Downlink Scheduling and Resource Allocation for Cognitive Radio MIMO Networks

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Downlink Scheduling and Resource Allocation forCognitive Radio MIMO Networks

Elmahdi Driouch,Student Member, IEEE, and Wessam Ajib,Member, IEEE

Abstract—Cognitive radio is regarded as the ideal candidate toenhance the efficiency of spectrum usage for the next generationof wireless systems. In fact, this emerging technology allows unli-censed cognitive users to transmit over frequency bands initiallyowned by license holders through the use of dynamic spectrumsharing. In this paper, we propose a novel algorithm that solvesefficiently the problem of spectrum sharing and user schedulingin a cognitive downlink MIMO system. We study the scenariowhere primary receivers do not allow any interference froma multi-antenna cognitive base station which serves cognitiveusers. Using graph theory, we model, formulate and develop analgorithm that finds a near optimal spectrum sharing with theobjective of approaching the maximum achievable secondarysumrate. Since the formulated graph coloring problem is shown tobe NP−hard, we design a low complexity greedy algorithm.Following, we add the well-known proportional fairness to theproposed algorithm in order to ensure time-based fairness andto resolve efficiently the fairness/sum rate tradeoff. The problemis also formulated as a binary integer programming problem tofind the optimal coloring solution. Computer simulations showthat the proposed algorithm is able to achieve near-optimalperformances with low computational complexity.

Index Terms—Cognitive radio, spectrum sharing, zero forcingbeamforming, NP−hard, graph coloring, greedy algorithms.

I. I NTRODUCTION

The cognitive radio technology allows the design of dy-namic spectrum sharing techniques where unlicensed users(called also secondary users) can use frequency bands ownedby licence holders (called primary users) [1], [2]. Manycognitive radio paradigms [3] were proposed in the literatureto describe the coexistence of these two types of users. Forexample, the underlay paradigm [4] allows the secondaryusers to operate in a frequency band if the interferencecaused to its owner is below a given threshold. Whereasin the overlay paradigm [3], the existence of the secondaryusers must be transparent to the primary users and hence nointerference should be perceived by the licence holders. Forboth paradigms, the performances of the secondary users arehighly affected by the used spectrum sharing algorithms.

Recently, Spectrum sharing for cognitive multi-input multi-output (MIMO) systems has attracted increasing research inter-est. In fact, the MIMO technology offers several advantagesto

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

The authors are with the Department of Computer Science, Universityof Quebec at Montreal, Montreal, QC H2X 3Y7, Canada (e-mail:[email protected]; [email protected])

This paper has been presented in part at the Proc. of IEEE Globecom,Anaheim, CA, December 2012.

enhance system performances. It can be used either to increasethroughput and/or reliability of the secondary transmissions orto eliminate or reduce the interference caused to the primaryreceivers. However, the additional spatial resource inherent inMIMO systems makes the design of efficient spectrum sharingtechniques a challenging task. In this context, the authorsof [5] considers the problem of joint transmit beamforming andpower control in the downlink of a multiuser MIMO secondarynetwork. In the studied model, the cognitive multi-antennabase station (BS) has to satisfy the quality of service (QoS)constraints of the served secondary users while protectingoneprimary receiver from interference. Their work also assumesthat the number of single antenna users is less than the numberof antennas deployed at the BS. The same authors investigatethe uplink transmission in [6] while assuming multiple primaryreceivers. In [7], a user selection algorithm is proposed withthe objective of maximizing the secondary sum rate. Assumingthe use of transmit beamforming, the secondary BS has toprotect the primary user while selecting the best secondaryusers. Using a similar system model, [8] proposes an iterativerate maximization technique that ensures that each secondaryuser attains a specific portion of the total data rate.

In this work, we tackle the problem of spectrum sharingin a cognitive radio MIMO network. The studied system iscomposed by a multi-antenna cognitive BS serving severalsingle-antenna secondary receivers. The secondary networkshares several frequency bands with multiple primary trans-mitters and receivers. The BS has to assign efficiently theavailable frequency bands to a portion of the secondary userswith the objective of approaching the maximum achievablesum rate. The primary receivers are assumed to allow nointerference from the cognitive BS. Therefore, by sacrificingsome of its degrees of freedom, the multi-antenna BS protectsthe primary receivers by using an adequate beamformingtransmission technique such as the zero forcing beamforming(ZFBF) and thus the presence of the secondary network istotally transparent to the licensed network. The choice ofZFBF as a beamforming technique is motivated by its verylow complexity and near-optimal performances. Compared tothe optimal dirty paper coding, ZFBF offers a good tradeoffbetween implementation complexity and performance, espe-cially for large number of receivers [9]. The zero interferenceconstraint that must be respected by the cognitive BS wasstudied in many recent works. In [10], the authors designedlinear precoding and linear reception schemes for a point topoint MIMO secondary system in the presence of one primarycommunication. Their proposition ensures zero interference atboth primary and secondary receivers. Using a similar system



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model, [11] proposes to sacrifice one secondary antenna torespect the zero primary interference constraint. The remainingantennas are used to perform space time block coding inorder to enhance diversity. Another recent work [12] hasproposed projection-based precoders for a MIMO cognitivepoint to point system while mitigating interference to primarytransmissions.

