# AN-Aided Transmit Beamforming Design for Secured Cognitive Radio Networks with SWIPT

Authors: Weili Ge; Zhengyu Zhu; Zhongyong Wang; Zhengdao Yuan
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

We investigate multiple-input single-output secured cognitive radio networks relying on simultaneous wireless information and power transfer (SWIPT), where a multiantenna secondary transmitter sends confidential information to multiple single-antenna secondary users (SUs) in the presence of multiple single-antenna primary users (PUs) and multiple energy-harvesting receivers (ERs). In order to improve the security of secondary networks, we use the artificial noise (AN) to mask the transmit beamforming. Optimization design of AN-aided transmit beamforming is studied, where the transmit power of the information signal is minimized subject to the secrecy rate constraint, the harvested energy constraint, and the total transmit power. Based on a successive convex approximation (SCA) method, we propose an iterative algorithm which reformulates the original problem as a convex problem under the perfect channel state information (CSI) case. Also, we give the convergence of the SCA-based iterative algorithm. In addition, we extend the original problem to the imperfect CSI case with deterministic channel uncertainties. Then, we study the robust design problem for the case with norm-bounded channel errors. Also, a robust SCA-based iterative algorithm is proposed by adopting theS-Procedure. Simulation results are presented to validate the performance of the proposed algorithms.


---

## Body

## 1. Introduction

The fifth-generation (5G) wireless technology is expected to satisfy an increasing demand for wireless device, such as high data service and radio coverage [1, 2]. However, the explosive increase of mobile terminal has resulted in severe scarcity of radio spectrum resources, which has become an outstanding problem [3–5]. As one of the most efficient ways to alleviate the problem of spectrum scarcity for green communications and networks, cognitive radio (CR) is a promising approach for heightening the spectrum utilization ratio [6, 7]. In CR networks, so long as the interference produced by the secondary transmitter (ST) is tolerable to each primary user (PU), a secondary user (SU) can employ the licensed spectrum of the primary system [8]. Although CR technology can significantly increase in the spectrum efficiency, the energy scarcity still give rise to a major bottleneck problem for the quality of service (QoS) and the long lifetime of wireless users [9].Recently, a promising technique, simultaneous wireless information and power transfer (SWIPT), has been proposed to use the radio frequency- (RF-) enabled signal harvesting power in wireless networks, which can greatly help to solve the bottleneck problem of energy scarcity [10–18]. Compared to the conventional energy harvesting techniques, such as solar power and wind, there is an advantage for SWIPT in providing more stable and controllable energy to portable wireless devices [10]. For the SWIPT operation, the work [16] examined colocated receivers which employ a power splitter for energy harvesting (EH) and information decoding (ID). Therefore, it is really important to research the combining SWIPT with CR, which has the dual function of improving both the energy efficiency and spectrum efficiency, and it has attracted more attention in [19–28]. In SWIPT-based CR system, the energy efficiency optimization was considered in [19, 20]. Moreover, the SWIPT approach for CR schemes was investigated in cooperative relay networks [21], a thresholding-based antenna selection multiple-input multiple-output (MIMO) systems [22], multiuser MISO system [23–25, 27], cooperative nonorthogonal multiple access networks [26], and sensing-Based wideband CR [28].On the other hand, secrecy transmission has gained attentions in communication systems [29]. Unlike traditional cryptographic methods which are normally adopted in the network layer, physical layer security was developed from information theoretical aspects to improve the secrecy capacity of wireless transmission systems [30]. In the conventional SWIPT systems, since the energy-harvesting receivers (ERs) are normally assumed to be closer to the transmitter compared with information receivers (IRs), this results in a new information security issue. In such a situation, ERs have a possibility of eavesdropping the information sent to the IRs and thus can become potential eavesdroppers [31]. As a result, physical layer security has been recognized as an important issue for SWIPT systems [32]. Moreover, a few techniques have been proposed for multiple antenna secrecy systems to cause more interference to eavesdroppers [31, 33]. In SWIPT operation, artificial noise (AN) was embedded in the transmit beamforming signal to confuse the eavesdroppers and harvest power simultaneously [33].In addition, due to the inherent characteristics of CR with SWIPT, ERs may illegitimately access the PU bands and change the radio environment. In this case, the legitimate SU is unable to use frequency bands of the PU. Thus, in order to satisfy secure communication and EH requirement, the security of CR SWIPT with ERs is also of great importance [34–38]. In [34], an outage-constrained secrecy rate maximization (SRM) problem has been investigated in an underlay MIMO CR network where the secondary transmitter (ST) provides SWIPT to all receivers. In order to guarantee secure communication and energy harvesting in MISO CR network with SWIPT, [35] studied a robust secure AN-aided beamforming and power splitting (PS) design under imperfect channel state information (CSI). For a CR network consisting of a PS-based to decode information and harvest energy simultaneously, [36] analyzed the behavior of the nodes by using game theoretic techniques and proved the existence of a unique Nash equilibrium (NE) strategy. Considering a system with ERs acting as potential eavesdroppers in CR-based SWIPT, [37] has derived a closed-form analytical expression for the exact secrecy outage probability. In a CR MIMO-SWIPT broadcast channel, [38] studied the SRM problem by designing secrecy AN-aided precoding.Motivated by the above observations, in this paper, we study secrecy transmission over a MISO CR system with SWIPT, which consists of one primary transmitter-receiver pair (denoted as PT and PU), one multiantenna ST, multiple SUs, and multiple ERs (potentially eavesdropper). Under the perfect CSI case, we formulate a transmit power of the information signal minimization (TPISM) problem subject to the secrecy rate constraint for SUs and PU, the harvested energy constraint for ERs, and total transmit power constraint. We seek to jointly design strategies of secure beamforming and AN for MISO CR with SWIPT wiretap channels. Then, the framework is extended to robust designs for the imperfect CSI case by adopting deterministic CSI uncertainties. The main contributions for this paper are summarized as follows:(i)
For the two different types of channel models, the secrecy rate constraint with linear fractional programming is equivalently converted into several constraints with exponential form by introducing exponential variables.(ii)
For the perfect CSI case at ST and PT, unlike the conventional semidefinite relaxation (SDR), we propose a novel reformulation of the TPISM problem. A successive convex approximation- (SCA-) based iterative algorithm is proposed, where the nonconvex constraint is approximated as a convex one.(iii)
For the imperfect CSI case, we use the norm-bounded channel uncertainty to model channel. First, by applying the triangle inequality, the original constraints can be transformed as the infinitely inequality constraints. Then, by employingS-Procedure we convert these infinitely inequality constraints into finite linear matrix inequalities (LMIs). At last, by utilizing a SCA method, the recast TPISM problem is transformed to a semidefinite programming (SDP) problem, which can be directly solved to obtain a local optimal solution. A robust SCA-based iterative algorithm is proposed.The rest of this paper is organized as follows: Section2 presents the system model and problem formulation. In Section 3, the SCA-based iterative algorithm is proposed with perfect CSI case. Robust SCA-based iterative algorithm is developed with imperfect CSI case in Section 4. Section 5 illustrates the simulation results. Finally, we conclude the paper in Section 6.Notation. Vectors and matrices are denoted by bold lowercase and uppercase letters, respectively. ·H represents the Hermitian transpose. For a vector x, x indicates the Euclidean norm. · defines the absolute value of a complex scalar. CM×L and HM×L describe the space of M×L complex matrices and Hermitian matrices, respectively. For a matrix A, A⪰0 means that A is positive semidefinite, and trA indicate the trace, respectively. E· describes the mathematical expectation. R· stands for the real part of a complex number. [x]+ equals max⁡x,0. I denote the identity matrix with appropriate size.


