Novel Interference Alignment in Multi-Secondary Users Cognitive Radio System

Bassant Abdelhamid1, Maha Elsabrouty1, Salwa Elramly2

Electronics and Communications Engineering1Egypt-Japan University of Science and Technology

New Borg El Arab, Egypt2Ain Shams University, Cairo, EgyptEmail: bassant.ahmed@ejust.edu.eg,

maha2000 eg@yahoo.com, sramlye@netscape.net

Abstract—New interference alignment (IA) design for sec-ondary users’ transceivers in multi-secondary users cognitiveradio system is presented. The proposed scheme allows oppor-tunistic transmitters (secondary users) equipped with multipleantennas (MIMO) to use the same frequency band alreadyoccupied by the pre-existing primary user. This can be doneby aligning and cancelling the interference on the primaryuser when local channel state information (CSI) is known. Anovel closed form solution for precoding, postcoding, powerallocation designed for the secondary users is derived. Theproposed solution eliminates the interference of secondaryusers on primary user while whitening the interference causedby the primary user and other secondary users on eachsecondary user in the system. Numerical results prove theeffectiveness of our novel IA scheme compared to other iterativemethods that are used for the same purpose.

Keywords-interference alignment; cognitive radio; MIMOinterference channel; multi-users.

I. INTRODUCTION

Interference management is an important key in any multi-user system. Interference decreases the rate of users sharingthe same spectrum. One of the most promising multi-usersystems is cognitive radio. Cognitive radio users can begenerally classified into two types. The primary user (PU)who has the license to access the spectrum, and the sec-ondary user (SU) who is allowed to use the spectrum undercondition that there is no or limited generated interferenceat the primary user receiver.

Interference Alignment (IA) is one of the most effectivecandidates to solve interference problem in multiuser sys-tems that proves its effectiveness in the case of underlaycognitive radio [1]. IA is basically a combination of linearprecoding at transmitters and interference suppression atthe receiver with the basic goal of achieving optimal ormaximum multiplexing gain that can be achieved in suchinterference network.

Interference Alignment (IA) is considered as a new inter-ference mitigating technique [2]. It basically depends onaligning the signal received by non-intended transmitters oneigen modes that are orthogonal to those used by other users.

Opportunistic interference alignment (OIA) is used to allowopportunistic transmitters (secondary users) to opportunisti-cally use the same channel of a pre-existing primary linkwithout generating any interference. The primary (licensed)transmitter maximizes its rate by water filling over thesingular values of its channel matrix. Frequently, many eigenmodes are left unused by the primary user due to PUpower limitations. Hence, the secondary user can transmitat a significant rate without causing any interference to theprimary user by aligning its signal along the free eigenmodes of the channel [3].

In this paper, we consider the case of multiple secondaryusers opportunistically exploiting the same frequency bandutilized by a licensed user. This work can be consideredan extension of that in [4] to the case when there aremultiple cognitive secondary transmitters. We develop IA-based cognitive scheme that can exploit the free spatialdimensions left by the primary user. In this scheme, theprecoding matrices of the secondary users are designed suchthat no interference is generated at the primary receiver. Fur-thermore, each secondary receiver whitens any interferencefrom the primary transmission and from the other secondaryusers. Moreover, this paper modifies the postcoding matrixin [4] at each secondary user’s receiver such that it allowsthe receiver to recover its transmitted data which is one ofthe basic goals in any receiver. Furthermore, our methodis a new superior non iterative method which can be usedto support higher rates than the iterative methods such as[3]. Simulation results show the improved performance ofsecondary users’ rates of our method compared with theexisting one.

The rest of this paper is organized as follows. SectionII presents the cognitive radio system model, section IIIpresents the related works to interference alignment method.Sections IV and V are illustrating the primary user andmultiple secondary users transceivers designs, respectively.Section VI presents the numerical results for cognitive radiosystem, and finally, the paper is concluded in section VII.

