## I Introduction

Recently, wireless power transfer (WPT) has been a promising paradigm to scavenge energy from the radio frequency (RF) signals [1]. As a key technology for really perpetual communications, simultaneous wireless information and power transfer (SWIPT) has been an promising of interests in RF-enabled signal to provide power supplies for wireless networks, which have been studied in various scenarios [2]-[5]. On the other hand, secrecy transmission, especially physical-layer security (PLS), has extracted more and more attentions in 5G wireless networks [6].

Specifically, PLS has been recognized as a important issue for SWIPT system due to its inherent characteristics make the wireless information more vulnerable to eavesdropping [7]-[11]. Moreover, the secure transmission with SWIPT schemes has also been investigated in the multi-user multiple-input-multiple-output (MU-MIMO) broadcasting [9].

It is noted that the assumption that perfect channel state information (CSI) is available at the transmitter in [2]-[4] [7]-[9]. In practice, it is not always possible to obtain perfect CSI at the transmitter due to the channel errors. Secure communication with SWIPT would be more challenging with imperfect CSI at the transmitter. Some robust optimization techniques have been constructed to secrecy SWIPT transmission under imperfect channel realization in [10][11] [13]-[18]. Considering the SWIPT scheme, an optimal transmit covariance matrix robust design has been proposed for MIMO secure channels with multi-antenna eavesdroppers [11].

In addition, some of state-of-art techniques have been developed to introduce more interference to the eavesdroppers [12]-[21]. Artificial noise (AN) technique has been used to embed the transmit beamforming to confuse the eavesdropper [12]. In secrecy SWIPT systems, AN plays both the roles of an energy-carrying signal for WPT and protecting the secrecy information transmission, which has been considered as interfering the eavesdropper and harvesting power simultaneously in [13]-[15]. In addition, to further increase the secrecy rates, jamming node has been introduced in the secrecy networks, which has the capability to improve the legitimate user’s performance and prevent the eavesdroppers from intercepting the intended messages [16]-[20]. Based on the worst-case scheme, cooperative jamming signal was generated by a external cooperative jammer (CJ) node to interfere the eavesdropper and improve the secrecy rate in multiple-input-single-output (MISO) secure SWIPT system with wiretap channels[18].

When information receivers (IRs) and energy-harvesting receivers (ERs) are placed in a same cell, the ERs are normally assumed to be closer to the transmitter compared with IRs. This gives rise to a new information security issue in the SWIPT systems. In such a situation, ERs have a possibility of eavesdropping the information sent to the IRs, and thus can become potential eavesdroppers [8] [14] [15].

In this paper, considering SWIPT, we investigate a secure transmission design problem in MU-MIMO secrecy system with one multi-antenna transmitter, one multi-antenna CJ, multiple single-antenna co-located receiver (CR), and multiple multi-antenna ER. The CR employs power splitting (PS) scheme to split the received signals into two streams for ID and energy harvesting (EH) simultaneously. Unlike [17] [18]

, our objective is to maximize the minimum of harvested energy (max-min HE) of both ERs and CRs subject to secrecy rate constraints for each CR and total transmit power constraint. Assuming the imperfect CSI case, we seek to the optimal transmission strategy to jointly optimize the AN-aided beamforming, the AN vector, the CJ vector and the PS ratio design.

Considering the worst-case scheme by incorporating the norm-bounded channel uncertainty model, we derive equivalent forms of secrecy rate constraint and the minimum of harvested energy. An iterative algorithm based on sequential parametric convex approximation (SPCA) is also addressed to recover a high quality rank-one beamforming solution to the original problem. Finally, the performance analysis are provided to verify that the proposed algorithm outperforms the conventional scheme.

*Notation:*
defines the Kronecker product. and describe the space of complex matrices and Hermitian matrices, respectively. equals the set of positive semi-definite Hermitian matrices, and denotes the set of all nonnegative real numbers. For a matrix , means
that is positive semi-definite, and , , and denote the Frobenius norm, trace, determinant, and the rank, respectively. stacks the elements of in a column vector. is a null matrix with size. describes expectation, and stands for the real part of a complex number. represents and

indicates the maximum eigenvalue of

.## Ii System Model and Problem Formulation

In this section, we consider a MU-MIMO secrecy channel, which consists of one multi-antenna transmitter, one multi-antenna CJ, single-antenna CRs, and multi-antenna ERs, where the CR employs the PS scheme to decode information and exploit power simultaneously. It is assumed that the transmitter and the CJ are equipped with and transmit antennas, and each ER has receive antennas. We denote as the channel vector between the transmitter and the -th CR, as the channel matrix between the transmitter and the -th ER, as the channel vector between the CJ and the -th CR, as the channel matrix between the CJ and the -th ER, respectively.

