JOINT ROBUST BEAMFORMING IN MULTI-USERS COGNITIVE RADIO SYSTEM VIA SEMIDEFINITE PROGRAMMING

Betty Nagy1, Maha Elsabrouty2, Salwa Elramly1

1: Ain Shams University (bettynagygiurgius@yahoo.com, sramlye@netscape.net) 2: Egypt -Japan University of Science and Technology (maha2000_eg@yahoo.com)

ABSTRACT

In cognitive radio (CR) networks, unlicensed users known as secondary users (SUs) can share the same spectrum with the licensed users known as primary users (PUs) on the condition that the SUs interference must not exceed a certain level on PUs. Our target is to provide the SUs with a minimum acceptable quality-of-service (QoS), while keeping the interference to the PUs below a given threshold. However, it is not easy to know the channel state information on the links to PUs, thus uncertainty should be considered on PUs links.

In this paper, we present a study of the optimal transmit beamforming weights of the SUs network in the presence of uncertainty on PUs links that is modeled by two different methods with exploiting the effect of multiple antennas at the PUs receivers. Moreover the two models of the problem are solved using semidefinite programming (SDP) optimization technique and they are both compared to other models using computer simulations.

Keywords-beamforming; cognitiveradio ;semidefinite programming; multi-users.

I. INTRODUCTION

CR networks allow SUs to share the same frequency spectrum of the licensed PUs by one of three different cognitive behaviors [1]: spectrum interweave, underlay and overlay. The first one avoids interference as SUs are allowed to operate only when PUs are not active. Underlay scenario controls interference by allowing the SUs all time in all frequencies but under the condition of not exceeding a certain level of interference on PUs. The third case, namely overlay, mitigates interference by allowing the SUs to know more information about the PU links to decrease interference. Our study is concerned with the second cognitive behavior where the SUs share the same spectrum of the PUs but must not exceed a certain level of interference on PU links i.e. the SUs transmit in such a fashion that they appear to be noise under the primary signals.

One of the methods of controlling interference is beamforming where the multiple antennas are given different gains (beamforming weights) to steer the signal towards a certain desired direction. Thus the weights of SUs’ antennas are designed in such a way to achieve two main targets: 1) limiting the interference on the PUs to be below a certain

threshold, 2) assuring the signal to interference and noise ratio (SINR) of the SUs to be above a certain level.

In order to achieve the first target, the SUs need to know the CSI of the PUs links .There is no direct communication between the secondary and primary networks so the CSI at the SUs has some level of uncertainty or may not be known at all. Consequently, to ensure robustness of the SUs’ beamforming, in the following sections we model the two levels of uncertainty and solve the beamforming problem in each case then we compare the results using computer simulations.

The beamforming problem has been recently studied in [2] - [5]. A solution using SDP has been formulated in [2] with the conditions when this SDP will be acceptable. However, cognitive radio conditions have not been considered i.e. the solution proposed was for an ordinary network. A MIMO multiple users’ cognitive radio system has been studied in [3]. However, the uncertainty in CSI has not been considered. A MISO multiple users’ cognitive radio system has been studied in [4] and the CSI imperfections has been considered. However, the channels has been assumed to be partially known i.e. known with some error and in cognitive radio network with no direct communication between the SUs and PUs we may not be able to know the channel completely. Furthermore, uncertainty has been added on hybrid channels only i.e. channels between SUs and PUs and no study of the effect of uncertainty on PUs links.

In this paper, we study the problem of optimal secondary link beamforming. Specifically, we aim to maximize the total throughput of SUs under the constraint that the interference to PU receivers is below a certain threshold. In contrast to the previous work in [2], we consider the beamforming problem in cognitive radio network and unlike [3] we study the effect of uncertainty by two different models. Furthermore, multiple antennas has been added to the PUs links unlike [4], and the uncertainty has been to the PUs links which are definitely the most difficult to be known by the SUs network.

We have adopted the uncertainty model of the channel that is presented in [5], which studied robust beamforming in the case of MIMO with only one SU in a CR network. It is worth noting that our case study is MISO multi-SUs which is completely different beamforming problem formulation in which the multi-SUs introduce the problem of interference on each other.

