document: the need for the efficient use of the scarce spectrum in wireless applications has led to significant interest in the analysis of cognitive radio systems . 
   one possible scheme for the operation of the cognitive radio network is to allow the secondary users to transmit concurrently on the same frequency band with the primary users as long as the resulting interference power at the primary receivers is kept below the interference temperature limit @xcite . 
   note that interference to the primary users is caused due to the broadcast nature of wireless transmissions , which allows the signals to be received by all users within the communication range . 
   note further that this broadcast nature also makes wireless communications vulnerable to eavesdropping . 
   the problem of secure transmission in the presence of an eavesdropper was first studied from an information - theoretic perspective in @xcite where wyner considered a wiretap channel model . in @xcite , 
   the secrecy capacity is defined as the maximum achievable rate from the transmitter to the legitimate receiver , which can be attained while keeping the eavesdropper completely ignorant of the transmitted messages . 
   later , wyner s result was extended to the gaussian channel in @xcite . 
   recently , motivated by the importance of security in wireless applications , information - theoretic security has been investigated in fading multi - antenna and multiuser channels . 
   for instance , cooperative relaying under secrecy constraints was studied in @xcite@xcite . in @xcite , for amplify and forwad relaying scheme , not having analytical solutions for the optimal beamforming design under both total and individual power constraints , an iterative algorithm is proposed to numerically obtain the optimal beamforming structure and maximize the secrecy rates . 
   although cognitive radio networks are also susceptible to eavesdropping , the combination of cognitive radio channels and information - theoretic security has received little attention . 
   very recently , pei _ et al . 
   _ in @xcite studied secure communication over multiple input , single output ( miso ) cognitive radio channels . in this work , finding the secrecy - capacity - achieving transmit covariance matrix under joint transmit and interference power constraints is formulated as a quasiconvex optimization problem .    in this paper , we investigate the collaborative relay beamforming under secrecy constraints in the cognitive radio network . 
   we first characterize the secrecy rate of the amplify - and - forward ( af ) cognitive relay channel . 
   then , we formulate the beamforming optimization as a quasiconvex optimization problem which can be solved through convex semidefinite programming ( sdp ) . 
   furthermore , we propose two sub - optimal null space beamforming schemes to reduce the computational complexity . 
   we consider a cognitive relay channel with a secondary user source @xmath0 , a primary user @xmath1 , a secondary user destination @xmath2 , an eavesdropper @xmath3 , and @xmath4 relays @xmath5 , as depicted in figure [ fig : channel ] . 
   we assume that there is no direct link between @xmath0 and @xmath2 , @xmath0 and @xmath1 , and @xmath0 and @xmath3 . 
   we also assume that relays work synchronously to perform beamforming by multiplying the signals to be transmitted with complex weights @xmath6 . 
   we denote the channel fading coefficient between @xmath0 and @xmath7 by @xmath8 , the fading coefficient between @xmath7 and @xmath2 by @xmath9 , @xmath7 and @xmath1 by @xmath10 and the fading coefficient between @xmath7 and @xmath3 by @xmath11 . in this model 
   , the source @xmath0 tries to transmit confidential messages to @xmath2 with the help of the relays on the same band as the primary user s while keeping the interference on the primary user below some predefined interference temperature limit and keeping the eavesdropper @xmath3 ignorant of the information . 
   it s obvious that our channel is a two - hop relay network . in the first hop 
   , the source @xmath0 transmits @xmath12 to relays with power @xmath13=p_s$ ] . 
   the received signal at the @xmath14 relay @xmath7 is given by @xmath15 where @xmath16 is the background noise that has a gaussian distribution with zero mean and variance of @xmath17 . 
   in the af scenario , the received signal at @xmath7 is directly multiplied by @xmath18 without decoding , and forwarded to @xmath2 . 
   the relay output can be written as @xmath19 the scaling factor , @xmath20 is used to ensure @xmath21=|w_m|^2 $ ] . 
   there are two kinds of power constraints for relays . 
   first one is a total relay power constraint in the following form : @xmath22 where @xmath23^t$ ] and @xmath24 is the maximum total power . @xmath25 and @xmath26 denote the transpose and conjugate transpose , respectively , of a matrix or vector . in a multiuser network such as the relay system we study in this paper , it is practically more relevant to consider individual power constraints as wireless nodes generally operate under such limitations 
   . motivated by this , we can impose @xmath27 or equivalently @xmath28 where @xmath29 denotes the element - wise norm - square operation and @xmath30 is a column vector that contains the components @xmath31 . 
