Wireless Powered Cooperative Jamming for Secure OFDM System

# Wireless Powered Cooperative Jamming for Secure OFDM System

Guangchi Zhang, Jie Xu, Qingqing Wu, Miao Cui, Xueyi Li, and Fan Lin G. Zhang, J. Xu, M. Cui, and X. Li are with the School of Information Engineering, Guangdong University of Technology, Guangzhou, China (e-mail: gczhang@gdut.edu.cn, jiexu@gdut.edu.cn, single450@163.com, leexueyi@gdut.edu.cn). J. Xu is the corresponding author. Q. Wu is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: elewuqq@nus.edu.sg). F. Lin is with Guangzhou GCI Science & Technology Co., Ltd., Guangzhou, China (e-mail: linfan@chinagci.com).
###### Abstract

This paper studies the secrecy communication in an orthogonal frequency division multiplexing (OFDM) system, where a source sends confidential information to a destination in the presence of a potential eavesdropper. We employ wireless powered cooperative jamming to improve the secrecy rate of this system with the assistance of a cooperative jammer, which works in the harvest-then-jam protocol over two time-slots. In the first slot, the source sends dedicated energy signals to power the jammer; in the second slot, the jammer uses the harvested energy to jam the eavesdropper, in order to protect the simultaneous secrecy communication from the source to the destination. In particular, we consider two types of receivers at the destination, namely Type-I and Type-II receivers, which do not have and have the capability of canceling the (a-priori known) jamming signals, respectively. For both types of receivers, we maximize the secrecy rate at the destination by jointly optimizing the transmit power allocation at the source and the jammer over sub-carriers, as well as the time allocation between the two time-slots. First, we present the globally optimal solution to this problem via the Lagrange dual method, which, however, is of high implementation complexity. Next, to balance tradeoff between the algorithm complexity and performance, we propose alternative low-complexity solutions based on minorization maximization and heuristic successive optimization, respectively. Simulation results show that the proposed approaches significantly improve the secrecy rate, as compared to benchmark schemes without joint power and time allocation.

Physical layer security, wireless powered cooperative jamming, OFDM system, joint power and time allocation.

## I Introduction

With recent technical advancements in Internet of things (IoT), future wireless networks are envisioned to incorporate billions of low-power wireless devices to enable various industrial and commercial applications [1]. How to ensure the confidentiality of these devices’ wireless communication against illegitimate eavesdropping attacks is becoming an increasingly important task for cyber-physical security. However, this task is particularly challenging, as conventional key-based cryptographic techniques are difficult to be implemented due to the broadcast nature of wireless communications. To overcome this issue, physical layer security has emerged as a viable anti-eavesdropping solution at the physical layer [3, 2, 4]. The key design objective in physical-layer security is to maximize the so-called secrecy rate, which is defined as the communication rate of a wireless channel, provided that eavesdroppers cannot overhear any information from this channel.

In the literature, there have been various approaches proposed to improve the secrecy rate. For example, one widely adopted approach is based on the idea of artificial noise (AN) (see, e.g., [5, 6]). In this approach, wireless transmitters send a combined version of both confidential information signals and AN, where the AN acts as jamming signals to interfere with eavesdroppers, thus avoiding the information leakage. Another celebrated approach is called cooperative jamming (see, e.g., [7, 8, 9]), where external network nodes cooperatively send jamming signals to disrupt the eavesdropping, thus helping protect the confidential information communication. As compared to the AN-based approach, cooperative jamming is able to further improve the secrecy rate by exploiting the cooperation diversity among different nodes. Cooperative jamming is also expected to have more abundant applications in the IoT era, where massive low-power wireless devices can cooperate in jamming to improve the network security. For instance, some idle devices in wireless networks can act as cooperative jammers to help ensuring the secrecy communication of other actively communicating devices.

