.

� Preliminaries: let xk;qj denote the binary assignmentvariable indicating the sj is connected with the cur-rent BS at channel k of which the beamwidth is q�min.

Let yk;qj denote the binary coverage variable indicat-ing sj is covered by the current BS at channel k ofwhich the beamwidth is q�min. ˇj denotes the anglemeasured counter-clockwise from the horizontal axisto the directional line from the current BS to sj . 'kdenotes the angle measured counter-clockwise fromhorizontal axis to the first side of the beam on the kthchannel of the current BS. We introduce a binary vari-able tq

kindicating if the current BS’s beam on channel

k has beamwidth q�min when tqkD 1. Obviously, it

is required thatPQqD1 t

qk� 18k. Similar as (2), we

have the following:

xk;qj � y

k;qj

xk;qj � t

qk

(19)

As a note, yk;qj is decided by ˇj and 'k similarly as(3) and (4), which can be mathematically formulatedas a linear constraint as in Section 3.1.

� Protecting existing links: we first introduce somenotation. Let us denote the received power at sjfrom the current BS on channel k when the antenna’sbeamwidth is q�min as pk;qj . We use E to denotethe set of existing links. Let il ; jl ; kl denote theindex of the BS, SU and channel of the l th existinglink, respectively. The beamwidth of the l th existinglink is of ql�min. Ikj denotes the overall interferencepower of the assigned links at sj on the kth chan-nel. We then denote the set of existing links as E . The

value of pk;qj can be easily calculated on the basisof (11). From (13), we can obtain the following con-straint (20) to protect existing links against excessiveinterference.

�

0@ QXqD1

pkl ;qljl

ykl ;qljl

tqlklCN0C Ijl

1A

� pkl ;qlil ;jl

;8l 2 E (20)

As a note, the nonlinear term ykl ;qljl

tqlkl

can be sub-

stituted by a binary variable ukl ;qljland then linearized

as follows:

ykl ;qljl

C tqlkl� 2u

kl ;qljl

ykl ;qljl

C tqlkl� u

kl ;qljl

� 1(21)

� Interference constraint for new links: as (13), theSINR at the receiver end should be larger than � fornewly established links, which is shown as follows:

pk;qj � �

�N0C I

kj

�xk;qj 8j 2R (22)

where R represents the set of SUs withoutconnections.

� Protecting primary users: for ease of demonstration,we first introduce some notation. kj� denotes theindex of the channel on which pj is operating. Ij�denotes the overall interference power of the assignedlinks at pj at the kj�th channel. ˇj� denotes theangle measured counter-clockwise from the horizon-tal axis to the directional line from the current BSto pj . The received power at pj from the currentBS on channel k of beamwidth q�min is denoted as

pkj�;q

j� . Obviously, the value of pkj�;q

j� can be eas-ily calculated on the basis of (11). The set of all PUs

is denoted as P . ykj�;q

j� denotes the binary coveragevariable indicating if the current BS is covering pj

at beamwidth of q�min. pkj�;q

j� denotes the receivedpower of the current BS on the kj�th channel whenthe beamwidth is q�min at pj .

XQ

qD1pkj�;q

j� tqkj�

ykj�;q

j� C Ij� � � 8j� 2 P(23)

As a note, ykj�;q

j� is decided by ˇj� and 'kj� sim-ilarly as (3) and (4), which can be mathematicallyformulated as a linear constraint as in Section 3.1. Inaddition, the term t

qkj�

ykj�;q

j� can be substituted by a

new variable vkj�;q

j� and then linearized as follows:

tqkj�C y

kj�;q

j� � 2vkj�;q

j�

tqkj�C y

kj�;q

j� � vkj�;q

j� � 1(24)

� Rate calculation: we use ck;qj to denote the rate ofthe link between sj and the current BS on channel kwhen the antenna beamwidth is q�min, which can becalculated as follows:

ck;qj D log2

0@1C p

k;qj

N0C Ikj

1A 8j 2RI (25)

Note that the rate of the existing links could proba-bly be reduced because of newly introduced interfer-

ence. 8l 2 E , its rate cl can be changed to ckl ;ql

if

ykl ;qjl

tqklD 1, which is calculated as follows:

ckl ;q

lD log2

0@1C p

kl ;qlil ;jl

N0C IkljlC p

kl ;qjl

1A I (26)



� Objective function: recall that the greedy algorithmiteratively increases the network throughput, thus theBS should try to maximize the sum rate of newlyestablished links as well as probably deterioratedexisting links, which can be mathematically statedas (27).

maxPj2R

PCkD1

PQqD1 x

k;qj c

k;qj CP

l2E�ykl ;qjl

tqklckl ;q

lC�1�y

kl ;qjl

tqkl

�cl

�(27)

It should be noted that the nonlinear term ykl ;qjl

tqkl

can be linearized as (24).

