Reweighted Nuclear Norm Minimization ofInterference Alignment for Cognitive Radio

NetworksAnizamariah Daud∗†, Mahamod Ismail∗, Nordin Ramli†, Ha zal Mohamad†

∗Faculty of Engineering, Universiti Kebangsaan Malaysia, Selangor, Malaysia†Wireless Communication Cluster, MIMOS Berhad, Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia

Email: aniza.daud@mimos.my

Abstract—A cognitive radio network allows the primary users(PU) and secondary users (SU) transmit simultaneously given thatthe primary user’s transmission is not disrupted. The secondaryusers are able to transmit their signals by aligning the signaldirection to the primary user’s unused direction. However, theperformance of SU is heavily affected by the degree of freedom(DoF) of the cognitive radio network in a static at-fadingmultiple input multiple output (MIMO) interference channel. Arank constraint rank minimization (RCRM) method has beenused to maximize the DoF but the optimization problem happensto be a non-convex problem thus, a tighter convex approximationwill help to solve this problem. One of the most popular methodsis the nuclear norm minimization method which provides aconvex envelope of the approximation but discovered to be notoptimal in nding the maximum achievable DoF. This paperproposes a reweighted nuclear norm minimization method in acognitive radio network with the presence of PU and multipleSU with the interest at the SU’s receiver side. The proposedmethod allows the PU and SU to coexist in the same frequencyband and transmit simultaneously without disturbance from SUto PU while avoiding degradation of performance for SU atthe same time. The weight matrix is placed at the receiverand updated iteratively according to the current environment,resulting in a tighter convex approximation and thus, enhancesthe performance of SU.

I. INTRODUCTIONIn order to sustain the increasing number of wireless users

and applications, various researches were made to fully utilizethe wireless resource since it has become scarcer nowadays.Although by law the spectrum is fully allocated to its primaryusers (PU), it is in actuality, severely underutilized since theyare only occupied at a certain time and location while the freeband is facing heavy traf c due to a large number of secondaryusers (SU) accessing the band. Recently, many studies madein proposing cognitive radio (CR) as a solution to utilize thespectrum in a more ef cient manner [1]–[4]. The CR allowssharing of the allocated spectrum between PU and SU whereSU will opportunistically transmit their signal at the portionsthat is unused by PU, given that PU’s performance is notaffected by the transmission of SU [1]. Transmitting signalssimultaneously between PU and SU, also known as underlay,could lead to a fully utilized spectrum. However, simultaneoustransmission causes undesirable interference to both partiesthus, interference management is very crucial to ensure thatPU and SU have the best performance in a CR environment

[2]. In this paper, our main focus is on the SU performancerather than the PU as previously studied by many.In the environment with concurrent transmission, interfer-

ence to either party is inevitable which has prompted manystudies made to solve such problem. One of the many methodsproposed is the interference alignment (IA) method. Theconventional IA scheme will align the interference signal inthe direction of the unused eigen modes, leaving more freedimension for the desired signal. While the conventional IAmethod has been well developed, only recently the study ofIA with cognitive radio networks has been conducted [5]–[10].The IA scheme which aligns the interference signal into thenullspace of the desired channel matrix would t perfectlyin a CR network which enacts the idea of simultaneoustransmission without affecting the performance of one another.The idea of allowing the SU to exploit unused portions ofPU in the form of IA has been made in [5]–[8] where themain goal is to maximize the interference free dimension,also known as the degree of freedom (DoF). A higher DoFwould let SU to transmit at a more signi cant rate. This isdone by designing the transmit and receive strategies, alsoknown as precoding and postcoding matrices. However, thesematrices are NP-hard to obtain and many alternatives werestudied vigorously.In [9] and [10], both studied for the case of multiple SU

transmitting their signal by aligning their signal directions tothe unused singular vectors left by PU when PU maximizes itsrate and proposed an interference cancellation method placedat the receiver to whiten the interference received. This is alsoknown as the interference minimization method. However, thismethod is not the optimal method in nding DoF as it is onlyachievable when the condition for the perfect IA is met whichis a highly non-trivial task as shown by the authors in [11],[12]. Hence, without the optimality of DoF, the performance ofSU degrades as they have lesser interference free dimensions.The authors in [11], developed the rank constrained rank

minimization (RCRM) method with the main idea to minimizethe rank of the subspace spanned by interference signals thatis subject to a full-rank signal space to have a maximized DoF.In this method, a nuclear norm minimization method is intro-duced for the approximation of the rank function. The nuclearnorm minimization has been considered as one of the most