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II. SYSTEM MODEL

Notations: matrices and vectors are respectively denotedby boldface upper case symbols and boldface lower casesymbols. An N × K matrix with ones on its main diagonaland zeros on its off-diagonal entries is denoted by IN×K,while the identity matrix of size N × N is simply denotedby IN . An N × K matrix with zeros in all its entries (nullmatrix) is denoted by 0N×K . Matrices AT and AH are thetranspose and Hermitian transpose of matrix A, respectively.The determinant of matrix A is denoted by |A|. Tr(A) andN (A) are the trace and the null space of matrix A. (q)+ isthe max (q,0) where q is a constant. A−1 is the inverse ofmatrix A. E[.] is the expectation.

Consider a (K+1) user MIMO interference channel in acognitive radio system, consisting of one primary user andthe K secondary users as shown in figure (1).

Figure 1. System Model

Let di be the degree of freedom (DoF) for user i or thenumber of symbols that can be transmitted simultaneouslyby user i. Mi and Ni are the number of antennas for user i atthe transmitter and receiver side, respectively. Ti and Ri arean Mi × di precoding and an di × Ni postcoding matricesfor user i at the transmitter and receiver sides respectively.Hij is an Ni × Mj channel matrix between transmitter j andreceiver i.

The received vector can be written as r0...

rK

=

H00 · · · H0K

.... . .

...HK0 · · · HKK

T0x0

...TK xK

+

n0

...nK

(1)

where xi is di × 1 vector contains the symbols transmitted

simultaneously by user i with E[xixHi ] = Pi where Pi repre-sents an di × di power allocation matrix for the transmittedsymbols. ni is an Ni × 1 vector which represents theadditive white Gaussian noise (AWGN) at receiver i withzero mean and variance σ2

i .It is assumed that the secondary users have full knowledge

about all the channels in the system similar to [2] [3] [4].

III. RELATED WORK

Recently, interference alignment has been proposed toachieve optimal degrees of freedom for interference channelsin [5]. Interference alignment for X MIMO channel is firstproposed in [6], a bound on M × N MIMO for K multi-users system is proposed. In cognitive radio specific scenario[7] proposed X channel interference alignment system. Thismodel was extended in [3] to multi-secondary users case.However, the extension is iterative and takes several cycles tosettle on the appropriate precoding and postcoding matricesfor the secondary users. In our work, we extend the workin [4] to develop a non iterative multi secondary usersinterference alignment technique. The following two sec-tions present the transceivers design of the primary user andsecondary users using the concept of interference alignment.

IV. PRIMARY USER SYSTEM DESIGN

Primary user is the user who has the right to use thespectrum whenever required and it is oblivious about theexistence of secondary users. This user designs its precodingand postcoding matrices by knowing only its own channelwith the goal of maximizing its own rate.

A. Precoding and postcoding matrices

The goal in designing these matrices is to diagonalize theprimary user’s channel matrix, this can be achieved by usingsingular value decomposition (SVD) [8] where

H00 = UH00ΛH00VHH00(2)

where UH00 , VH00 are N0 × N0 and M0 × M0 unitarymatrices, respectively, and ΛH00 is a N0 × M0 diagonalmatrix with main diagonal {λH00,1,· · · ,λH00,min(N0,M0)}and zeros on its off diagonal.To achieve the diagonalization criteria, the (M0 ×M0) pre-coding and (N0×N0) postcoding matrices will be designedto be

T0 = VH00; R0 = UHH00

(3)

These matrices will be used as initial selection for theprecoding and postcoding. They will be modified in the nextsubsection after knowing the PU degrees of freedom (d0) tosatisfy the system model and dimensions in figure (1).

