Secondary User Access Control (SUAC) via Quadratic Programming in Massive MIMO Cognitive Radio Networks

In cognitive radio (CR) networks, a secondary user access control (SUAC) technique has been designed to enhance spectrum efficiency, in which a jamming signal is deliberately injected to maintain reliable sensing performance of authorized secondary users (A-SUs) and degrades unauthorized secondary users (UA-SUs) spectrum sensing results. We consider the problem of jamming signal design in massive multiple-input multiple-output (MIMO) CR networks wherein each primary user has a larger number of antennas that coexists with multiple secondary users. In this paper, we propose a jamming signal design framework that combines maximizing the jammer’s influences on UA-SUs and minimizing on A-SUs. The resulting problem is a non-convex quadratically constrained quadratic programming (QCQP) problem, and a semidefinite relaxation (SDR) method can be one of the approximate solutions but cannot meet our stringent constraints and lacks jamming efficiency. We propose a novel optimization algorithm based on the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-best methodology to design the jamming signal. Simulation results show the effectiveness of the proposed <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-best based SUAC method in improving the spectrum sensing performance of A-SUs.


I. INTRODUCTION
I N TRADITIONAL wireless communications systems, there exist significant spectrum under-utilization problems, where some spectrum bands are heavily occupied while others are kept vacant for a long time. Cognitive radio (CR) technology has been introduced to address this issue [1]. Specifically, there are two types of users in the CR network: primary user (PU) and secondary user (SU). Secondary users are allowed to access the spectrum only when the PUs are not utilizing that spectrum. Therefore, one of the crucial tasks in CR is spectrum sensing in order to enhance the spectrum efficiency [2], [3].
The cognitive radio network is very vulnerable to malicious user attacks due to its open and dynamic nature [4], [5], [6]. For example, [4] proposes a primary user emulation attack model, where attackers imitate the PU's behavior and mislead other SUs' spectrum access decisions. In [6], a most active band (MAB) attack model is presented where an attacker selects and performs a denial-of-service (DoS) attack on a spectrum band with the most activities. Thus, it is important to design a robust CR system that maintains PUs/SUs communications quality under severe malicious user attacks. In [7], a secondary user access control (SUAC) framework has been introduced. Two types of secondary users are investigated in SUAC: authorized secondary users (A-SUs) and unauthorized secondary users (UA-SUs). When PU is inactive and vacant in the spectrum, only A-SUs are authorized to utilize the spectrum. To assure a reliable spectrum sensing performance for A-SUs and atrocious sensing results for UA-SUs, a carefully designed jamming signal is injected at the transmission end, where such jamming pattern can only be obtained by A-SUs to eliminate the jamming signal influence when performing spectrum sensing task. To consider A-SUs with different spectrum access priority levels, [8] proposes a prioritized SUAC (P-SUAC) mechanism. Reference [9] presents the secondary user access control technique in a massive multiple-input multiple-output (MIMO) communications environment, where a jamming signal is generated from a PU transmitter based on the estimated channel state information (CSI) which has a small or negligible influence on A-SUs. The proposed jamming signal design optimization technique is within the quadratically constrained quadratic programs (QCQP) domain with a semi-definite relaxation (SDR) solution provided in [10], [11].
Sparse representation provides a powerful emerging model for signals, where a data source is approximated with a linear combination of a few atoms from an over-complete dictionary [12]. The theory of compressive sensing (CS) has shown that a sparse signal can be reconstructed exactly from many few measurements [13]. Since then, intensive research on sparse signal and data processing have been investigated in the area of wireless communications [14], such as compressive channel estimation in massive MIMO [15], compressive spectrum sensing in CR networks [16], and compressive sensing enabled large-scale wireless sensor networks (WSNs) [17]. Massive data and connectivity are expected to be supported in the next-generation wireless communications systems, such a huge amount of data transmission and storage brings tremendous challenges in signal processing in massive MIMO communications systems.
Motivated by the proposed research in SUAC and the sparse representation research for wireless communications, the objective of this paper is to explore sparse jamming signal design for SUAC in cognitive radio networks by exploiting quadratic programming. The proposed jamming signal design is a sparse representation technique, where the majority of the jamming signal weights are zero. This implies that the proposed sparse jamming signal design approach is greener. In this paper, we consider the secondary user access control model in a massive MIMO communications environment and propose a jamming signal design approach that utilizes uplink CSI between the PU and SUs. To design the jamming signal, an optimization problem is considered and we show that such a non-convex optimization problem can be optimally solved in some special cases. In addition to that, we show that an alternative solution to solve our nonconvex problem can be determined through a K-best method based technique [18]. Specifically, K is a tunable parameter in the K-best method that controls performance and complexity tradeoffs. Consider this optimization problem, the original problem is decomposed into optimally solvable subproblems. We then formulate this jamming signal design to a non-convex p -norm optimization problem with 0 ≤ p ≤ 1. It has been shown that such p -minimization problem can be effectively solved via an iteratively reweighted algorithm [19], [20], [21], which consists of solving a sequence of weighted 2 -norm minimization problems where the weights used for the next iteration are computed based on the values of the current solution. Finally, we evaluate the proposed technique under practical scenarios with software simulation, in terms of the SU's spectrum sensing results.
In summary, there are the following contributions presented in this paper.
• We propose an effective sub-optimal approach based on the K-best method to address the jamming signal design of our non-convex QCQP problem at each iteration.
The original problem has been decomposed into small sub-problems which can be optimally solved. • We propose a jamming signal design approach in CR networks with sparse weight representation, which leads to a greener system design. The sparsity is obtained by solving the proposed non-convex p -minimization (0 ≤ p ≤ 1) problem with an iterative reweighted approach. • We then evaluate our approach with practical scenarios simulation. We show that a sparse jamming signal weight can be effectively generated and the A-SUs spectrum sensing performance is well-maintained while UA-SUs' sensing performance is significantly impacted. This paper is organized as follows. Section II presents the system model and problem formulation. Section III shows the jamming signal design and detailed K-best method. Section IV describes the jamming signal with the general model considered. Section V presents an alternate optimization approach based on the SDR technique. Section VI describes the iteratively reweighted algorithms for sparse jamming signal design. Section VII presents simulation results and related discussions. Finally, conclusions are given in Section VIII.

