## I Introduction

In recent years, the idea of energy harvesting (EH) has been introduced to power electronic devices by energy captured from the environment. [1]. Based on this idea, simultaneous wireless information and power transfer (SWIPT) schemes have been proposed to extend the lifetime of wireless networks [2]-[5]. For SWIPT operation in multiple antenna systems [3], co-located receiver architecture employing a power splitter for EH and information decoding (ID) has been studied [5].

On the other hand, in the literature we see increasing research interest in secrecy transmission through physical layer (PHY) security designs [6] [7]. By adding artificial noise (AN) and projecting it onto the null space of information user channels, the eavesdroppers would experience a higher noise floor and thus obtain less information about the messages transmitted to the legitimate receivers [8]. The AN-aided beamforming for SWIPT operation has been investigated in various multiple-input multiple-output (MIMO) channels [10]-[12].

In SWIPT systems, a new information security issue is raised because the energy-harvesting receivers (ERs) can potentially eavesdrop the information transmission to the information receivers (IRs) with relatively higher received signal strength [10]. In order to guarantee information security for the IRs, it is desirable to implement some mechanism to prevent the ERs from recovering the confidential message from their observations.

In this paper, we study a AN-aided secrecy transmission over a multi-user MIMO secrecy channel which consists of one multi-antenna transmitter, multiple legitimate single antenna co-located receivers (CRs) with power splitter and multiple multi-antenna ERs.
The design objective is to jointly optimize the transmit beamforming matrix, the AN covariance matrix, and the power splitting (PS) ratio
such that the AN transmit power is maximized^{1}^{1}1AN power maximization is equivalent to minimizing the transmit power of the information signal [11]. subject to constraints on the secrecy rate, the total transmit power, and the energy harvested by both the CRs and the ERs.
Because of the coupling effect in the joint optimization problem, determination of the AN covariance matrix and the PS ratio makes the derivation of the secrecy rate and the harvested energy at the CRs more complicated.

The formulated power minimization (PM) problem for AN-aided secrecy transmission is shown to be non-convex, which cannot be solved directly [13]. First, the PM problem is thus transformed into a two-layer optimization problem. The inner loop of the PM problem is solved through semi-definite relaxation (SDR), while the outer loop is shown to be a single-variable optimization problem, where a one-dimensional (1-D) line search algorithm is employed to find the optima. To reduce computational complexity, a sequential parametric convex approximation (SPCA) method is also investigated [16]. The tightness of the SDR is verified by showing that the optimal solution is rank-one.

*Notation:* stacks the elements of A

in a column vector.

is a zero matrix of size

. is the expectation operator, and stands for the real part of a complex number. represents anddenotes the maximum eigenvalue of

A.## Ii System Model

In this section, we consider a multi-user MIMO secrecy channel which consists of one multi-antenna transmitter, single-antenna CRs and multi-antenna ERs. We assume that each CR employs the PS scheme to receive the information and harvest power simultaneously. It is assumed that the transmitter is equipped with transmit antennas, and each ER has receive antennas.

We denote by the channel vector between the transmitter and the -th CR, and the channel matrix between the transmitter and the -th ER. The received signal at the -th CR and the -th ER are given by

where is the transmitted signal vector, and and are the additive Gaussian noise at the -th CR and the -th ER, respectively.

In order to achieve secure transmission, the transmitter employs transmit beamforming with AN, which acts as interference to the ERs, and provides energy to the CRs and ERs. The transmit signal vector can be written as

(1) |

where defines the transmit beamforming vector, with is the information-bearing signal intended for the CRs, and represents the energy-carrying AN, which can also be composed by multiple energy beams.

As the CR adopts PS to perform ID and EH simultaneously, the received signal at the -th CR is divided into ID and EH components by the PS ratio . Therefore, the signal for information detection at the -th CR is given by

where is the additive Gaussian noise at the -th CR.

Denoting as the transmit covariance matrix and as the AN covariance matrix, the achieved secrecy rate at the -th CR is given by

(2) |

The harvested power at the -th CR and the -th ER is therefore

(3) |

where and represent the EH efficiency of the -th CR and the EH efficiency of the -th ER, respectively. In this paper we set for simplicity.

## Iii Masked Beamforming Based Power Minimization

In this section, we study transmit beamforming optimization under the assumption that perfect CSI of all the channels is available at the transmitter.

