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Joint Beamforming, Power and Channel Allocationin Multi-User and Multi-Channel Underlay MISO

Cognitive Radio NetworksSuren Dadallage, Changyan Yi and Jun Cai, Senior Member, IEEE

Abstract—In this paper, we consider a joint beamforming,power, and channel allocation in a multi-user and multi-channelunderlay multiple input single output (MISO) cognitive radionetwork (CRN). In this system, primary users’ (PUs’) spec-trum can be reused by the secondary user transmitters (SU-TXs) to maximize the spectrum utilization while the intra-userinterference is minimized by implementing beamforming at eachSU-TX. After formulating the joint optimization problem asa non-convex, mixed integer nonlinear programming (MINLP)problem, we propose a solution which consists of two stages.In the first stage, a feasible solution for power allocation andbeamforming vectors is derived under a given channel allocationby converting the original problem into a convex form with anintroduced optimal auxiliary variable and semidefinite relaxation(SDR) approach. After that, in the second stage, two explicitsearching algorithms, i.e., genetic algorithm (GA) and simulatedannealing (SA)-based algorithm, are proposed to determinesuboptimal channel allocations. Simulation results show thatbeamforming, power and channel allocation with SA (BPCA-SA)algorithm can achieve close-to-optimal sum-rate while having alower computational complexity compared with beamforming,power and channel allocation with GA (BPCA-GA) algorithm.Furthermore, our proposed allocation scheme has significantimprovement in achievable sum-rate compared to the existingzero-forcing beamforming (ZFBF).

Index Terms—Cognitive radio network, beamforming, semidef-inite relaxation, genetic algorithm, simulated annealing.

I. INTRODUCTION

RECENT studies reveal that the existing static spectrumallocation is the key reason for highly inefficient spec-

trum utilization [1], [2], which in turn leads to a prob-lem of spectrum scarcity. To overcome this issue, a novelconcept called cognitive radio (CR) was introduced [3]–[5],which allows unlicensed users or secondary users (SUs) toinitiate transmissions with the licensed communications. Foran underlay CR network (CRN) coexisting with a multi-channel primary user (PU) network, managing interference isa critical issue since spectrum reusing among multiple usersmay cause negative effects on received signals at both PUsand SUs. By exploiting multiple antennas, a signal processingtechnology called beamforming [6] has been introduced to CR

Copyright (c) 2015 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 pubs-permissions@ieee.org.

This work was supported by the Natural Sciences and Engineering ResearchCouncil of Canada under Discovery Grant.

The authors are with the Department of Electrical and Computer Engineer-ing, University of Manitoba, Winnipeg, MB, Canada R3T 5V6. E-mail: dadal-las@myumanitoba.ca, changyan.yi@umanitoba.ca, jun.cai@umanitoba.ca.

for directional signal transmission, so as to effectively mitigatethe mutual interference and improve the signal-to-interference-plus noise ratio (SINR) [7].

In the literature, joint beamforming and resources allocationhave been widely studied for multiple-antenna CRNs. Forexample, Xie et al. in [8] considered a sum-rate maximizationproblem in a single PU channel CRN. In this work, the mutualinterference between SUs are nullified by deploying the zero-forcing beamforming (ZFBF). The authors in [9] discussed ajoint beamforming and single PU channel assignment problemto maximize the uplink throughput of the CRN while guaran-teeing a SINR constraint at each secondary user receiver (SU-RX) and interference cancellation at the primary user receiver(PU-RX). However, ZFBF does not consider the potentialinterference tolerance at SUs, which in turns results in adegradation on overall achievable sum-rate of the secondarynetwork. Recent studies [10]–[12] showed that both PU andSU receivers can tolerate some amount of interference. As aresult, it is not necessary to null the co-channel interference.Jiang et al. in [13] employed a zero-gradient based iterativeapproach to determine the local optimal beamforming vectorswhile maximizing the energy efficiency of the CRN. In [14],beamforming vectors were calculated by using an iterativealgorithm based on semidefinite programming to maximizethe sum-rate with a total power constraint and co-channelinterference constraints at both PU and SU receivers. Thiswork was further extended in [15] by adding an extra qualityof service (QoS) constraint. However, the assumption of asingle PU channel as used in [13]–[15] reduces the degreeof freedom available at the secondary base station (SBS) onchannel allocation for SUs. Therefore, joint beamforming andresource allocation with multiple PU channels were studied.For example, in [7], a single secondary user transmitter (SU-TX)/SU-RX pair was considered with uniformly distributedprimary user transmitters (PU-TXs) and PU-RXs in a circulardisc area. Beamforming was implemented by the SU-TX tominimize the interference to the PU-RXs while the receivedsignal strength was maximized at the SU-RX. Gharavol etal. in [16] discussed a transmit power minimization problemwith a guarantee of SUs’ QoS and total power constraints in amultiple channels multiple-input-single-output (MISO) CRN.Some other works in this area include an adaptive intercellinterference cancellation (ICIC) technique for MISO downlinkcellular systems with channel allocation and beamforming tomaximize the weighted sum-rate [17]. Hamdi et al. in [18]considered joint beamforming with a near-orthogonal user

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selection method to maximize the downlink throughput of theCRN, subject to constraints on SINR at each SU, interferenceat the PU-RX and total transmission power. The authors in[19] proposed an algorithm based on branch and bound (BnB)method to allocate PU frequency bands with beamforming toserve a maximum number of SUs.

However, most of these works [14]–[21] assumed a cellulararchitecture for secondary networks, where either SU-Tx orSU-Rx is the secondary base station. Recently, device-to-device (D2D) communication based secondary networks [22],[23] are attracting more attentions from researchers due to theiradvantages in offering greater coverage with spatial diversity,higher data rates and lower energy consumption. Differentfrom the traditional cellular architecture, D2D communicationscause co-channel receivers interfered by multiple transmittersat different locations. This challenge requests the considerationof individual SINR constraint at each SU-Rx, and makes thejoint beamforming and resource allocation more complicatedthan the traditional case.