## 2. System Model and Problem Formulation

In this section, we consider a MISO secured SWIPT cognitive radio networks, which consists of one PT, one PU, one multiantenna ST,K SU, and L ERs, as shown in Figure 1. We assume that the ST is equipped with NS transmit antennas and each ER and each SU has one receive antenna. This system operates as follows: the primary network shares their frequency spectrum with the secondary network, whereas the confidential messages from the PT and the ST are intercepted by the eavesdroppers.Figure 1
MISO Secured cognitive radio networks with SWIPT.In order to achieve secure transmission, the ST employs transmit beamforming with AN, which acts as interference to the ERs and provides energy to the SUs. The transmit signal vectorxs from the ST can be written as(1)xs=∑j=1Kwjsj+z,where wj∈CNs defines the transmit beamforming vector, sj with Esj2=1 is the information-bearing signal intended for the SUs, and z represents the energy-carrying AN.We assume a frequency-flat slow-fading channel. Therefore, the received signal at the PU, thekth SU, and the lth ER can be given by, respectively,(2)yp=Ppgpsp+hpHxs+np,(3)yk=Ppgs,ksp+hs,kHxs+ns,k,k=1,…,K,(4)ye,l=Ppge,lsp+he,lHxs+ne,l,l=1,…,L.where gp∈C and hp∈CNS are denotes by the channel between the PT and PU as well as that between the ST and PU, gs,k∈C and hs,k∈CNS indicate the channel between the PT and the kth SU as well as that between the ST and the kth SU, ge,l∈C and he,l∈CNS are the channel between the PT and the lth ER as well as that between the ST and the lth ER, sp is confidential information-bearing signal for the PU from the PT satisfying Esp2=1, and Pp represents the transmitting power from the PT. In addition, np~CN(0,σp2), ns,k~CN0,σs,k2, and ne,l~CN(0,σe,l2I) denote the complex Gaussian noise at the PU, the kth SU, and the lth ER, respectively.Thus, the channel capacity of thekth SU can be written as(5)Rs,k=log⁡1+hs,kHwkwkHhs,khs,kH∑j≠kwjwjH+Zhs,k+Ppgs,k2+σs,k2.where Z=zzH. According to [3, 25], it is assumed that the kth SU can successfully decode the information from the PT by exploiting the successive interference cancellation and, thus, (5) can be rewritten as(6)R~s,k=log⁡1+hs,kHwkwkHhs,khs,kH∑j≠kwjwjH+Zhs,k+σs,k2.Moreover, the channel capacity of thelth ER for decoding the desired signal of the kth SU can be represented as(7)Re,lk=log⁡1+he,lHwkwkHhe,lσe,l2+he,lH∑j≠kwjwjH+Zhe,l+Ppge,l2.Thus, the secrecy capacity of thekth SU can be written as(8)Rsu,k=R~s,k-maxl⁡Re,lk+,∀k,l.In order to achieve the more reliable secrecy capacity (i.e., the minimum channel capacity at thek-th SU), (7) can be thus rewritten as (9)R~e,lk=log⁡1+he,lHwkwkHhe,lσe,l2+he,lHZhe,l.Then, we can obtain a lower bound of the secrecy capacity of thekth SU as follows:(10)Rsu,kmin=R~s,k-maxl⁡R~e,lk+,∀k,l.Additionally, the channel capacity of the PU can be expressed as(11)Rpu=log⁡1+Ppgp2hpH∑k=1KwkwkH+Zhp+σp2.The channel capacity of the lth ER for decoding the PU is given as(12)Re,lpu=log⁡1+Ppge,l2he,lHZhe,l+σe,l2.Hence, the secrecy capacity of the PU under the considered worst case scenario can be written as(13)Rspu=Rpu-maxlRe,lpu+,∀l.Moreover, the harvested power at thelth ER is expressed as, respectively,(14)Ele=ηe,lhe,lH∑j=1KwjwjH+Zhe,l+Ppge,l2+σe,l2,∀l,where 0≤ηe,l≤1 denote the energy conversion efficiency at the lth ER.In this paper, our aim is to minimize the transmit power of the information signal subject to the secrecy rate constraint at the SUs, the harvested energy constraint at the ERs, and the total transmit power constraint. Based on the system models, the TPISM problem is formulated as(15a)minwk,Z∑k=1Kwk2(15b)s.t.minkRsu,kmin≥R-su,∀k,(15c)Rspu≥R-pu,∀l,(15d)minlEle≥E-e,∀l,(15e)∑k=1Kwk2+trZ≤P,∀k,(15f)Z⪰0.where P is the total transmit power, R¯su and R¯pu denote the secrecy capacity requirement of the kth SU and the PU, and E¯e mean the harvested power requirement of the lth ER, respectively. The constraints (15b) and (15c) guarantee that the minimum secrecy rate should be achieved by the kth SU and the PU, respectively. The constraint (15d) guarantees that the minimum harvested power at the lth ER is no less than E¯e, respectively. The constraint (15e) limits the total transmit power of the ST. It is seen that problem (15a) are nonconvex, difficult to be solved directly.