Putting together the objective and all the constraintsdescribed earlier, we formulate a MILP problem, whichcan be solved by the branch and bound algorithm.

4.3.3. Base station sorting.

This module is aimed to identify the BS, which can bringthe maximum throughput benefit, and hence, the networkthroughput can be increased greedily. After each BS isassociated with a maximum throughput, they can exchangetheir results with neighboring BSs in a distributed fashion.At the end of this process, each BS should keep the maxi-mum network throughput along with the ID of the BS thatproduces this amount.

The implementation of this module entails a certainamount of information exchange. Once a BS receivesits knowingly best network throughput, it propagates thisdatum to its neighboring BSs exactly once. In particu-lar, each BS is only concerned if its maximum networkthroughput is larger than any other BS in this iteration.Thus, they would discard their own maximum networkthroughput along with the associated beam configurationonce they realize some other BS produces larger through-put. For the case of equal throughput, the BS would alsodiscard its own results if the BS that produces the sameamount of throughput is indexed smaller. This sorting pro-cedure terminates when any BS has not been notified ofany larger throughput for a preset amount of time.

4.3.4. Channel usage implementation.

Finally, we discuss the method of channel usage imple-mentation and TAL updating. At first, this module isapplied at the BS whose maximum throughput is thelargest among all BSs in each iteration. The BS updatesthe calculated channel assignment, antenna orientation andbeamwidth in its TAL. This BS also has to inform otherBSs to update their TALs for calculating the maximumthroughput during the next iteration.

4.4. Proof of convergence

We now show that the algorithm must converge. Weshow that in each iteration, the algorithm increases the

throughput performance of CRN. Because the overallthroughput is upper bounded, this implies that the algo-rithm must converge.

At the beginning of each iteration, each BS’s beamconfiguration represents a feasible solution to the max-imum throughput estimation problem formulated inSection 4.3.2. After solving the maximum throughput esti-mation problem, we obtain the optimal solution for eachBS. Therefore, no matter which BS is chosen to imple-ment new settings for antenna orientation and beamwidth,the network throughput is expected to grow larger than thelower bound at the beginning of the iteration.

Because the network throughput performance monoton-ically increased after every iteration, convergence of thegreedy algorithm is guaranteed.

4.5. Complexity analysis

The complexity analysis of the greedy algorithm is pre-sented as follows. During each iteration, the majority com-putation mainly resides in the step of maximum throughputestimation. The solution space contains all combinationsof binary variables and continuous variables. The binary

variables include xk;qj ; yk;qj ; y

kj�;q

j� ; tqk

(as explained inTable II), which totals as many as .N � C � Q C N �C � Q C jPj2 � Q C C � Q/. The continuous vari-ables include 'k , which contains at most C variables.The MILP problem can be solved by branch and boundalgorithm, which could potentially search all 2L solutions,where L denotes the number of binary variables. There-fore, the running time for the greedy algorithm for one

iteration equals to O.2.2NC1/CQCjPj2Q/. As all BSs are

required to compute the MILP problem in one iteration,and the total number of iterations is measured as O.B/,the total times to solve the MILP algorithm can be approx-imated asO.B2/. Therefore, the overall computation com-plexity for the greedy scheme can be approximated as

O.B22.2NC1/CQCjPj2Q/DO.2.2NC1/CQCjPj

2Q/.

4.6. Notes on implementation issue

During the first few iterations, the majority of SUs can beconnected with only one BS under many circumstances.One of them is that each BS has abundant beams andlarge beamwidth such that most SUs can be covered toachieve high network throughput. The other one is that thenumber of SUs is comparatively small and they are notfar away from one BS. When one BS is covering mostof the SUs, these links are protected as existing links infuture iterations, which would cause the greedy algorithmto converge very fast. To avoid this situation, we imposedifferent procedures at the beginning of the greedy algo-rithm: the set SUs are divided into B clusters according totheir distances to the BSs; then, each BS sequentially esti-mates the maximum throughput and implement the result



considering the cluster of SUs to which the current BS isthe nearest.