Nevertheless, the practical implementation of cooperative jamming in IoT networks is hindered by the low-power nature of wireless devices, since cooperative jamming will consume energy on these devices and thus they may prefer keeping idle to save energy instead of involving in the cooperation. To overcome this issue, a new efficient method, namely wireless powered cooperative jamming, has been proposed in [10, 11, 12, 13] motivated by the recent success of wireless information and power transfer via radio frequency (RF) signals [14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 23, 24].111It is worth noting that in addition to the far-field RF-based wireless power transfer, magnetic induction is a widely used near-field wireless power transfer technique for charging electronic devices [22, 26]. However, the magnetic induction has a limited operating range of less than one meter in general, which is much shorter than that of the RF-based wireless power transfer in the order of several meters. Therefore, RF-based wireless power transfer is expected to have more abundant applications to charge low-power IoT devices in a wide range, and thus is considered here in the wireless powered cooperative jamming systems. In this method, the cooperative jamming is powered by the wireless energy transferred from external wireless transmitters, and does not require cooperative jammers to consume their own energy. Therefore, wireless powered cooperative jamming is a promising solution to inspire low-power IoT devices to cooperate in the jamming. In [10, 11], wireless powered cooperative jamming was employed to secure a point-to-point communication system in the presence of an eavesdropper, where a cooperative jammer operates in an accumulate-and-jam protocol by first harvesting the wireless energy and storing in the battery over multiple blocks and then using the accumulated energy for cooperative jamming. The long-term secrecy performance is optimized by adjusting jamming parameters while taking into account the channel and battery dynamics over time. In [12, 13], wireless powered cooperative jamming was used in a secrecy two-way relaying communication system, where an eavesdropper aims to intercept the communicated information at the second hop, and more than one cooperative jammers operate in a harvest-then-jam protocol for cooperative jamming: in the first slot, the jammers harvest the wireless energy from the source, while in the second slot, they use the harvested energy to cooperatively jam the eavesdroppers. As the harvested energy is immediately used in the following slot, the harvest-then-jam protocol does not require large-capacity energy storages nor sophisticated energy management at cooperative jammers. For this reason, it is generally much easier to be implemented in practice than the accumulate-and-jam protocol.

In this paper, we consider wireless powered cooperative jamming to secure a point-to-point communication system from a source to a destination with the presence of a potential eavesdropper. Different from prior works considering single-carrier systems, we focus on the multi-carrier orthogonal frequency division multiplexing (OFDM) system, which offers the following advantages. First, note that the wireless transmission must meet the transmit power spectrum density constraints imposed by regulatory authorities. In this case, the transferred power over a narrow-band system is often limited. By contrast, using OFDM over a wideband wireless power transfer system and exploiting the channel diversity over frequency can help deliver more power to intended receivers. On the other hand, as OFDM has been widely adopted in major existing and future wireless communication networks, using it here can also help better integrate wireless power transfer and wireless communication for future wireless networks (see, e.g., [26, 27, 28, 29, 30] and references therein). The cooperative jammer works in a harvest-then-jam protocol to help the secrecy communication by dividing each transmission block into two time-slots: in the first slot, the source sends dedicated energy signals to power the jammer; while in the second slot, the jammer uses the harvested energy to interfere with the eavesdropper to protect the confidential information transmission.

In general, there exists a tradeoff in the time allocation between the two slots to optimize the performance of secrecy communication, i.e., while a longer WPT time in the first slot can transfer more energy to increase the jamming power for better confusing the eavesdropper, it can also reduce the efficient wireless information transmission (WIT) time in the second slot for delivering confidential data. Therefore, in order to improve the secrecy rate at the destination by maximally exploring the benefit of wireless power cooperative jamming, it is important to jointly design the time allocation, together with the transmit power allocation at the source and the jammer over sub-carriers, by taking into account the energy harvesting constraint at the jammer. We maximize the secrecy rate via joint time and power allocation by particularly considering two types of receivers at the destination, namely Type-I and Type-II receivers, which do not have and have the capability of canceling the (a-priori known) jamming signals, respectively (see Section II for the details). Under both receiver types, however, the two joint time and power allocation problems are non-convex and usually difficult to be solved. To tackle such challenges, we propose to recast each problem into a two-layer form, in which the outer layer corresponds to a single-variable time allocation problem and the inner layer is a sub-carrier transmit power allocation problem under given time allocation. The outer layer time allocation problem is solved via a one-dimension search. As for the inner-layer power allocation problem, we first present the globally optimal solution via the Lagrange dual method, which, however, is of high implementation complexity. Next, to balance the tradeoff between the implementation complexity and the performance, we further develop two suboptimal solutions based on minorization maximization and heuristic successive optimization, respectively. Simulation results show that the proposed approaches achieve significantly higher secrecy rate than benchmark schemes without joint time and power allocation, and the minorization maximization based suboptimal solution achieves a near optimal performance as compared to the optimal solution.