2013 IEEE 9th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

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popular algorithm as it replaces rank and guarantees to yield anexact solution [13]. The nuclear norm minimization accountsfor the sum of singular values in a matrix and has been ableto provide a considerably good convex approximation for theRCRM problem. Various studies were made to enhance thenuclear norm minimization problem such as weighted nuclearnorm minimization (WNNM) and the reweighted nuclear normminimization (RNNM) [14]. The studies for RNNM methodhas been made in many areas but to our best of knowledge,none has been made in a cognitive radio environment.In this paper, we present a reweighted nuclear norm min-

imization (RNNM) with rank constrained rank minimization(RCRM) of IA for a cognitive radio network with multipleSU. Previously, authors in [15] applied the method in a noncognitive environment whereas in this work, we consider theunderlay cognitive radio network and in the case of multipleSU opportunistically exploiting the same frequency band thatis utilized by PU. In the cognitive environment, SU will beable to initialize the precoding from what it had learned ofthe knowledge of PU’s transmission instead of initializingthe postcoding as had done previously in [15]. The intendedSU enhances its rates by striving for a minimum rank of thetotal interference combined from PU and other SU rather thanchasing a perfect IA at its receiver. Weights matrices are placedat the receiver and reweighted iteratively with accordanceto the precoding and postcoding matrices to nd the bestweight for the optimal solution. This method provides a tighterapproximation for the rank of interference received at thedesired SU receiver while the transmitter avoids any disruptionto PU’s transmission. In this scheme, the precoding matricesof the SU are designed in a way such that no interference isgenerated at the PU receiver yet maintains certain QoS for SU.The postcoding matrix at each SUs receiver are able to producethe optimized minimum interference that does not deteriorateits achievable rate. Furthermore, the cognitive environmentpresumes all information are known to users which helps SUto update and adapt its precoding and postcoding matricesiteratively. Simulation results show this method has a higherachievable rate compared to other conventional method.The rest of the paper is organized as follows. Section II

describes the system model of the problem while section IIIis on reweighted nuclear norm minimization in cognitive radionetworks. The simulation results and discussion are made insection IV while section V concludes the whole paper.

II. SYSTEM MODEL

In this paper, we consider a cognitive radio network in astatic at fading (K + 1) user MIMO interference channelwith a single PU and K SUs transmitting simultaneously. Thescenario is shown in Fig.1 but the model is extendable to anynumber of users in a system. PU−Tx and SUk−Tx indicatethe transmitter for PU and SU while PU −Rx and SUk−Rxrepresent the PU and SU receivers where each transmitterand receiver are equipped with Mk

t transmit antennas andNkr receive antennas respectively and k is the user index

of a K-secondary user network with k = {1, · · · ,K}. For

PU-Tx

SU1-Tx

SUk-Tx

PU-Rx

SU1-Rx

SUk-Rx

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H10

Hk0

H01

H11

Hk1

H0k

H1,k

Hkk

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#1

#1

#1 #1

#1

#1

kXsu

1Xsu

0Xpu0Ypu

1Ysu

kYsu

Primary User (user 0)

Secondary User (user 1~k)

0# tM

1# tM

ktM# k

tN#

1# tN

0# tN

Fig. 1. The propagation channel model in a cognitive radio network.

simplicity, we consider a scenario of three user system witheach of them are equipped with the same number of antennasbut the model is not constrained to any maximum number. Inavoiding disruption to PU, SU is assumed to transmit belowa power threshold and each transmitting user are assumed tosynchronize their transmission. In this paper, the PU is indexedas 0 and the remaining K users are secondary users thatare present in the system. It is assumed that all informationabout PU is known to every SU in the system while PU isoblivious to SU’s presence in the network and assumes allSU as interference. In this paper, our main focus is on theSU’s receiver side rather than the PU. In general, the receivedsignal at the intended k-th SU’s receiver in this model can berepresented as;