## 3. Proposed Algorithm with Perfect CSI

In this section, we investigate a joint design of the transmit beamforming and AN for systems under the assumption that the PT and ST can obtain perfect CSI. Problems (15a)–(15f) can be rewritten as(16a)minwk,Z∑k=1Kwk2(16b)s.t.log⁡1+hs,kHwkwkHhs,khs,kH∑j≠kwjwjH+Zhs,k+σs,k2-log⁡1+he,lHwkwkHhe,lσe,l2+he,lHZhe,l≥R-su,∀l,k,(16c)log⁡1+Ppgp2∑j=1KhpHwjwjHhp+hpHZhp+σp2-log⁡1+Ppge,l2he,lHZhe,l+σe,l2≥R-pu,∀l,k,(16d)ηe,lhe,lH∑j=1KwjwjH+Zhe,l+Ppge,l2+σe,l2≥E-e,∀l,(16e)15e,15f.In order to solve the nonconvex problems (16a)–(16e), we firstly define Wk≜wkwkH and denote WZ=∑j=1KwjwjH+Z. Then, the constraints (16b) and (16c) can be reformulated as(17a)log⁡hs,kH∑j≠kWj+Zhs,k+σs,k2+hs,kHWkhs,khs,kH∑j≠kWj+Zhs,k+σs,k2-log⁡σe,l2+he,lHZhe,l+he,lHWkhe,lσe,l2+he,lHZhe,l≥R-su,(17b)log⁡hpHWZhp+σp2+Ppgp2hpHWZhp+σp2-log⁡he,lHZhe,l+σe,l2+Ppge,l2he,lHZhe,l+σe,l2≥R-pu,where they can be also rewritten as (18a)hs,kHWZhs,k+σs,k2σe,l2+he,lHZhe,lhs,kH∑j≠kWj+Zhs,k+σs,k2σe,l2+he,lHZ+Wkhe,l≥2R-su,∀l,k,(18b)hpHWZhp+σp2+Ppgp2he,lHZhe,l+σe,l2hpHWZhp+σp2he,lHZhe,l+σe,l2+Ppge,l2≥2R-pu,∀l,k.For solving linear fractional programming (18a) and (18b), we can introduce the following exponential variables to equivalently convert. Then, we introduce slack variables xk, yl, tk, rl,k, xp, ul, tp, and sl. Equations (18a) and (18b) can be transformed as follows, respectively:(19a)exk+yl-tk-rl,k≥2R-su,(19b)hs,kHWZhs,k+σs,k2≥exk,(19c)σe,l2+he,lHZhe,l≥eyl,(19d)hs,kH∑j≠kWj+Zhs,k+σs,k2≤etk,(19e)σe,l2+he,lHZ+Wkhe,l≤erl,k.(20a)exp+ul-tp-sl≥2R-pu,(20b)hpHWZhp+σp2+Ppgp2≥exp,(20c)he,lHZhe,l+σe,l2≥eul,(20d)hpHWZhp+σp2≤etp,(20e)he,lHZhe,l+σe,l2+Ppge,l2≤esl.It is noted that the aforementioned constraints (19a), (19d), (19e), (20a), (20d), and (20e) are not still convex. Firstly, (19a) and (20a) can be reshaped respectively as the following convex constraints:(21a)e-xk-yl+tk+rl,k≤2-R-su,∀k,∀l,(21b)e-xp-ul+tp+sl≤2-R-pu,∀l.Secondly, an SCA method is used to jointly design the secure beamforming and AN matrix. Let us define tk(n), rl,k(n), tp(n), and sl(n) as the variables tk, rl,k, tp, and sl at the nth iteration for the SCA method. By adopting a Taylor series expansion exl(n)(xl-xl(n)+1)≤exl, we can convert the nonconvex constraints (19d), (19e), (20d), and (20e) to their corresponding convex approximations as(22a)hs,kH∑j≠kWj+Zhs,k+σs,k2≤etkntk-tkn+1,(22b)σe,l2+he,lHZ+Wkhe,l≤erl,knrl,k-rl,kn+1,(22c)hpHWZhp+σp2≤etpntp-tpn+1,(22d)he,lHZhe,l+σe,l2+Ppge,l2≤eslnsl-sln+1.At last, we consider the constraints (16d) and (15e), which can be converted as(23a)he,lHWZhe,l+Ppge,l2+σe2≥E-eηe,l,∀l,(23b)∑k=1KtrWk+trZ≤P,∀k.According to equations from (16a) to (23b), an SCA-based iterative algorithm is proposed. At the (n+1)th iteration, by removing the nonconvex rank-one constraint rank(Wk)=1,∀k, problems (16a)–(16e) can be thus reformed as (24)minΩ∑k=1KtrWks.t.15f,19b,19c,20b,20c,21a,21b,22a,22b,22c,22d,23a,23b,23c,Wk⪰0,Ω=Wk,Z,xk,yl,tk,rl,k,xp,ul,tp,sl.For givenΨ(n)=tkn,rl,kn,tpn,sln as the optimal solution obtained at the nth iteration, problem (24) is convex by removing the nonconvex rank(Wk)=1 constraint, which can be solved by using an CVX tools [39]. From SCA method, the convex approximation with the current solution is iteratively updated until the constraints (22a)–(22d) hold with equality, which implies that (16a)–(16e) can be optimally solved. The optimal solution obtained by the proposed SCA-based iterative algorithm at the nth iteration is assumed to be (tk(n), rl,k(n), tp(n), and sl(n)), which can achieve a stable point until the SCA-based iterative algorithm converges [35], and this is summarized in Algorithm 1.Algorithm 1:SCA-based iterative algorithm with perfect CSI.
Setn=0 and initialize Ψ(0)=tk0,rl,k0,tp0,sl0. Repeat (i) Solve problem (24) with {tk(n),rl,k(n),tp(n),sl(n)} and denote a solution by {tk(n+1),rl,k(n+1),tp(n+1),sl(n+1)}. (ii) Update n←n+1. UntilConvergence


