Scaled SLNR Precoding for Cognitive Radio

Scaled SLNR Precoding for Cognitive Radio Yiftach Richter Faculty of Engineering Bar-Ilan University Ramat-Gan, Israel Email: yifric@gmail.com Itsik Bergel Faculty of Engineering Bar-Ilan University Ramat-Gan, Israel Email: bergeli@biu.ac.il Abstract In this paper, we propose and analyze a lowcomplexity precoding scheme for cognitive radio networks. We consider a secondary user, equipped with multiple transmit antennas, that is allowed to access the spectrum of the primary network only if it does not interfere to the reception of the primary users. The proposed precoder is based on the signal to leakage plus noise ratio SLNR) criterion, with additional scaling to comply with the cognitive constraint. The proposed Scaled SLNR SSLNR) scheme is attractive for practical cognitive wireless networks as it combines near optimal performance with low implementation complexity. The SSLNR parameter is optimized using stochastic geometry analysis of an alternative gated zero-forcing scheme. The optimal parameter value is shown to depend only on the system parameters and not on the primary network density. Simulations results demonstrate the accuracy of the optimization, and show that the performance of the resulting SSLNR scheme is close to the performance of the optimal solution. I. INTRODUCTION The electromagnetic spectrum is a scarce and precious resource; Most of the spectrum is already in use by the military or by the private sector. Cognitive radio CR) is a promising paradigm for a better utilization of the frequency resource, by allowing the coexistence of primary users PUs) and secondary users SUs) in the same bandwidth [1]. Multiple antennas in CR opens up the spatial dimension, and adds opportunities where the SU can transmit at the same time as the PUs without causing any interference to the PU. The SU needs to configure its beamforming patterns in a way that balances between the maximization of its own throughput and the minimization of the interference to primary receiver. The optimal precoding vector in CR networks can be computed by a convex optimization, e.g. [2], [3]. But, such an optimization is too complex for implementation in many real-time systems in particular in systems with many primary users). To reduce the implementation complexity, several works proposed suboptimal precoding schemes e.g., [4] [8]). Most works rely the on zero-forcing ZF) principle or its block version, commonly termed block diagonalization, BD) to mitigate the interference from the SU transmitter to the PU receivers e.g., [4] [6]). However, in many situations, allowing a negligible amount of interference from the SU to the PU can significantly increase the SU performance. Thus, these This research was supported by the Israel Ministry of Labor, Trade and Commerce, as part of the CORNET consortium with Tadiran Spectralink as direct supervisor. ZF scheme can be far from optimal due to their zero interference constraint. Furthermore, in scenarios with many primary receivers, these schemes require an additional non-trivial step of choosing the significantly interfered PU receivers. A simpler and more efficient precoding scheme is based on the signal-to-leakage-plus-noise ratio SLNR) method. The SLNR method was presented by Sadek et al. [9] for multi-user multiple-input multiple-output MU-MIMO) wireless systems as an alternative optimization metric. The obective of the SLNR precoding is to increase the power ratio between the desired signal power and the sum of the interference power that the transmitter causes to other receivers leakage). The main advantage of the SLNR method is that it allows a direct computation of the precoding vector in multi-user environment. Recently, several works have considered the use of SLNR in multi-antenna CR. Some of these e.g., [5],[6]) adopted a ZF or BD scheme to completely remove the co-channel interference caused by SU to PU, and then used SLNR maximization algorithm to mitigate the interference between SUs. Another approach [7],[8] used the SLNR criterion also to eliminate the interference to the primary user and not only to optimize the secondary network). Both works consider a single PU receiver, a single SU base-station and multiple SU receivers, and find the precoding vector by the simple SLNR solution. In order to guarantee the compliance with the PU interference constraint, Zhao et al. [7] added a power optimization stage, in which the precoding vector is scaled to match