II. SYSTEM MODEL AND PROBLEM FORMULATION
The proposed SUAC system model in a massive MIMO cognitive radio network is presented in Fig. 1. Each PU is composed of M antennas to serve multiple single-antenna SUs. Assume there are total L SUs with K a A-SUs and K ua UA-SUs (K a + K ua = L). h ∈ C M×1 represents the channel link between the PU and SU, s ∈ C M×1 is the transmitted PU message signal vector, g ∈ C M×1 denotes the jamming signal sent by the PU, and n denotes the white Gaussian noise with 0 mean and σ 2 variance. H 0 /H 1 denotes PU absent/present status. The jamming signal is only generated when the PU is absent, and the received signal r at the SU side can be expressed as To further simplify our explanations, the CSI between PU and A-SUs is represented as H a [h a,1 , h a,2 , . . . , h a,K a ] ∈ C M×K a , where h a,i ∈ C M×1 represents the channel link between the PU and i th A-SU. The channel link between the primary and UA-SUs is H ua [h ua,1 , h ua,2 , . . . , h ua,K ua ] ∈ C M×K ua , and h ua,i ∈ C M×1 represents the channel link between the PU and i th UA-SU. Given H a and H ua , the PU plans to generate jamming signal g to minimize g H h a,i and simultaneously maximize g H h ua,i . In the following sections, we present K-best method based algorithms to create the jammer. For convenience, we present some of the mathematical symbols used in this paper in Table 1.