Yk = HkkVkXk +K∑

j �=k

Hk,jVjXj + Nk (1)

The rst part of the equation indicates the desired signal fromthe intended SU transmitter to the intended SU receiver whilethe second part is the interference from other transmittersbesides the intended transmitter that is denoted as j wherek = {1, · · · ,K} and j = {0, 1, · · · ,K}. Other transmittersare made up of PU and other SU that is present in the system.In the equation, Hkk ∈ CNk

r×Mkt , indicates the channel

between intended k-th SU transmitter to the intended k-thSU receiver while Hkj ∈ CNk

r×Mjt is the channel from j-

th transmitter to intended k-th receiver. The transmitted datavector from intended SU k-th transmitter is denoted as Xk

and Xj is the transmitted data vector from another transmitterincluding from PU and they are independently identicallydistributed (i.i.d) with E[XkXH

k ] = I. The Nk represents thezero-mean complex additive white Gaussian noise (AWGN)vector at the k-th receiver with σ2kINk

ras its covariance matrix.

The precoding matrix for the k-th transmitter is denoted as

758

Vk ∈ CMk

t ×dk for desired SU, while Vj is the precoder forother transmitter in the system with dk indicating the avail-able DoF at the intended k-th SU. Since SU is transmittingsimultaneously with PU, the intended SU receiver will haveto take account of both interference from PU and other SUas well. Should the network be free from transmission ofother transmitter, the receiver of the intended SU can omit theinterference from the party. In this work, we assume that PU’stransmitter will choose its own precoding matrix, V0 and thePU’s receiver will choose its postcoding matrix, U0, in a waythat the primary user channel transfer matrix diagonalized suchas UH

0 H00V0 = Λ0 where Λ0 shows the diagonal matrix forPU with diagonal elements ofmin {N0

r ,M0t } nonzero singular

matrix of the H00. It is also assumed that the PU is obliviousto SU and will maximize its own primary user rate with water-lling solution and suppresses interference at its own receiveras to ensure that the PU rate is maintained. Previously in [11],the model without PU consideration, assumes the dk as;

dk � min(Mkt , N

kr ) (2)

Since this work considers the system is in a CR networkwhere the presence of PU needs to be accounted for and itsDoF, d0, is known at the intended SU. In general, the DoF ofintended k-th transmitter is de ned as dk and in the form of;

dk � (Mkt +Nk

r − 2d0)/K (3)

This equation provides an upper bound that is useful for theintended SU to ensure the QoS of its performance. Each of theintended receiver k will process the received signal linearly bythe postcoding matrix Uk ∈ C

Nkr×dk , denotes the orthonormal

basis of the intended k-th receiver subspace which will obtainthe following;

UHk Yk = UH

k HkkVkXk + UHk

K∑

j �=k

HkjVjXj + UHk Nk (4)

In [5], the purpose of the IA is to design precoder matricesand the postcoder matrices that will span in a manner that eachreceiver will be able to decode its own signal with enforcinginterfering users to share a reduced dimensional subspace.Firstly, should the conventional perfect IA condition be

considered, the intended SU receiver will take the form of;

rank(UHk HkkVk) = dk, ∀k = {1, 2, ...,K} (5)

K∑

j �=k

UHk HkjVj = 0, ∀j = {0, 1, 2, ...,K} (6)