the cognitive interference constraint. This scheme will be termed herein scaled SLNR SSLNR). Lee and Sung [8] showed that multiplying the interference by a carefully chosen constant before applying the SLNR scheme results with optimal performance in mean square error sense) in the case of a single PU receiver. The optimal value of this interference gain constant is obtained by a line search. In this paper we adopt the SSLNR approach of Zhao et al. [7], and improve it by adding the interference gain constant of Lee and Sung [8]. The use of scaling relaxes the need of a line search for the gain optimization, while the use of interference gain constant allows us to control the trade-off between interference mitigation and desired signal gain even in multiple PUs scenarios). The optimal value of the interference gain constant depends on the system parameters the cognitive constraint and the available transmission power). In

this paper, we will evaluate the optimal constant value through a statistical geometry performance analysis [1], assuming that the PUs are distributed as a Poisson point process PPP). Notations: Throughout this paper, matrices and vectors are denoted by boldface symbols. The conugate of complex number is marked by ), and ) H denotes the conugate transpose of matrix or vector. The absolute value of x is denoted by x. The expectation and probability of r.v., are denoted by E ) and P ), respectively. I N is the N N identity matrix, and x is the Frobenius norm of vector x. II. SYSTEM MODEL Consider CR system as depicted in Figure 1, where a single SU pair transmitter and receiver) is located within a network of primary users. The SU transmitter is equipped with N t antennas, and communicates with SU receiver, which is equipped with a single antenna. An adaptation of these scheme to multi-antenna SU receiver is described in the full ournal version of this article [11]. Without loss of generality, we assume that the SU transmitter is located at the origin of the coordinate system, and that the SU receiver is located at a normalized distance of 1 from the origin. The received signal in the SU receiver is given by: y = h s fx + n 1) where x CN, 1) and n CN, 1) are the transmitted symbol and the additive noise sample, respectively, both are complex Gaussian random variables with zero mean and unit variance. The vector h s C 1 Nt denotes the channel between the SU transmitter and the SU receiver, and the vector f C Nt 1, with f 2 P max, is the precoding vector. In this work we assume that the SU has perfect channel state information, and the transmitter uses the precoding vector, f, both in order to avoid interference and in order to maximize the received signal power at the desired) SU. In this work we consider the average signal to noise ratio SNR) 1 measured by the SU receiver: SNR = E h s fx 2}. 2) The SU transmitter is allowed to access the spectrum of the primary network only if it does not interfere with the communication of the PUs. We assume the spectrum is used by the PUs all the time and that each PU receiver is equipped with a single antenna. The interference power measured by the -th PU receiver is r h c f 2, where r and h c C 1 Nt denote the distance and the channel vector between the SU transmitter and the th PU receiver, respectively, and is the channel exponential decay factor. Allowing an interference of at most to each PU, the cognitive interference constraint is stated as: r hc f 2,. 3) 1 Note that the single antenna SU receiver cannot mitigate the interference from the PU transmitters. Hence, in this work we ignore the PU transmitters, and their effect is identical to an increase in the white noise level. Fig. 1. System Model: SU transmitter Tx) and SU receiver Rx) are located within a primary network Typically, will be chosen to be small enough so that even in the presence of multiple SUs, the probability to interrupt the PU reception will still be small. To allow statistical analysis, we assume that the locations of the PU receivers are modeled by a homogeneous Poisson point process PPP) with density λ i.e., the number of users in any area of size A has a Poisson distribution with a mean of λa). Also, all elements of the channel vectors h s, h ci are statistically independent and identically distributed iid) complex Gaussian random variables with zero mean and unit variance. III. THE SSLNR SCHEME The proposed SSLNR direction vector is defined by: f = arg max f and the SSLNR precoding vector is f H h H s h s f ) f H I Nt + γ N t r h H c h c f 4) f = c f/ f 5) where c the is a scaling factor guarantees the compliance with the cognitive interference constraint: c = min Pmax, max P max r h c f 2. 