III. APPROACH I: JAMMING SIGNAL DESIGN
We formulate designing the jamming signal problem as where R a,i = h a,i h H a,i ∈ C M×M (i = 1, . . . , K a ) denotes the covariance of i th A-SU. R ua,j = h ua,j h H ua,j ∈ C M×M (j = 1, . . . , K ua ) denotes the covariance matrix of j th UA-SU. τ is a large-magnitude scalar. We start by noting that the first constraint can be eliminated if the jamming vector g is in the null space of A-SU's corresponding channel covariance matrix R a,i . Let us define g = Bε, where B = ∩ K a i=1 null(R a,i ) is the intersection of vector space null(R a,i ) with dimension B ∈ C M×S (0 < S ≤ M) and satisfies B H B = I. The objective of (2) can be expressed as and the first constraint in (2) is eliminated since B H R a,i = 0, and we always have The second constraint of (2) can be further written as We can now equivalently express problem (2) as with dimension S × K ua , we provide the following proposition. Proposition 1: Any optimal solution of problem (5) must lie in Range(C ).
Proof: We first provide the following QR de-compositions where Q is a semi-unitary matrix whenever it is non-zero, and R = [r i,j ] is an upper triangular matrix.
The objective in problem (5) can be further written as From (8) we can infer that none of the constraints depend onβ, it is optimal to setβ equals to zero.
Define as a K ua ×K ua permutation matrix and C = C . Specify a QR de-composition C = QR, where R is a K ua ×K ua upper triangular matrix with non-negative diagonal elements. Q is an S×K ua matrix that contains a zero-column corresponding to each diagonal element of R that is zero. With Proposition 1 in hand, we can expand the jamming signal vector we seek as ε = Qβ. From (8) and (9), problem (5) can be rewritten as

A. K -BEST ALGORITHM
In this section, we present the detailed steps of the proposed We present the steps of the K-best algorithm in the following paragraphs.
• Step 1: There are K ua selection choices and each choice corresponded descendant is one column from matrix C . Through two-step sub-problems solving which is described in the following sections, we compute each descendant's metrics and sort the metric's value in ascending order. By selecting the K smallest values, we keep their descendants in the first step. • Step j(2 ≤ j ≤ K ua − 1): We expand each K survivor from the j − 1 step by considering all possibilities that are not been selected in previous steps. Thus, each survivor will be expanded with a new column from matrix C that corresponds to the newly selected descendant. In step j, there are K survivors and each survivor with K ua − j + 1 descendants. There are (K ua − j + 1)K updated optimization sub-problems that need to be solved in this step. Given the calculated value, we sort the metrics in descending order and choose those K lowest values corresponding to descendants in this step.
• Step K ua (last step): There is only one remaining descendant for each one of the K remaining nodes from the previous K ua − 1 steps. There are K expanded optimization subproblems in the last step. By solving those sub-problems and sorting the value from low to high, the final survivor will be selected as the descendant with the smallest value. Fig. 2 shows a K-best algorithm diagram where there are 5 steps in this example, and we keep K = 2 descendants in each of the first K ua − 1 steps. For instance, after calculating all 5 sub-problems in step 1, we sort the calculated metrics from low to high and keep the user 2 and 4 as the survivors. In the last step, the descendant with the minimum metrics will be selected and the selected path is 4 → 3 → 2 → 1 → 5.

B. SUCCESSIVE STEP
In j th step, the associated matrix C has j − 1 columns from C and we have C = QR. The computation conducted from this survivor determined vector β = [β 1 , . . . , β j−1 ] T such thatε = Qβ is a sub-optimal solution to a relaxation of problem (10) which only has constraints corresponding to j − 1 columns of C. Let c denotes any column from the remaining K ua − j + 1 un-selected columns from C .
denotes the updated and expanded β. Notice that γ and β are two complex variables that are determined by solving the following sub-problem.
Notice that only the last constraint depends on the phases of β and γ , the other constraint and the objective function are invariant to the phases. Without loss of optimality, we can suppose that the phase of β equals to the phase of r j,j , and the phase of γ equals the phase of where we have a convex quadratic objective and convex constraint conditions. We can explicitly solve this optimization problem through Karush-Kuhn-Tucker (KKT) conditions. We start by noting that either Detailed proof can be found in Appendix B. (12) is given as follows:

IV. APPROACH II: GENERAL MODEL
In previous Approach I analysis, jamming vector g is calculated by maximizing impact on UA-SUs (g H R ua g ≥ τ ) and minimizing the influence on A-SUs (g H R a g = 0). However, under the condition that A-SUs channel vector h a and UA-SUs channel vector h ua are highly correlated, it will be challenging to obtain g satisfies the constraints shown in Approach I at the same time. To address this issue, consider the following problem, where w i ∈ R + (i = 0, 1, . . . , K a ) are given weights. Specifically, if we want to find a jamming signal g has a negligible effect on A-SUs, we could change w i (i = 1, 2, . . . , K a ) to be a large value, in order to force the jamming signal within the desired subspace. Define a matrix and a matrix H a = [h a,1 , . . . , h a,K a ] ∈ C M×K a . Consider following QR de-composition where Q is a semi-unitary matrix whenever it is non-zero, and R = [r i,j ] is an upper triangular. Suppose any candidate solutions for problem (13) can be represented as g = Qβ + Qβ, where Range(Q) ∪ Range(Q) spans the whole space, Q H Q = I &Q H Q = 0. We extend Proposition 1 to obtain the following Proposition 3. The detailed proof can be found in Appendix C. Proposition 3: Any optimal solution of problem (13) must lie in Range([C , H a ]).
From (60) we have h H a,iQβ = 0. We then rewrite the objective function in problem (13) as follows From Proposition 3, we can further simplify the objective by settingβ to zero. Problem (13) can be expressed as We then present a solution to solve the problem (16) via a sub-problem approach based on the K-best method.

A. SUCCESSIVE STEP
In j th step, assume the associated matrix C has j − 1 columns from C . The computation conducted from this survivor determined vector β = [β 1 , . . . , β j−1 ,β 1 , . . . ,β K a ] T . Let c denotes any column from the remaining K ua − j + 1 un-selected columns from C . The updated de- denotes the updated and expanded β . Here γ, β,β 1 , . . . ,β K a are K a + 2 variables that are determined by solving following optimization problem min γ,β,β 1 ,...,β Ka ∈C which can be expressed as the following compact form r k,j+i β * k , r j,j+i , r j+1,j+i , . . . , r j+i,j+i , 0, . . . , 0 Denote y D 1 2 x with dimension y ∈ C (K a +2)×1 , and we then have x = D − 1 2 y. Optimization problem (18) can be rewritten as Define a matrixC = [D − 1 2 v, D − 1 2 u] ∈ C (K a +2)×2 , and the corresponded QR de-composition isC = UL, where U ∈ C (K a +2)×2 is a semi-unitary matrix satisfies U H U = I and L ∈ C 2×2 is an upper triangular matrix which can be represented as Without loss of optimality, denote y = U[ν, ζ ] T , the new formulation can be expressed as Notice that only the last constraint depends on the phase of ν whereas the other constraint and the objective function are invariant to them. Thus, we can suppose that the phase of ν equals to the phase of l 1,2 . Suppose |ν| = (ã +b) with a = 1 l 1,1 , and |ζ | =c, problem (23) is equivalent to miñ b,c∈R + ã +b 2 +c 2 s.t.ã = 1 l 1,1 , l 1,2 ã +b +cl 2,2 ≥ 1 (24) such formulation is a convex optimization problem and can be solved in closed form, which is provided in the following Proposition 4.
Proposition 4: Any optimal solution of problem (23) is given as follows: • ν opt = 1 When > 0, the optimal solution is obtained by investigating relationships between the circle, (b + 1 l 1,1 ) 2 +c 2 = C, and the line, |l 1,2 |b + l 2,2c = . Specifically, the line is tangential to the circle at the point denoted as S. In Fig. 3, the optimal solution is marked by a red dot. As shown in Fig. 3 (left), the tangent point S is the optimal solution if S is located in the first quadrant. Otherwise, the optimal solution is obtained at the intersection point (shown in Fig. 3