In this paper, we readdress the joint transmit beamforming,power and channel allocation problem in an underlay CRNwith independent D2D communications among multiple PUand SU pairs. Specifically, in our work, beamforming isperformed by each SU-TX instead of a single SBS so as topromote the spatial diversity and curtail the interference. Ourmain objective is to maximize the sum-rate of the secondarynetwork while minimizing the intra-user interference subjectto the constraints of total power budget, interference on PUsand SINR requirement at each SU. We formulate the jointoptimization problem as a non-convex, mixed integer nonlin-ear programming (MINLP) problem [14], which is NP-hard.In order to solve this problem with reasonable computationalcomplexity, a two-stage solution approach is proposed. In thefirst stage, an iterative algorithm is proposed to determine thefeasible beamforming vectors and power allocation for a givenchannel allocation. After that, two explicit searching algo-rithms based on genetic algorithm (GA) [24] and simulatedannealing (SA) [25] are proposed to find out the suboptimalchannel allocation. The main contributions of this paper aresummarized as follows.

• We consider a multi-channel underlay CR system withthe capability of beamforming at each SU-TX to mitigateinterference, allow more transmission opportunities, andexploit the benefits of spatial diversity.

• We develop a sum-rate maximization problem to joint-ly optimize beamforming vectors, power allocation andchannel allocation.

• Two different algorithms are proposed to solve the for-mulated non-convex MINLP problem.

• Simulation results show that the proposed system modeloutperforms the existing ZFBF model by exploiting inter-ference tolerance capacities. The proposed algorithms canachieve close-to-optimal performance in terms of sum-rate with low computation complexity so that they aremore suitable for practical applications.

The rest of this paper is organized as follows. The systemmodel and the problem formulation are discussed in Section II.

PU-RXn

PU-TXn

SU-TXk

SU-RXk

SU-TXm

SU-RXm

gnk

hkn

hmk

hk

Central

Entity

PU-TXSU-TX

PU-RX

SU-RX

PU-RXPU-TX

Desired Channel

Interference Channel

Fig. 1. System model and channel responses.

The solution approaches and their computational complexitiesare analyzed in Section III. After that, the simulation resultsare presented in Section IV, followed by the conclusions inSection V.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a CRN with N PU transceivers and K SUtransceivers randomly distributed in the coverage area of theprimary network. Each PU transceiver occupies one separatedlicensed channel so that there are N PU channels in total.Each SU-TX is equipped with J antennas and its receiverwith a single antenna, while each PU (transmitter or receiver)has a single antenna. Each SU-TX is allowed to communi-cate with its corresponding receiver in the underlay mannerwhile satisfying a pre-defined interference constraint at thecorresponding PU-RX. There is a central entity who performsall control functions (e.g., resource allocations) for SUs. Anillustration of the system model is shown in Fig. 1. We definetwo sets S and P to indicate all possible SU pairs and PUpairs in the network, respectively.

The following notations are used in this paper. Boldfaceuppercase and lowercase letters will be used for matrices andvectors, respectively. (·)†, (·)T , E{·} and ‖x‖ denote conju-gate transpose, transpose, expectation and Euclidean norm ofvector x, respectively. In addition, Tr(A) indicates the traceoperation of a square matrix A. For convenience, Table I listssome important symbols used in this paper.

A. Signal Model

Define the transmitted signals from the kth SU-TX, i.e.,SU-TXk, to its receiver SU-RXk and from nth PU-TX, i.e.,PU-TXn, to its receiver PU-RXn as sk and un, respectively.We assume that each modulated transmitted signal has unitenergy and is uncorrelated with each other, i.e., E{|sk|2} =E{|un|2} = 1, E{sks†m} = E{unu†d} = 0,∀k,m ∈ S andn, d ∈ P , k 6= m and d 6= n. With antenna array, eachSU-TX performs transmit beamforming to direct the signalto the intended receiver while at the same time controlling theinterference to other users (PUs and SUs). The received signalat the SU-RXk using the nth PU channel can be representedas the aggregation of desired signal, interference from other

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TABLE ILIST OF NOTATIONS

Symbol DefinitionSn Set of SU pairs using the nth PU channelK Number of SU pairsN Number of PU channels

sk/un kth SU transmit signal/nth PU transmit signalhk Channel response between the SU-TXk and the SU-RXk

hmk Channel response between the SU-TXm and the SU-RXkhmn Channel response between the SU-TXm and the PU-RXngnk Channel response between the PU-TXn and the SU-RXkgn Channel response between the PU-TXn and the PU-RXn

ηk/ηn Noise at the SU-RXk/PU-RXnQn PU-TXn’s transmit power

αki, αkn, αn, βk Large-scale fading coefficientsϑki, ϑkn, ϑn, bk Small-scale fading coefficients

J Number of transmit antennasvk(θ, φ) SU-TXk’s steering vector having a counter-clockwise azimuth angle of θ and φ elevationrnk Rate of the kth SU-TX on nth PU channelB Transmission bandwidth of each PU channelInth Maximum interference tolerance threshold at the PU-RXnΓk Minimum QoS threshold at the kth SU-RXPmax Power budget of the secondary networkϕ Auxiliary vector to indicate the intra-user interference thresholds of each SU-RXϕk kth SU-RX intra-user interference thresholdwk kth SU-TX beamforming vectorWk kth SU-TX positive semidefinite beamforming matrixxnk Binary variable to indicate the nth PU channel allocation on kth SU-TX

SU-TXs using the same PU channel for transmission, theinterference from the nth PU-TX, and noise, i.e.,

ynk = (wksk)†hk +∑

m∈Sn,m 6=k

(wmsm)†hmk +√Qngnkun + ηk,

(1)where wk is the kth SU-TX’s beamforming vector with size ofJ×1, Qn is the transmit power of the nth PU-TX, hk denotesthe J × 1 channel response between SU-TXk and SU-RXk,hmk is the J × 1 channel response between SU-TXm andSU-RXk, gnk denotes the channel response between PU-TXnand SU-RXk, and Sn denotes the set of SU-TXs using the nthPU channel. The noise ηk is Gaussian distributed with zeromean and variance σ2. Similarly, the received signal at the nthPU-RX consists of desired PU signal, co-channel interferencefrom SUs and noise. Thus, it can be formulated as

yn =√Qngnun+

∑k∈Sn

(wksk)†hkn+ηn, n = 1, . . . , N (2)

where gn and hkn are channel responses from the nth PUtransmitter and the kth SU-TX to the nth PU-RX, respectively,and ηn denotes the noise.