It is worth noting that in the literature, there have been several existing works [28, 29, 30] investigating the physical layer security over OFDM systems. For example, the secrecy rate of OFDM systems was investigated in [28] under a Rayleigh fading channel setup without using AN or cooperative jamming. In [29] and [30], the AN-based approach and cooperative jamming were considered to improve the secrecy rate of OFDM systems, respectively. Different from these prior studies, in this paper the cooperative jamming is powered by WPT, and thus requires a more sophisticated design with joint time and power allocation for both WPT and jamming. This is new and has not been addressed.

The remainder of the paper is organized as follows. Section II presents the system model and problem formulation. Sections III and IV propose three efficient approaches to obtain solutions to the two joint time and power allocation problems with Type-I and Type-II destination receivers, respectively. Section V presents simulation results to validate the performance of our proposed joint design as compared to other benchmark schemes. Finally, Section VI concludes this paper.

## Ii System Model and Problem Formulation

### Ii-a System Model

As shown in Fig. 1, we consider secrecy communication in an OFDM system with a source communicating with a destination in the presence of a potential eavesdropper. We employ wireless powered cooperative jamming to secure this system, where a cooperative jammer uses the transferred energy from the source to help jam the eavesdropper against its eavesdropping. Suppose that the OFDM system consists of a total of orthogonal sub-carriers, and denote the set of sub-carriers as . We consider a block-based quasi-static channel model by assuming that the wireless channels remain constant over each transmission block and may change from one block to another. We focus on one particular block with a length of , and denote , , , , as the vectors collecting the channel coefficients of all the sub-carriers from the source to the jammer, from the source to the destination, from the source to the eavesdropper, from the jammer to the destination, from the jammer to the eavesdropper, respectively. Here, the superscript denotes the transpose operation. It is assumed that the source, destination, and the cooperative jammer perfectly know the global channel state information (CSI) , , , , and in order to obtain the performance upper bound of the wireless powered cooperative jamming system. Specifically, the CSI , , and associated with these users can be obtained via efficient channel estimation and feedback among them, while and can be obtained by monitoring the possible transmission activities of the eavesdropper, as commonly assumed in the physical-layer security literature [3, 4, 5, 6, 7]. Note that in practice the CSI acquisition may consume additional energy at the cooperative jammer, and the obtained CSI may not be perfect due to channel estimation and feedback errors. However, how to address these issues in practice is left for future work.

We consider a harvest-then-jam protocol for the cooperative jammer by dividing each transmission block into two time-slots with lengths and , respectively, where and denote the portions of the two time-slots with

 α1+α2=1,0≤α1≤1,0≤α2≤1. (1)

In the first time-slot, the source sends wireless energy to power the cooperative jammer; while in the second time-slot, the source transmits confidential information to the destination and simultaneously the jammer uses the harvested energy in the first time-slot to cooperate in jamming the eavesdropper against its eavesdropping. The detailed operation in the two slots is presented in the following, respectively.

First, consider the WPT from the source to the jammer in the first time-slot. Over each sub-carrier , let denote the energy signal transmitted by the source, which is assumed to be a random variable with variance . Here, denotes the transmit power for WPT at the source over the sub-carrier , and denotes the statistic expectation. The harvested energy by the jammer is

 EEH=α1TηN∑n=1pPT,n|hJ,n|2, (2)

where denotes the energy harvesting efficiency at the jammer. Note that similarly as in [14, 15, 16, 17, 18, 19, 20, 21], we adopt a linear energy harvesting model in (2) by considering the harvested power at the jammer lies in the linear regime of the energy harvester. In the literature, there have been various works [33, 34, 35, 36] investigating the wireless power transfer by considering the non-linearity of the energy harvester, while how to extend the wireless powered cooperative jamming into such a scenario is left for future work.