The rest of this paper is organized as follows. Section 2 presents the system model; section 3 presents the robust CR beamforming problem for MISO multi-SUs cognitive radio system. Section 4 presents the numerical results for cognitive radio system, and finally, the paper is concluded in section 5.

II. SYSTEM MODEL

II.1 Setup

Notations: matrices and vectors are respectively denoted by boldface upper case symbols and boldface lower case symbols. AH is the Hermitian transpose of matrix A, A-1 is the inverse of matrix A, tr(A) is the trace of matrix A, [ ] i is the ith entry of a vector.

In this paper we consider a CR network in which S secondary users share the spectrum with P primary users. Let NS or NP denote the number of secondary or primary transmit antennas, respectively. Let M denote the number of primary receive antennas. We assume a Rayleigh fading, so that the entries of the channel matrices are independently and identically distributed complex Gaussian random variables with zero mean and unit variance. In our study we assume that Ns=Np=N, which is merely a simplification of the notation that has no consequence on the generality of the derived algorithm. We use Hp,s ϵ C

MxN to denote the channel from the pth primary transmitter to pth primary receiver and from the sth secondary transmitter to pth primary receiver, hs,p

ϵ C1xN, hs,j ϵ C1xN and hs,s ϵ C

1xN to denote the channel from the pth primary transmitter to sth secondary receiver, from the j th secondary transmitter to sth secondary receiver and from the sth secondary transmitter to sth secondary receiver.

Let ts denote the Nx1 beamforming vector at the sth secondary transmitter. Likewise, let tp and rp be the Nx1 and Mx1 beamforming vectors at the pth primary transmitter and receiver, respectively.

The received signal by the sth SU is:

y��t� = ��,�[x��t� �] + � ��,�[x��t� �]�������

+ n��t� (1)

����� is the noise of the sth SU that is assumed to be a circular complex Gaussian noise with zero mean and σ

2s

variance and ����� ��� ����� are the signals transmitted to the sth SU, jth SU and pth PU respectively.

The received SINR at the sth SU can be expressed as in [6] as follows:

SINR� = $��,� �$%∑ $��,� �$% + σ(�)��*���

(2)

Assuming that +[|��|(] = 1 and + .$�/$(0 = 1.

II.2 Objective and Constraints

In this paper, we aim to find the optimal beamforming vectors 1� of the SUs with minimal total transmitted power of all SUs in the network meanwhile keeping the 2345� per SU above a certain threshold to maintain QoS and keeping the interference on each PU in the network below a tolerable threshold. This can be expressed mathematically by the following optimization problem:

min89 �: �:%���� (3)

s.t. (means subject to): $��,� �$%∑ $��,� �$% + σ(��������

≥ γ� ∀s = 1,2, … S (4)

�$ABB,� �$(���� ≤ ϵE ∀p = 1,2, … P (5)

where γ� is the minimal acceptable SINR of the sth SU, HB is the upper limit on the interference power caused by the secondary network base station at the pth PU, and : : denotes the Frobenius norm of a matrix or Euclidean norm of a vector. Notice that:

$��,� �$% = ���,���,� � = AI��,���,� � �J= AI��,���,�K�J (6)

Where K� ( � � ) is a rank one Hermitian positive semidefinite matrix. Thus, the problem can be turned convex by relaxing the rank constraint; rank (K�) = 1∀� = *, %, … ) as in [8], we obtain the following SDP relaxation

LM�89 � tr�K������ (7)

s.t.:

γ�tr�� ��,�O��,�K��)��*���

− trI��,�O��,�K�J + γ��σ(�� ≤ 0 ∀s = 1,2, … S

(8)

tr�� E,�K�E,�OAE AEO ��

��� ≤ ϵE ∀p = 1,2, … P (9)

The solution of this SDP problem can be done using interior point methods with the aid of optimization toolboxes [9] or [10] (note that these toolboxes are well known solvers for beamforming problems used in [3], [4] and [5].