The channel responses hki, gkn and gn in (1) and (2) canbe further defined as

hki =

{hki = [h1ki . . . h

Jki]T , if i 6= k ∈ Sn, i ∈ Sn ∪ P,(3a)

βkbkvk(θ, φ), if i = k (3b)

hlki = αkiϑki, i 6= k ∈ Sn, i ∈ Sn ∪ P and l = 1, . . . , J (4)

gkn = αknϑkn, k ∈ Sn and n ∈ P (5)

gn = αnϑn, n ∈ P (6)

where αki, αkn, αn and βk denote the large-scale fading, whileϑki, ϑkn, ϑn and bk represent the small-scale fading which aremodeled as Rayleigh distributed random variables. Consideruniform circular array (UCA) [26] of antenna configuration ateach SU-TX. Then, vk(θ, φ) is the steering vector for the kthSU-TX, which indicates the relative responses of the isotropicarray elements with respect to the signal impinging at thecenter of the array from a particular direction. Here, φ denotesthe counter-clockwise azimuth angle measured from X-axisand θ denotes elevation measured from Z-axis. In this paper,we assume that all users are located in the same plane, sothat θ becomes zero and mutual coupling [27] among arrayelements is not considered.

Similar to [15], we assume that perfect knowledge of thechannel state information (CSI) is available at each SU-TXbefore transmission.

B. Problem Formulation

Given the kth SU transceiver is working on the nth PUchannel, the SINR at the kth SU-RX, denoted as SINRk, canbe represented as

SINRk =|w†khk|2∑

m∈Sn,m6=k|w†mhmk|2 +Qn|gnk|2 + σ2

(7)

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where |w†khk|2 is the desired received signal power at theSU-RXk.

∑m∈Sn,m 6=k

|w†mhmk|2 denotes the total intra-user

interference received at the SU-RXk from other SU-TXs’who are using the same channel. Qn|gnk|2 and σ2 are theinterference power from PU-TXn and the noise power at theSU-RXk, respectively.

Let xnk , k = 1, . . . ,K, n = 1, . . . , N, denote the channelallocation for SUs. xnk = 1 means that the nth PU channel isassigned to the SU-TXk. Otherwise, xnk = 0. Then, from (7),the achievable rate of SU pair k on the nth PU channel, rnk ,can be calculated as

rnk = B log2

1 +xnk |w

†khk|2∑

m∈Sn,m6=k|w†mhmk|2 +Qn|gnk|2 + σ2

(8)

where B is the transmission bandwidth of a single PU channel.

Our major objective is to maximize the sum-rate of the SUnetwork while allowing concurrent transmission with PUs. Theoptimization problem can be formulated as

P1 : maxw,X

K∑k=1

N∑n=1

rnk (9)

s.t.K∑k=1

|w†khkn|2xnk ≤ Inth, ∀n = 1, . . . , N, (10)

SINRk ≥ Γk, ∀k = 1, . . . ,K, (11)K∑k=1

N∑n=1

‖wk‖2xnk ≤ Pmax, (12)

N∑n=1

xnk ≤ 1, ∀k = 1, . . . ,K, (13)

xnk ∈ {0, 1}, ∀k ∈ S, ∀n ∈ P, (14)

where Inth is the maximum interference tolerance thresholdof the nth PU-RX. Γk is the minimum QoS threshold atthe kth SU-RX. Pmax is the available power budget of thesecondary network. w = [w1 . . .wK] of size J ×K indicatesthe beamforming weights matrix, where its kth column definesthe beamforming vector for the SU-TXk. X is a K×N channelallocation matrix, which consists of all xnk , k ∈ S and n ∈ P .The constraint (10) limits the total interference power fromSU-TXs to a specific PU-RX below a pre-defined threshold.Constraint (11) guarantees the required SINR level at eachSU-RX. Constraint (12) keeps the total power consumptionunder the available power budget. Constraint (13) implies thateach SU-TX can access at most one PU channel. Note that thesetting of single-channel access has been widely used in mostof related works [10]–[12]. Even though the sum-rate may befurther improved by adopting multiple channels for a singleSU-TX, it will significantly increase the complexity of thesystem due to the difficulties involved in the power allocation.Since the power allocation of SU-TXk is determined by‖wk‖2, the problem P1 in fact integrates beamforming, powerallocation and channel allocation.

III. A TWO-STAGE SOLUTION APPROACH

The problem P1 consists of both continuous and discretevariables, and there are nonlinear terms in both the objectivefunction and constraints. Hence, it is a non-convex, mixedinteger non-linear programming problem, which has beenproved to be NP-hard [28]. In order to balance performanceand computational complexity, in the following a two-stagesolution approach is proposed. The idea is to separate themain problem into two sub-problems. In the first sub-problem,the power and beamforming vectors are calculated basedon a given channel allocation. After that, the second sub-problem, which determines a suboptimal channel allocation,will be solved. For the second sub-problem, two algorithmsare proposed with different computational complexity.

A. Power and beamforming vector determination based on agiven channel allocation

In this section, the beamforming vector and power allocationfor each SU-TX will be determined based on a given channelallocation, X. Given X, constraints (10) and (12) can betransformed to a summation of quadratic terms and normsso that they become convex. However, the original problemP1 is still non-convex because neither the objective function(9) nor the constraint (11) are convex. To overcome this issue,we use semidefinite programming (SDP) approach [29], whichallows to express the quadratic terms with some equivalentaffine expressions. With SDP, the quadratic terms, |w†khkn|2,|w†mhmk|2 and ‖wk‖2, can be equivalently represented as

|w†khkn|2 = Tr(w†khknh†knwk)

= Tr(WkHkn), ∀k ∈ S,∀n ∈ P (15)‖wk‖2 = Tr(Wk), ∀k ∈ S (16)

|w†mhmk|2 =

{Tr(WmHmk), ∀m 6= k,m ∈ SnTr(WkHk), m = k

(17)

where Wk = wkw†k, Hkn = hknh†kn, Hk = hkh

†k, and

Hmk = hmkh†mk. From (15) - (17), we have i) Wk is a rank

one positive semidefinite (PSD) matrix, i.e., Wk � 0, ∀k;ii) Tr(WkHkn),Tr(WkHk) and Tr(WmHmk) are all affineexpressions with respect to the symmetric PSD matrices Wk

and Wm.