Next, consider the cooperative jamming in the second time-slot. Over the sub-carrier , let and denote the confidential information signal transmitted by the source and the jamming signal transmitted by the jammer, respectively. The received signals by the destination and the eavesdropper over the sub-carrier are respectively denoted as

 yD,n=hD,nsIT,n+gD,nsJ,n+nD,n, (3)
 yE,n=hE,nsIT,n+gE,nsJ,n+nE,n, (4)

where and denote the Gaussian noise at the receivers of the destination and the eavesdropper with mean zero and variances and , respectively. Assume that Gaussian signaling is employed for both and , which are thus cyclic symmetric complex Gaussian (CSCG) random variables with mean zero and variances and , with and denoting the transmit power of the source and the jamming power of the jammer over the sub-carrier , respectively. Let denote the maximum transmit sum power of the source over all sub-carriers, and denote the peak transmit power of the source over each sub-carrier. Then we have

 α1N∑n=1pPT,n+α2N∑n=1pIT,n≤PS, (5a) 0≤pPT,n,pIT,n≤PS,peak,n∈N. (5b)

As for the jammer, as it uses the harvested wireless energy in (2) in the first time-slot to supply the cooperative jamming in the second time-slot, it is subject to the energy harvesting constraint: the total energy used for jamming in the second time-slot cannot exceed , i.e.,

 α2TN∑n=1pJ,n≤EEH=α1TηN∑n=1pPT,n|hJ,n|2, (6a) 0≤pJ,n≤PJ,peak,n∈N, (6b)

where denotes the peak transmit power of the jammer over each sub-carrier.

In particular, we consider two types of receivers at the destination, namely Type-I and Type-II receivers [16], which do not have and have the capability of canceling the jamming signals ’s from the jammer, respectively. In order for a Type-II receiver to successfully cancel the jamming signals, such signals should be securely shared between the jammer and the destination before the cooperative jamming [29, 16, 31, 32]. This can be practically implemented as follows [32]. First, the same jamming signal generators and seed tables are pre-stored at both the jammer and destination (but not available at the eavesdropper). Next, before each transmission phase, one seed is randomly chosen from the seed table and the index of this seed is shared between the jammer and destination. In particular, the two-step phase-shift modulation-based method in [32] can be applied for the seed index sharing as follows. In the first step, the destination sends a pilot signal for the jammer to estimate the channel phase between the destination and jammer. In the second step, the jammer randomly chooses a seed index, and modulates it over the phase of the transmitted signal after pre-compensating the channel phase that it estimated in the previous step. The destination is able to decode the seed index sent by the jammer from the phases of the received signal. Since the length of this seed index sharing procedure is very short and the channel phase between the destination and jammer is different from that between the destination/jammer and the eavesdropper, the eavesdropper does not know the channel phase between the destination and jammer, and thus is not able to decode the signal containing the seed index in such a short time period. For Type-I and Type-II receivers, the secrecy rates of the secure OFDM system over the sub-carriers are respectively given by

 R(I)sec=N∑n=1[R(I)SD,n−RSE,n]+, (7)
 R(II)sec=N∑n=1[R(II)SD,n−RSE,n]+, (8)

where . Here, and are the achievable rates over the sub-carrier from the source to the destination for Type-I and Type-II receivers, respectively, and denotes the achievable rate from the source to the eavesdropper over the sub-carrier , given by

 R(I)SD,n=α2log2⎛⎜⎝1+pIT,n|hD,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠, (9)
 R(II)SD,n=α2log2⎛⎝1+pIT,n|hD,n|2σ2D⎞⎠, (10)
 RSE,n=α2log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠. (11)

### Ii-B Problem Formulation

Our objective is to maximize the secrecy rates in (7) and in (8) for both types of destination receivers, subject to the transmit power constraint in (5) at the source, the energy harvesting constraint in (6) at the jammer, and the time constraint in (1). The decision variables include the transmit power allocation ’s (for WPT) and ’s (for WIT) at the source, and the jamming power allocation ’s at the jammer, as well as the time allocation and . For Type-I receiver, we mathematically formulate the secrecy rate maximization problem as

 (P1):maxα1,α2,pPT,pIT,pJ α2N∑n=1[log2⎛⎜⎝1+pIT% ,n|hD,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠ −log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠] (12) s.t.

where , , and . Note that in the objective function of problem (P1) we have omitted the positive operation , which is due to the fact that the optimal value of each summation term of the objective of problem (P1), i.e. , must be non-negative, and thus the problems with and without the positive operation have the same optimal value and the same optimal solution.222This fact can be proved by contradiction. If , we can increase its value to zero by setting without violating the constraints.