III. ROBUST CR BEAMFORMING PROBLEM

III.1 Robust Model 1

As we mentioned earlier in section 1, PUs will not provide their beamforming weights to the SUs as they are not aware of their presence. Accordingly, it is expected that the PUs’ receiver beamforming weights will have some error; so in our first model we will assume some added error term ∆/ in equation (9) that represents imperfections in the covariance matrix �S/S/T� of the receive beamforming weights vector where ∆/ is the error matrix with an upper bound such thatU∆/U ≤ V/.

max � trIE,�K�E,�O�AE AEO + ∆E� J���� ≤ ϵE ∀p

= 1,2, … P U∆/U ≤ V/ �AB AB + ∆B� ≥ X (10)

Equation (10) can be treated as an optimization problem as in [4], the LaGrange dual function is:

gIαE, μEJ= inf]^�−tr{�� E,�K�E,�O�

��� ��AE AEO+ ∆E�} + αE aU∆EU( − βE(c− tr d�AE AEO + ∆E�μE e�

(11)

Differentiating (11) with respect to ∆f and equating to

zero, we get the optimal∆E= �E,�K�E,�O + μE�/2αE.

By substituting with ∆/ then the problem turns to:

gIαE, μEJ = −tr{�� E,�K�E,�O����+ μE��AE AEO�}

− U�E,�K�E,�O + μE�U(4αE −αEβE(

(12)

Equation (12) can be solved by solving its dual function which in turn is solved by differentiating with respect to i/and equating to zero, we get the optimal i/ as follows:

αE = j� B,�K�B,����� + μEj /2βE

The resulting problem turns to:

gIμEJ =

−tr{�� E,�K�E,�O + μE�

��� ��AE AEO�}

−βE j�� E,�K�E,�O���� + μE�j

(13)

As Ts is positive Semidefinite then according to the congruence theorem ∑ k/,�l�k/,�k ≥ 0m��� and n/ ≥ 0 thus a trivial solution n/ = 0 solves equation (13), more details can be found in [4].Thus constraint (9) turns to:

tr�� E,�K�E,����� AE AEO� + βE o� E,�K�E,��

��� o ≤ ϵE ∀p = 1,2, … P

Our optimization beamforming problem (7-9) after adding the imperfections in constraint (9):

min89 � tr�K������ (14)

s.t.:

γ�tr�� ��,�O��,�K���������

−trI��,�O��,�K�J + γ��σ(�� ≤ 0 ∀s = 1,2, … S

(15)

tr�� E,�K�E,����� AE AEO�

+βE o� E,�K�E,����� o ≤ ϵE ∀p = 1,2, … P

(16)

K� ≥ 0, rank�K�� = 1 ∀s = 1,2, … S

Using SDP the problem can be turned convex by relaxing the non-convex rank constraint and then solved using optimization toolbox in [10].

III.2 Robust Model 2

Model 2 studies the second case of uncertainty where the

SUs don’t have the luxury of knowing the PUs’ receive beamforming weights. Thus constraint (5) turned probabilistic as follows:

Pr q�$AEE,� �$(���� ≤ ϵEr ≥ 1 − δE ∀p = 1,2, … P

(17)

As t/ is unknown, we will assume that it is a normalized

complex Gaussian vector (t/ = u:u:) proved in corollary 1 in

[5]. Thus t/ is rotationally invariant then we can assume E,� � = vE,� = [UvE,�U, 0, 0, . . ]x .

Then, we have

Pr y�$AEOvE,�$(���� ≤ ϵEz =

Pr y�UvE,�U(. $[AE]� $(���� ≤ ϵEz =

Pr y��UvE,�U( − ϵE�|[{]� |(���� ≤ �|[{]| |(}

|�( z

(18)

Following the same steps in [5] but with considering Multiple SUs i.e. summing all the interference terms from all SUs on a PU instead of dealing with one interference term of SU only , we get from (18) that the probabilistic constraint (17) is equivalent to:

�UvE,�U(���� =

�UE,� �U( = � �OE,�OE,� � ����

����

= � trIE,�OE,� � �OJ����

= � trIE,�OE,� � �OJ���� ≤ ϵE1 − δE}

(19)

And our optimization beamforming problem (3-5) after adding the imperfections in constraint (5) then using SDP relaxation and neglecting the rank constraint like model 1, the problem is reformulated to:

min89 � tr�K������ (20)

s.t.:

γ�tr�� ��,���,�K���������

− trI��,���,�K�J+ γ��σ(�� ≤ 0 ∀s= 1,2, … S

(21)

tr�� E,�OE,�K� � ≤ ϵE1 − δE}�

���

(22)

Problem (20-22) can be solved using optimization toolbox in [10].