4.7. Other algorithm

Because the existing schemes can not be directly appliedto solve the ADCA problem, we design an Reformulation-Linearization Technique (RLT)-based heuristic scheme tocompare with the proposed greedy scheme. RLT[24] hasbeen widely used to solve non-linear optimization prob-lems. Because the ADCA problem is a non-linear opti-mization problem, involving both integer and non-integervariables, the RLT technique can be employed to relax theoriginal problem to linear form and obtain the non-integervariables. Then, on the basis of acquired solution for non-integer variables, a greedy scheme is employed to obtain

the solution for integer variables. Specifically, tk;qa yk;qa;j

in (14) and tkj�;q

i ykj�;q

i;j in (15) are polynomial terms.By using RLT, we can substitute these terms with newvariables, thus relaxing nonlinear terms into linear terms.For instance,

tkj�;q

i C ykj�;q

i;j� �wkj�;q

i � 1

tkj�;q

i C ykj�;q

i;j� � 2wkj�;q

i

(28)

Moreover, we can relax (12) as follows:

ck;qi;j D log2

0@1C p

k;qi;j

N0

1A I (29)

After relaxation, we can obtain the upper bound by solv-ing a linear programming problem. We fix the non-integervariables 'ki .1 � i � B; 1 � k � C/ and employthe following heuristic algorithm to solve the assignmentvariables X. The heuristic algorithm increases the over-all network throughput iteratively and terminates until theoverall network throughput can not be further increased.At the beginning of an iteration, each BS pretends thatthe existing links that connect itself are annihilated. Then,each BS proceeds calculating the best possible through-put increase under the interference constraints imposedby both PUs and existing links within the CRN. Aftercalculating the potentially maximum throughput increase,each BS exchanges its result with other BSs, and finally,the globally largest is identified associated with one BS.Then, the chosen BS is required to implement its chan-nel assignment, antenna orientation and beamwidth to bestincrease the overall network throughput. At the last stepof each iteration, new results of assigned links shouldbe broadcasted to all BSs. One BS solves the follow-ing binary integer programming problem during one iter-

ation. Note that ykl ;qljl, y

kj�;q

j� and yk;qj are all knownbecause angle of beams on each BS are fixed. The objec-tive is the same as (27), and constraints are extracted from(19), (20), (23) and (22).

MaxPj2R

PCkD1

PQqD1 x

k;qj c

k;qj CP

l2E�ykl ;qjl

tqklckl ;q

lC�1� y

kl ;qjl

tqkl

�cl

�s:t PQ

qD1 tqk� 1

xk;qj � y

k;qj 8j 2R

xk;qj � t

qk8j 2R

��PQ

qD1 pkl ;qljl

ykl ;qljl

tqlklCN0 C Ijl

�� p

kl ;qlil ;jl

;8l 2 EPQqD1 p

kj�;q

j� tqkj�

ykj�;q

j� C Ij�

� � 8j� 2 P

pk;qj � �

�N0C I

kj

�xk;qj 8j 2R

1� k � C; 1� q �Q(30)

5. PERFORMANCE EVALUATION

In this section, we present simulation results to demon-strate the performance of our greedy algorithm. Ideally, thebest performance measure of our distributed optimizationalgorithm is the solution to the centralized optimizationproblem. However, as is shown in the Appendix section,the problem formulation is in the form of MINLP andis NP-hard. Although the exact solution to the MINLPproblem is not obtainable, we are able to develop a tightupper bound for the objective function of the problem. Asa result, we will compare the performance of our greedyalgorithm with the upper bound.

5.1. Simulation setup

We consider a square service area of size 100 km�100 kmin which a CRN is deployed. Three BSs are placed at thecoordinates fŒ3; 4�I Œ4; 7�I Œ7; 4�g�104. Fifteen SUs and fivePUs are randomly deployed within the service area. Weassume a free-space path loss model with the path-lossexponent n D 2. The ambient noise power at each PU andSU isN0 D�100 dB. The channel bandwidth is of 1 MHz.The threshold power for PUs is � D�90 dB. We arbitrarilyset the channels where PUs are receiving signals. For eachset of system parameters, we generate 20 instances of thenetwork scenarios with randomly distributed user nodes toobtain the average performance.

5.2. Evaluation metric

To evaluate the performance of the greedy algorithm, themetric we utilize is the average throughput of CRN overthe 20 instances of different network scenarios.