In this paper, we make use of graph theory in orderto model, formulate and develop heuristic solutions to theformulated spectrum sharing problem. This theory is widelyused for resource allocation in wireless systems especiallythose using the MIMO technology [13], [14]. Hence, thespectrum sharing problem is formulated as a vertex coloringproblem for weighted graphs. First, the overall network ismodeled as an undirected weighted graph. Second, sincethe formulated coloring problem isNP−hard, an efficientand very low computationally complex greedy algorithm isproposed to solve heuristically the problem. This algorithmuses one of the four designed vertex selection criteria. Thensome changes are brought to the initial greedy algorithm usingthe well known proportional fair scheduler in order to improvelong term fairness without sacrificing sum rate performance.We also formulate the coloring problem as a binary integerprogramming (BIP) problem in order to find the optimalcoloring solution for comparison and performance evaluationpurposes. Computer simulations show that the performancesof the proposed greedy algorithm approach the optimal BIPsolution. In addition, the computational complexity of theproposed algorithm is computed for the worst case and isshown to be largely lower than the complexity of finding theoptimal solution. This makes the proposed greedy algorithmpractical for real systems.

The organization of the paper is the following. In Section II,the system model and the spectrum sharing problem areformulated. Then, in Section III the weighted graph adoptedtomodel the studied system is outlined and the spectrum sharingproblem is formulated as a vertex coloring problem. SectionIVdescribes the proposed algorithm and the vertex selection cri-teria. In Section V, a binary integer programming formulationof the coloring problem is presented. Simulation results andcomplexity computation are presented in Section VI. Finally,conclusions are provided in Section VII.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We consider a secondary network which sharesN frequencybands (FBs) with a primary network. The secondary networkconsists of aM−antenna cognitive BS andK single antennasecondary users (SUs) whereas the primary network consistsof several transmitter and receiver nodes. It is assumed thateach FB is used by exactly one primary transmitter servingone or multiple primary receivers. The number of primaryreceivers on bandn is denoted byN (n)

P with N(n)P < M .

These receivers tolerate no interference from the secondarytransmissions. Also, FBs are assumed to be orthogonal and sothere is no interference between simultaneous transmissionsusing different bands. The system model is illustrated in Fig. 1.

Furthermore, it is assumed that the secondary BS has theability to transmit over all theN FBs used by the primarylinks.

Let us denote the channel coefficients between the sec-ondary BS antennas and thekth SU antenna on thenth FBby the 1 × M vector hk,n while the channel coefficientsbetween the secondary BS antennas and the primary receiverpn (pn = 1, . . . , N

(n)P ) as the1 ×M vector gn(pn). These

channel coefficients are assumed to be perfectly known at thecognitive BS in order to perform efficient spectrum sharingand to design beamforming vectors. Note that the assumptionof perfect channel knowledge is widely used when proposingscheduling algorithms for ZFBF systems. Anyhow, the impactof imperfect channel knowledge or outdated CSI on similarsystems is widely investigated in the literature, e.g. [15], [16].Also, note that the channel coefficients between the cognitiveBS and the secondary users can be estimated using pilot aidedestimation techniques, for instance. Channel coefficientsfromthe BS to the primary receivers can be obtained through thedeployment of sensors near the receiving points as discussedin [4]. The BS can also estimate these channel coefficients ifthe primary receivers send pilot signals. In that case, the BShas to know these pilot signals.

Let us also denote the channel coefficient between thekthSU and the primary transmitter operating on bandn by fk,n.The channels are assumed to experience flat fading over eachFB but their coefficients vary from one FB to another. Allthe channel coefficients remain invariant during each timeslot (TS) but may vary from one TS to another. Therefore,assuming that userk is served in thenth FB, its receivedsignal is given as

yk,n = hk,nx(s)n + fk,nx

(p)n + zk,n, (1)

wherex(s)n is theM × 1 vector containing the precoded data

symbols transmitted to the scheduled SUs on thenth FB,x(p)n is the primary signal transmitted on thenth FB andzk,n

is the independent and identically distributed (i.i.d.) complexGaussian noise with varianceσ2

zk.

Prior to transmission, the secondary BS processes each datavector intended to the users scheduled to be served on the bandn by a weight matrix and a power vectorPn. Thus, the signalreceived by userk will be written as

yk,n =

√

P(s)k,nhk,nwk,nu

(s)k,n +

∑

i∈Sn

i6=k

√

P(s)i,n hk,nwi,nu

(s)i,n

+

√

P(p)n fk,nu

(p)n + zk,n, (2)

where Sn is the set of secondary users served on thenthFB, P (s)

k,n is the portion of power allocated to userk, P (p)n is

the transmit power of the primary transmittern, wk,n is theweight vector corresponding to userk, u(s)

k,n is the data symbol

transmitted to secondary userk andu(p)n is the data symbol

transmitted from primary transmittern with |u(s)k,n| = 1 and

|u(p)n | = 1. Note that the secondary BS has a limited powerP

that must be shared efficiently among the scheduled SUs.



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Fig. 1. System Model -M -antenna BS servesK SUs in 2 FBs (band 1 is used by one primary transmitter (PT) andtwo receivers (PR) and band 2 is usedby one PT and one PR).

The channel coefficients are modeled as

hk,n =(

d(s)k

)−α

2

β(s)k,n (3)

fk,n =(

d(sp)k,n

)−α

2

β(sp)k,n (4)

gn(pn) =(

d(p)pn

)−α

2

β(p)n (pn) (5)

whered(s)k , d(sp)k , d(p)pnare the distances between thekth SU

and the BS, thekth SU and thenth primary transmitter andthe primary receiverpn and the BS, respectively andα isthe path loss exponent. The elements of the two vectorsβ

(s)k,n

andβ(p)n (pn) and the scalarβ(sp)

k,n are assumed to be modeledas i.i.d. complex Gaussian variables with zero mean and unitvariance.