(right)) of the straight line and the vertical axis (c-axis).
Denote the coordinate of point S as (Sb, Sc), and after solving the following two equations,

V. ALTERNATE METHOD
In this section, we briefly show an alternative jamming signal design method based on the semidefinite relaxation (SDR) [10] technique, which has been discussed in [9].
Consider the constraint problem shown in Eq. (2), which we have rewritten as follows Such QCQP problem can be solved via the semidefinite relaxation approach. Denote G = gg H , and G is a positive symmetric semidefinite matrix. Without considering the rank of matrix G equals to one, the problem of (27) can be represented as where H M×1 denotes the set of M × M complex Hermitian matrices. ψ t is a vector randomly distributed with ψ t ∼ CN (0, G * ), t = 1, . . . , T, where T denotes the randomization trials number. Consider each trail (say t th trail), we obtain the feasible solution The best solution after T trails can be determined by solving the problem g * = arg min g t (ψ) g t (ψ) H g t (ψ). The worstcase computation complexity of the SDR problem (28) is O(max{L, M} 4 M 1/2 log(1/δ)) [10], where δ > 0 is a given solution accuracy.

VI. ITERATIVELY REWEIGHTED ALGORITHMS FOR SPARSE JAMMING SIGNAL DESIGN
From the perspective of a jammer in the cognitive radio network, the jammer intends to generate a jamming signal with a sparse configuration in order to mitigate the influence added to the CR network. To design a sparse jamming signal, instead of formulating a 0 optimization problem (NP-hard), we consider 1 optimization as an approximation solution. In addition, it was shown empirically that using p with p < 1 can obtain more sparse results than with p = 1 [19].
In this research, we plan to consider p -norm optimization problem with p ∈ [0, 1] in the jamming signal g design. In particular, similar to the problem shown in (13), we re-design our problem in the following format: where the objective function is a p -norm of the jamming signal g, with 0 ≤ p ≤ 1. Early papers considered iteratively reweighted algorithms for solving this p (0 ≤ p ≤ 1) norm optimization problem [20], specifically, we replace the p objective function in (29) by a weighted 2 -norm g p = M k=1 α k g * k g k , where g k represents the k th component in vector g, and weights α k are computed from the previous iterate g (t−1) . After considering the damping approach, the weight α k is given as α k = (g * k g k + φ) p 2 −1 . The parameter φ > 0 is introduced in order to provide stability [21]. Appropriate determination of the value of φ is very crucial in order to perform a fast, accurate, and stable algorithm process. Given ϒ = diag{α 1 , . . . , α M }, we have g p = ϒ 1 2 g 2 . Problem (29) can now equivalently be expressed as The above non-convex optimization problem can be solved via an iterative algorithm that alternates between estimating g and updating the weight α k . During each iteration, g will be calculated using the proposed K-best algorithm. The steps of the algorithm are as follows: (1). Initialize the iteration number t to zero, determine the parameter φ's initial value, and set α (2). Solve the following non-convex QCQP minimization problem via the K-best approach: Detailed procedures to solve this problem can be seen in Appendix D.
Assume there are total T r iterations, the worst-case (i.e., the convergence condition during the iteration never satisfied) computation complexity of the proposed iteratively reweighted algorithm is O(T r K ua ML 2 ).

VII. SIMULATION RESULTS AND RELATED DISCUSSIONS
In this section, we present the simulation results of the proposed method. We consider a single-cell scenario in which the PU equipped with M = 64 transmission antennas, communicates with L single antenna-equipped SUs. We adopt the popular one-ring channel model in our simulation, which has been extensively studied to characterize massive MIMO channels [22], [23], [24]. Denote the total number of the propagation paths as P, the channel vector h can be expressed as h = 1 ] T is the steering vector, χ represents the signal wavelength, d denotes the distance between neighboring antenna elements, and θ p ∈ [0, π] represents the p th path azimuth angle of arrival (AoA). In our simulation, we have P = 100, d = 1 2 χ , and = 1. The mean angle of i th SU,θ i , is assumed to be uniformly distributed over an interval [ π 4 − π 6 , π 4 + π 6 ]. Denote the angle of i th SU's p th path as θ i,p , which is assumed uniformly distributed over an interval [θ i − π 10 ,θ i + π 10 ]. In the following figures, the key simulation results are presented as follows. There are two important measures in cognitive radio: PU detection probability (P d ) and false alarm probability (P f ). Notice that, P f should be a very small value in order to let SUs find the empty spectrum efficiently, and P d is a large value (i.e., P d ≥ 90%) in system design in order to avoid significant interference to the PUs.