6) This scheme is similar to the scheme of Zhao et al. [7], except for the multiplication of the interference term in the denominator of 4) by the factor γ/n t. In the following we will show that this constant has a significant effect on the secondary user performance. To get an insight on the effect of the constant γ on the precoding vector, consider the simple case of a system with a single PU receiver. Since h c1 C 1 Nt, then h H c 1 h c1 is a rank one Hermitian matrix with one non-zero eigenvalue λ 1 = h c1 h H c 1, and E λ 1 } = N t. Thus, we can characterize the two extremes:

Weak interference: If γ/r1 1, the optimal precoder can be approximated by the beam forming solution f = c h H s, i.e., the precoder ignores the PU interference. Strong interference: If γ/r1 1, the optimal precoder will satisfy by the ZF condition: h c1 f =, i.e., perfect cancellation of the interference to the PU. Thus, the parameter γ determines which PUs will be considered important and which will be ignored. The solution to the SLNR formula is well known 1 f = I Nt + γ N t r h H c h c IV. PERFORMANCE ANALYSIS h H s. 7) In this section we will show that the optimal value of the constant γ is well approximated by: γ const Pmax logn t ). 8) In the next section we will demonstrate by simulation that const =.4 gives good performance in all tested scenarios. To find the optimal value of γ we need to derive an expression for the average SNR in the SSLNR scheme. However, the SSLNR scheme is quite complicated for analysis. Instead, in this paper we derive the optimal value of γ through the analysis of an alternative scheme with similar characteristics. As noted above, the SSLNR operation can be considered intuitively) as placing zeros in the directions of some PUs, while ignoring the others. Thus, in the following we analyze a partial ZF PZF) scheme with the same characteristics. In the PZF scheme, the SU transmitter identifies the group of the M nearest PU receivers. Then the SU transmitter calculates the precoding vector that achieves maximal desired signal gain subect to a zero interference constraint to all PUs in the selected group. Yet, although the PZF scheme zeros the interference to the nearest receivers, it cannot guarantee the interference level to any other PU. In particular, the interference level depends on the selection of the precoding vector, and hence the most interfered PU is known only after the selection of the precoding vector. To avoid the need for an iterative selection algorithm, the PZF scheme selects the important PUs only by their distance, and use gating to avoid interference conflicts. The resulting scheme, termed gated-pzf GPZF) uses the PZF precoding vector if the resulting interference to all PU satisfy the interference constraint. Otherwise, the transmission is gated, and the transmitter does not transmit anything. Denote by r M the distance between the SU transmitter and the M-th nearest PU receiver, and let F 1 be the set of the M PU receivers that are closest to the SU transmitter: F 1 = : r r M }. 9) We define the combined channel matrix, H F1, in which each row is the channel vector of one of the PUs in F 1. By construction, the size of the combined channel matrix is M N t, and we limit the discussion to M N t 1. In the GPZF scheme, the SU transmitter zeros the interference to the nearest M PU receivers. Thus, the GPZF precoding vector, f GPZF, lies in the null space of the channel matrix H F1. Using singular value decomposition, H F1 = UΛV H, where U C M M and V C Nt M contain the left and right singular vectors of H F1, respectively, and Λ R M M is a diagonal matrix that contains the singular values of H F1. The PZF vector f PZF is given by h = h s I Nt VV H) 1) f PZF = P max h H h 11) which guarantees that H F1 f PZF = M where M is M 1 zero vector). Using the zero-forcing vector, f PZF, the interference caused to the -th PU receiver is I r hc f PZF 2 12) and the GPZF scheme aborts the transmission if max I >. Define the indicator function 1 max I i tr = o.w. the precoding vector by the GPZF scheme is 13) f GPZF = i tr f PZF. 14) A. Performance analysis of the GPZF scheme Noting that the PZF vector, f PZF, depends only on the channel vectors of the M nearest PU receivers, it is statistically independent of the parameters of all other PUs, and hence also statistically independent on i tr. Hence, the average SNR, 2), can be decomposed to: E h s f GPZF 2} = P i tr = 1) E h s f PZF 2 } i tr = 1 = P i tr = 1) E h s f PZF 2}. 