Since the BS is assumed to have perfect knowledge of thechannel coefficients between its antennas and both secondaryand primary receivers, it is able to use ZFBF as a MIMOtransmit technique. Thus, it is able to serve simultaneouslymultiple secondary receivers without originating any interfer-ence to the primary receivers. The BS is able to transmit overtheN available FBs and hence has to share these bands amongthe SUs. Therefore, the chosen users are disposed in at mostN sets. Using ZFBF forces the BS to schedule at mostM

users in each set of users. Furthermore, in order to generateno interference to the primary receivers in a given FB, the BShas to sacrifice some of its degrees of freedom. More precisely,each set of usersSn should contain at mostM − N

(n)P SUs

because of sacrificingN (n)P degrees of freedom. Thanks to

ZFBF, the secondary transmission is complectly transparentto the primary receivers and do not cause any interference tothem. Hence, there is no need to perform any power controlfor limiting the interference. In the case whenM −Np ≤ 0,ZFBF can no longer be applied.

For each FB, the BS must design a ZFBF weight matrixdenoted byWn, n = 1, . . . , N . Let us denote byH(Sn), the

(

(|Sn|+N(n)P )×M

)

matrix made up of the channel vectorsof the users scheduled inSn and the channel vectors of theprimary receivers on FBn where | · | is the cardinality of aset. The weight matrixWn can be simply given by

Wn = H†(Sn), (6)

where(·)† denotes the pseudo-inverse matrix.Therefore, the received signal at userk can be rewritten as

yk,n =

√

P(s)k,nu

(s)k,n +

√

P(p)n fk,nu

(p)n + zk,n. (7)

Using matrixWn as a precoding matrix, The second termin (2) corresponding to the intra-set interference among thesecondary users sharing the same FB is eliminated thanks toZFBF. The matrixWn ensures also that the primary receiversoperating in FBn receive no interference from the BS sincetheir channel coefficients vectorsgn(pn) are orthogonal to thecolumn vectors (corresponding to the SUs) inWn.

Hence, the signal-to-interference-plus-noise ratio (SINR)experienced by userk can be written as

γk,n =P

(s)k,n

σ2zk

+ P(p)n ‖fk,n‖2

. (8)

Therefore, the maximum achievable rate of userk scheduledin the bandn, if interference is considered as noise, is givenby:

Rk,n = log2 (1 + γk,n) . (9)

B. Spectrum Sharing Problem Formulation

The secondary BS implements a spectrum sharing algorithmthat is responsible of sharing theN available FBs among thesecondary users. In fact, at the beginning of each new TS, theBS has to perform a new user selection, to arrange the selectedusers in at mostN sets corresponding to the available FBs



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and finally to allocate optimally the available power amongthese users. This sharing should be based on the instantaneouschannels states and has the objective of maximizing the sumrate of the secondary network while generating no interferenceto the primary receivers.

The spectrum sharing problem can be written as

MaximizeRT =

N∑

n=1

K∑

k=1

αk,nRk,n (10)

subject to

N∑

n=1

K∑

k=1

αk,n‖wk,n‖2P

(s)k,n ≤ P (11)

αk,n ∈ {0, 1} ∀k, n (12)

whereαk,n is equal to one if userk is scheduled in FBn andto zero otherwise andP is the fixed power budget availableat the secondary BS. It is assumed that each secondary usercan receive data over only one FB. Hence, the solution mustalso satisfy the following constraint:

∑N

n=1 αk,n ≤ 1, for allthe secondary users. In other words, the set of users have tobe necessarily disjoint, i.e.Si

⋂

Sj = ∅, (i, j) ∈ {1, . . . , N}2

(the case when SUs can use multiple FBs will be discussedlater in Section VI).

After assigning the FBs to the selected secondary users,the cognitive BS performs an optimal power allocation amongthese users by applying the well-known water-filling algo-rithm. We assume that the BS knows the power of theinterference caused by the primary transmitter to the SUs.These users can estimate this information and send it to theBS. Therefore, an optimal power allocation must satisfy

P(s)k,n =

(

µ

‖wk,n‖2 − σ2

zk− P (p)

n ‖fk,n‖2

)+

, (13)

where(X)+ is equal tomax(0, X) andµ is the solution of

N∑

n=1

∑

k∈Sn

(

µ−(

σ2zk

+ P (p)n ‖fk,n‖

2)

‖wk,n‖2)+

= P. (14)

III. G RAPH BASED PROBLEM FORMULATION

A. System Graph Building

The cognitive radio MIMO network is formulated as aweighted graphG = (V,E,C) whereV denotes the set ofvertices,E denotes the set of edges andC is the K × N

matrix where each row represents theN different weightscorresponding to each vertex inV . The different componentsof graphG can be obtained as follows:

1) The set of verticesV : Each secondary userk in thesystem is represented by a vertexvk ∈ V . The primarytransmitters and receivers are not represented as vertices.

2) The set of edgesE: The edges represent the degrees oforthogonality between the channel vectors of the secondaryusers on each FB. Hence, each two vertices will have at mostN edges. Each edge corresponds to exactly one FB. Graphsthat have multi-edges are also referred in the graph theoryliterature as multi-graphs or pseudo-graphs [17], [18].