## 4. Proposed Algorithm with Imperfect CSI

Because of channel estimation and quantization errors, it may not be possible to have perfect CSI in practice. In this section, we extend the proposed algorithm to more practical scenarios with imperfect CSI. First, we introduce one scenario of the norm-bounded channel uncertainty and then provide the problem formulation based on the norm-bounded channel uncertainty. Moreover, we consider a joint robust design of transmit beamforming and AN.


### 4.1. Norm-Bounded Channel Uncertainty

Now, we adopt imperfect CSI based on the deterministic model [31, 35]. In particular, we assume that the actual channel vector hs,k lies within a ball with the radius εs,k around the estimated channel vector h¯s,k from the BS to the kth user, i.e.,(25)hs,k∈Hs,k=h¯s,k+Δhs,k∣Δhs,k≤εs,k,∀l,he,l∈He,l=h¯e,l+Δhe,l∣Δhe,l≤εe,l,∀l,hp∈Hp=h¯p+Δhp∣Δhp≤εp,∀l,gp∈G^p=g¯p+Δgp∣Δgp≤γp,gs,k∈G^s,k=g¯s,k+Δgs,k∣Δgs,k≤γs,k,∀k,ge,l∈G^e,l=g¯e,l+Δge,l∣Δge,l≤γe,l,∀l.where h¯s,k, h¯e,l, h¯p, g¯p, g¯s,k, and g¯e,l are the estimated channel available, the channel estimation errors Δhs,k, Δhe,l, Δhp, Δgp, Δgs,k, and Δge,l are bounded by εs,k, εe,l, εp, γp, γs,k, and γe,l, respectively.By taking the norm-bounded channel uncertainty model into account, the TPISM problem can be rewritten as(26a)minwk,Z∑k=1Kwk2(26b)s.t.log⁡1+h-s,k+Δhs,kHwkwkHh-s,k+Δhs,kh-s,k+Δhs,kH∑j≠kwjwjH+Zh-s,k+Δhs,k+σs,k2-log⁡1+h-e,l+Δhe,lHwkwkHh-e,l+Δhe,lσe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,l≥R-su,∀l,k,(26c)log⁡1+Ppg-p+Δgp2h-p+ΔhpH∑j=1KwjwjH+Zh-p+Δhp+σp2-log⁡1+Ppg-e,l+Δge,l2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2≥R-pu,∀l,k,(26d)ηe,lh-e,l+Δhe,lHWZh-e,l+Δhe,l+Ppg-e,l+Δge,l2+σe2≥E-e,∀l,(26e)15e,15f.Utilizing a similar methodology in (17a)–(18b), the constraints (26b) and (26c) can be reformulated as(27a)h-s,k+Δhs,kH∑j=1KWj+Zh-s,k+Δhs,k+σs,k2σe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,lh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k+σs,k2σe,l2+h-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≥2R-su,(27b)h-p+ΔhpHWZh-p+Δhp+σp2+Ppg-p+Δgp2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2h-p+ΔhpHWZh-p+Δhp+σp2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2+Ppg-e,l+Δge,l2≥2R-pu.Then, by introducingx^k, y^l, t^k, r^l,k, x^p, u^l, t^p, and s^l as slack variables, we equivalently convert the linear fractional programming (27a) and (27b) as follows, respectively:(28a)ex^k+y^l-t^k-r^l,k≥2R-su,∀k,∀l,(28b)minΔhs,k≤εs,kh-s,k+Δhs,kHWZh-s,k+Δhs,k+σs,k2≥ex^k,∀k,(28c)minΔhe,l≤εe,lσe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,l≥ey^l,∀l,(28d)maxΔhs,k≤εs,kh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k+σs,k2≤et^k,∀k,(28e)maxΔhe,l≤εe,lσe,l2+h-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≤er^l,k,∀k,∀l.(29a)ex^p+u^l-t^p-s^l≥2R-pu,∀l,(29b)minΔhp≤εph-p+ΔhpHWZh-p+Δhp+σp2+Ppg-p+Δgp2≥ex^p,(29c)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2≥eu^l,∀l,(29d)maxΔhp≤εph-p+ΔhpHWZh-p+Δhp+σp2≤et^p,(29e)maxΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2+Ppg-e,l+Δge,l2≤es^l,∀l.By employing the slack variables (i.e.,dk, wl, bk, and fl,k) for (28b)–(28e), respectively, (28a)–(28e) can be equivalently modified as(30a)minΔhs,k≤εs,kh-s,k+Δhs,kHWZh-s,k+Δhs,k≥dk,(30b)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≥wl,(30c)maxΔhs,k≤εs,kh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k≤zk,(30d)maxΔhe,l≤εe,lh-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≤fl,k,(30e)dk+σs,k2≥ex^k,(30f)σe,l2+wl≥ey^l,(30g)zk+σs,k2≤et^k,(30h)σe,l2+fl,k≤er^l,k.For the worst-case based design in (26b)–(26d), the PT channel gains are upper-bounded and lower-bounded using the following triangle inequality properties:(31)x+y2≤x+y2=x2+y2+2xy,x+y2≥x-y2=x2+y2-2xy.Applying (31), it follows that(32)ge,lmax≜maxΔge,l≤γe,lge,l2=maxΔge,l≤γe,lg¯e,l2+Δge,l2+2g¯e,lΔge,l≤g¯e,l2+εe,l2+2εe,lg¯e,l,ge,lmin≜minΔge,l≤γe,lge,l2=minΔge,l≤γe,lg¯e,l2+Δge,l2-2g¯e,lΔge,l≥g¯e,l2+εe,l2-2εe,lg¯e,l,gpmin≜minΔgp≤γpgp2=minΔgp≤γpg¯p2+Δgp2-2g¯pΔgp≥g¯p2+εp2-2εpg¯p,gs,kmin≜minΔgs,k≤γs,kgs,k2=minΔgs,k≤γs,kg¯s,k2+Δgs,k2+2g¯s,kΔgs,k≥g¯s,k2+εs,k2-2εs,kg¯s,k.Similarly, we