With (15), (16), (17) and the assumption of a known channelallocation, the optimization problem P1 can be rewritten as

P2 : maxW1,...,WK

K∑k=1

N∑n=1

rnk (18)

s.t.K∑k=1

xnkTr(WkHkn) ≤ Inth, ∀n = 1, . . . , N (19)

SINRk ≥ Γk, ∀k = 1, . . . ,K (20)K∑k=1

N∑n=1

xnkTr(Wk) ≤ Pmax, (21)

Wk � 0, ∀k = 1, . . . ,K, (22)Rank(Wk) = 1, ∀k = 1, . . . ,K (23)

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where,

rnk = B log2

1 +xnkTr(WkHk)∑

m∈Snm6=k

Tr(WmHmk) +Qn|gnk|2 + σ2

(24)

Note that two additional constraints, (22) and (23), areadded in P2. Obviously, the former has a convex expression,but the later does not. In addition, the constraints (19) and (21)have been transformed to linear functions without affectingtheir convexity. The optimization problem P2 can be furthertransformed into a convex form by doing the following twosteps. First, the non-convex rank constraint of (23) can berelaxed by using the SDR approach again [30]. Then, follow-ing the similar approach in [14], intra-user interference fromother SUs to the SU-RXk, i.e.,

∑m∈Snm6=k

Tr(WmHmk), can

be constrained by introducing a new auxiliary variable. Afterthese two steps, the revised optimization problem P3 can bewritten as

P3 : maxW,ϕ

K∑k=1

N∑n=1

B log2

[1 +

xnkTr(WkHk)

ϕk +Qn|gnk|2 + σ2

](25)

s.t. constraints (19), (21), (22),

Tr(WkHk) ≥ Γk

( ∑m∈Snm6=k

Tr(WmHmk)

+Qn|gnk|2 + σ2

),∀k ∈ Sn (26)∑

m∈Snm6=k

Tr(WmHmk) ≤ ϕk, ϕk ≥ 0,∀k = 1, . . . ,K (27)

where W = [W1 . . .WK ] is a J×KJ matrix which consistsof all the beamforming matrices, and ϕ = [ϕ1, . . . , ϕK ]T isa K × 1 non-negative auxiliary vector which represents intra-user interference thresholds of all SU-RXs.

Given ϕ, the objective function becomes concave since it isjust a logarithmic function of an affine expression. In addition,the convexity of the constraint (26) can be proven as in [15].As a result, the problem P3 turns into a form of convexoptimization problem. It is similar to the standard form of SDP,but with a logarithmic rather than a linear objective function.There are many tools available in literature (e.g., CVX [31],a Matlab based modeling software package) to solve such aconvex optimization problem. Thus, we only need to determinethe optimal value for ϕ. Following the similar method in [14],we introduce the following iterative algorithm for P3.

(i) Initialization: Relax the intra-user interference constraint(27) by assigning a feasible non-negative large val-ue for ϕ. Then, solve the problem P3 to find outa feasible value of W, called W(0). Set the intra-user interference thresholds for each user as ϕ

(0)k =∑

m∈Snm6=k

Tr(W(0)m Hmk),∀k ∈ S and calculate the sum-

rate, R(W(0),ϕ(0)), based on (25). Define SU pairindex, k = 1, 1 ≤ k ≤ K and the iteration index,a = 1.

(ii) Update: Update ϕ(a) by ϕ(a) = ϕ(a−1) − (1 −δ)ϕ

(a−1)k Ik where δ is a fixed step size, 0 < δ < 1,

and Ik is the kth column of an K ×K identity matrix.(iii) Iteration results: Calculate the new R(W(a),ϕ(a)) by

using W(a).(iv) Check improvements: Calculate ∆R = R(W(a),ϕ(a))−

R(W(a−1),ϕ(a−1)). The nonnegativity of ∆R can beproved as in [15]. If ∆R is greater than a predefinedthreshold, let a = a+ 1 and repeat steps (ii) and (iii) till∆R below a pre-defined threshold. Then, set W(a) =W(a−1), R(W(a),ϕ(a)) = R(W(a−1),ϕ(a−1)) andupdate the intra-user interference for each user as ϕ(a)

k =∑m∈Snm6=k

Tr(W(a)m Hmk), ∀k ∈ S.

(v) Continue iterations and pick the next user: Set k = k+1and continue steps (ii) - (iv) for the newly selected user.

(vi) Termination: If k > K, stop the iterations.

At each iteration, the problem P3 needs to be solved. Sincemost of the convex optimization toolboxes (e.g., CVX) use theinterior-point algorithm [32] as a basic solution platform, con-sidering the worst-case scenario as in [30], the computationalcomplexity of a SDR problem P3 can be expressed as

O(max{ξ, κ}4κ(1/2)log(1/ε)) (28)

where κ and ξ describe the problem size (i.e., number ofPSD matrices) and the number of constraints involved in theoptimization problem P3, respectively. ε is the given accuracyof the solution defined by the solver. Let tT denote the totalnumber of iterations taken by the iterative algorithm to producea feasible solution for total K SU pairs. Then, the overallcomplexity to find beamforming and power vectors for a givenchannel allocation can be derived as

O(max{ξ, κ}4κ(1/2)log(1/ε))× tT (29)

Let W = [W1 . . .WK ] and ϕ = [ϕ1, . . . , ϕK ]T be thefeasible solutions of the problem P3. Then, W is optimal ifand only if the rank of each Wk,∀k ∈ S, is equal to one, i.e.,rank(Wk) = 1. If such condition is not satisfied, appropriaterank one approximation methods, e.g., eigen-decompositionmethod [30], can be deployed to get the final solution ofW. By using eigen-decomposition method, each beamformingmatrix Wk can be equivalently represented as

Wk =

J∑j=1

λkjckjc†kj (30)

where λkj and ckj denote the jth eigenvalue and the cor-responding eigenvector of the kth beamforming matrix Wk,respectively. If Wk is a rank one matrix, then there existsexactly one non-zero eigenvalue, say λkJ . As a result, using(30), the kth user beamforming matrix Wk can be written as

Wk = λkJckJc†kJ = (√λkJckJ)(

√λkJckJ)† (31)

= wkw†k

Thus, the beamforming vector for the SU-TXk is,

wk =√λkJckJ (32)

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B. Suboptimal channel allocation

In the previous section, we have determined the optimalpower and beamforming vectors for a known channel allo-cation. In order to find the optimal channel allocation, anexhaustive searching algorithm can be used, which needs tocompute beamforming vectors, power allocations and sum-rates for all possible channel allocations. With N PU channelsand K SU pairs, the searching space size of the exhaustivesearching algorithm is (N + 1)K , which increases exponen-tially with K. Obviously, this searching method is practicallyinfeasible. Hence, for practical applications, we propose twochannel allocation algorithms based on genetic algorithm (GA)[24] and simulated annealing (SA) [33], [34] algorithm to findsuboptimal channel allocations.