We define the degree of orthogonality, between the channelsof secondary usersk andk′ on FB n, as

en(k, k′) =

∣

∣

∣hk,nh

∗k′,n

∣

∣

∣

‖hk,n‖‖hk′,n‖, (15)

where()∗ denotes the conjugate transpose of a vector.Hence, we draw an edge corresponding to the bandn

between two secondary verticesvk and vk′ where (k, k′) ∈{1, . . . ,K}2 if and only if

en(k, k′) > εs, (16)

i.e. the channels of usersk andk′ are notεs-orthogonal whereεs is a constant orthogonality threshold. The thresholdεs hasto be chosen carefully in order to achieve high system perfor-mances. Finding the optimal threshold values is analyticallyintractable, and hence it is performed by means of simulations.The impact of this choice will be discussed in Section VI.

In order to differentiate edges having the same endpointsbut corresponding to different FBs, the FB index is added toeach edge. For example, if the channels of usersk1 and k2on bandn are notεs-orthogonal, then an edge{vk1 , vk2 , n}is added to the setE. In this case, the two verticesvk1 andvk2 are calledn−adjacent.

3) The weight matrixC: We define the degree of orthog-onality between a secondary userk, k = 1, . . . ,K and aprimary receiverpn, pn = 1, . . . , N

(n)P on each bandn as

e(k, pn) =|hk,ng

∗n(pn)|

‖hk,n‖‖gn(pn)‖. (17)

For each vertex, we defineN positive weights. For vertexvk, the weight corresponding to FBn is defined as follows

ck,n =

{

‖hk,n‖2, if ∀pn : e(k, pn) ≤ εp,

0, otherwise(18)

i.e. the weight of a vertexvk is equal to the channel gain ofthe corresponding secondary user if and only if the channelof this user isεp-orthogonal to the channels of the primaryreceivers operating in the same FBn. Otherwise, this weightis equal to zero.

4) The availability vectorb: In our model, the numberof possible users sharing the same FB is limited due to thenumber of primary receivers on this FB and to the use ofZFBF. Hence, we define aN×1 availability vectorb where thenth element corresponds to the number of degrees of freedomavailable for scheduling secondary users in bandn which isequal tobn = M −Npn

.Fig. 2 illustrates a system graph example. This

graph denotedGe = (Ve, Ee, Ce) is made of fourvertices Ve = {A,B,C,D} representing four SUs.



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Fig. 2. A graph example with four vertices (SUs) and three colors (FBs).

Its set of edges is Ee = {(A,B, 2), (B,C, 1),(B,C, 2), (B,C, 3), (B,D, 1), (B,D, 2), (C,D, 1)} andthe elements of matrixCe is given by the different weightsof each vertex.

B. Graph Theory Definitions

Let G = (V,E,C) be an undirected weighted graph. Wedenote byG(V ′) the subgraph induced byV ′ ⊆ V . Theset of edges ofG(V ′) is E′ = {(vk, vk′ , n) ∈ E : vk ∈V ′ andvk′ ∈ V ′}.

Definition 1. For a positive integerN , G(V ′)is said to beN -colorable if we can assign to eachvertex inV ′ a color from the set{1, . . . , N} suchthat n-adjacent vertices do not receive the samecolor n.

Definition 2. For a positive integerN and agraphG, the color sensitive maximum N -colorablesubgraph problem consists of finding a setV ′ ⊆ V

that induces aN -colorable subgraph and has max-imum weight

∑

vk∈V ′

ck,n(k) wheren(k) is the color

assigned to vertexvk.

C. Graph Problem Formulation

As previously reported by many works dealing with userscheduling for MIMO systems using ZFBF [9], [19], theachievable sum rate approaches its optimal value if and onlyif the BS schedules simultaneously near orthogonal users. Inthis paper, instead of evaluating all the possible combinationsof users, the search is limited to near-orthogonal users. Inourgraph representation, near-orthogonal users are clearly charac-terized by the absence of edges between their correspondingvertices in a given FB. Such users can be scheduled in the sameset and hence can share the same FB without penalizing the

achievable sum rate. Furthermore, the weight vector associatedwith each vertex (user) represents the sum rate gain achievedwhen scheduling the corresponding user. At the same time,our graph forbids scheduling the users that are not quasi-orthogonal to primary receivers in a given FB.

Therefore, based on this graph formulation, it can beconcluded that the objective of finding a spectrum sharingthat approaches the optimal achievable sum rate is similar tosolving the color sensitive maximumN -colorable subgraphproblem on the system graph. In fact, the formulated spectrumsharing problem is similar to coloring a subset of the verticesof G (V ′ ⊆ V ) with N colors while maximizing the totalweight of the colored vertices, i.e.

∑

vk∈V ′

ck,n(k) wheren(k)

is the color given to vertexvk. In addition, the number ofvertices colored with each color has to respect the constraintimposed by the availability vectorb.

Since vertices correspond to users, colors correspond toFBs and weight vectors correspond to channel gains, coloringa vertexvk using a colorn is equivalent to scheduling thecorresponding userk in bandn.

IV. ZFBF GREEDY GRAPH BASED ALGORITHM

The coloring problem formulated in this paper can be shownto be NP−hard based on the proof of Lemma 1 in [20].In fact, it suffices to takeN = 1 and our coloring problembecomes similar to the maximum stable set problem whichis known to beNP−hard [20]. Therefore, this motivates thedesign of heuristic algorithms and the sacrifice of optimality inreturn of a considerable reduction of computational complex-ity. In this section, we present a heuristic greedy algorithmthat obtains near optimal solutions to the spectrum sharingproblem.