5.3. Simulation results

(1) Impact of transmit power: We evaluate the impactof transmit power on the performance of the CRN.Figure 7 shows the performance comparison when wechange the transmit power from 0.2 to 1 W. Because theoptimal performance lies between the feasible solution viathe greedy algorithm and the upper bound, it is obvious thatthe throughput performance of the greedy algorithm pro-duces near-optimal solution. Besides, as the transmit powerincreases from 0.2 to 1 W, the performance yielded by boththe greedy algorithm and the upper bound increases, whichagrees with the rate model (12). In addition, by comparingwith the RLT-based scheme, the proposed greedy algorithmdemonstrates very good performance.(2) Impact of minimal beamwidth: Next, we look into theimpact of minimal beamwidth �min on the throughput per-formance of CRN. From Figure 8, it can be drawn that thegreedy algorithm produces close-to-optimal performance.Moreover, we can see that the average performance forboth via the greedy algorithm and the upper bound degradeas �min increases. The reason lies in the fact that thereceived power of each link is inversely proportional to thebeamwidth. As a result, the rate of each link is reducedbecause of the increase of �min on the basis of the ratemodel (12). By comparing the performance of the proposedgreedy algorithm with the RLT-based approach, it can beseen clearly that our proposed greedy scheme outperformsthe RLT-based approach.(3) Impact of interference threshold: In Figure 9, we evalu-ate the impact of interference threshold on the performanceof the CRN. We can make two observations: first, theupper bound is close to the optimal performance; second,the greedy algorithm produces near-optimal performance.Besides, as the interference threshold � increases from 30to 50, both performances yielded by the greedy algorithmand the upper bound decrease. The rationale behind is that

0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Transmit Power P (watt)

Ave

rage

thro

ughp

ut o

f CR

N (

Mbp

s)

Upper BoundGreedyRLT

Figure 7. Simulation results when P changes. Twenty random-generated network topologies are tested for each parameter

set.�min D �=4. � D 10, QD 3. C=3.

1 2 3 40

20

40

60

80

100

120

θmin

(×π/6)

Ave

rage

thro

ughp

ut o

f CR

N (

Mbp

s)


Figure 8. Simulation results when �min changes. Twentyrandom-generated network topologies are tested for each

parameter set. � D 10, QD 3, P D 0:5 W,C=3.

30 35 40 450

10

20

30

40

50

60

70

80

90

100

110

γ

Ave

rage

thro

ughp

ut o

f CR

N (

Mbp

s)


Figure 9. Simulation results when � changes. Twenty random-generated network topologies are tested for each parameter

set.�min D �=4. pD 0:5 W, qD 3. cD 3.

with higher interference threshold, less links would be ableto succeed. In addition, our proposed greedy algorithmgenerates much smaller gap between its performance andupper bound than the RLT-based algorithm.(4) Impact of Q: Then, we look into the impact of thenumber of angles Q on the throughput performance inFigure 10. From Figure 10, it is clear that the average ofboth of the performances yielded by the greedy algorithmand the upper bound increase when we change Q from 1to 4. This is because more number of angles would giverise to more opportunities that one beam covers more SUs,which thereby increases the total throughput performance.This phenomenon demonstrates the advantage of variablebeamwidth of directional antennas in network throughputof multicast communication (for unicast communication,the optimal solution usually consists of beams with mini-mal angle to maximize throughput). Furthermore, we can



1 2 3 40

20

40

60

80

100

120

Q

Ave

rage

thro

ughp

ut o

f CR

N (

Mbp

s)Upper BoundGreedyRLT

Figure 10. Simulation results when Q changes. Twenty random-generated network topologies are tested for each parameter

set. �min D �=8. P D 0:5 W, � D 10, C D 2.

see that the proposed greedy algorithm outperforms theRLT-based algorithm.

6. IMPACT OF POWER THRESHOLDFOR PUs:

As for the impact of power threshold � for PUs, wecarry out numerical investigation in Figure 11. FromFigure 11, first of all, the greedy algorithm yields perfor-mance close to the upper bound. In addition, we can seea slight increase as the threshold power increases from�90 to �50 db for the performance of the greedy algo-rithm. This is because the constraint of protecting the PUsis relaxed as � increases. In particular, as the constraint (23)shows, when � is enlarged, more interference to each PU is

−90 −80 −70 −60 −500

20

40

60

80

100

120

Power Threshold for PUs(db)

Ave

rage

thro

ughp

ut o

f CR

N (

Mbp

s)

Upper BoundGreedyFontsize

Figure 11. Simulation results when � changes. Twenty random-generated network topologies are tested for each parameter

set. �min D �=4. P D 0:5 W, QD 3, � D 10.

allowed from CRNs, which leads to more possible estab-lishment of secondary links or larger transmit power ofindividual downlink secondary connections. Therefore, theoverall throughput of the cognitive network is increased.Besides, the proposed greedy algorithm yields much betterperformance than the RLT-based approach.