### Iii-a Problem Formulation

In this problem, the transmit power of the information signal is minimized subject to the total transmit power constraint, the secrecy rate constraint, and the EH constraints of the CRs and the ERs such that the AN transmit power is maximized for secrecy consideration. The AN-aided PM problem is thus formulated as

(4e) | |||||

where is the target secrecy rate, is the total transmit power, and and denote the predefined harvested power at the -th CR and the -th ER, respectively. The constraint (4e) guarantees that a minimum energy harvested power should be achieved by the -th ER.

### Iii-B One-Dimensional Line Search Method (1-D Search)

Problem (4) is non-convex due to the secrecy rate constraint (4e), and thus cannot be solved directly. In order to circumvent this issue, we convert the original problem by introducing a slack variable for the -th ER’s rate. Then we have

(5a) | |||

(5b) | |||

Problem (5) is still non-convex in constraints (5a) and (5b), which can be addressed by reformulating (5) into a two-layer problem. For the inner layer, we solve problem (5) for a given , which is relaxed as

(6a) | |||

(6b) | |||

where is defined as the optimal value of problem (6), which is a function of . Even though the function cannot be expressed in closed-form, numerical evaluation of is feasible. Note that the LMI constraint (6b) is obtained from [9, Proposition 1], and is based on the assumption that , which will be shown later.

By ignoring the non-convex constraint , problem (6) becomes convex and thus can be solved efficiently by an interior-point method for any given [13]. The outer layer problem, whose objective is to find the optimal value of , is then formulated as

(7) |

where and are the upper and lower bounds of , respectively. The solution to problem (7) can be found by one-dimensional line search. For the line search algorithm, we need to determine the lower and upper bounds of the searching interval for . It is straightforward that can be used as the upper bound due to the feasibility of (5b), while a lower bound is calculated as

where the first inequality is based on the secrecy rate , the third inequality follows from (4e).

Utilizing the results in Remark 1, next we show the tightness of the AN-aided PM problem (4) by the following theorem.

*Theorem 1:* Provided that problem
(6) is feasible for a given , there
exists an optimal solution to (4) such that the rank of is always equal to 1.

*Proof: * See Appendix A.

### Iii-C Low-Complexity SPCA Algorithm

In this subsection, we propose an SPCA based iterative method to reduce the computational complexity. By introducing two slack variables and , the constraint (4e) can be rewritten as

(9a) | |||

(9b) | |||

(9c) |

which can be further simplified as

(10a) | |||

(10b) | |||

(10c) |

The inequality constraint (10a) is equivalent to , which can be converted into a conic quadratic-representable function form as

(11) |

By transforming inequality constraints (10b) and (10c) into

(12a) | |||

(12b) |

where and , we observe that these two constraints are non-convex, but the right-hand side (RHS) of both (12a) and (12b) have the function form of quadratic-over-linear, which are convex functions [13]. Based on the idea of the constrained convex procedure [17], these quadratic-over-linear functions can be replaced by their first-order expansions, which transforms the problem into convex programming. Specifically, we define

(13) |

where and . At a certain point , the first-order Taylor expansion of (13) is given by

(14) |

By using the above results of Taylor expansion, for the points and , we can transform constraints (12a) and (12b) into convex forms, respectively, as

(15a) | |||

(15b) |

Denoting and , (15a) and (15b) can be recast as the following second-order cone (SOC) constraints

(16a) | |||

(16b) |

Next we employ the SPCA technique for the SOC constraints (4e) and (4e) [18] to obtain convex approximations. By substituting and into the left-hand side (LHS) of (4e) and (4e), we obtain

(17) |

where the inequality is given by dropping the quadratic terms and . Similarly, in the LHS of (4e), we have

(18) |

According to (17) and (18), we obtain linear approximations of the concave constraints (4e) and (4e) as

(19) |

and

(20) |

Finally, by rearranging (4e) as

(21) |

the original problem (4) is transformed into

(22) |

Given , , , and , problem (22) is convex and can be efficiently solved by convex optimization software tools such as CVX [15]. Based on the SPCA method, an approximation with the current optimal solution can be updated iteratively, which implies that (4) is optimally solved. In Section V, we will show that the proposed SPCA method achieves the same performance as the 1-D search scheme, but with much lower complexity.

## Iv Computational Complexity

In this section, we evaluate the computational complexity of the proposed robust methods. As will be shown in Section VI, the proposed SPCA algorithm achieves substantial improvement in complexity for the same performance compared with the method based on 1-D search. Now we compare complexity of the algorithms through analyses similar to that in [18] [19]. The complexity of the proposed algorithms are shown in Table I on the top of next page. We denote , , and as the number of decision variables, the 1-D search size, and the SPCA iteration number, respectively. The complexity analysis is given in the following.