A. Main Algorithm

The algorithm takes as input matricesH andG denoting thechannel coefficients between the BS and the secondary users,and between the BS and the primary receivers, respectively.Ituses also the orthogonality thresholdsεs andεp. The first stepof the algorithm is the system graph construction. Then, thealgorithm starts assigning colors to vertices one at a time in agreedy fashion. The picked vertex on each iteration must be thebest vertex to color (which is considered as locally optimal)based on a given selection criterion. Each time a vertex isassigned its best choice color, the algorithm updates the weightvectors of its adjacent uncolored vertices by reducing to zerothe weights corresponding to colorn. Also, the availabilityvectorb has to be updated by decrementing its element thatcorresponds to the assigned color.

The algorithm terminates when the availability vectorb isequal to the null vector or when there are no more verticesthat can be colored. The main steps of the proposed greedyalgorithm are outlined in Fig. 3 where0N and0K,N are theN × 1 zero vector andK ×N zero matrix respectively.

B. Selection Criteria

The most important phase in the proposed greedy scheduleris the way the next vertex to be colored is picked. In the case of



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Fig. 3. Flowchart of the proposed greedy algorithm.

unweighted graphs, most vertex coloring heuristic algorithmscolors at each iteration the vertex having the larger degree, i.e.the vertex having the larger number of adjacent vertices [21],[22]. They are based on the intuition that a vertex having alarger degree will be more difficult to color if it is delayed.Thus, we propose new selection criteria since we are dealingwith new parameters which are vertices’ weights:

1) Cd(vk) is the weight degree ofvk. It is the sum ofthe weights of vertexvk. Note that when the algorithmchooses the vertex having the largerCd(vk), the totalweight of the solution increases considerably. However,the total weight of the final solution may be penalizedif the chosen vertex has a large number of adjacentvertices.

2) Ed(vk) is the edge degree ofvk. It is the number ofedges whose endpoints are vertexvk and a non coloredvertex. As it will be shown through simulations, thiscriterion results in poor performance since it does nottake into account the vertices’ weights.

3) Dd(vk) is the value of the difference between the twolargest weights in the weight vector ofvk. Using thiscriterion, the algorithm will color a vertexvk (the onehaving the largest value ofDd) with its first choicecolor before its adjacent vertices. Hence, it will avoid

Algorithm 1: Steps of the weight update phase

i← Current algorithm iteration;

Proj(n)(i) = Proj(n)(i−1) +q(i−1)∗

k,nq(i−1)

k,n

|q(i−1)

k,n|2

;

foreach vk non n-adjacent tovk doq(i)k,n = hk,n

(

I− Proj(n)(i))

;

if c(i−1)k,n > 0 then

c(i)k,n = ‖q

(i)k,n‖;

endend

penalizing vertices by a long delay.4) Md(vk): Let defineM (n)

d (vk) as the difference betweenthe nth weight of vertexvk (i.e. ck,n) and the largestnth weight among its adjacent non-colored vertices, i.e.

M(n)d (vk) = max

k′

(ck,n−ck′,n) ∀{vk, vk′} ∈ E, (19)

wherevk′ is a not yet colored vertex. Hence, we defineMd(vk) as the largestM (n)

d (vk) for each vertexvk.The rationale behind this criterion is that each vertexis only compared to its adjacent vertices. Thus, a vertexhaving a largerMd(vk) = max

nM

(n)d (vk) will receive

the corresponding color before its adjacent vertices.

Note that new selection criteria can be defined by combiningtwo or more of the proposed criteria.

C. Weight Update Phase

Instead of assigning weights to vertices in a static manner,the proposed greedy algorithm can implement a weight updatephase. It is based on the well known semiorthogonal userselection algorithm (SUS) proposed in [9]. This new phase isexecuted each time a new vertex is colored, that is, betweenphase 2 and phase 4 of the initial algorithm and can besummarized as follows:

1) Initialization: Initialize a (M ×M ) matrix for each FB(color) n ∈ {1 . . .N} as follows

Proj(n)(0) =

N(n)P∑

pn=1

gn(pn)∗ gn(pn)

|gn(pn)|2. (20)

Hence, for each vertexvk, its weight corresponding to FBn is now defined as

c(0)k,n =

{

∣

∣

∣q(0)k,n

∣

∣

∣, if ∀pn : e(k, pn) ≤ εp.

0, otherwise(21)

whereq(0)k,n is a vector defined as the projection ofhk,n on

the subspace spanned by the channel vectors between the BSand the primary receivers

q(0)k,n = hk,n

(

I− Proj(n)(0))

, (22)



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2) Weight Update:Each time a new vertexvk is coloredusing a colorn, the weights of its nonn-adjacent verticescorresponding to colorn are updated as in Algorithm 1.

D. Secondary Feedback Reduction

In the proposed system, all the secondary users mustfeedback their channel state information to the secondaryBS. Though such approach gives high system performance,it is resource hungry. Hence, we propose a simple approachthat reduces secondary users feedback at the expense ofintroducing a short delay. The feedback reduction approachcan be summarized as follows:

1) The BS receives the channel coefficients between itsantennas and the primary receivers;

2) The BS transmits the received coefficients to the SUs;3) Each SU evaluates the degree of orthogonality (as in

equation (17)) between its channel and each primaryreceiver channel (received in step 2);

4) A SUk feeds back its channel coefficients correspondingto bandn to the BS if and only if∀pn : e(k, pn) ≤ εp.

Compared to the conventional approach where all the SUsfed back their CSI to the BS, only SUs having a channel whichis near orthogonal to the primary receivers’ channels have totransmit their CSI to the secondary BS. Hence, the feedbackis considerably reduced for large numbers of SUs. However,a short delay is introduced corresponding to the transmissionof the primary receivers’ channels from the BS to the SUs.