1) Genetic Algorithm (GA): GA is a searching algorithm,which can be applied to find out near optimal solution to anoptimization problem without the knowledge of the objectivefunction’s derivatives or any gradient related information. Thekey idea of GA is to first select a set of feasible values for thedecision variables and then design new solutions based on theprevious set to improve the objective function [35]. Differentfrom standard GA, in this thesis, we define a K×N matrix as achromosome instead of a single string chromosome as in [24],where the kth row and nth column entry of the chromosomeindicates whether the nth channel is allocated to the kth SU-TX or not. In fact, a chromosome describes one realization ofchannel allocation. An example of a chromosome for two PUchannels and four SU-TXs can be written as

Chromosome =

1 00 10 11 0

(33)

Define Φ as the search space, which includes all possiblechannel allocations, and has a size of (N + 1)K . Define thesize of each generation as ν. Let ρ (0 < ρ < ν) and Gmaxdenote the number of best chromosomes being selected andthe maximum number of generations, respectively. The detailsof GA is as follows.

Initially, ν chromosomes are randomly selected from Φ,called the initial generation, Φ(0). We define R(i)(W,ϕ), thesum-rate achieved for chromosome i, as the fitness functionfor GA. Then, the chromosomes in Φ(0) can be ordered withthe descending order of their sum-rates. From the sorted chro-mosome list (G(g)sorted), the first ρ chromosomes are selectedand put into the set G(g)best, while the last ρ chromosomes areremoved and inserted into the set G(g)worst at the gth generation.The set of remaining chromosomes, G(g)luckies, is named asluckies.

Next, a new generation of chromosomes is formed. Thenew generation consists of the set G(g)best and ν − ρ newchromosomes generated from the two sets G(g)best and G(g)luckies,through Children Generation Process (CGP). CGP consists oftwo major operations called Mutation and Cross-over. Thedetails are explained as follows.• At first, two chromosomes, P1 and P2, are randomly

selected from the two sets G(g)best and G(g)luckies, respectively.

Algorithm 1 : GA-based channel allocation algorithm1: Initialization: Given K,N,Φ, ν, ρ and Gmax2: Start : randomly pick ν channel realizations from Φ to

define the initial generation Φ(0), Φ(0) ⊂ Φ3: Fitness : find R(i)(W, ϕ) ∈ R(0) for each chromosome i

within the set Φ(0)

4: Set g = 0,5: while g < Gmax do6: if g = 0 then7: G(0) = Φ(0), the set of corresponding rates are R(0)

8: else9: Fitness : find R(g) ← R(i)(W, ϕ),∀chromosome i ∈

G(g)10: end if11: [R(g)

sorted,G(g)sorted]←sort(R(g),G(g),‘Descending’)

12: [R(g)best,G

(g)best]←select(R(g)

sorted,G(g)sorted,,‘Best’)

13: [R(g)worst,G

(g)worst]←select(R(g)

sorted,G(g)sorted,,‘Worst’)

14: [R(g)luckies]← (R(g)

best −R(g)worst)

15: [G(g)luckies]← (G(g)best − G(g)worst)

16: for c = 1 : (ν − ρ) do17: P1← select(G(g)best, 1,’Random’)18: P2← select(G(g)luckies, 1,’Random’)19: [TempCH1, T empCH2]← Crossover(P1, P2)20: [CH1, CH2]← Mutation(TempCH1, T empCH2)21: [G(g)child]← [CH1, CH2]22: end for23: G(g+1) ← {G(g)best + G(g)child}24: g ← g + 125: end while26: Output: Optimal channel allocation (or chromosome),

optimal beamforming matrices, W?

P1 and P2 are called parents.• Next, the cross-over operation is initiated between the

two selected parents to generate two new children,called TempCH1 and TempCH2. Here, we use theSingle Point Cross-over technique [36] as an example.Specifically, entries of a randomly selected column of thetwo parents will be swapped to perform the single pointcross-over. Note that swapping may produce infeasiblechromosomes such as multiple channel allocation to asingle user. In this case, except the swapped column,we keep the parent columns unchanged and let onlythe troublesome entries of the swapped column of bothchildren to be zero to ensure that channel allocationconstraint (13) is always satisfied.

• At the end, two randomly selected entries, xn1k , xn2k , ofan arbitrary selected kth row will be swapped, calledmutation. Note that, when selecting those two entries,either of them should have a value of one. The importanceof mutation is to avoid GA converging to a local optimalvalue.

The termination condition for GA will be activated whenthe number of generations produced is beyond the predefinedvalue Gmax. Eventually, the chromosome in the last genera-tion, which has the maximum fitness, will be the best solution.

7

Furthermore, the corresponding fitness value will be the nearoptimal sum-rate.

At each iteration, the complexity of GA results from threemajor operations, i.e., sorting, mutation and cross-over, whichintroduce complexity of O(ν log(ν)), O(K) and O(K), re-spectively. Beside the first generation, (ν − ρ) fitness valueshave to be determined at each generation. Thus, the overallcomplexity of GA can be computed as

Gmax ×

[O(nlog(n)) + d(ν − ρ)/2e(O(K))

+ (ν − ρ)× tT ×O(max{ξ, κ}4κ(1/2)log(1/ε))

](34)

The implementation steps of the GA is summarized inAlgorithm 1. We call the proposed beamforming, power, andchannel allocation with GA as BPCA-GA.