Similarly, for Type-II receiver, the secrecy rate maximization problem is formulated as

 (P2):maxα1,α2,pPT,pIT,pJ α2N∑n=1[log2⎛⎝1+pIT% ,n|hD,n|2σ2D⎞⎠ −log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠] (13) s.t.

Note that problems (P1) and (P2) are non-convex as their objective functions are non-concave. As a result, they are difficult to solve in general. In the following two sections, we tackle such difficulties for (P1) and (P2), respectively.

## Iii Solution to Problem (P1) with Type-I Destination Receiver

First, consider problem (P1) with Type-I destination receiver. We solve this problem by formulating it in a nested form:

 maxα2α2R(I)(α2),s.% t.0≤α2≤1, (14)

where

 R(I)(α2)= maxpPT,pIT,pJN∑n=1[log2⎛⎜⎝1+pIT,n|h%D,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠ −log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠] (15a) s.t.(1−α2)N∑n=1pPT,n+α2N∑n=1pIT,n≤PS, (15b) 0≤pPT,n,pIT,n≤PS,% peak,n∈N, (15c) α2N∑n=1pJ,n≤(1−α2)ηN∑n=1pPT,n|hJ,n|2, (15d) 0≤pJ,n≤PJ,peak,n∈N. (15e)

Here, the outer layer problem (14) corresponds to the time allocation via optimizing , while the inner layer problem (15) corresponds to the joint power allocation optimization under given time allocation. We solve problem (P1) by first solving (15) under any given , and then adopting a one-dimensional search over the interval to find the optimal to solve (14). In the following, we focus on solving the non-convex inner layer problem (15) under given .

### Iii-a Optimal Solution to Problem (15) Via The Lagrange Dual Method

First, we present the optimal solution to problem (15). Despite the non-convexity, problem (15) can be shown to satisfy the “time-sharing” condition defined in [37] as the number of sub-carriers tends to infinity, and the duality gap is zero in this case.333It is observed in our simulations that when , the duality gap for problem (15) is negligibly small and thus can be ignored. Hence, we apply the Lagrange dual method [39] to find its optimal solution.

The partial Lagrangian of problem (15) is

 L(I)(pPT,pIT,pJ,λ,μ) = N∑n=1[log2⎛⎜⎝1+pIT,n|hD,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠ −log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠] +λ[PS−(1−α2)N∑n=1pPT,n−α2N∑n=1pIT,n] +μ[(1−α2)ηN∑n=1pPT,n|hJ,n|2−α2N∑n=1pJ,n], (16)

where and are the dual variables associated with the constraints (15b) and (15d), respectively. The dual function is defined as

 g(λ,μ)=maxpPT,pIT% ,pJ L(I)(pPT,pIT,pJ,λ,μ) s.t. 0≤pPT,n≤PS,peak,∀n, 0≤pIT,n≤PS,peak,∀n, 0≤pJ,n≤PJ,peak,∀n. (17)

Then, the dual problem of (15) is

 minλ,μg(λ,μ)s.t.λ≥0,μ≥0. (18)

Due to the strong duality between problem (15) and the dual problem (18), in the following we solve problem (15) by first obtaining under given and via solving problem (17), and then find the optimal and to minimize for solving (18).

First, consider problem (17) under any given and . In this case, problem (17) can be decomposed into subproblems as follows by removing irrelevant terms, where each subproblems in (19) and (20) are for one sub-carrier .

 maxpPT,n −λ(1−α2)pPT,n+μ(1−α2)η|hJ,n|2pPT,n s.t. 0≤pPT,n≤PS,peak, (19)
 maxpIT,n,pJ,n log2⎛⎜⎝1+pIT,n|hD,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠ − log2⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠−λα2pIT,n−μα2pJ,n s.t. 0≤pIT,n≤PS,peak, 0≤pJ,n≤PJ,peak. (20)

As for subproblem (19), as the objective function is linear over , it is evident that the optimal solution is

 (21)

Note that if , is not unique, and can take any arbitrary value within . In this case, we set only for solving problem (17), which may not be the optimal solution of to problem (15) in general.