The condition above ensures that no interference from otherusers at the output of the intended receiver and guaranteesthe desired signal of the intended transmitter to an intendedreceiver achieves dk degree of freedom. In this work, whichconsiders the PU and SU transmit simultaneously, while theperfect IA is possible for PU, the perfect IA is consideredas impractical at SU as it is receiving high interference fromPU and a minimization of rank of interference at the receiveris considered instead. The SU will look into maximizing the

available spatial DoF in order to have a better performancesince more DoF will translate to a higher achievable rate forSU. Unfortunately, nding the maximum available spatial DoFis an NP-hard problem when there is more than two antennas ateach of the transmitter and receiver respectively as mentionedin [14].Many alternatives have been proposed to approach such

problem and one of the most popular is by minimizing theinterference leakage which takes account of low interferencepower instead of the low dimension solutions [16]–[18]. How-ever, minimizing interference leakage has found not to be theoptimal solution. A study made in [19] suggests RCRM withNNM is able to obtain a certain amount of DoF per user but itis not able to provide an optimal DoF. In this paper, we proposethe method of reweighted nuclear norm heuristic approach in acognitive radio network to improve the approximation of rankminimization of interference thus, improving the achievablerate for the SU yet still maintaining the rate of PU wherepreviously [15] the method was not made to function in acognitive environment.

III. REWEIGHTED NUCLEAR NORM MINIMIZATION INCOGNITIVE RADIO NETWORK

Various studies were made to further improve the conven-tional heuristic method to reduce the rank of the solution. Oneof the variations is the reweighted nuclear norm minimization(RNNM), a further improvement of the weighted nuclearnorm minimization (NNM) method. The RCRM method wasdeveloped to have a tighter approximation and to guaranteethat the interference will fall into a low-dimensional subspacewhile the desired signal spaces span all available spatialdimensions. While the method had previously applied in anon cognitive network, this work considers it in a cognitiveradio network.In this paper, we assume that the PU maximizes its transmit-

ting rate and leaves some eigenmodes unused that is availablefor SU’s transmission and will not affect PU achievable ratejust as long as the interference from SU falls into these unusedeigenmodes [14].The purpose of enhancement from the NNM to RNNM is

to have a convex approximation that is as close as the rankoperator to solve the NP-hard optimization problem that isa non-convex problem. Previously in [15], the model onlyconsiders coordinated interference and ignores uncoordinatedinterference as it does not have the suf cient information whilein CR environment, all information are made available to allparties involved and all interference will be used to determinethe weight matrix which is inversely related to the interferencematrix. In determining the most minimum rank possible, thedesired signal, Sk and interference matrices from PU and otherSU at the receiver, Jk for all receivers is de ned as follows;

Sk(Uk,Vk)�

= UHk HkkVk (7)

Jk(Uk, {Vj}Kj �=k)

�

= Uk({HkjVj}Kj �=k) (8)

759

The achievable spatial DoF of the intended k-th SU receiverin terms of ranks is de ned as;

dsuk = [rank(Sk)− rank(Jk)]+ (9)

The proof for the achievable DoF is given in [9].The maximum spatial DoF is achievable when the differ-

ence in the equation (9) which shows in terms of ranks ismaximized. This is in actuality, the same as minimizing thesum of interference dimensions at the intended receiver withsubject to the full rank signal space constraints [11] which canbe represented as;

min{Vj}Kj �=k,Uk

rank(Jk) (10)

s.t.:rank(Sk) = dk (11)

The rank function is a non-convex problem which requires aconvex approximation which previous authors uses the NNMapproach in solving the problem. Previous method of NNMprovides a convex envelope for the rank function and is ableto recover suf ciently low rank matrices [10], [11], [13].Although it is considered as an effective method, however,the pairing of NNM in the RCRM method is unbalanced asthe NNM only considers the sum of the magnitudes of singularvalues while the rank operator takes in the sum of the numberof positive values [15] and this made the pairing not suitablefor the CR network as combined interference from PU andother SU would be high. Thus, the RCRM with NNM methodin a CR network is only able to achieve the conservative DoFper user which would be low for the intended SU. To coun-terbalance the problem faced in the previous RCRM-NNMmethod, a reweighted nuclear norm minimization (RNNM) incognitive radio network is introduced. In this method, with thecognitive radio network, the information of all interference areknown, which makes SU able to determine the bounds of thesystem without the exhaustive search for the set of optimalweight matrices thus, providing a closer approximation tothe problem easier. Furthermore, the SU needs to observeits power transmission is below a certain threshold to ensureno disruption to PU. The weights are made available at thereceiver. The rank minimization problem in the equation (12)and (13) after the introduction of RNNM is rewritten to;

min{Vj}Kj �=k,Uk,Wsuk�0

K∑‖WsukJk‖∗ (12)

s.t. : σmin(UHk HkkVk) ≥ γ (13)