15) where the last inequality results from the statistical independence between h s and f PZF, i tr. We start the analysis by evaluating the GPZF transmission probability, Ei tr }, using tools of stochastic geometry [1]. Assume that we use the precoder f PZF without gating. Some of the PU receivers may suffer interference above the allowed CR threshold,. Define the set F 2 = : I >, r > r M } 16) which contains the PU receivers that suffer interference power which is not zeroed) greater than the CR threshold. Consider the indicator function 1 F 2 i F2} = o.w. 1 = o.w. h c hh 2 > r P max, r > r M. 17)

-25-2 -15-1 -5 5 1 15 SNR db) 8 6 4 2-2 -4-6 -8-1 N=4 db) Optimal SSLNR.4 Zhao et al[1] GPZFsimulation) GPZFtheory) Fig. 2. Average SNR versus the cognitive constraint power ) for various transmission schemes, for a secondary transmitter with N t = 4 antennas and = 3.3. The distribution of 2 h c hh 2 is a Chi-square distribution with 2 degrees of freedom. Hence, h c hh 2 has an exponential distribution with a mean of 1 [12]. Using Slivnyak s Theorem [1], the mean number of points in F 2 given r M is } λ F2 r M ) r E i F2} r > r M dr r M r Pmax r M Pmax r e x dxdr r e r Pmax dr ) 2/ 2 Γ, ) rm 18) P max where Γ, ) is the upper incomplete gamma function, given by Γa, x) = t a 1) e t dt. The number of users in the set x F 2 is Poisson distributed. The transmission probability is the probability that this set is empty, i.e., ) ) P F 2 = r M = e λ F2 rm. 19) The expectation of 19) with respect to r M is quite complicated. However, we can get a good approximation using R M = r M 2, and taking the first-order Taylor } approximation to 19) around the point RM E RM = M πλ. The resulting approximated transmission probability is: P F 2 = ) = P i tr = 1) = E e λf 2 )} RM e λ F 2, πλ) M. 2) Next, we evaluate the PZF signal power loss, E h s f P ZF 2}. As shown in 1), the PZF vector can be written using the proection matrix I Nt VV H). Thus, the resulting precoder is the best beamforming vector over the remaining N t M dimensions, and its power is: E h s f P ZF 2} = N t M). 21) The average SNR of the GPZF scheme is given by substituting 2) and 21) in 15). To maximize this SNR, we need to find the optimal value of M, the number of zeroed directions PU receivers with zero interference). Taking a high value of M will force many zeros and can result in a significant loss of signal power. On the other hand, a low value of M can cause the transmission probability to be low. Therefore, the GPZF maximization problem can be written as M opt = arg max M Pi tr) N t M). 22) Differentiation of 22) with respect to M, the optimum value must satisfy: N t M opt ) = e P M opt ) /2 πλ. 23) Focusing on the case that N t M opt, we will use hereafter the approximation 2/ M opt Pmax πλ logn t )). 24) B. Implication to the SSLNR As noted above, the operation of the SSLNR can be described intuitively as zero forcing toward the most interfered PUs while ignoring all other PUs. Considering the two cases in the single interference case as described in the previous section) the SSLNR threshold can be taken as γ r 1. Comparing to 24), we choose the SSLNR constant in 4) so > 1 is equal to M opt. In a PPP with density λ, the number of users in a circle with radius r is πλr 2. Therefore, the SSLNR constant is given by: that, the average number of users with γ r γ P max logn t ). 25) Interestingly, the 25) depends only on the system parameters, and not on the network density. This is convenient, because the transmitter design does not take into account the statistical nature of the primary network which is not necessarily known). Yet, the GPZF scheme and the SSLNR scheme are not identical the most obvious example in the SSLNR scaling as opposed to gating in the GPZF). Thus, we take 25) to describe the behavior of the optimal SSLNR constant, and multiply it by a constant for better optimization. Thus, in the this work, the best SSLNR constant is chosen to be: γ = const Pmax logn t ). 26)