A. PERFORMANCE OF VARIOUS K
In this section, we present the spectrum sensing performance of SUs with various numbers of K considering the K-best optimization algorithm. Number of SUs is L = 30, among which there are K a = 15 A-SUs and K ua = 15 UA-SUs. Fig. 4 shows A-SU and UA-SU's false alarm probability P f versus signal-to-noise ratio (SNR) under the condition that the detection probability is P d = 99% consider Approach I. Perfect CSI scenario is considered and K = 1, 5, 15. From the simulation results, we have the following observations. 1) For A-SUs with a different number of K, P f performance improves with the increasing SNR values.
2) The spectrum detection performance of UA-SU is significantly degraded by the proposed method (Approach I).
3) For A-SUs, the detection performance with various K is very close. 4) For UA-SUs, the detection performance becomes worse given larger K. Fig. 5 presents A-SU and UA-SU's P f versus SNR under the condition that P d = 99% consider Approach II. K = 1, 15, and perfect CSI is considered. We set w i = 100(i = 1, . . . , K a ) and w 0 = 1 in Eq. (13). From the simulation results we can see that: 1) A-SUs spectrum sensing performance improves with the increasing SNR value and is irrelevant to the K value.
2) UA-SUs spectrum sensing performance was significantly degraded by the proposed Approach II algorithm. A larger K value corresponds to better jamming performance.

B. PERFORMANCE OF VARIOUS W I IN APPROACH II
In this section, we show the SUs spectrum sensing performance in Approach II with a various number of weights w i (w i = 10, 1000) in Eq. (13). We consider K a = 15 A-SUs and K ua = 15 UA-SUs in our simulation. Fig. 6 shows A-SU and UA-SU's P f versus SNR given P d = 99%, K = 5, and perfect CSI. We have the following observations from the simulation results.
1) For A-SUs, larger weight w i value (w i = 1000) shows better spectrum sensing performance.
2) For UA-SUs, smaller weight w i value (w i = 10) have a better jamming performance.

C. DEPENDENT CHANNEL VECTOR BETWEEN A-SUs AND UA-SUs
We consider the spectrum sensing performance when the channel vector of A-SUs (h a ) and UA-SUs (h ua ) are highly dependent for Approach I and II. Specifically, consider there are 25 A-SUs (K a = 25) and 5 UA-SUs (K ua = 5). We first randomly generate the A-SUs channel vector based on the one-ring model and then generate the UA-SUs channel vector by taking a linear combination of the generated A-SUs channel. Fig. 7 presents the A-SU/UA-SU spectrum sensing performance versus SNR given P d = 99%, K = 5, w i = 100, and perfect CSI. From the simulation results, we have the following observations. 1) For Approach I, both A-SUs and UA-SUs spectrum sensing performance improves with increasing SNR value. There will be no secondary user access control given the larger SNR value. Performance is significantly affected due to the channel dependence between A-SU and UA-SU.
2) For Approach II, with increasing SNR value, only A-SUs spectrum sensing performance improved. UA-SU sensing performance is significantly degraded by the proposed algorithm. The channel dependence between A-SUs and UA-SUs will not affect Approach II algorithm effectiveness.
The simulation results verified our design purpose for Approach II. Specifically, if the A-SUs channel vector h a and UA-SUs channel vector h ua are dependent, it will be challenging to generate the jamming signal satisfies the constraints shown in Approach I at the same time. Approach II is designed to address this issue, where the objective function is to minimize the weighted sum of the jamming signal vector squares and the jamming signal influence on K a A-SUs. Thus, if the channel vectors of A-SUs and UA-SUs are highly correlated or dependent, the spectrum sensing result of Approach II has a much better performance compared to that of Approach I, as shown in Fig. 7.