introduce the slack variablesdp, vl, bp, and ql for (29b)-(29e). Then, by substituting the above results (32) into (29b) and (29e), respectively, we can equivalently express (29a)–(29e) as(33a)minΔhp≤εph-p+ΔhpHWZh-p+Δhp≥dp,(33b)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≥vl,(33c)maxΔhp≤εph-p+ΔhpHWZh-p+Δhp≤bp,(33d)maxΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≤ql,(33e)dp+σp2+Ppgpmin≥ex^p,(33f)vl+σe,l2≥eu^l,(33g)bp+σp2≤et^p,(33h)ql+σe,l2+Ppge,lmax≤es^l.In order to make (26b)–(26d) more tractable, by applying S-procedure [40], we can convert the infinitely inequality constraints (30a)-(30d) and (33a)-(33d) into finite linear matrix inequalities (LMIs). For completeness, the S-procedure is presented in Lemma 1 in the following.Lemma 1 (S-procedure [40, Appendix B.2]).
Let a functionfm(x) with x∈CN×1(m=1,2) be defined as(34)fmx=xHAmx+2RebmHx+cmwhere Am∈HN×N, bm∈CN×1, and cm∈RN×1. Then, fm(x)≤0 holds if and only if there exists θ≥0 such that (35)θA1b1b1Hc1-A2b2b2Hc2⪰0,provided that there is a point x^ which satisfies fm(x^)<0.To employ theS-procedure, we rewrite the constraint (30a) as(36)Δhs,kHWZΔhs,k+2Rh¯s,kHWZΔhs,k+h¯s,kHWZh¯s,k≥dk.According to Lemma 1, by using a slack variable λs,k, (36) can be expressed as(37)λs,kI+WZWZh¯s,kh¯s,kHWZh¯s,kHWZh¯s,k-dk-λs,kεs,k2⪰0.Using the same approach for the constraints (30b)-(30d) and (33a)-(33d), we have(38a)λe,lI+ZZh-e,lh-e,lHZh-e,lHZh-e,l-wl-λe,lεe,l2⪰0,(38b)αs,kI-∑j≠kWj-Z-∑j≠kWj+Zh-s,k-h-s,kH∑j≠kWj+Z-h-s,kH∑j≠kWj+Zh-s,k+zk-αs,kεs,k2⪰0,(38c)αe,lI-Z-Wk-Z+Wkh-e,l-h-e,lHZ+Wk-h-e,lHZ+Wkh-e,l+fl,k-αe,lεe,l2⪰0,(39a)βpI+WZWZh-ph-pHWZh-pHWZh-p-dp-βpεp2⪰0,(39b)βe,lI+ZZh-e,lh-e,lHZh-e,lHZh-e,l-vl-βe,lεe,l2⪰0,(39c)αpI-WZ-WZh-p-h-pHWZ-h-pHWZh-p+bp-αpεp2⪰0,(39d)νe,lI-Z-Zh-e,l-h-e,lHZ-h-e,lHZh-e,l+ql-νe,lεe,l2⪰0,where λe,l, αs,k, αe,l, βp, βe,l, αp, and νe,l are slack variables.By using the above results of Taylor series expansion in (22a)–(22d), we can transform the nonconvex constraints (30g), (30h), (33g), and (33h) into the corresponding convex forms, respectively, as(40a)zk+σs,k2≤et^kmt^k-t^km+1,(40b)σe,l2+fl,k≤er^l,kmr^l,k-r^l,km+1,(40c)bp+σp2≤et^pmt^p-t^pm+1,(40d)ql+σe,l2+Ppge,lmax≤es^lms^l-s^lm+1.where t^k(m), r^l,k(m), t^p(m), and s^l(m) are the variables t^k, r^l,k, t^p, and s^l at the nth iteration for the SCA method.Here, we consider the EH constraint (30c). By use the inequalities (32) and (26d) can be rewritten as(41)h¯e,l+Δhe,lHWZh¯e,l+Δhe,l+Ppge,lmin+σe2≥E¯eηe,l,∀l.Introducing slack variablesϑe,l, we can transform (41) as(42)ϑe,lI+WZWZh¯e,lh¯e,lHWZh¯e,lHWZh¯e,l+Ppge,lmin+σe2-E¯eηe,l-ϑe,lεe,l2⪰0,Finally, similarly to (21a) and (21b), we need to recast the nonconvex constraints (28a) and (29a) as follows:(43a)e-x^k-y^l+t^k+r^l,k≤2-R-su,∀k,∀l,(43b)e-x^p-u^l+t^p+s^l≤2-R-pu,∀l.Combining all the results in (27a)–(34) and (36)–(43b), (26b)–(26d) can be reformed as(44)minΦ∑k=1KtrWks.t.23b,30e,30f,33e,33f,37,38a,38b,38c,39a,39b,39c,39d,40a,40b,40c,40d,42,43a,43b,Φ=Wk,Z,x^k,y^l,t^k,r^l,k,x^p,u^l,t^p,s^l,dk,wl,zk,fl,k,dp,vl,bp,ql,λs,k,λe,l,αs,k,αe,l,βp,βe,l,αp,νe,l,ζs,k,ϑe,l,Wk⪰0,Z⪰0.For given {t^k(m),r^l,k(m),t^p(m),s^l(m)} at the mth iteration, problem (44) is a convex SDP form without the nonconvex rank-one constraint rank(Wk)=1,∀k, which can be solved by convex optimization packages such as CVX [39], to update the solution for the m+1-th iteration until the algorithm converges. In addition, the proposed robust SCA-based iterative algorithm with imperfect CSI is summarized in Algorithm 2.Algorithm 2:Robust SCA-based iterative algorithm with imperfect CSI.
Setm=0 and initialize Ψ(0)=t^k0,r^l,k0,t^p0,s^l0. Repeat (i) Solve problem (44) with t^km,r^l,km,t^pm,s^lm and denote a solution by t^km+1,r^l,km+1,t^pm+1,s^lm+1. (ii) Update m←m+1. UntilConvergence