*1)* PM with 1-D Search in problem (6) involves LMI constraints of size , two LMI constraints of size , and linear constraints.

*2)* PM with SPCA in problem (22) has SOC constraints of dimension , SOC constraints of dimension , SOC constraints of dimension , one SOC constraints of dimension , and linear constraints.

For example, for a system with , , and , the complexity of the PM with 1-D search, the PM with SPCA, are , and , respectively. Thus, the complexity of the proposed SPCA method is only 1% compared to the scheme based on 1-D search.

Algorithms | Complexity Order |
---|---|

## V Numerical Results

In this section, we present numerical results to validate performance of the proposed transmit beamforming schemes. In the simulations, we consider a system with , , , and . Both large-scale and small-scale fading are considered in the channel model. The simplified large-scale fading model is given by where represents the distance between the transmitter and the receiver, is a reference distance equal to m in this work, and is the path loss exponent. We define m as the distance between the transmitter and the CRs, and m as the distance between the transmitter and the ERs, unless otherwise specified.

Because all the receivers are are expected to harvest energy from the RF signal, we consider line-of-sight (LOS) communication scenario where the Rician fading model is adopted for small scale fading coefficients. The channel vector is expressed as , where indicates the LOS deterministic component with , represents the Rayleigh fading component as , and is the Rician factor. It is noted that for the LOS component, we use the far-field uniform linear antenna array model [20]. In addition, we set , , , , , , and 0.3.

Fig. 1 illustrates the convergence of the SPCA method with respect to iteration numbers for dBm, , bps/Hz, and 0.01. It is easily seen from the plots that convergence is achieved for all cases within just 8 iterations.

Fig. 2 illustrates the average transmit power of the information signal in terms of different target secrecy rates with dBm and dBm, . It is observed that the transmit power increases with the secrecy rate target. Here, the no-AN scheme is set with . In addition, the SPCA algorithm achieves the same performance as the 1-D search method, but with much lower complexity. Compared with the scheme without AN, the power consumption of the proposed AN-aided scheme is dB lower. Moreover, we can check that the proposed scheme performs better than the scheme with , and the performance gap becomes larger as the target secrecy rate increases. This indicates that optimizing the PS ratio is important, especially when the target secrecy rate is high.

In Fig. 3, we plot the average transmit power in terms of different target harvested power at the CR with dBm, dBm and bps/Hz. We can check that the curves of the 1-D search method and SPCA method increase with the same slope. Moveover, when the harvested power target decreases, the performance gap between the no-AN PM scheme and the proposed PM schemes become wider. This indicates that AN is essential in achieving the performance gains. Furthermore, the 1-D search method require dB lower power than the 1-D search method with fixed , respectively.

## Vi Conclusion

In this paper, we have proposed AN-aided secure transmission scheme in multi-user MIMO SWIPT Systems where power splitters are employed by the receivers for SWIPT operation. The original problem, which was shown to be non-convex, was relaxed to formulate a two-layer problem. The inner layer problem was recast as a sequence of SDPs. Then the optimal solution to the outer problem has been obtained through one-dimensional line search. Moreover, tightness of the relaxation scheme has been investigated by showing that the optimal solution is rank-one. To reduce the computational complexity, an SPCA based iterative algorithm has been proposed, which achieved near-optimal solution. Finally, numerical results have been provided to validate the performance of the proposed transmit beamforming schemes.

## Appendix A Proof of Theorem 1

We first consider the Lagrange dual function of (6) as

where , , , , , , and are the dual variables of , , (6a), (6b), (4e), (4e), and (4e), respectively. Then, some of the related KKT conditions are listed as

(23a) | |||

(23b) | |||

(23c) |

From the Lagrangian function and the KKT conditions, we have and the KKT condition and . Now, we will show these conditions via the dual problem of (6) as

Since problem (6) is convex and satisfies the Slater’s condition, the duality gap between (6) and (A) is zero, and the strong duality holds. Therefore solving problem (6) is equivalent to solving (A). In addition, the constraint can be satisfied as

Also the optimal variable , and the dual variables are related by

From the above inequality, we will show that and by contradiction. Suppose that and/or . Then there are two cases (i.e., or ), which violate the constraints (4e) and (4e). Thus, it follows and . Now, subtracting (23b) from (23a) yields

(25) |

We post-multiply by both sides of (25) and use (23c) as

Then, it becomes