IV. NUMERICAL RESULTS

MATLAB software is used to simulate the proposed robust techniques in model 1 and 2 using the toolbox in [10]. The simulated system considers P primary users (P=3) and S secondary users (S=4).The transmit antennas Ns=Np=6 and the primary receive antennas M =6. The channels are considered to be Rayleigh with identity variance ~��0, 3�. The algorithms developed for model 1 and 2 are compared to the robust models with MISO PUs receiver weights in [4] and [7]. Moreover, the results in [4] shows that the robust model in [4].outperforms the two models in [7] that is why we will limit our comparison to the robust model in [4]. Each point in the figures is an average of 500 independent simulation runs.

For Model 1, we assume βp= β=0.25, ϵ p=ϵ =5 dB and γs=γ. For Model 2, we assume�/ = � = 1% . Figure 1 indicates the percentage of feasible solutions (the percent of optimal solutions that could be found after running 500 independent simulations each with a different channels’ response) versus the target SINR and feasible means that an optimal solution can be found with all the constraints satisfied, i.e. the optimization problem can find optimal SUs’ beamforming weights. It is clear from figure 1 that as the target SINR (γ) increases the percentage of feasible solutions decreases. We simulated 5 different cases; 1) Model 1 in which uncertainty is added to the PUs’ receiver weights without considering the PUs’ interference in the sth SU SINR, 2) Model 2 in which the PUs’ receiver weights are random variable without considering the PUs’ interference in the sth SU SINR, 3) Model 3 presented in [4] in which uncertainty is added to the channel response. From the figure, it is clear that proposed model 1 in this paper outperforms model 3

presented in [4] as the target SINR (γ) increases as the multiple antenna at the primary receiver suppresses the interference caused by SUs. Model 2 is the least conservative with the assumption that t/ is unknown completely, that’s why model 1 outperforms model 2 and yet we can find feasible solutions to our beamforming problem.

Fig. 1: The Percentage of feasible solutions versus target SINR

Figure 2 shows the total transmitted power by SUs in the system, and we mean by the total power the objective function (equation (3)), as we mentioned earlier the problem targets optimal beamforming weights that minimizes the total power of all the SUs with minimum acceptable SINR per SU and limited interference on PUs.

Fig. 2: Total transmitted power by 4 SUs versus the target SINR

Figure 2 shows the total transmitted power using the optimal beamforming weights (∑s=1,…S||ts-optimal||

2) in the 3 models stated above, as the total transmitted power decreases the beamforming weights are considered to be more optimal. We consider only the cases which yields feasible solutions in the 3 models, the total power transmitted by the SUs in model 1 and 2 is less than that of model 3 which means that the proposed model 1 and 2 outperforms model 3 presented in [4] in terms of transmitted power.

Although the total transmitted power in the 3 models is close to each other, model 1 and 2 include uncertainty in the beamforming vector of the PUs’ receivers; the term which is the most likely to be unknown as it depends on the channel between the PUs’ transmitters and receivers while model 3 include uncertainty but in the channel between the SUs’ transmitters and PUs’ receivers which is less likely to be unknown.

V. CONCLUSION AND FUTURE WORK In this paper we developed a model for the MISO-

Multiple secondary user scenario operating in underlay cognitive radio setup along with multiple primary users through utilizing beamforming. The model is studied with the realistic effect of uncertainty in the receive beamforming vector of the PUs. It is the effect of adding uncertainty by two different methods on multiple SUs in a CR network with MIMO PUs. Model 1 considered adding error on the beamforming weights of the PUs’ receivers while Model 2 considered those beamforming weights are completely unknown. We can see that multiple antennas at the PUs’ receivers improve the optimal solution and make it more feasible.

Studying the MIMO SUs will be more challenging case as the problem will not be convex anymore as the SINR constraint (equation 4) will be non-convex so, we will consider this case in the future.

ACKNOWLEDGMENT

Thanks to Egyptian National Telecom Regulatory Authority (NTRA) for funding our project "Enhancement Proposals for DVB-T2 Systems and Cognitive Radio Networks Sharing the Same Frequency Band".

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