## 4.1. Norm-Bounded Channel Uncertainty

Now, we adopt imperfect CSI based on the deterministic model [31, 35]. In particular, we assume that the actual channel vector hs,k lies within a ball with the radius εs,k around the estimated channel vector h¯s,k from the BS to the kth user, i.e.,(25)hs,k∈Hs,k=h¯s,k+Δhs,k∣Δhs,k≤εs,k,∀l,he,l∈He,l=h¯e,l+Δhe,l∣Δhe,l≤εe,l,∀l,hp∈Hp=h¯p+Δhp∣Δhp≤εp,∀l,gp∈G^p=g¯p+Δgp∣Δgp≤γp,gs,k∈G^s,k=g¯s,k+Δgs,k∣Δgs,k≤γs,k,∀k,ge,l∈G^e,l=g¯e,l+Δge,l∣Δge,l≤γe,l,∀l.where h¯s,k, h¯e,l, h¯p, g¯p, g¯s,k, and g¯e,l are the estimated channel available, the channel estimation errors Δhs,k, Δhe,l, Δhp, Δgp, Δgs,k, and Δge,l are bounded by εs,k, εe,l, εp, γp, γs,k, and γe,l, respectively.By taking the norm-bounded channel uncertainty model into account, the TPISM problem can be rewritten as(26a)minwk,Z∑k=1Kwk2(26b)s.t.log⁡1+h-s,k+Δhs,kHwkwkHh-s,k+Δhs,kh-s,k+Δhs,kH∑j≠kwjwjH+Zh-s,k+Δhs,k+σs,k2-log⁡1+h-e,l+Δhe,lHwkwkHh-e,l+Δhe,lσe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,l≥R-su,∀l,k,(26c)log⁡1+Ppg-p+Δgp2h-p+ΔhpH∑j=1KwjwjH+Zh-p+Δhp+σp2-log⁡1+Ppg-e,l+Δge,l2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2≥R-pu,∀l,k,(26d)ηe,lh-e,l+Δhe,lHWZh-e,l+Δhe,l+Ppg-e,l+Δge,l2+σe2≥E-e,∀l,(26e)15e,15f.Utilizing a similar methodology in (17a)–(18b), the constraints (26b) and (26c) can be reformulated as(27a)h-s,k+Δhs,kH∑j=1KWj+Zh-s,k+Δhs,k+σs,k2σe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,lh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k+σs,k2σe,l2+h-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≥2R-su,(27b)h-p+ΔhpHWZh-p+Δhp+σp2+Ppg-p+Δgp2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2h-p+ΔhpHWZh-p+Δhp+σp2h-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2+Ppg-e,l+Δge,l2≥2R-pu.Then, by introducingx^k, y^l, t^k, r^l,k, x^p, u^l, t^p, and s^l as slack variables, we equivalently convert the linear fractional programming (27a) and (27b) as follows, respectively:(28a)ex^k+y^l-t^k-r^l,k≥2R-su,∀k,∀l,(28b)minΔhs,k≤εs,kh-s,k+Δhs,kHWZh-s,k+Δhs,k+σs,k2≥ex^k,∀k,(28c)minΔhe,l≤εe,lσe,l2+h-e,l+Δhe,lHZh-e,l+Δhe,l≥ey^l,∀l,(28d)maxΔhs,k≤εs,kh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k+σs,k2≤et^k,∀k,(28e)maxΔhe,l≤εe,lσe,l2+h-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≤er^l,k,∀k,∀l.(29a)ex^p+u^l-t^p-s^l≥2R-pu,∀l,(29b)minΔhp≤εph-p+ΔhpHWZh-p+Δhp+σp2+Ppg-p+Δgp2≥ex^p,(29c)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2≥eu^l,∀l,(29d)maxΔhp≤εph-p+ΔhpHWZh-p+Δhp+σp2≤et^p,(29e)maxΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l+σe,l2+Ppg-e,l+Δge,l2≤es^l,∀l.By employing the slack variables (i.e.,dk, wl, bk, and fl,k) for (28b)–(28e), respectively, (28a)–(28e) can be equivalently modified as(30a)minΔhs,k≤εs,kh-s,k+Δhs,kHWZh-s,k+Δhs,k≥dk,(30b)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≥wl,(30c)maxΔhs,k≤εs,kh-s,k+Δhs,kH∑j≠kWj+Zh-s,k+Δhs,k≤zk,(30d)maxΔhe,l≤εe,lh-e,l+Δhe,lHZ+Wkh-e,l+Δhe,l≤fl,k,(30e)dk+σs,k2≥ex^k,(30f)σe,l2+wl≥ey^l,(30g)zk+σs,k2≤et^k,(30h)σe,l2+fl,k≤er^l,k.For the worst-case based design in (26b)–(26d), the PT channel gains are upper-bounded and lower-bounded using the following triangle inequality properties:(31)x+y2≤x+y2=x2+y2+2xy,x+y2≥x-y2=x2+y2-2xy.Applying (31), it follows that(32)ge,lmax≜maxΔge,l≤γe,lge,l2=maxΔge,l≤γe,lg¯e,l2+Δge,l2+2g¯e,lΔge,l≤g¯e,l2+εe,l2+2εe,lg¯e,l,ge,lmin≜minΔge,l≤γe,lge,l2=minΔge,l≤γe,lg¯e,l2+Δge,l2-2g¯e,lΔge,l≥g¯e,l2+εe,l2-2εe,lg¯e,l,gpmin≜minΔgp≤γpgp2=minΔgp≤γpg¯p2+Δgp2-2g¯pΔgp≥g¯p2+εp2-2εpg¯p,gs,kmin≜minΔgs,k≤γs,kgs,k2=minΔgs,k≤γs,kg¯s,k2+Δgs,k2+2g¯s,kΔgs,k≥g¯s,k2+εs,k2-2εs,kg¯s,k.Similarly, we introduce the slack variablesdp, vl, bp, and ql for (29b)-(29e). Then, by substituting the above