E. ZFBF Proportional Fair (PF) Algorithm

The proposed algorithm aims to approach the maximumachievable sum rate by scheduling in each TS the users havingthe best channel conditions. Therefore, it is thus clear that suchan opportunistic algorithm is unfair and it has to be adaptedin order to ensure fairness while achieving high sum rate. Agood tradeoff between achievable sum rate and fairness can beobtained through the use of the well known proportional fairscheduling (PFS) [23]. Hence, we brought some changes intothe greedy algorithm in order to ensure proportional fairness.

All the steps of the ZFBF PF are similar to the proposedgreedy algorithm. The only modification lies in the definitionand update of the weight matrixC. In fact, the ZFBF PFhas to keep track of the average channel gain of each userover a determined time period. Then, instead of using (18)to assign weights to vertices, the new algorithm defines theweight matrix as follows

ck,n(t) =

‖hk,n(t)‖2

Tk,n(t), if ∀pn : e(k, pn) ≤ εp,

0, otherwise(23)

where t is the TS index andTk,n(t) denotes the averagechannel gain of userk on bandn which is updated as follows

Tk,n(t+1) =

(

1−1

ω

)

Tk,n(t) +ck,n(t)

ω, if k ∈ S(t),

(

1−1

ω

)

Tk,n(t), otherwise

(24)

whereω is the averaging window time andS(t) =⋃N

n=1 Sn(t)is the set of users scheduled in slott.

Therefore, instead of always picking the users with highchannel gains (as the greedy algorithm do), the ZFBF PFalgorithm schedules a SU only when its instantaneous channelquality is near to its average channel gain. Furthermore, byperforming such a selection, multiuser diversity benefits canstill be extracted because users’ channels fluctuate indepen-dently [23].

To evaluate the long-term fairness of the proposed algo-rithms, the fairness index proposed in [24] and known as theJain’s fairness index is used.

V. B INARY INTEGERPROGRAMMING FORMULATION

In this section, we formulate the presented graph coloringproblem into a BIP problem in order to find optimal coloringsolutions. Although, finding such solutions can be obtainedonly for problems involving small number of variables andconstraints. This formulation is still very interesting inorderto assess the performances of the proposed heuristics.

Let

ak,n = 1 if vertex vk receives colorn

= 0 otherwise.

Then the problem can be formulated as a BIP problem asfollows

MaximizeK∑

k=1

N∑

n=1

ck,nak,n (25)

subject to

N∑

n=1

ak,n ≤ 1 ∀k (26)

K∑

k=1

ak,n ≤ bn ∀n (27)

ak,n + ak′,n ≤ 1 ∀n and∀{vk, vk′} ∈ E (28)

ak,n ∈ {0, 1} ∀k, n (29)

Constraint (26) ensures that each user is scheduled in atmost one FB. This constraint is equivalent to the one thatforces each vertex to be colored with at most one color inthe graph based formulation. Constraint (27) ensures that thenumber of secondary users sharing the same FB do not exceedthe number of available degrees of freedom at the BS inthat band. Constraint (28) ensures that secondary users whosechannels are notεs-orthogonal, do not share the same FB.This is equivalent to the requirement that adjacent vertices arecolored using different colors.

VI. SIMULATION RESULTS

In this section, we first study the worst case computa-tional complexity of the proposed greedy algorithm. Thiscomplexity is then compared to the one of the optimal col-oring solution provided through BIP in terms of execution



8

time. Second, we analyze the performance of the proposedalgorithm using the designed four selection criteria in termsof the maximum achievable sum rate. The performances ofthe proposed greedy algorithm are compared with the optimalcoloring performances obtained by solving the formulatedBIP problem usingbintprog matlab solver. This solver uses alinear programming (LP)-based branch-and-bound algorithmin order to find optimal solutions. Note thatbintprog replacesthe binary integer requirement on the variables (a ∈ 0, 1) bythe weaker constraint (0 ≤ a ≤ 1). A complete description ofthe algorithm used by the solver is given in [25].

A. Complexity Comparison

We evaluate in the following the computational complexityof the proposed algorithm. The complexity is given usingasymptotic notations in order to assess how the algorithm re-sponds (in terms of execution time) to input size changes [26].Here the input size is expressed in terms ofM , N , andK.

The graph building phase constructs edges with a compu-tational complexity ofO(MNK2) and then assigns weightsto vertices with a complexity ofO(MNK). Therefore, thisphase has a computational complexity of

C1 = O (MNK(K + 1)) . (30)

In each iteration, the main algorithm computes the selectioncriterion for each vertex. For the criterionCd, this computationhas a complexity ofO(NK). The algorithm then picks thevertex having the maximum selection criterion value. Thecomplexity of finding this maximum isO(K). Therefore, themain algorithm has a computational complexity of

C2 = O (Sb(K +NK)) , (31)

whereSb is the sum of the elements of matrixb. Therefore,the computational complexity of the overall greedy algorithmwithout including the channel inversion and power assignmentcomplexities is given by

C = C1 + C2. (32)

In order to compare the complexity of the greedy algorithmto the one of the BIP optimal solution, we plot in Fig. 4 bothcomplexities as a function of the number of secondary userswhile varying the values ofM and N . The complexity isgiven by the amount of time needed to find the sets of usersto serve during one TS. It can be observed that the optimalalgorithm has very high computational complexity comparedto the greedy algorithm for different values ofM , N andK.In fact, whereas the complexity of the proposed scheme growsalmost linearly withK, the complexity of the BIP is far frombeing linear.