D. PERFORMANCE OF ITERATIVELY REWEIGHTED ALGORITHMS FOR JAMMING SIGNAL DESIGN
In this section, we show the spectrum sensing performance considering the iterative reweighted algorithms. We consider L = 30 SUs (15 A-SUs and 15 UA-SUs), K = 15, and P d = 99% in our simulation.  and M = 64. The damping parameter φ has been investigated with various values, say φ = 0.2 and φ = 0.8. As we discussed in Section VI, the damping parameter φ is designed to regularize the optimization process. Without the damping parameter, the weight α k is undefined whenever g k = 0. Our simulation result shows that a larger value of φ (φ = 0.8) corresponds to better jamming performance: A-SU has better spectrum sensing performance while UA-SU has worse sensing performance. Decreasing φ too soon results in poorer performance. In our future research, we plan to perform experiments with the φ-regularization strategy [19] of using a relatively large φ and then repeating the process by decreasing the damping parameter values after convergence to recover the sparse jamming signal. Fig. 9 and Fig. 10 present the false alarm probability P f of A-SU/UA-SU versus SNR given φ = 0.5 with various values of p. The BS antenna number is M = 64 for Fig. 9 and M = 128 for Fig. 10. From the simulation results, we can see that there is no significant performance difference among different p values (p = 0.2, 0.5, and 0.8). Sparse jamming signal representation (lower p values) will maintain good spectrum sensing performance for A-SUs while introducing a significant impact on UA-SUs with a much lower power cost. Fig. 11 and Fig. 12 present the non-zero element ratio of the jamming signal versus p value given φ = 0.5. The BS antenna number is M = 64 for Fig. 11 and M = 128 for Fig. 12. In our simulation, we set the jamming signal element equal to zero whenever the element magnitude is smaller than 10 −4 , and count the non-zero element number, divide the antenna number M in order to calculate the nonzero element ratio. From Fig. 11 and Fig. 12, we can see that the non-zero element ratio rises with p value increasing for both M = 64 and M = 128.
In Fig. 13, we present the false alarm probability of A-SU/UA-SU versus SNR given p = 0.5 and φ = 0.5 with 2) For UA-SUs, the spectrum sensing performance is significantly degraded. Larger BS antenna number M (i.e., M = 256) shows better jamming performance.
From the above observations, we can see that there is a trade-off between improving spectrum sensing performance on A-SUs and having a powerful jammer on UA-SUs when selecting large or small M values. A larger value of M corresponds to a better jammer while the spectrum sensing performance of A-SUs degrades.  ratio decreases (the jamming signal becomes sparser) with the M value increasing.
Notice that our proposed iteratively reweighted jamming signal design not only shows equally good or better performance compared with the traditional SUAC method, but we also present a sparse jamming signal weight which leads to a greener system design. The visualization of the jamming signal weights generated from our proposed iteratively reweighted algorithm and traditional SUAC is illustrated in Fig. 15, where we plot the weight magnitude of a 16 × 100 matrix with M = 16 antennas and 100 trails. The figure colors correspond to the data values of the weight matrix and the colorbar on the right indicates the mapping of data values into the colors. From the figure, we can see that our iteratively reweighted approach shows a very sparse jamming signal weight solution.

VIII. CONCLUSION
We proposed a new formulation for jamming signal design that uses the channel state information between SU and PU in a massive MIMO communication architecture. We consider designing the jamming signal process as a non-convex quadratic programming optimization problem and solve this problem via the K-best method. We have shown that A-SUs are able to achieve reliable spectrum sensing performance while UA-SUs suffer significant performance degradation given the carefully designed jamming signal.