7. DISCUSSION

Although the current network model assumes that thechannel usage pattern remains static, the proposed algo-rithm can be easily extended to handle the case whenPUs dynamically change their channel usage pattern. Weassume when PUs update operating channels, the BSs inCRN can sense which PUs have moved to which chan-nels. Because each BS maintains one database recordingposition information of all PUs and SUs, current chan-nel assignment, antenna orientation and beamwidth, theycan update the channel usage pattern information accord-ingly. Therefore, each BS checks if its supported down-link multicast connections have caused interference tothe PUs who newly updated their operating channel. Theproposed greedy algorithm can be implemented involv-ing only the BSs that have caused interference to PUsdue to their dynamic channel usage. In particular, eachBS pretends that the existing links that connect itself areannihilated. Then, each BS proceed in calculating thebest possible throughput increase under the interferenceconstraints imposed by both PUs with updated channelusage and existing links within the CRN. After calculat-ing the potentially maximum throughput increase, eachBS exchanges its result with other BSs, and finally, theglobally largest is identified as associated with one BS.Then the chosen BS is required to implement its chan-nel assignment, antenna orientation and beamwidth to bestincrease the overall network throughput. At the last stepof each iteration, new results of the assigned links shouldbe updated in the TALs of all BSs. With PUs dynami-cally changing channel usage pattern, all the approacheswould be extended to apply in time dimension. Thus,results are also time-based. Until a specific time, the pro-posed greedy algorithm still keeps increasing the overallthroughput performance iteratively every time after sens-ing channel usage change from the primary network. Theadaptation to the CRN is incremental by implementing thegreedy algorithm. In contrast, the RLT-based approach hasto tear down all connections and rebuild them again onceinterference is caused to primary network due to channelusage update, which is very expensive. Besides, if scruti-nized after a random primary network change of channelusage pattern, the greedy algorithm works by the sametoken as if the channel usage pattern were static amongPUs. Therefore, given a specific time, the greedy algo-rithm should still yield performance that is relatively closeto upper bound performance and much better than RLT-based approach.



In real applications, different BSs usually exert differenttransmit power levels in CRNs. Although BSs use iden-tical transmit power for secondary downlink connectionsin the proposed network model, the greedy algorithm canbe easily be applied to scenarios where BSs use differ-ent transmit power. Transmit powers play their role in thesecond step of greedy algorithm, when each BS calcu-lates its biggest throughput increase. Because this step isimplemented by each BS independently, different transmitpowers on BSs will not cause any effect to the procedureof the BS’s execution. Therefore, it is safe to claim thatthe proposed greedy algorithm is seamlessly applicable toCRNs where BSs use different transmit power levels. Theresults of proposed method would change though. Each BSwould generate new throughput values with transmit powerchanged. Although it is hard to predict the change of thefinal absolute performance for the greedy algorithm withBSs using different transmit powers, the greedy algorithmshould still yield performance that is close to the upperbound and much better than the RLT-based approach. Thereason lies in the fact that the change of BSs’ transmitpowers will only affect the performance amplitudes of allapproaches to the same extent and keep their relative per-formance unchanged. The impact of the parameters on theapproaches described in Section 5 should remain valid.

The current network model assumes homogeneous SUsin CRNs. It is worth noting that the proposed greedy algo-rithm can easily address heterogeneous SUs in CRNs. Inparticular, each BS will maintain one database recordingposition information of all PUs and SUs, current chan-nel assignment, antenna orientation and beamwidth, aswell as SINR threshold of all SUs. When calculating themaximal throughput increase during the greedy algorithm,each BS modifies constraint (22) using the SINR thresh-old value based on the current SU, which the constraintapplies to. By incorporating the modified constraint (22),each BS can calculate the potentially maximal through-put increase. After calculating the potentially maximumthroughput increase, each BS exchanges its result withother BSs, and finally, the globally largest is identified asassociated with one BS. Then the chosen BS is required toimplement its channel assignment, antenna orientation andbeamwidth to best increase the overall network throughput.