(UHk HkkVk) � 0 (14)

which σmin(UHk HkkVk) is the smallest singular value of the

matrix of UHk HkkVk with γ as the minimum rate that SU’s

receiver needs to achieve. By having a minimum rate for SU,the QoS for SU is ensured to be maintained and that the ratewill not be degraded. The weightsWsuk for user k is a positivesemide nite matrix. The weights are inversely related to theinterference matrices. Since SU has the signi cant information

of PU and other SU present in the system, the weights can bedetermined as;

Wsuk = ΨHk ΘkΨk = ΨH

k Σ−1

k Ψk (15)

Ψk and Σk denotes the right singular vectors and singularvalues of Jk while Θk represents the diagonal matrix of dsukeigenvalues which is the inverse of the singular values of theaccumulated interference matrix.The optimization problem is solved with alternating mini-

mization [11] which in this work takes a different approachwhereby the initialization for precoding is done in a cognitivemanner. Firstly, the transmitter Vk will initialize its precodersafter gaining knowledge of PU’s transmission by learning itsenvironment and this too will ensure that the interference toPU is below a threshold level. Holding the Vk xed for atemporary time and initialize the weight, the objective functionfor Uk is then optimized and alternated between the twovariable to be xed and optimized. The main difference isby initializing the precoders for the rst time from the gainedknowledge, the optimization for the variables becomes fasterand easier. The same goes for the intended SU receiver andis able to estimate Uk by listening to PU’s and other SU’stransmitter where the same method are used for the estimationof Vk. The postcoding is chosen such that each receivercan decode its own signal where interference from othersare forced to share a reduced-dimensional subspace whilethe precoding allows the transmission of desired signal whilekeeping an allowable interference level to PU all based on theknowledge gained in the CR network. The weights are thenupdated iteratively until the algorithm reaches a convergenceor until the maximum number of the iteration is reached.

IV. SIMULATION AND RESULTIn order to validate the effectiveness of the proposed method

and evaluate its performances, a monte carlo computer sim-ulation has been conducted for the proposed rank constraintrank minimization with reweighted nuclear norm minimization(RCRM-RNNM) method along with other existing methodsall modeled in a cognitive radio network (CRN). Here, weconsider an underlay cognitive radio network where PUs andSUs transmit simultaneously. The scenario is set with threeusers which PU and SUs coexisting where each node isequipped with Mt transmit antenna and Mr receive antennas.The model is applicable for any number of PU and SU. In thispaper, we run the simulations for (6 × 6, dk = 1, 2, 3)3 and(8× 4, dk = 1, 2, 3)3 MIMO interference system where dk isthe DoF for intended SU. The power for each precoding ma-trices are allocated where P ∈ {0, 5, 10, 15, 20, 25, 30, 35, 40}dB and the signal-to-noise ratio (SNR) per user is P dB. Theperformance results are computed as the achievable rate forthe intended SU in the system. The transmit power for eachPU is higher compared to SU which transmits at a lowerpower as to avoid any disruption towards PU. The variancefor AWGN is set as σ2k = 1. Three other existing methods thathas been studied thorughly [20], namely, the conventional rankconstraint rank minimization with nuclear norm minimization

760

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

P (dB)

Ach

ieva

ble

Rat

e (b

its/s

ec/H

z)

PUSU with RCRMSU with RCRM RNNM

Fig. 2. Achievable sum-rate for PU and SU with proposed method andconventional method.

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

P (dB)

Ach

ieva

ble

Rat

e (b

its/s

ec/H

z)

Proposed RCRM RNNMMax SINR, iter=1000Leakage Minimization, iter=1000

Fig. 3. Achievable sum-rate for SU with different methods in a 3-user system,Mr = 6,Mt = 6, and dk = 1.