B. Power allocation matrix

This matrix must be designed to maximize the primaryuser rate and at the same time achieve the power constraint

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for the primary link. Then the optimization problem for theprimary user can be written as

maxP0

log2 |IN0+

1

σ20

ΛH00P0ΛHH00

|

s.t. Tr{T0P0TH0 } ≤ P0,max (4)

where P0 is (M0×M0) power allocation matrix and P0,max

is the maximum power that can be supported by PU trans-mitter.This optimization problem in equation (4) is a well knownproblem that can be solved using the water filling method[9]. The resultant power allocation matrix P0 = diag{P0,1,· · · , P0,M0} can be written with respect to each elementP0,n, n = 1, · · · , M0 as:

P0,n = (µ0 −σ20

λHH00H00,n

)+ (5)

where λHH00H00,n is the element n on the diagonal of matrix

ΛHH00H00

= ΛHH00ΛH00

and µ0 is a constant which representsthe water level that saturates the power constraint.According to equation (5), it is shown that the primary usercan leave some of its eigen modes unused due to powerconstraint, i.e.; P0,n= 0 for some values of n. The numberof diagonal elements in P0 which equals to zero are (M0-d0).The primary user’s Degrees of Freedom (d0) varies between1 and min(M0, N0). The reason is that the maximum degreesof freedom at any MIMO system with M0 transmit antennasand N0 receive antennas equals to min(M0, N0) [10].

As a result, the precoding and postcoding matrices, asthey are unitary matrices, can be written as

T0 = VH00; R0 = UHH00

(6)

where VH00 and UH00 are the d0 columns of VH00 and UH00 ,respectively. In addition, the power allocation matrix P0 isthe upper left d0 × d0 submatrix of P0.

V. IA-BASED MULTIPLE SECONDARY USERS DESIGN

These users have to design their own matrices withoutaffecting the primary user’s performance and at the sametime maximizing their own rates.

A. Precoding matrix

This matrix is designed to eliminate the interference fromthe secondary users on the primary user. In other words, theprimary user’s rate is unchangable in both cases (with andwithout the existence of the secondary users) i.e.;

log2 |Id0 +1

σ20

ΛH00P0ΛH

H00| = log2 |Id0 + Q−10 ΛH00P0Λ

H

H00|

(7)where ΛH00

is the upper left d0×d0 submatrix of ΛH00and

Q0 is the covariance matrix of the co-channel interference

plus noise signal at the primary receiver and can be writtenas

Q0 = σ20Id0 +

K∑j=1

R0H0jTjPjTHj HH0jR

H0 (8)

One of the solutions to achieve equation (7) is nulling theinterference done by each secondary user on the primaryuser separately, this means that

R0H0iTiPiTHi HH0iR

H0 = 0d0×d0 , ∀i ∈ {1, ...,K} (9)

Interference alignment can be used to achieve the aboveequation, where the precoding matrix is designed to beorthogonal only to the used eigen modes by the primaryuser

Ti = N{R0H0i} ,∀i ∈ {1, ...,K} (10)

It is worth noting that zero forcing beamforming (ZFBF)can also be used to achieve equation (7). However, ZFBFdesigns the precoding matrix such that it orthogonalizes theSU space to the all possible eigen modes that can be usedby the primary user, i.e.

Ti = N{R0H0i} ,∀i ∈ {1, ...,K} (11)

Comparing the dimensions of R0 and R0, It is obvious thatinterference alignment can produce higher secondary usersrates than zero forcing beamforming, therefore, this paperconcentrates on interference alignment.

B. Postcoding matrix

This matrix should be designed to maximize the rate of thesecondary user and to recover the transmitted data streams.The design will be done over two stages. The first stageis whitening the interference at secondary users receiver,similar to [4]. The second stage estimates transmitted datastreams from an additive white Gaussian noise (AWGN).As a result, the postcoding block at the secondary user’sreceiver can be decomposed into two blocks. The first blockis a whitening filter (Fi) and the other block is the estimatorblock (Ei) as shown in figure (2). The postcoding matrixcan be written as

Ri = EiFi (12)

Figure 2. Postcoding block decomposition.