B. Performance evaluation

Fig. 5 illustrates the impact of the orthogonality thresholdεs value on the achievable secondary sum rate. We use thegreedy algorithm with selection criterionMd. We setN = 3,b = (M − 2,M − 1,M − 1), P = 10dB and εp = 0.55.

8 10 12 14 16 18 2010

−1

100

101

102

103

104

Number of users

Exe

cutio

n tim

e

M=3, N=2M=3, N=3M=4, N=2M=4, N=3

BIP Solution

Greedy algorithm

Fig. 4. Complexity comparison between the greedy algorithmwith Cd andthe optimal coloring BIP.

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6511

12

13

14

15

16

17

18

Orthogonality threshold εs

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

K = 14K = 16K = 18K = 20

M=3

M=4

Fig. 5. The achievable sum rate vs. the orthogonality threshold εs.

Whenεs is relatively small, the sum rate is penalized becausethe first phase of the greedy algorithm builds a dense graph (aquasi-complete graph where the number of edges is close toits maximal number). Using such a graph, the selection phaseis forced to form small sets of users. Contrarily to the firstcase, whenεs tends to one, the constructed graph is sparse.The selection phase forms larger sets containing users withchannels that are far from being quasi-orthogonal.

Using the same system parameters as in Fig. 5, we illustratein Fig. 6 the impact of the orthogonality thresholdεp value onthe proposed algorithm performances. The value ofεp affectsthe weight assignment step on graph building. In fact, whenεpis small, the weight vectors are sparse (populated primarily byzeros) and thus the algorithm forms small sets which penalizesthe sum rate. On the other hand, whenεp approaches one thealgorithm schedules secondary users even if they are not quasi-orthogonal to the primary receivers.

Therefore, in the next simulations we pick the values ofεs



9

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.79

10

11

12

13

14

15

16

17

18

Orthogonality threshold εp

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

K = 14K = 16K = 18K = 20

M=3

M=4

Fig. 6. The achievable sum rate vs. the orthogonality threshold εp.

8 9 10 11 12 13 14 15 16 17 189

10

11

12

13

14

15

16

17

18

Number of Secondary users

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

BIP Cd(v) D

d(v) E

d(v) M

d(v) Optimal ES

M=4

M=3

Fig. 7. The achievable sum rate vs. the number of secondary usersK.

and εp that achieves the best performances according to thevalues ofM andK. For instance, for the parameters used inFig. 5,εs is chosen in the interval[0.4, 0.45] andεp is chosenin [0.5, 0.6]. Also, the path loss exponent is set toα = 4.

In Fig. 7, we plot the achievable sum rate as a function ofthe number of secondary users forM = 3 (lower curves) orM = 4 (upper curves). The figure compares the performancesof the greedy algorithm using the four designed selectioncriteria to the sum rate achieved by optimal coloring foundthrough BIP. The plotted rates are also compared (in the caseof M = 3 andK ≤ 16) to the optimal ZFBF sum rate obtainedthrough a highly complex exhaustive search (ES) over all thepossible combinations of users and groups. We setN = 3,b = (M − 2,M − 1,M − 1) and P = 10dB. We noticethat the greedy algorithm achieves different performancesaswe change the selection criterion used to pick the next vertexto be colored. As expected, the edge degreeEd is the worstselection criterion since it does not take into account vertices’

2 3 4 5 68

10

12

14

16

18

20

22

Number of frequancy bands

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

BIP optimal solutionGreedy with C

d

Greedy with Dd

Greedy with Ed

Greedy with Md

Fig. 8. The achievable sum rate vs. the number of frequency bandsN .

weights. The performances of criteriaMd andDd are closeespecially when the value ofK increases beyond 14 users.They both outperform clearly the other two criteria since theygive priority to vertices that can be penalized by a long delay.Using the best selection criterion, the performances of thegreedy algorithm are very close to those of the BIP solutionthat uses a complex branch-and-bound algorithm. In fact,the proposed algorithm achieves more than 97% of the BIPsolution when usingMd. Furthermore, compared to the sumrate obtained by the highly complex ES, the greedy algorithmusingMd achieves almost 94% of the optimal ZFBF solution.

In Fig. 8, we plot the achievable sum rate as a functionof the number of available FBs. We setM = 3, K = 20,P = 10dB andNpn

= 1 for all n ∈ {1, . . . , N}. It can beobserved that the selection criteriaMd andDd outperform theother two ones. However, we notice that when the number ofFBs increases,Md outperformsDd. In fact, asN increases, thenumber of elements in the weight vectors increases and thusthe values of the criterionDd for different vertices becomealmost similar. Whereas, the criterionMd is not affected by theincrease on the number of FBs. Using this selection criterion,the proposed algorithm achieves near-optimal performances.It allows the BS to use efficiently each new spectrum oppor-tunity. In terms of computational complexity, the execution ofthe greedy algorithm is more than104 faster than the algorithmthat finds the BIP solution whenN = 6, K = 20 andM = 3.

The maximum achievable sum rate of the proposed greedyalgorithm with and without the weight updating phase as afunction of the number of secondary users is shown in Fig. 9.The used selection criteria isCd and we setM = 3, P =10dB andNpn

= 1 for all n. It is noticed that the weightupdating phase yields a sum rate gain of almost 1% over theinitial algorithm (without weight updating). The disadvantageof including this phase in the main algorithm is that this smallrate gain comes at the expense of computational complexity.Therefore, the fact of adding this phase or not must take intoaccount the performance/complexity tradeoff.