As for subproblem (20), the optimization variables and couple together, thus making (20) difficult to solve. To handle this issue, we first obtain the optimal under any given , and then apply a one-dimension search to find the optimal within . To find the optimal to solve problem (20) under given , we define

 an≜|hD,n|2pJ,n|gD,n|2+σ2D, (22)
 bn≜|hE,n|2pJ,n|gE,n|2+σ2E. (23)

When , the objective function of (20) is non-increasing with respect to , and the optimal solution of should be zero. When , the objective function of (20) is concave with respect to , and the optimal solution can be obtained by checking its first-order derivative. Therefore, the optimal for problem (20) under given is

 p∗IT,n(pJ,n)={0,an≤bn,min([p∗n]+,PS,peak),an>bn, (24)

where

 p∗n= √(12bn−12an)2+1λα2ln2(1bn−1an) −12bn−12an. (25)

In addition, let denote the optimal to problem (20), obtained via the one-dimensional search. Then becomes the optimal solution of for (20), denoted by . By combining them with for (19), the optimal solution to (17) under given is found.

Next, we solve the dual problem (18). As this problem is convex but may not be differentiable in general, we find the optimal by applying the ellipsoid method [39]. The required subgradients of with respect to and are respectively given by

 PS−(1−α2)N∑n=1p∗PT,n−α2N∑n=1p∗IT,n, (26)
 (1−α2)ηN∑n=1p∗PT,n|hJ,n|2−α2N∑n=1p∗J,n. (27)

Therefore, the optimal solution of (18) can be obtained as .

With the optimal dual variable at hand, the corresponding ’s and ’s, which are obtained by solving problem (20), become the optimal solution to problem (15). Now, it remains to obtain the optimal solution of ’s for problem (15). In general, the optimal solution of ’s, denoted as ’s, cannot be obtained from (21), since the solution is not unique if . Fortunately, it can be shown that, given , , ’s, and ’s, any ’s that satisfy the constraints (15b), (15c), and (15d) are the optimal solution to problem (15). Thus we can find ’s by solving the following feasibility problem:

 find pPT (28a) s.t. (1−α2)N∑n=1pPT,n+α2N∑n=1p∗IT,n≤PS, (28b) 0≤pPT,n≤PS,peak,n∈N, (28c) α2N∑n=1p∗J,n≤(1−α2)ηN∑n=1pPT,n|hJ,n|2. (28d)

The solution of problem (28) can be obtained by solving the following problem.

 maxpPT N∑n=1pPT,n|hJ,n|2 (29) s.t.

This is because any solution to problem (28) is a feasible solution to problem (29), and thus the optimal solution to (29) must be a solution to problem (28). Let , where denotes the largest integer lower than , and denote as the th largest value in . The optimal solution to problem (29) is

 p∗PT,n=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩PS,peak,|hJ,n|>|~hJ,^k+1|,PS−α2∑Nn=1p∗IT,n1−α2−^kPS,peak,|hJ,n|=|~hJ,^k+1|,0,|hJ,n|<|~hJ,^k+1|. (30)

Using (30), we obtain the closed-form optimal solution of ’s to problem (15).

In summary, the overall algorithm is presented in Algorithm 1. Denote the required accuracy for the one-dimension search in finding and the convergence accuracy of the ellipsoid method as and , respectively. The complexity of the Algorithm 1 for finding the optimal solution is , where and are the radius and Lipschitz constant of the initial ellipsoid, respectively [40].

### Iii-B Minorization Maximization (MM)

Although the Lagrange dual method can find the optimal solution, it needs an exhaustive search of to find the optimal power and for each sub-carrier . As a result, the computational complexity is rather high and even prohibitive for large . Here, we propose a suboptimal approach to solve problem (15) based on the MM approach [41] to avoid exhaustive search, which obtains the power allocation solution iteratively. To facilitate the description, we rewrite (15) as

 maxpPT,pIT,pJ N∑n=1[ln(pIT,n|hD,n|2+pJ,n|gD,n|2+σ2D) −ln(pJ,n|gD,n|2+σ2D) −ln(pIT,n|hE,n|2+pJ,n|gE,n|2+σ2E) +ln(pJ,n|gE,n|2+σ2E)] (31) s.t.

where the property is used. The MM approach solves this problem iteratively as follows: in each iteration, this approach first constructs a surrogate function that is a concave lower bound of the objective function of the original problem, then maximizes the surrogate function within the feasible region of the original problem to obtain a feasible solution. The iteration terminates until the series of the obtained feasible solution converges.