   @xmath32 is the maximum power for the @xmath14 relay node . 
   the received signals at the destination @xmath2 and eavesdropper @xmath3 are the superposition of the messages sent by the relays . 
   these received signals are expressed , respectively , as @xmath33 where @xmath34 and @xmath35 are the gaussian background noise components with zero mean and variance @xmath36 , at @xmath2 and @xmath3 , respectively . 
   it is easy to compute the received snr at @xmath2 and @xmath3 as @xmath37 where @xmath38 denotes the mutual information . 
   the interference at the primary user is latexmath:[\ ] ] where superscript @xmath43 denotes conjugate operation . 
   then , the received snr at the destination and eavesdropper , and the interference on primary user can be written , respectively , as @xmath44 with these notations , we can write the objective function of the optimization problem ( i.e. , the term inside the logarithm in ( [ srate ] ) ) as @xmath45 if we denote @xmath46 , @xmath47 , define @xmath48 , and employ the semidefinite relaxation approach , we can express the beamforming optimization problem as @xmath49 the optimization problem here is similar to that in @xcite . 
   the only difference is that we have an additional constraint due to the interference limitation . 
   thus , we can use the same optimization framework . the optimal beamforming solution that maximizes the secrecy rate in the cognitive relay channel 
   can be obtained by using semidefinite programming with a two dimensional search for both total and individual power constraints . for simulation 
   , one can use the well - developed interior point method based package sedumi @xcite , which produces a feasibility certificate if the problem is feasible , and its popular interface yalmip @xcite . 
   it is important to note that we should have the optimal @xmath50 to be of rank - one to determine the beamforming vector . 
   while proving analytically the existence of a rank - one solution for the above optimization problem seems to be a difficult task , we would like to emphasize that the solutions are rank - one in our simulations . 
   thus , our numerical result are tight . also , 
   even in the case we encounter a solution with rank higher than one , the gaussian randomization technique is practically proven to be effective in finding a feasible , rank - one approximate solution of the original problem . 
   details can be found in @xcite . 
   obtaining the optimal solution requires significant computation . to simplify the analysis 
   , we propose suboptimal null space beamforming techniques in this section . 
   we choose @xmath51 to lie in the null space of @xmath52 . with this assumption , 
   we eliminate @xmath3 s capability of eavesdropping on @xmath2 . 
   mathematically , this is equivalent to @xmath53 , which means @xmath51 is in the null space of @xmath54 . 
   we can write @xmath55 , where @xmath56 denotes the projection matrix onto the null space of @xmath54 . 
   specifically , the columns of @xmath56 are orthonormal vectors which form the basis of the null space of @xmath54 . in our case 
   , @xmath56 is an @xmath57 matrix . 
   the total power constraint becomes @xmath58 . 
   the individual power constraint becomes @xmath59    under the above null space beamforming assumption , @xmath60 is zero . 
   hence , we only need to maximize @xmath61 to get the highest achievable secrecy rate . 
   @xmath61 is now expressed as @xmath62 the interference on the primary user can be written as @xmath63 defining @xmath64 , we can express the optimization problem as @xmath65 this problem can be easily solved by semidefinite programming with bisection search @xcite .      in this section , 
   we choose @xmath51 to lie in the null space of @xmath52 and @xmath66 . 
   mathematically , this is equivalent to requiring @xmath67 , and @xmath68 . 
   we can write @xmath69 , where @xmath70 denotes the projection matrix onto the null space of @xmath54 and @xmath71 . 
   specifically , the columns of @xmath70 are orthonormal vectors which form the basis of the null space . in our case 
   , @xmath70 is an @xmath72 matrix . 
   the total power constraint becomes @xmath73 . 
   the individual power constraint becomes @xmath74 . 
   with this beamforming strategy , we again have @xmath75 . 
   moreover , the interference on the primary user is now reduced to @xmath76 which is the sum of the forwarded additive noise components present at the relays . 
   now , the optimization problem becomes @xmath77 again , this problem can be solved through semidefinite programming . with the following assumptions , 
   we can also obtain a closed - form characterization of the beamforming structure . 
   since the interference experienced by the primary user consists of the forwarded noise components , we can assume that the interference constraint @xmath78 is inactive unless @xmath41 is very small . with this assumption 
   , we can drop this constraint . 
   if we further assume that the relays operate under the total power constraint expressed as @xmath79 , we can get the following closed - form solution : @xmath80 where @xmath81 is the largest generalized eigenvalue of the matrix pair @xmath82 . and positive definite matrix @xmath83 , @xmath84 is referred to as a generalized eigenvalue 
   eigenvector pair of @xmath82 if @xmath84 satisfy @xmath85 @xcite . 