2) Simulated Annealing (SA)-based algorithm: From (34),the complexity of GA mainly depends on the values of ν andρ. Since these two parameters also decide the accuracy and theconvergence of GA, sufficiently large values of ν and ρ have tobe set, which causes high computational complexity for GA.In order to further reduce the complexity, a new allocationalgorithm based on SA [33] is introduced.

Let Φ = {X1, . . . ,XL} denote the feasible set of all chan-nel allocations available with K SU pairs and N PU channels,where a K×N matrix Xl, l = 1, . . . , L = (N+1)K , indicatesa certain channel allocation with binary entries and a row sum,i.e.,

∑Nn=1 x

nk ≤ 1. Then, the optimization problem for X can

be equivalently expressed as

X? = arg maxXl∈Φ

R(Xl) (35)

where R(Xl) indicates the sum-rate associated with Xl. TheSA-based algorithm uses neighborhood searching to determinea suboptimal solution. Specifically, the SA-based algorithmstarts with a control parameter and an initial channel allocationthat is used to generate new neighbor channel allocation.Then, the new channel allocation is clearly selected if itshows any performance improvement. Otherwise, it may stillbe accepted with a certain probability, which allows SA-based algorithm to escape from local optimal configurations.The cooling schedule manages the control parameter duringthe optimization process. The details of the algorithm is asfollows.

At the initial state, we set the iteration index l = 0 andrandomly pick a channel allocation, X0, from Φ. Using X0, anew neighbor channel allocation, i.e., X0 ∈ Φ, is formed byswapping randomly selected row of X0 with an entry in A,which denotes the set of all feasible combinations of channelallocation to a single user. Given X0 and X0, we compute thecorresponding sum-rates, which are denoted as R(X0) andR(X0), respectively. The acceptance rule at the lth iterationcan be formulated as

P{Accept Xl} =

1, if R(Xl) ≥ R(X0)

exp(

∆R

Tl

), if R(Xl) < R(X0),

(36)

Algorithm 2 : SA-based channel allocation algorithm1: Initialization: Given K,N,Φ, cooling-rate and Smax2: Set l = 0,3: Start : Initial channel allocation X0, X0 ⊂ Φ and computeR(X0)

4: while l < Smax do5: Generate new channel allocation, Xl from X0, Xl ⊂ Φ6: Calculate sum-rate, R(Xl)7: ∆R := R(Xl)−R(X0)8: if l = 0 then9: Compute T0

10: end if11: if ∆R ≥ 0 then12: X0 ← Xl and R(X0)← R(Xl)

13: else if exp(

∆R

Tl

)> random [0, 1] then

14: X0 ← Xl and R(X0)← R(Xl)15: end if16: Update Tl+1 = cooling-rate ∗ Tl17: l← l + 118: end while19: Output: suboptimal channel allocation and suboptimal

beamforming matrices, i.e., X0 and W?

where ∆R = R(Xl) − R(X0). It implies that a candidateneighbor channel allocation, i.e., Xl, is accepted with theprobability of one if its sum-rate is greater than that of the

current channel allocation, or exp(

∆R

Tl

)if the new neighbor

channel allocation has a smaller achievable sum-rate than thecurrent allocation. If accepted, we replace X0 by Xl, andupdate the value of the control parameter for the next iterationas Tl+1 = cooling-rate∗Tl, where cooling-rate takes a value in[0.50; 0.99] [37]. Note that T0 can be determined by followingthe similar way as in [38]. We increase l by one and repeatthe aforementioned process. When the maximum number ofallowable iterations is reached, the algorithm is terminated andthe resulting outputs give the suboptimal channel allocation,sum-rate and the beamforming vectors.

The SA-based algorithm has two major steps, i.e., thegeneration of a neighbor channel allocation and the deter-mination of sum-rate. A new neighbor channel allocationcan be generated from the current channel allocation witha complexity of O(N), and a sum-rate calculation processinvolves a complexity of O(max{ξ, κ}4κ(1/2)log(1/ε)) × tTas in (29). If Smax is the maximum number of iterations forconvergence, the complexity of the algorithm can be computedas

Smax × {O(N) +O(max{ξ, κ}4κ(1/2)log(1/ε))× tT } (37)

Compared with (34), the computational complexity of SA-based algorithm is much smaller than the GA. The SA-basedalgorithm is summarized as in Algorithm 2. In this paper, wecall the proposed beamforming, power, and channel allocationwith SA-based algorithm as BPCA-SA.

8

IV. SIMULATION RESULTS

In this section, the performance of both BPCA-GA andBPCA-SA algorithms is evaluated by using computer simu-lations.

A. Simulation environment

Consider a CRN with three PU-TX/PU-RX pairs (i.e.,N = 3). The locations of the PU-TXs are given by the x-yplane coordinates (300, 0), (−400, 0) and (0,−100), and theirassociated PU-RXs are situated at (600, 0), (−400, 300) and(0,−400), respectively, where all the distances are measuredin meters. There are six SU-TX/SU-RX pairs (i.e., K = 6)randomly located within a square area of 600m × 600m.For each SU-TX, its associated SU-RX is randomly locatedin a circle centered at the SU-TX with a radius of 100m.Each SU-TX with J = 3 antennas is oriented in a UCAconfiguration having a radius of λ =

c

fc, where the carrier

frequency fc = 900MHz and the speed of light c = 3 ×108m/s. The transmit power budget, Pmax = 30dBm andQn = 23dBm, ∀n ∈ P . Furthermore, each PU-RX has aninterference threshold, Inth,∀n ∈ P , of −40dBm and eachSU-RX’s minimum QoS threshold, Γk,∀k ∈ S, is 6dB. Thenoise power at any location is considered as −90dBm and thebandwidth of each channel, B, is set as 10MHz. The small-scale fading coefficients are modeled as a Circular SymmetricComplex Gaussian (CSCG) random variables with zero meanand unit variance. We consider the free space propagationmodel to simulate the path-loss, i.e., PL =

(λ

4πd

)2, where

d is the distance between the transmitter and the receiver.Normally, the generation size ν in GA should be neither toosmall nor too large. A small generation size tends to end upwith a local optimal value, while a large generation size leadsto a high computational complexity though it may be closerto the optimal value. Hence, in order to balance this tradeoff,we use the size of the searching space (i.e., Φ = (N+1)K) indetermining ν, and set ν as the ceiling integer of

√(N + 1)K .