(RCRN-NNM), leakage minimization and maximum SINRare included in the simulation for comparison purpose.First, we investigate the performance of the achievable ratefor SU and PU simultaneous transmission over CRN. Here,we consider the intended SU with two different methods,the conventional rank constraint rank minimization (RCRM)method and the proposed RCRM with reweighted nuclearnorm minimization (RCRM-RNNM) method. From the curvesshown in Fig.2, it is clearly seen that the proposed method ina CR network outperforms the conventional RCRM methodwhile avoiding any disruption to the PU. The RCRM-RNNM

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

P (dB)

Ach

ieva

ble

Rat

e (b

its/s

ec/H

z)

Proposed RCRM RNNMMax SINR, iter=1000Leakage Minimization, iter=1000

Fig. 4. Achievable sum-rate for SU in a 3-user system, Mr = 8,Mt = 4,and dk = 1.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

90

P (dB)

Ach

ieva

ble

Rat

e (b

its/s

ec/H

z)

RCRM, dk=1

RCRM, dk=2

RCRM, dk=3

Proposed RCRM RNNM, dk=1

Proposed RCRM RNNM, dk=2

Proposed RCRM RNNM, dk=3

Fig. 5. Achievable sum-rate for SU with different DoF in a 3-user system,Mr = 8,Mt = 4, and dk = 1, 3.

is able to outperform the conventional RCRM method due tothe weighted matrix that provides a tighter convex relaxationwhich is not available in the RCRM method. By assumingthat PU suppresses interference at their receiver also helps toincrease the DoF of the SU.Next, we simulate the proposed method with two other

methods, speci cally, the leakage minimization and max-SINRoperating in a CR network. Fig.3 and Fig.4 shows the perfor-mance of the achievable rate for the intended SU in a threeuser system. In Fig.3 shows the model with the same numberof antennas at the transmitter and receiver while Fig.4 is when

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different number of antennas are applied at each transmitterand receiver. Based from the gure, max-SINR outperformsleakage minimization since max-SINR method considers thestrength of the direct signal but the proposed method, RCRM-RNNM in CR network outperforms both methods in all twoscenarios where it achieves the highest rate at all power levels.The RCRM-RNNM has an edge to the other method as it isable to provide a better approximation for the rank functionand it is feasible even with the same number of antennas inthe system.In Fig.5 shows the performance of the achievable rate

for SU using the proposed RCRM-RNNM method and theconventional RCRM in a CR network at different value ofDoF for SU. From the gure, the proposed RCRM-RNNMmethod outperforms the RCRM method at all power levelsand DoF. Their rate for SU increases steadily at each powerlevel since the DoF does not fall to zero which allows theSU to achieve a higher rate. Although both method transmitswith the same number of DoF, the RCRM-RNNM is able tooutperform the RCRM due to its closer approximation withthe help of the weight matrix. Having suf cient informationabout the interference in the whole system helps the proposedmethod in CR network, RCRM-RNNM to place appropriateweight which is used to form the directional pattern hence,allowing a higher achievable rate without disrupting the PU.

V. CONCLUSIONS

In this paper, we have proposed a rank constraint rank mini-mization with reweighted nuclear norm minimization (RCRN-RNNM) algorithm to be applied in a cognitive radio networkwith the primary user (PU) and multiple secondary user (SU)transmitting simultaneously. A weight is placed at the receiverand updates its matrix iteratively after SU has gained suf cientinformation of the interferences in the network. The RCRN-RNNM method with cognitive ability produces an algorithmthat is able to provide a tighter convex relaxation thus, betterperformance than the previous method and is able to be furtherextended to multiple PU and SU. Simulation result shows thatthe proposed method outperforms any other methods includingthe previous conventional method modeled in a cognitive radionetwork where previously the cognitive environment was notconsidered.

ACKNOWLEDGMENT

This work was supported by Universiti KebangsaanMalaysia under Grant DPP-2013-006 and E-Science Fundproject (Project number: 01-03-04-SF0015) from Malaysia’sMinistry of Science, Technology and Innovation (MOSTI),Malaysia.

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