In order to improve the reliable transmission, the transmitter employs the transmit beamforming with AN, which acts as interference to the ERs and simultaneously provides energy to the CRs and ERs. Thus, the transmitter sends the confidential message by using transmit beamforming with AN to the -th CR as

(1) |

where denotes the linear beamforming vector for the -th CR at the transmitter, represents the information-bearing signal intended for the -th CR satisfying , and is the energy-carrying AN vector, which can also be composed by multiple energy beams. The received signal at the -th CR and the -th ER can be expressed as

(2) |

where is the cooperative jamming signal introduced by the CJ satisfying , indicates the CJ vector, and stand for the additive Gaussian noise by the receive antenna at the -th CR and the -th ER. In addition, each CR considers the PS scheme to manage the processes of ID and EH simultaneously. Based on this reason, the received signal at the -th CR is divided into ID and EH by PS ratio . Thus, the received signal for ID at the -th CR can be given as

(3) |

where is the antenna noise introduced by signal process of the ID at the -th CR [2].

We denote as the transmit covariance matrix, as the AN covariance matrix, and as the CJ covariance matrix. Hence, the achieved secrecy rate can be calculated by

(4) |

where

Thus, the harvested power at the -th CR and the -th ER are written as

(5) |

where and denote the energy conversion efficiency of the -th CR and the -th ER.

Due to channel estimation and quantization errors, it may not be possible to achieve the perfect CSI at the transmitter in practice. In this section, our aim is to jointly optimize the max-min worst-case HE formulation at the imperfect CSI case. Now, we adopt the imperfect CSI case under the norm-bounded channel uncertainty model

[11][13][14]. Specifically, the actual channels , , , and can be given as(6) |

where , and denote the estimated channel available at the transmitter and the CJ, respectively, and , and are the channel errors, bounded as , , , and , respectively.

In this paper, our aim is to maximize the minimum of the total harvested power among the all CRs and ERs subject to the secrecy rate constraint and the total transmit power constraint at the transmitter and the CJ power constraint. By taking the above channel model into account, the transmit beamforming design is formulated as a multi-object optimization problem which can be given by

(7a) | |||

(7b) | |||

(7c) | |||

(7d) |

where

is the priority parameter, stands for a given secrecy rate threshold, and and denote the available power budget at the transmitter and the CJ, respectively. Variable reflects the preference of the system operator. Problem (7) is non-convex due to the secrecy rate constraint and the objective function, and thus cannot be solved directly.

## Iii Proposed Robust Design Method

In order to circumvent the roust max-min HE problem, we transform problem (7) by introducing two slack variables and into

(8a) | |||

(8b) | |||

(8c) | |||

Problem (8) is still non-convex in terms of (8b), (8c) and (7b).

Now, let us consider another formulation of problem (7) based on SPCA method [22]. The optimization framework can also be recast as a convex form by incorporating channel uncertainties. First, by applying the matrix inequality [23], the robust secrecy rate constraint (7b) can be relaxed as

(9) |

To make the constraint (9) tractable, we introduce two slack variables and , the robust secrecy rate (9) can be equivalently relaxed as

(10a) | |||

(10b) | |||

(10c) |

Then, we can be further simplify (10) as

(11a) | |||

(11b) | |||

(11c) |

The inequality constraint (11a) is equivalent to , which can be converted into a conic quadratic-representable function form as

(12) |

Design , , , and . The inequalities in (11b) and (11c) can be rearranged, respectively, which give

(13a) | |||

(13b) |

where , , , and . which stand for the CSI uncertainty. It is straightforward to show that

(14) |

(15) |

(16) |

(17) |

Note that , , , and are norm-bounded matrices as , , , and , where , , , and .

According to [24], we can minimize constraint (11b) by maximizing the left-hand side (LHS) of (13a) while minimizing its the right-hand side (RHS). Then (13a) and (13b) can be approximately rewritten as, respectively,

(18) |

(19) |

In order to minimize the RHS of (18) and (19), a loose approximation [24] is applied, which gives

(20) |

From (18)-(22), (13a) and (13b) can be given as, respectively,

(23) |

(24) |

where , and . We observe that these two constraints are non-convex, but the RHS of both (23) and (24) have the function form of quadratic-over-linear, which are convex functions [25]. Based on the idea of the constrained convex procedure [22], these quadratic-over-linear functions can be replaced by their first-order expansions, which transforms the problem into convex programming. Specifically, we define

(25) |

where and . At a certain point , the first-order Taylor expansion of (25) is given by

(26) |

By using the above results of Taylor expansion, for the points , and , we can transform (23) and (24) into convex forms, respectively, as

(27a) | |||

(27b) |

In order to approximate the EH constraint (8b) and (8c) to convex one, we apply an SCA-based method. First, by using a loose approximation approach for (8b) and (8c), we have

(28a) | |||

(28b) |

where .

It is observed that , and are the concave part of constraints (29a) and (29b). In order to make (29a) and (29b) more tractable, we employ the SCA technique for (29a) and (29b) to obtain convex approximations.

Firstly, we take as an example. Let be an initial feasible point. We substitute into as follows

(30) |

where (30) are derived by dropping the quadratic form .