Accordingly, we let ρ be the ceiling integer of√ν. The

stopping criteria for SA algorithm is based on the maximumnumber of iterations Smax. We define Smax as the number ofiterations after which a given minimum temperature level, i.e.,Tlast = 0.3×T0 can be reached. Thus, Smax can be calculatedbased on the equation Tlast = (cooling-rate)Smax × T0,where the cooling-rate is a constant of 0.99. Note that someparameters may vary according to evaluation scenarios.

B. Convergence of BPCA-GA and BPCA-SA algorithms

Fig. 2 shows the convergence of the two proposed algo-rithms, i.e., BPCA-GA and BPCA-SA. The optimal sum-rate, 42.1765 bits/Hz, is determined by adopting the exhaus-tive searching method. From the figure, we can see thatBPCA-GA approaches a close-to-optimal sum-rate value of41.9634 bits/Hz. BPCA-SA can find a suboptimal channelallocation with a sum-rate value of 41.0557 bits/Hz whichis only 2.16% worse than BPCA-GA. By considering thecomputational complexity, we can conclude that BPCA-SAis more feasible for practical applications. Note that, though

10 20 30 40 50 60 70 80 90 100 110 120 130 140 15020

25

30

35

40

45

Iterations (Generations)

Sum

−ra

te (

bits

/Hz)

Optimal sum−rateBPCA−GABPCA−SA

Fig. 2. Convergence of the BPCA-GA and BPCA-SA algorithms with K =4, N = 3, Pmax = 1W, J = 3.

TABLE IIAVERAGED COMPUTATION TIMES OF DIFFERENT SCHEMES.

Scenario BPCA-SA (s) BPCA-GA (s)K = 3, N = 2 293.65 688.04K = 3, N = 3 334.69 1518.50K = 3, N = 4 548.08 3660.75

BPCA-GA converges much faster than BPCA-SA in terms ofthe number of iterations, each iteration takes more time inBPCA-GA than BPCA-SA because of the high computationalcomplexity involved in GA. As a proof for such fact, Table IIillustrates the numerical values of the average computationtimes taken by the two proposed schemes with respect todifferent configurations (i.e., the number of SU-pairs (K) andchannels (N )). From this table, we can clearly observe thatthe computation time of BPCA-SA algorithm is always shorterthan that of the BPCA-GA, which proves that the BPCA-SAhas lower computational complexity than the BPCA-GA. Inaddition, such superiority becomes more obvious for largervalues of K and N .

C. The achievable sum-rate with increased number of SUpairs and PU channels

Fig. 3 illustrates a comparison of the achievable sum-rateby the BPCA-GA and BPCA-SA algorithms. From the figure,we can see that when the number of SU pairs is small, e.g.,K = 2, both algorithms show similar performance. This isbecause the search space is small in this case so that both GAand SA-based algorithms have an equal chance to obtain aclose-to-optimal value. However, as the number of SU pairsincreases, the BPCA-GA outperforms BPCA-SA.

Fig. 4 depicts the performance of BPCA-GA and BPCA-SAwith the number of PU channels. From the figure, a similarobservation as in Fig.3 can be obtained. As N increases,achievable sum-rate of both algorithms tend to increase due tothe increased degrees of freedom on channel allocation. Formost of the cases, BPCA-SA is lagged behind BPCA-GA witha small performance gap. By comparing Figs 3 and 4, we canfurther observe that the number of SU pairs has more effectson the performance than the number of PU channels.

9

2 3 4 5 60

10

20

30

40

50

60

Number of SU pairs (K)

Sum

rat

e (b

its/H

z)

BPCA−GABPCA−SA

Fig. 3. Optimal sum-rate with an increased number of SU pairs for N = 3PU channels, Pmax = 1W and J = 3.

1 2 3 4 510

15

20

25

30

35

40

Number of PU channels (N)

Sum

rat

e (b

its/H

z)

BPCA−GABPCA−SA

Fig. 4. Optimal sum-rate with an increased number of PU channels forK = 3 SU pairs, Pmax = 1W and J = 3.

D. Comparison with Zero-Forcing Beamforming (ZFBF)

We compare the proposed beamforming method with thetraditional ZFBF scenario where the beamforming vectorsfor each SU-TX is determined by confining the interferenceamong the SUs to become zero. We call the secondaryuser zero-forcing beamforming, power and channel allocationwith GA and SA-based algorithm as ZFBPCA-GAS andZFBPCA-SAS, respectively.

Fig. 5 shows that our proposed model outperforms the ZFBFscenario irrespective to the channel allocation algorithms.It is because the degrees of freedom available on channelallocation in our proposed system model are much greaterthan those in the ZFBF model. Moreover, after K > 3, thecurve for ZFBPCA-GAS increases gradually while that forZFBPCA-SAS decreases. It is because once K reached themaximum number of channels (i.e., N = 3 in our simulation),it is very difficult to find the beamforming vectors with ZFBFdue to the increased channel access requirement of the SUs.Furthermore, after K > 3, the ZFBPCA-SAS tends to gener-ate more infeasible channel allocation, and hence eventuallyconverges to a value which is away from the optimal. The

2 3 4 5 60

10

20

30

40

50

60

Number of SU pairs (K)

Sum

rat

e (b

ps/H

z)

BPCA−GABPCA−SAZFBPCA−GA

s

ZFBPCA−SAs

Fig. 5. Comparison of sum-rate between SU ZFBF and proposed systemmodels with an increased number of SU pairs for N = 3 PU channels,Pmax = 1W and J = 3.

2 3 410

15

20

25

30

35

40

45

Number of Antennas (J)

Sum

rat

e (b

ps/H

z)

BFCA−GABFCA−SAZFBFCA−GA

s

ZFBFCA−SAs

Fig. 6. Comparison of sum-rate between SU ZFBF and proposed systemmodels with an increased number of antennas with N = 2 PU channels,K = 4 SU pairs and Pmax = 1W.

performance of achievable sum-rate with respect to the numberof transmit antennas is shown in Fig. 6. The figure depicts thatthe sum-rates are increased with the number of antennas. It isbecause once J increases, the transmitter can direct the signalwith better intensity to the intended receiver and at the sametime further suppress the interference to the other users whoare using the same channel.