2 4 6 8 1 12 14 16 12 1 8 6 1 5 N=16 SSLNR N=16 Zhao et al[1] N=16 N=4 SSLNR N=4 Zhao et al[1] N=4 SNR db) 4 2 SNR db) -5-2 -1-4 Optimal -15-6 -8 Number of antennas SSLNR.4 Zhao et al[1] 2.8 3 3.2 3.4 3.6 3.8 4 Fig. 3. Average SNR vs. number of transmit antennas, N t, = 16dB and = 3.3. Fig. 4. Average SNR vs. the path loss exponent,, = 26dB. V. NUMERICAL RESULT In this section, we provide simulation results to demonstrate the efficiency of the proposed SSLNR precoding scheme in terms of the received SNR. In particular, we show that choosing const =.4 in 26) results in good SSLNR performance in all tested cases. We consider a square with an area of 1, and randomly located PU receivers with density λ =.1 i.e., the number of PUs has Poisson distribution with a mean of 1). Fig. 2 depicts the average SNR as a function of the cognitive interference power constraint, for a system with N t = 4 transmit antennas, P max = 1 and a path loss exponent of = 3.3. The figure depicts the optimal performance and the performance of the proposed SSLNR scheme with const =.4. In this case, the loss of the SSLNR scheme compared to the optimal precoding is at most 2dB, which is a reasonable price for a significant reduction in implementation complexity. For reference, Fig. 2 also presents the curve, which depicts the SSLNR performance when γ is numerically optimized not using 26)). The gain of this numerical optimization is negligible and can hardly be seen in the figure. On the other hand, the figure also show the performance of the Zhao et al. scheme [7] which is given by 7) with γ = N t. This scheme suffers additional loss of up to 3dB from the proposed SSLNR scheme. This loss demonstrates the importance of the optimization of the SSLNR scheme with respect to the value of γ. The figure also depicts for reference the performance of the GPZF scheme, and demonstrates the accuracy of the theoretical performance analysis. This scheme is inferior to the SSLNR scheme, but, as was shown, is very helpful in the optimization of the SSLNR scheme. Fig. 3 depicts the average SNR as a function of the number of antennas, for the same path loss exponent and = 16dB. Fig 4 depicts the average SNR as a functions of for the cases of 4 and 16 transmit antennas and = 26dB. Both figures show again that the SSLNR scheme is close to optimal and much better than the Zhao scheme. VI. CONCLUSIONS In this paper, we presented and analyzed an SLNR based precoding scheme termed SSLNR. The SSLNR scheme is attractive for practical cognitive wireless networks as it combines near optimal performance with low implementation complexity. The SSLNR parameter was optimized using stochastic geometry analysis, and it was shown that the optimal parameter value depends only on the system parameters and not on the primary network density. Future research will extend these results also to the case of multiple receive antennas and to the case that the SU transmitter has only partial imperfect) channel state information. REFERENCES [1] S. Haykin et al., Cognitive radio: brain-empowered wireless communications, IEEE ournal on selected areas in communications, 25. [2] R. Zhang and Y.-C. Liang, Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks, IEEE Journal of Selected Topics in Signal Processing, 28. [3] R. Zhang, Y.-C. Liang, and S. Cui, Dynamic resource allocation in cognitive radio networks, Signal Processing Magazine, IEEE, 21. [4] J. Zhou and J. Thompson, Linear precoding for the downlink of multiple input single output coexisting wireless systems, Communications, IET, 28. [5] C. Sun, J. Ge, X. Bao, and X. Shi, A leakage-based precoding scheme for cognitive multiuser mimo systems, 4th International Conference on Intelligent Networking and Collaborative Systems INCoS), 212. [6] P. Jaehyun, P. Yunu, S. HWANG, and B. JEONG, Low-complexity gsvd-based beamforming and power allocation for a cognitive radio network, IEICE transactions on communications, 212. [7] K. Zhao, H. Zhang, and D. Yuan, Optimizing sum-capacity through power allocation for slnr-precoding-based cognitive networks, 212. [8] I. L. Kyoung-Jae Lee, Hakea Sung, Linear precoder designs for cognitive radio multiuser mimo downlink systems, 211 IEEE International Conference on Communications ICC), 211. [9] M. Sadek, A. Tarighat, and A. Sayed, A leakage-based precoding scheme for downlink multi-user mimo channels, IEEE Transactions on Wireless Communications, 27. [1] F. Baccelli and B. Blaszczyszyn, Stochastic geometry and wireless networks. Now Publishers Inc, 29, vol. 1. [11] Y. Richter and I. Bergel, Optimization of the Scaled SLNR precoder for cognitive radio, In preparation. [12] N. Jindal, J. G. Andrews, and S. Weber, Multi-antenna communication in ad hoc networks: Achieving mimo gains with simo transmission, IEEE Transactions on Communications, 211.