For designing the whitening filter which maximizes thesecondary user’s rate, consider the ith secondary user inter-

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ference covariance matrix which can be written as

Qi = σ2i INi

+K∑

j=0,j 6=i

HijTjPjTHj HHij (13)

and the secondary user’s rate can be written as

RSUi = log2 |INi+RiHiiTiPiTHi HHiiRHi (RiQiR

Hi )−1|

≤ log2 |INi + Q−1/2i HiiTiPiTHi HHiiQ−1/2i | (14)

The equality can be achieved if the postcoding matrix equalsto

Fi = Q−1/2i (15)

Estimation theory will be used for designing the esti-mation block. From estimation theory [11], if the receivedsignal can be written as a linear model with an AWGN thenthe estimated data can be extracted as follows

x̂i = (H̃H

H̃)−1H̃H

yi (16)

where yi = Firi is a signal vector at the output of whiteningfilter, Wi = Fini is the AWGN, H̃ = FiHiiTi. and

Ei = (H̃H

H̃)−1H̃H

(17)

A problem that will appear while designing the postcodingmatrix Ri, is that the whitening filter at user i (Fi) dependson covariance matrix Qi of this user which in turn is afunction of the power allocation of users (j, j 6= i) denotedby Pj . These power allocation matrices are not availableso we propose that the secondary user assumes a uniformpower allocation matrix for other secondary users i.e.

Pj = Idj ; ∀j 6= i (18)

This proposed method in designing is applicable and doesnot require any iterations among the secondary users, so itis called non iterative method.

C. Power allocation matrix

Secondary user power allocation matrix should be de-signed according to the following optimization problem

maxPi

log2 |Idi + RiHiiTiPiTHi HHiiRHi (RiQiR

Hi )−1|

s.t. Tr{TiPiTHi } ≤ Pi,max (19)

where Pi,max is the maximum power that can be supportedby any secondary user i transmitter. Substituting Ri = EiFiin equation (19) will lead to a similar solution of thefollowing optimization problem:

maxPi

log2 |INi + Q−1/2i HiiTiPiTHi HHiiQ−1/2i |

s.t. Tr{TiPiTHi } ≤ Pi,max (20)

It is worth noting that minimizing the interference atthe PU receiver is not included as an explicit constraint

in equation (20). This is because the precoding design isbased on nulling this interference completely. Let Ki =

Q−1/2i HiiTi = UKiΛKi

VHKi, where UKi

, VKiare Ni×Ni

and di×di unitary matrices, respectively. ΛKiis an Ni×di

diagonal matrix with main diagonal {λKi,1,· · · ,λKi,di} andzeros on its off diagonal. Then the optimal power allocation(OPA) matrix can be written as

Pi,OPA = VKi P̃iVHKi

(21)

where P̃i = diag{P̃i,1, · · · , P̃i,di}, P̃i,n = (µi − 1λKH

iKi,n

)+

considering λKHi Ki,n is the element n on the diagonal

of matrix ΛKHi Ki

= ΛHKiΛKi

and µi is a constant whichrepresents the water level that saturates the power constraint.

VI. NUMERICAL RESULTS

MATLAB software is used to simulate the proposed inter-ference alignment system. The simulated system considersone primary user and K secondary users (K=3). The channelsare considered to be Rayleigh i.e. random matrix withindependent identically distributed complex Gaussian circu-larly symmetric entries with zero mean and unity variance[CN (0, I)]. The proposed IA algorithm referred to “IA-OPA,non iterative Method” is compared to the iterative methodillustrated in [3] which is referred to “IA-OPA, iterativeMethod”.In the simulation, we consider the same Signal to NoiseRatio (SNR) for the primary and secondary users, i.e.;SNR = SNR0 = SNRi for all i = 1:K. For the ease ofpresentation, we consider that both primary and secondarydevices are equipped with the same number of antennas atall transmitters (Mt = M0 = Mi for all i = 1: K). Similarly,the number of antennas at all receivers are equal (Nr = N0

= Ni for all i = 1: K). The simulation presents two cases(Mt = Nr = 3), and (Mt = Nr = 6).