When the SUs have the ability to receive data simultane-



10

10 12 14 16 18 20 22 24 26 28 309

10

11

12

13

14

15

16

17

Number of secondary users

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

Without weight updatingWith weight updating

N=3

N=2

Fig. 9. The achievable sum rate vs.K with and without weight updating.

10 12 14 16 18 20 22 24 2616

16.5

17

17.5

18

18.5

19

19.5

20

Number of secondary users

Ach

ieva

ble

su

m r

ate

(b

ps/

hz)

Greedy with Cd(v)

Greedy with Md(v)

MultipleFBs Per SU

OneFBs Per SU

Fig. 10. The achievable sum rate vs.K (one FB vs. multiple FBs per SU).

ously on multiple FBs, minor changes have to be made to theproposed algorithm. Specifically, the algorithm should allowa vertex to be colored using multiple colors. Hence, once avertex is selected in phase 2, its weight corresponding to itsfirst choice color is reduced to zero while its other weightsare not updated. The weight vectors of its adjacent verticesare updated such as the elements corresponding to the usedcolor are reduced to zero (in order to prevent them from beingchosen later). In Fig 10, we show the sum rate versusK wheneach SU is allowed to use only one FB (lower curves) andwhen SUs can use multiple FBs (upper curves). If the SUsare able to make use of multiple FBs for reception, the sumrate is slightly ameliorated and this amelioration becomeslesssignificant as the number of SUs increases. This is because,when the number of SUs gets larger, the algorithm assignsrarely multiple FBs to one SU thanks to the multiuser diversity.

Fig. 11 and Fig. 12 illustrate and compare the degree offairness of the greedy algorithm and the proportional fair

10 11 12 13 14 15 16 17 18 19 200.4

0.5

0.6

0.7

0.8

0.9

1

Number of Users

Fa

irn

ess

Ja

in In

de

x

Greedy with Cv

Greedy with Dv

Greedy with Mv

PFS

Fig. 11. Jain’s fairness index [24] vs. the number of secondary usersK.

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

User index

Ave

rag

e r

ate

(b

ps/

hz)

Greedy with Dv

PFS

Fig. 12. The average rate per secondary user.

algorithm. We can see from Fig. 11 that the proposed PFalgorithm reaches a Jain index of 96% which is much betterthan the one provided by the greedy algorithm for the threeused selection criteria (between 67% and 49%). Furthermore,when using the greedy algorithm, the Jain index decreasesconsiderably if the number of secondary users increases. Itisnot the case for the PF algorithm where the Jain index remainsalmost the same for different values ofK. Fig. 12 plots theaverage rate for each user in the secondary network. ThereareK = 50 secondary users randomly located according toa uniform distribution with distances ranging from 0.5 to 1.5from the BS. We setM = 3 andb = (2, 2). It can be observedthat contrarily to the greedy algorithm which favors users nearthe BS (i.e. users with a higher signal to noise ratio), the PFalgorithm gives almost equal chances to all secondary usersand thus no starvation is caused even for the farthest users.



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VII. C ONCLUSION

This paper considered the problem of secondary userscheduling and spectrum sharing in cognitive radio MIMOsystems. The studied system assumes the coexistence of a sec-ondary network made up of a multi-antenna base station andseveral secondary receivers with multiple primary transmittersand receivers. The cognitive BS transmits over the same FBsowned by the primary users with the strict constraint of caus-ing zero interference to the license holders. Therefore, basedon zero forcing beamforming as a simple MIMO transmittechnique, we have proposed a spectrum sharing algorithmwith the objective of approaching the maximum achievablesum rate. The proposed algorithm starts by constructing aweighted multigraph taking as input all the system parameters(channel gains, FBs, orthogonality between the channels ofall the primary and secondary users, etc). Afterwards, thealgorithm solves a special case of vertex coloring problemsthat we proved to beNP−hard in a greedy fashion. Thealgorithm uses one of the four designed vertex selectioncriteria to color the best vertex in each new iteration. Inaddition, since the first objective of the proposed algorithmmakes it unfair, we make use of proportional fairness to bringsome changes to the weight assignment phase in order toensure a time-based fairness. We have also presented a binaryinteger formulation of the coloring problem in order to find itsoptimal solution using branch-and-bound techniques. Finally,we have compared through simulations the performances ofthe proposed algorithm to the optimal solution and we haveshown that it presents a good trade-off between complexityand system performances in terms of achievable sum rate.

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Elmahdi Driouch received the B.Eng. degree fromthe Ecole Nationale des Sciences Appliques, Mo-rocco, in 2006 and the M.Sc. degree in computerscience from the University of Quebec at Montreal(UQAM), Montreal, QC, Canada, in 2009. He is cur-rently working toward the Ph.D. degree in UQAM,under the supervision of Prof. W. Ajib. His cur-rent research interests include multiuser MIMO net-works, cognitive radio, and cross-layer and heuristicalgorithm design for communication networks.

Wessam Ajib received the Engineer Diploma degreein physical instruments from the Institute NationalPolytechnique de Grenoble, Grenoble, France, in1996 and the Diplome d’Etudes Approfondies de-gree in digital communication systems and the Ph.D.degree in computer sciences and computer networksfrom the cole Nationale Superieure des Telecommu-nication, Paris, France, in 1997 and 2000, respec-tively. Since June 2005, he has been with the De-partment of Computer Science, University of Quebecat Montreal, Montreal, where he is currently an As-

sistant Professor of computer networks. His research interests include wirelesscommunications and wireless networks, multiple and medium-access controldesign, traffic scheduling, MIMO systems, and cooperative communications.

Downlink Scheduling and Resource Allocation for Cognitive Radio MIMO Networks

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