We further compare the proposed beamforming methodwith another ZFBF scenario with respect to the number ofSU pairs and the number of antennas, as shown in Figs. 7and 8, respectively. The beamforming vectors in this ZFBFscenario are determined by eliminating the interference ateach PU-Rx. We call the GA and SA-based algorithms as-sociated with primary user zero-forcing beamforming, powerand channel allocation as ZFBPCA-GAP and ZFBPCA-SAP,respectively. Compared to Figs. 5 and 6, similar observationscan be obtained. It is obvious that our proposed algorithm canoutperform the primary user ZFBF scenario regardless of the

10

2 3 4 5 60

10

20

30

40

50

60

Number of SU pairs (K)

Sum

rat

e (b

ps/H

z)

BPCA−GABPCA−SAZFBPCA−GA

p

ZFBPCA−SAp

Fig. 7. Comparison of sum-rate between PU ZFBF and proposed systemmodels with an increased number of SU pairs for N = 3 PU channels,Pmax = 1W and J = 3.

2 3 410

15

20

25

30

35

40

45

Number of Antennas (J)

Sum

rat

e (b

ps/H

z)

BFCA−GABFCA−SAZFBFCA−GA

p

ZFBFCA−SAp

Fig. 8. Comparison of sum-rate between PU ZFBF and proposed systemmodels with an increased number of antennas with N = 2 PU channels,K = 4 SU pairs and Pmax = 1W.

channel allocation algorithms.

E. The performance of BPCA-GA and BPCA-SA with differentpower budget

Fig. 9 illustrates the performance of sum-rates by varyingpower budget, Pmax. As we increase the power budget, theachievable sum-rate also increases as shown in the figure. Infact, the sum-rate performance gap between the BPCA-GAand BPCA-SA is small and keeps approximately unchangedas we increase Pmax. It is mainly because the convergenceproperties of both algorithms do not change with the power.With the increased power budget, the SUs are favored to havemore individual power allocation. Hence, they are rewardedwith more power as long as the interference constrains aresatisfied, which in return increases the sum-rate.

V. CONCLUSION

In this paper, a problem of joint beamforming, power andchannel allocation is considered for multi-user multi-channel

0.6 0.8 1 1.225

30

35

40

Power (W)

Sum

rat

e (b

its/H

z)

BPCA−GABPCA−SA

Fig. 9. Comparison of sum-rate variation for BPCA-GA and BPCA-SA withtotal power budget available for fixed N = 2 PU channels, K = 4 SU pairsand J = 3 antennas.

underlay cognitive radio networks. The problem is formulatedas a non-convex MINLP problem, which is NP-hard. In orderto reduce the computational complexity, we decouple the origi-nal problem into two sub-problems. At first, a feasible solutionfor beamforming vectors and power allocation is obtained fora known channel allocation by an iterative algorithm, whichuses the SDR approach with an auxiliary variable. After that,GA and SA-based algorithms have been applied to determinesuboptimal channel allocations. Simulation results show thatBPCA-GA can obtain close-to-optimal solution with a priceof high computation complexity. Whereas, BPCA-SA can sig-nificantly reduce the computational complexity with marginalperformance degradation compared to BPCA-GA. Moreover,beamforming with interference tolerance capability introducedby our system model can achieve better performance thantraditional ZFBF.

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Suren Dadallage received the B.Sc. degree in Elec-tronics and Telecommunication Engineering fromUniversity of Moratuwa, Sri Lanka, in 2010, andM.Sc. degree in Electrical and Computer Engineer-ing from University of Manitoba, Winnipeg, MB,Canada, in 2015. His research interests includebeamforming algorithms and radio resource manage-ment for cognitive radio networks.

Changyan Yi received the B.Sc. degree from GuilinUniversity of Electronic Technology, China, in 2012,and M.Sc. degree from University of Manitoba, Win-nipeg, MB, Canada, in 2014. He is currently workingtoward the Ph.D. degree in electrical and computerengineering, University of Manitoba. He was award-ed Edward R. Toporeck Graduate Fellowship in En-gineering in 2014, and University of Manitoba Grad-uate Fellowship (UMGF) in 2015-2018. His researchinterests include algorithmic game, optimization andqueueing theories for radio resource management,

prioritized scheduling and network economics in wireless communications.

Jun Cai (M’04, SM’14) received the B.Sc. andM.Sc. degrees from Xi’an Jiaotong University, X-i’an, China, in 1996 and 1999, respectively, and thePh.D. degree from the University of Waterloo, ON,Canada, in 2004, all in electrical engineering. FromJune 2004 to April 2006, he was with McMasterUniversity, Hamilton, ON, as a Natural Sciences andEngineering Research Council of Canada Postdoc-toral Fellow. Since July 2006, he has been with theDepartment of Electrical and Computer Engineering,University of Manitoba, Winnipeg, MB, Canada,

where he is currently an Associate Professor. His current research inter-ests include energy-efficient and green communications, dynamic spectrummanagement and cognitive radio, radio resource management in wirelesscommunications networks, and performance analysis. Dr. Cai served as theTechnical Program Committee Co-Chair for the IEEE Vehicular TechnologyConference 2012 Fall Wireless Applications and Services Track, the IEEEGlobal Communications Conference (Globecom) 2010 Wireless Communi-cations Symposium, and International Wireless Communications and MobileComputing (IWCMC) Conference 2008 General Symposium; the PublicityCo-Chair for IWCMC in 2010, 2011, 2013, and 2014; and the RegistrationChair for the First International Conference on Heterogeneous Networking forQuality, Reliability, Security and Robustness (QShine) in 2005. He also servedon the editorial board of the Journal of Computer Systems, Networks, andCommunications and as a Guest Editor of the special issue of the Associationfor Computing Machinery Mobile Networks and Applications. He receivedthe Best Paper Award from Chinacom in 2013, the Rh Award for outstandingcontributions to research in applied sciences in 2012 from the University ofManitoba, and the Outstanding Service Award from IEEE Globecom in 2010.