Figures (3) and (4) present the average rates of theprimary and secondary users (in bits per seconds) versusSNR using iterative [3] and non-iterative (proposed) methodswhen designing the precoding and postcoding matrices. It isobserved that our proposed (non-iterative) method alwayshas higher secondary users’ rates than the iterative methodin [3]. The reason for this is that the iterative method limitsthe degrees of freedom that the secondary user can use, whilein our method, the secondary user can use all the degreesof freedom left by the primary user. Also, it is shown thatthe secondary users have higher rates at intermediate SNRthan low and high SNR. At low SNR, the SU suffers fromhigh noise which does not allow the users to transmit withhigh rates. On the other hand, at high SNR, the PU occupiesall eigen modes in the space therefore the secondary usershave to decrease their rates to achieve the cognitive radiocondition (does not affect the PU performance).

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Figure 3. Average transmission rate for PU and SU links using iterativeand non-iterative (proposed) methods, Mt = Nr = 3.

Figure 4. Average transmission rate for PU and SU links using iterativeand non-iterative (proposed) methods, Mt = Nr = 6.

VII. CONCLUSION & FUTURE WORK

Interference Alignment is one of the new interferencemitigating techniques and is considered the most effectivecandidate until now. It can be used in multi-user system andachieve a good performance. A new design for secondaryuser’s link is proposed and proves its superiority over theiterative method from the secondary users rates perspective.As a future work, we suggest extending our transceiversdesign which assumes a cooperation between secondaryusers.

ACKNOWLEDGMENT

The authors would like to thank Egyptian National Tele-com Regulatory Authority (NTRA) for funding our project“Enhancement Proposals for DVB-T2 Systems and Cogni-tive Radio Networks Sharing the Same Frequency Band”.

REFERENCES

[1] A. Goldsmith, S.A. Jafar, I. Maric, and S. Srinivasa, “Break-ing spectrum gridlock with cognitive radios: An informationtheoretic perspective,” Proceedings of the IEEE, vol. 97, no.5, pp. 894–914, 2009.

[2] V.R. Cadambe and S.A. Jafar, “Interference alignment anddegrees of freedom of the k-user interference channel,” In-formation Theory, IEEE Transactions on, vol. 54, no. 8, pp.3425–3441, 2008.

[3] Mohamed Amir, Amr El-Keyi, and Mohammed Nafie, “Con-strained interference alignment and the spatial degrees offreedom of mimo cognitive networks,” Information Theory,IEEE Transactions on, vol. 57, no. 5, pp. 2994–3004, 2011.

[4] S.M. Perlaza, N. Fawaz, S. Lasaulce, and M. Debbah, “Fromspectrum pooling to space pooling: opportunistic interferencealignment in mimo cognitive networks,” Signal Processing,IEEE Transactions on, vol. 58, no. 7, pp. 3728–3741, 2010.

[5] Hui Shen, Bin Li, Meixia Tao, and Yi Luo, “The newinterference alignment scheme for the mimo interferencechannel,” in Wireless Communications and Networking Con-ference (WCNC), April 2010, pp. 1 – 6.

[6] T. Gou and S.A. Jafar, “Degrees of freedom of the k user m× n mimo interference channel,” Information Theory, IEEETransactions on, vol. 56, no. 12, pp. 6040–6057, 2010.

[7] S.M. Perlaza, M. Debbah, S. Lasaulce, and J.M. Chaufray,“Opportunistic interference alignment in mimo interferencechannels,” in Personal, Indoor and Mobile Radio Com-munications, 2008. PIMRC 2008. IEEE 19th InternationalSymposium on. Ieee, pp. 1–5.

[8] CD Meyer, “Matrix analysis and applied linear algebra.2000,” SIAM, Philadelphia.

[9] Simon Hykin, Communication Systems, John Wiley and Sons,2001.

[10] B. Vucetic, J. Yuan, and J. Wiley, Space-time coding, WileyOnline Library, 2003.

[11] S.M. Kay, “Fundamentals of statistical signal processing:Estimation theory, 1993,” Prentice-Hall.

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