8. CONCLUSION

In this paper, we presented a study on multicast commu-nications in CRN using directional antennas. We developa mathematical model for such problem with joint consid-eration of ADCA. The main contribution of this paper isthe development of a greedy optimization algorithm thatiteratively increases the overall CRN network throughput.Simulation results show that the achievable performancevia our greedy algorithm is close to the upper bound andalso outperforms the RLT-based approach. Because theunknown optimal solution lies between the upper boundand the lower bound (feasible solution obtained via our

greedy algorithm), we conclude that the results obtainedby the greedy algorithm are very close to the optimal solu-tion. Besides, it can also be drawn that the upper bound isvery tight.

APPENDIX A: ANTENNADIRECTIONALITY AND CHANNELASSIGNMENT PROBLEMFORMULATION AND UPPER BOUND

To develop a performance benchmark for the greedyalgorithm, we first formulate the cross-layer optimizationproblem. Putting together all the constraints described inSection 3, we have the formulation (A.1). The optimiza-tion problem is in the form of MINLP, which is NP-hard ingeneral. Because we are not able to obtain the exact solu-tion, we develop an upper bound to this problem, whichcan be used as a measure for the performance of the greedyalgorithm.

maxPBiD1

PNjD1

PCkD1

PQ

qD1 xk;q

i;jck;q

i;j

PQ

qD1 tk;q

i� 1

xk;q

i;j� y

k;q

i;j

xk;q

i;j� t

k;q

i

xk;q

i;j.�N0C�/C �

PBaD1

PNbD1;b¤j

PQ

qD1 pk;q

a;jtk;qa y

k;q

a;j

�pkq

ij� �

PBiD1

PQ

qD1 tkj�;q

iykj�;q

ij�pkj�;q

i;j��

ck;q

i;jD log2

�1C

pk;q

i;j

N0CPBaD1;a¤i

PNbD1

PQqD1 t

k;qa y

k;q

a;jpk;q

a;j

�

.7/ if ˇi;j � q�min; or .10/ if ˇi;j � q�min

.16/ if ˇi;j� � q�min; or .17/ if ˇi;j� � q�min

xk;q

i;j; yk;q

i;j; yk;q1

i;j; yk;q2

i;j; yk;q

i;j�; yk;q1

i;j�; yk;q2

i;j�; tk;q

i2 f0; 1g

1� i �B;1� j �N;1� k �C;1� q �Q

8j� 2P; 0� 'ki� 2�

(A.1)The upper bound for the MINLP problem can be obtainedby relaxation. First of all, according to (12), the calculation

of ckqij is very complicated; we can relax it as follows:

ck;qi;j D log2

�1C

pk;q

i;j

N0

�I (A.2)

Because the terms representing the interference poweris removed from the denominator, each individual valueof ck;qi;j becomes larger, at least no less than the originaldefinition (12). It is obvious that with the newly defined



objective, the upper bound will always be no less than thesolution to the original problem.

To relax the third inequality constraint, we can remove

the nonlinear term �PBaD1

PNbD1;b¤j

PQqD1 p

k;qa;j t

k;qa

yk;qa;j . Then, we get xk;qi;j .�N0C �/� p

k;qi;j � �.

For the nonlinear term tkj�;q

i ykj�;q

i;j� , we can substitute

it with a new binary variable wkj�;q

i ; then, we need to addtwo more constraints as follows:

tkj�;q

i C ykj�;q

i;j� �wkj�;q

i � 1

tkj�;q

i C ykj�;q

i;j� � 2wkj�;q

i

(A.3)

With the mentioned three steps, we can reformulate theMINLP problem as a MILP problem, which can be solvedby the branch and bound algorithm.

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AUTHORS’ BIOGRAPHIES

Wenxuan Guo received his BS degreein Computer Science and Technologyand his MS degree in InformationSecurity from Huazhong Universityof Science and Technology, Wuhan,China in 2004 and 2007, respectively,and subsequently his PhD degree inElectrical Engineering from Worcester

Polytechnic Institute, MA. He currently works at ActiveBroadband Networks, Inc. His research interest is inresource allocation of wireless networks.

Xinming Huang received his BS andMS degrees in Electrical Engineeringfrom Northwestern Polytechnic Uni-versity, Xi’an, China in 1994 and 1996,respectively, and his PhD degree inElectrical Engineering from VirginiaTech, Blacksburg, VA in 2001. He iscurrently an associate professor in the

Department of Electrical and Computer Engineering atWorcester Polytechnic Institute, Worcester, MA. He pre-viously worked at the Wireless Advanced TechnologyLaboratory, Bell Labs of Lucent Technologies. Hisresearch interests are in circuits and systems designfor embedded computing, including computer architec-tures, reconfigurable computing, embedded systems, andwireless networks.


Documents

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