results (32) into (29b) and (29e), respectively, we can equivalently express (29a)–(29e) as(33a)minΔhp≤εph-p+ΔhpHWZh-p+Δhp≥dp,(33b)minΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≥vl,(33c)maxΔhp≤εph-p+ΔhpHWZh-p+Δhp≤bp,(33d)maxΔhe,l≤εe,lh-e,l+Δhe,lHZh-e,l+Δhe,l≤ql,(33e)dp+σp2+Ppgpmin≥ex^p,(33f)vl+σe,l2≥eu^l,(33g)bp+σp2≤et^p,(33h)ql+σe,l2+Ppge,lmax≤es^l.In order to make (26b)–(26d) more tractable, by applying S-procedure [40], we can convert the infinitely inequality constraints (30a)-(30d) and (33a)-(33d) into finite linear matrix inequalities (LMIs). For completeness, the S-procedure is presented in Lemma 1 in the following.Lemma 1 (S-procedure [40, Appendix B.2]).
Let a functionfm(x) with x∈CN×1(m=1,2) be defined as(34)fmx=xHAmx+2RebmHx+cmwhere Am∈HN×N, bm∈CN×1, and cm∈RN×1. Then, fm(x)≤0 holds if and only if there exists θ≥0 such that (35)θA1b1b1Hc1-A2b2b2Hc2⪰0,provided that there is a point x^ which satisfies fm(x^)<0.To employ theS-procedure, we rewrite the constraint (30a) as(36)Δhs,kHWZΔhs,k+2Rh¯s,kHWZΔhs,k+h¯s,kHWZh¯s,k≥dk.According to Lemma 1, by using a slack variable λs,k, (36) can be expressed as(37)λs,kI+WZWZh¯s,kh¯s,kHWZh¯s,kHWZh¯s,k-dk-λs,kεs,k2⪰0.Using the same approach for the constraints (30b)-(30d) and (33a)-(33d), we have(38a)λe,lI+ZZh-e,lh-e,lHZh-e,lHZh-e,l-wl-λe,lεe,l2⪰0,(38b)αs,kI-∑j≠kWj-Z-∑j≠kWj+Zh-s,k-h-s,kH∑j≠kWj+Z-h-s,kH∑j≠kWj+Zh-s,k+zk-αs,kεs,k2⪰0,(38c)αe,lI-Z-Wk-Z+Wkh-e,l-h-e,lHZ+Wk-h-e,lHZ+Wkh-e,l+fl,k-αe,lεe,l2⪰0,(39a)βpI+WZWZh-ph-pHWZh-pHWZh-p-dp-βpεp2⪰0,(39b)βe,lI+ZZh-e,lh-e,lHZh-e,lHZh-e,l-vl-βe,lεe,l2⪰0,(39c)αpI-WZ-WZh-p-h-pHWZ-h-pHWZh-p+bp-αpεp2⪰0,(39d)νe,lI-Z-Zh-e,l-h-e,lHZ-h-e,lHZh-e,l+ql-νe,lεe,l2⪰0,where λe,l, αs,k, αe,l, βp, βe,l, αp, and νe,l are slack variables.By using the above results of Taylor series expansion in (22a)–(22d), we can transform the nonconvex constraints (30g), (30h), (33g), and (33h) into the corresponding convex forms, respectively, as(40a)zk+σs,k2≤et^kmt^k-t^km+1,(40b)σe,l2+fl,k≤er^l,kmr^l,k-r^l,km+1,(40c)bp+σp2≤et^pmt^p-t^pm+1,(40d)ql+σe,l2+Ppge,lmax≤es^lms^l-s^lm+1.where t^k(m), r^l,k(m), t^p(m), and s^l(m) are the variables t^k, r^l,k, t^p, and s^l at the nth iteration for the SCA method.Here, we consider the EH constraint (30c). By use the inequalities (32) and (26d) can be rewritten as(41)h¯e,l+Δhe,lHWZh¯e,l+Δhe,l+Ppge,lmin+σe2≥E¯eηe,l,∀l.Introducing slack variablesϑe,l, we can transform (41) as(42)ϑe,lI+WZWZh¯e,lh¯e,lHWZh¯e,lHWZh¯e,l+Ppge,lmin+σe2-E¯eηe,l-ϑe,lεe,l2⪰0,Finally, similarly to (21a) and (21b), we need to recast the nonconvex constraints (28a) and (29a) as follows:(43a)e-x^k-y^l+t^k+r^l,k≤2-R-su,∀k,∀l,(43b)e-x^p-u^l+t^p+s^l≤2-R-pu,∀l.Combining all the results in (27a)–(34) and (36)–(43b), (26b)–(26d) can be reformed as(44)minΦ∑k=1KtrWks.t.23b,30e,30f,33e,33f,37,38a,38b,38c,39a,39b,39c,39d,40a,40b,40c,40d,42,43a,43b,Φ=Wk,Z,x^k,y^l,t^k,r^l,k,x^p,u^l,t^p,s^l,dk,wl,zk,fl,k,dp,vl,bp,ql,λs,k,λe,l,αs,k,αe,l,βp,βe,l,αp,νe,l,ζs,k,ϑe,l,Wk⪰0,Z⪰0.For given {t^k(m),r^l,k(m),t^p(m),s^l(m)} at the mth iteration, problem (44) is a convex SDP form without the nonconvex rank-one constraint rank(Wk)=1,∀k, which can be solved by convex optimization packages such as CVX [39], to update the solution for the m+1-th iteration until the algorithm converges. In addition, the proposed robust SCA-based iterative algorithm with imperfect CSI is summarized in Algorithm 2.Algorithm 2:Robust SCA-based iterative algorithm with imperfect CSI.
Setm=0 and initialize Ψ(0)=t^k0,r^l,k0,t^p0,s^l0. Repeat (i) Solve problem (44) with t^km,r^l,km,t^pm,s^lm and denote a solution by t^km+1,r^l,km+1,t^pm+1,s^lm+1. (ii) Update m←m+1. UntilConvergence