   ] hence , the maximum secrecy rate is achieved by the beamforming vector @xmath86 where @xmath87 is the eigenvector that corresponds to @xmath88 and @xmath89 is chosen to ensure @xmath90 . 
   the discussion in section [ sec : op ] can be easily extended to the case of more than one primary user in the network . 
   each primary user will introduce an interference constraint @xmath91 which can be straightforwardly included into ( [ optimal ] ) . 
   the beamforming optimization is still a semidefinite programming problem . on the other hand , 
   the results in section [ sec : op ] can not be easily extended to the multiple - eavesdropper scenario . in this case , the secrecy rate for af relaying is @xmath92 , where the maximization is over the rates achieved over the links between the relays and different eavesdroppers . 
   hence , we have to consider the eavesdropper with the strongest channel . in this scenario , 
   the objective function can not be expressed in the form given in ( [ srate ] ) and the optimization framework provided in section [ sec : op ] does not directly apply to the multi - eavesdropper model . 
   however , the null space beamforming schemes discussed in section [ sec : null ] can be extended to the case of multiple primary users and eavesdroppers under the condition that the number of relay nodes is greater than the number of eavesdroppers or the total number of eavesdroppers and primary users depending on which null space beamforming is used . 
   the reason for this condition is to make sure the projection matrix @xmath93 exists . 
   note that the null space of @xmath94 channels in general has the dimension @xmath95 where @xmath4 is the number of relays . 
   we assume that @xmath96 , @xmath97 are complex , circularly symmetric gaussian random variables with zero mean and variances @xmath98 , @xmath99 , @xmath100 and @xmath101 respectively . in this section 
   , each figure is plotted for fixed realizations of the gaussian channel coefficients . 
   hence , the secrecy rates in the plots are instantaneous secrecy rates .    in fig . 
   [ fig:1 ] , we plot the optimal secrecy rates for the amplify - and - forward collaborative relay beamforming system under both individual and total power constraints . we also provide , for comparison , the secrecy rates attained by using the suboptimal beamforming schemes . 
   the fixed parameters are @xmath102 , @xmath103 , and @xmath104 . since af secrecy rates depend on both the source and relay powers , the rate curves are plotted as a function of @xmath105 . 
   we assume that the relays have equal powers in the case in which individual power constraints are imposed , i.e. , @xmath106 . 
   it is immediately seen from the figure that the suboptimal null space beamforming achievable rates under both total and individual power constraints are very close to the corresponding optimal ones . 
   especially , they are nearly identical in the high snr regime , which suggests that null space beamforming is optimal at high snrs . thus , null space beamforming schemes are good alternatives as they are obtained with much less computational burden . 
   moreover , we interestingly observe that imposing individual relay power constraints leads to small losses in the secrecy rates . in fig . 
   [ fig:11 ] , we change the parameters to @xmath107 , @xmath108 and @xmath104 . in this case , channels between the relays and the eavesdropper and between the relays and the primary - user are on average stronger than the channels between the relays and the destination . 
   we note that beamforming schemes can still attain good performance and we observe similar trends as before .    in fig . 
   [ fig:2 ] , we plot the optimal secrecy rate and the secrecy rates of the two suboptimal null space beamforming schemes ( under both total and individual power constraints ) as a function of the interference temperature limit @xmath41 . 
   we assume that @xmath109 . 
   it is observed that the secrecy rate achieved by beamforming in the null space of both the eavesdropper s and primary user s channels ( bnep ) is almost insensitive to different interference temperature limits when @xmath110 since it always forces the signal interference to be zero regardless of the value of @xmath41 . 
   it is further observed that beamforming in the null space of the eavesdropper s channel ( bne ) always achieves near optimal performance regardless the value of @xmath41 under both total and individual power constraints . 
   in this paper , collaborative relay beamforming in cognitive radio networks is studied under secrecy constraints . 
   optimal beamforming designs that maximize secrecy rates are investigated under both total and individual relay power constraints . 
   we have formulated the problem as a semidefinite programming problem and provided an optimization framework . 
   in addition , we have proposed two sub - optimal null space beamforming schemes to simplify the computation . 
   finally , we have provided numerical results to illustrate the performances of different beamforming schemes . 
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summary: in this paper , a cognitive relay channel is considered , and amplify - and - forward ( af ) relay beamforming designs in the presence of an eavesdropper and a primary user are studied . 
   our objective is to optimize the performance of the cognitive relay beamforming system while limiting the interference in the direction of the primary receiver and keeping the transmitted signal secret from the eavesdropper . 
   we show that under both total and individual power constraints , the problem becomes a quasiconvex optimization problem which can be solved by interior point methods . 
   we also propose two sub - optimal null space beamforming schemes which are obtained in a more computationally efficient way . 
   _ index terms : _ amplify - and - forward relaying , cognitive radio , physical - layer security , relay beamforming .