Without loss of generality, we consider the -th iteration with . Suppose that , , denote the solution obtained in the -th iteration. We show how to find , and in the -th iteration. Note that the first-order Taylor expansions of convex functions and around and are their respective global under-estimators [39]. Therefore, we have

 −ln(pJ,n|gD,n|2+σ2D)≥−|gD,n|2(pJ,n−p(k)J,n)p(k)J,n|gD,n|2+σ2D−ln(p(k)J,n|gD,n|2+σ2D), (32)
 −ln(pIT,n|hE,n|2+pJ,n|gE,n|2+σ2E)≥−|hE,n|2(pIT,n−p(k)IT,n)+|gE,n|2(pJ,n−p(k)J,n)p(k)IT,n|hE,n|2+p(k)J,n|gE,n|2+σ2E−ln(p(k)IT,n|hE,n|2+p(k)J,n|gE,n|2+σ2E). (33)

We construct a surrogate function of the objective function in (31) by replacing and with their respective first-order Taylor expansions. Then the maximization of the surrogate function within the feasible region of (31) is expressed as

 maxpPT,pIT,pJ N∑n=1[ln(pIT,n|hD,n|2+pJ,n|gD,n|2+σ2D) +ln(pJ,n|gE,n|2+σ2E)−|gD,n|2pJ,np(k)J,n|gD,n|2+σ2D −|hE,n|2pIT,n+|gE,n|2pJ,np(k)IT,n|hE,n|2+p(k)J,n|gE,n|2+σ2E] (34) s.t.

where the constant terms in the objective function are removed. Since the first and second summation terms in the objective function of (34) are concave with respect to and , and the third and fourth summation terms in the objective function are linear, the objective function of (34) is concave. Furthermore, the constraint functions in (15b)–(15e) are all convex, so the feasible region of (34) is convex. As a result, problem (34) is convex. We solve it by using the Lagrange dual method given in Appendix A, without requiring the one-dimension exhaustive search applied in the optimal approach, and thus the complexity is lower.

In summary, we have the MM approach as in Algorithm 2. Since problem (34) maximizes the surrogate function which is a lower bound of the objective function of problem (15), and the lower bound and the objective function of (15) are equal only at the given point , the objective value of problem (15) with the solution obtained by solving problem (34) is non-decreasing over iteration. As the optimal value of (15) is bounded from above, the MM approach is guaranteed to converge to at least a local optimum [41]. The complexity of the MM approach is , where is the iteration number.

### Iii-C Heuristic Successive Optimization

The previous two approaches are implemented iteratively and thus may have relatively high computation complexity. To overcome this issue, we further propose a low-complexity heuristic successive optimization by finding , , and successively without any iteration. To this end, we decouple the variables and in the constraint (15b), and have the following problem:

 maxpPT,pIT,pJ N∑n=1[ln⎛⎜⎝1+pIT,n|hD% ,n|2pJ,n|gD,n|2+σ2D⎞⎟⎠ −ln⎛⎜⎝1+pIT,n|hE,n|2pJ,n|gE,n|2+σ2E⎞⎟⎠] (35a) s.t. N∑n=1pPT,n≤PS,0≤pPT,n≤PS,peak,∀n (35b) N∑n=1pIT,n≤PS,0≤pIT,n≤PS,peak,∀n (35c) N∑n=1pJ,n≤1−α2α2PEH,0≤pJ,n≤PJ,peak,∀n. (35d)

where denotes the harvested power at the jammer. Problem (35) is obtained based on (15) by replacing the constraints (15b) and (15c) with (35b) and (35c). Since any variables , , and satisfying (35b) and (35c) must satisfy (15b) and (15c), the feasible region of problem (35) is a subset of that of (15). Therefore, solving (35) will result in a feasible solution to (15) and achieve its lower bound.

Next, we solve problem (35) by finding , and successively as follows.

1) Solution of . Note that the optimal value of (35) can be viewed as a function of , denoted by . It is evident that for any given , we have . This is due to the fact that the larger can admit a larger feasible region for , , and for problem (35), as compared to that admitted by (see (35d)). Therefore, is non-decreasing function of . As a result, although is not directly involved in the objective function (35a), increasing in (35d) can increase the objective value in (35a).

Hence, we propose to find the desirable by maximizing . This corresponds to allocating power over the sub-carriers with highest channel gains as follows. Sort the sequence in the descent order and form a new sequence