## 5. Simulation Results

In this section, we provide the simulation results to validate the performance of our proposed algorithms. We consider a system with two SUs and two ERs. The ST is equipped with six transmit antennas (NS=6). We define dSUkST=5m, dERlST=3m, and dPUST=10m as the distance between the ST and the kth SU, the lth ER, and the PU, respectively. Also, we fix dSUkPT=4m, dERlPT=3m, and dPUPT=2m as the distance between the PT and the kth SU, the lth ER, and the PU, respectively, unless otherwise specified. Thus, the estimated channels h¯s,k, h¯e,l, h¯p, g¯p, g¯s,k, and g¯e,l, are respectively, modelled as h¯s,k=H(dSUkST)h¯I, h¯e,l=H(dERlST)h¯I, h¯p=H(dPUST)h¯I, g¯s,k=H(dSUkST)g¯I, g¯e,l=H(dERlST)g¯I, and g¯p=H(dPUST)g¯I, where H(d)=c/4πfc(1/d)κ/2, h¯I~CN(0,I), and g¯I~CN(0,1), and we set the speed of light, the carrier frequency, and the path loss exponent as c=3×108ms-1, fc=900MHz, and κ=2.7, respectively. In addition, the noise power at the PU, SU, and ER is assumed to be σp2=σs,k2=σe,l2=-90dBm. Also the additional noise power of all the SUs is δa,k2=-50dBm,∀k. We fix the channel error bound for the deterministic model as ε=εs,k=εe,l=εp,∀k,∀l and γ=γp=γs,k=γe,l,∀k,∀l. The EH efficiency coefficients are set to ηc,l=ηe,k= 0.3. In our simulations, we compare the following transmit designs: the perfect CSI case, robust SCA-based scheme, no-AN scheme which is obtained by setting Z=0, and the nonrobust scheme which assumes no uncertainty in the CSI.Figure2 illustrates the convergence performance of the proposed SCA-aided iterative algorithm with respect to (w.r.t.) iteration numbers. Here, we set R¯su=R¯pu=0.5 bps/Hz, E=E¯e=1 dBm, Pp=20 dBm, and ε=γ=0.01. It is easily seen from the plots that the convergence of all perfect CSI cases can be quickly achieved within 5 iterations. It is observed that robust SCA-based scheme converges slower than the perfect CSI case regardless of E. This is due to the fact that the number of variables in the robust SCA-based scheme is greater than the perfect CSI case.Figure 2
Average transmit power of the information signal w.r.t. iteration numbers with variousE.Figure3 compares the average transmit power of the information signal w.r.t. the target secrecy rate at SU with R¯pu=0.5 bps/Hz, Pp=20 dBm and E=3 dBm. It is observed that the performance gaps of perfect CSI scheme with ε=0.01 and 0.1 over the robust SCA-aided iterative algorithm are 0.5dB and 1.1dB at all target secrecy rate region, respectively. Also, the transmit power for the no-AN scheme grows faster than the robust schemes as R¯su increases. We can see that, for ε=0.01 and 0.1, the robust SCA-aided iterative algorithm outperforms the no-AN scheme and nonrobust scheme. This is due to the help of the AN.Figure 3
Average transmit power of the information signal w.r.t. the target secrecy rate at SU.Moreover, the transmit power of the information signal performance is plotted in Figure4 with R¯su=0.5 bps/Hz, Pp=20 dBm, and E¯e=5 dBm. One can observe from Figure 4 that the average transmit power of the information signal increases as the target secrecy rate at PU becomes large. Also, the performance gap between the robust SCA-aided iterative algorithm and the no-AN scheme becomes large at high secrecy rate region. Also, there are 0.4dB and 0.9dB gaps between perfect CSI and robust SCA-aided iterative algorithm curves for ε=0.01 and 0.1, respectively.Figure 4
Average transmit power of the information signal w.r.t. the target secrecy rate at PU.Finally, in Figure5, the transmit power of the information signal w.r.t. the target harvested power is illustrated with R¯su=R¯pu=1 bps/Hz and Pp=30 dBm. We can see that the robust SCA-aided iterative algorithm with ε=0.01 outperforms the nonrobust scheme by 3.6dB and the performance gains of the robust SCA-aided iterative algorithm over the no-AN scheme become larger as the target harvested power increases. Similarly, when E¯e≤0 dBm, the performance of the proposed algorithms changes slowly. This is due to the introduction of AN for the CR system.Figure 5
Average transmit power of the information signal w.r.t. the harvested power.


## 6. Conclusion

In this paper, we have studied AN-aided secure beamforming designs for MISO Secured CR networks with SWIPT. Our aim is to minimize the transmit power of the information signal subject to the secrecy rate constraint, the harvested energy constraint, and total transmit power constraint. As the original problem is nonconvex, we recast the original problem as a convex form by using SCA method. Then, we have proposed an SCA-based iterative algorithm for the perfect CSI case. Moreover, we have extended the proposed algorithm to the norm-bounded channel uncertainty model. Finally, simulation results have been given to validate the performance of our proposed algorithms.



---

*Source: 6956313-2018-08-13.xml*