# Power-Efficient and Secure WPCNs with Hardware Impairments and Non-Linear EH Circuit

###### Abstract

In this paper, we design a robust resource allocation algorithm for a wireless-powered communication network (WPCN) taking into account residual hardware impairments (HWIs) at the transceivers, the imperfectness of the channel state information, and the non-linearity of practical radio frequency energy harvesting circuits. In order to ensure power-efficient secure communication, physical layer security techniques are exploited to deliberately degrade the channel quality of a multiple-antenna eavesdropper. The resource allocation algorithm design is formulated as a non-convex optimization problem for minimization of the total consumed power in the network, while guaranteeing the quality of service of the information receivers in terms of secrecy rate. The globally optimal solution of the optimization problem is obtained via a two-dimensional search and semidefinite programming relaxation. To strike a balance between computational complexity and system performance, a low-complexity iterative suboptimal resource allocation algorithm is then proposed. Numerical results demonstrate that both the proposed optimal and suboptimal schemes can significantly reduce the total system power consumption required for guaranteeing secure communication, and unveil the impact of HWIs on the system performance: (1) residual HWIs create a system performance bottleneck in the high transmit/receive power regimes; (2) increasing the number of transmit antennas can effectively reduce the system power consumption and alleviate the performance degradation due to residual HWIs; (3) imperfect CSI increases the system power consumption and exacerbates the impact of residual HWIs.

## I Introduction

Wireless charging of battery-powered devices in wireless communication networks via wireless power transfer (WPT) technology could prolong the lifetime of the networks. In fact, the concept of wireless-powered communication networks (WPCNs), where wireless devices are powered via radio frequency (RF) electromagnetic waves, has gained considerable attention recently in the context of enabling sustainability via WPT [2]. In particular, it is expected that the number of interconnected wireless devices will increase to up to billion by 2020 [3], due to the roll-out of the Internet-of-Things (IoT). A large portion of these wireless devices, some of which may be inaccessible for frequent battery replacement, could be powered wirelessly by dedicated power stations via RF-based WPT technology to facilitate their information transmissions [4]–[6]. Specifically, RF-based WPT offers a more stable and controllable source of energy compared to natural energy sources, such as solar, wind, and tidal, etc., which are usually climate and location dependent [7]–[9]. More importantly, RF-based WPT exploits the broadcast nature of the wireless medium which enables one-to-many simultaneous long-range wireless charging. On the other hand, the large number of wireless devices in future networks encourages the use of low-quality and low-cost hardware components in order to reduce deployment costs. However, RF transceivers equipped with cheap hardware components suffer from various kinds of hardware impairments (HWIs) resulting potentially in a performance degradation for communications. These HWIs are caused by non-linear power amplifiers, frequency and phase offsets, in-phase and quadrature (I/Q) imbalance, and quantization noise. Although the negative impact of HWIs on the system performance can be reduced by calibration and compensation algorithms, residual distortions at the transceivers that depend on the power of the transmitted/received signal are inevitable [10]–[13]. Hence, existing resource allocation algorithms for multiuser WPCNs, e.g. [4]–[6], designed based on the assumption of ideal hardware, may lead to substantial performance losses in practical systems.

The increasing number of wireless devices also poses a threat to communication security in future wireless networks due to the enormous amount of data transmitted over wireless channels [14]–[16]. Nowadays, wireless communication security is ensured by cryptographic encryption algorithms operating in the application layer. Unfortunately, these traditional security methods may not be applicable in future wireless networks with large numbers of transceivers, since encryption algorithms usually require secure secret key distribution and management via an authenticated third party. Recently, physical layer (PHY) security has been proposed as an effective complementary technology to the existing encryption algorithms for providing secure communication [14]–[19]. Specifically, PHY security exploits the unique characteristics of wireless channels, such as fading, noise, and interference, to protect the communication between legitimate devices from eavesdropping. In this context, the authors of [16] designed a resource allocation algorithm that jointly optimizes the transmit power, the duration of WPT, and the direction of spatial beams to facilitate security in WPCNs. In [17], beamforming design was studied for secrecy provisioning in distributed antenna systems with WPT. The authors of [18] investigated the design of secure transmission in wireless-powered relaying systems. In [19], the use of a wireless-powered friendly jammer was proposed to enable secure communication in a point-to-point communication system. However, most of the existing works on secure WPT systems were based on the assumption of ideal hardware [16]–[19] and are not applicable to practical systems with HWIs. Recently, the notion of secure communication under the consideration of HWIs has been pursued. For instance, the work in [20] considered the analysis and design of secure massive multiple-input multiple-output (MIMO) systems in the presence of a passive eavesdropper and HWIs at the transceivers. Besides, the authors of [21] studied the impact of residual HWIs on the performance of a two-way WPT-based cognitive relay network, where the relay is powered by harvesting energy from the signals transmitted by the source in the RF. In [22], the authors analyzed the impact of phase noise on downlink WPT in secure multiple antennas systems. However, the authors of [21, 22] assumed an overly simplified linear energy harvesting (EH) model for the end-to-end WPT characteristic. Yet, measurements of practical RF-based EH circuits demonstrate a highly non-linear end-to-end WPT characteristic [23], which implies that transmission schemes and algorithms designed based on the conventional linear EH model may cause performance degradation in practical implementations. Moreover, the transmission strategies in [20]–[22] were not optimized. Hence, the design of resource allocation for secure communication in WPCNs with the non-linear EH circuits suffering from HWIs is an important open problem.

To address the above issues, we propose a resource allocation algorithm design, which aims at providing power-efficient and secure communication in WPCNs in the presence of a multiple-antenna eavesdropper. The resource allocation algorithm design is formulated as a non-convex optimization problem taking into account the non-linearity of the EH circuits, the existence of residual HWIs at the transceivers, and the imperfectness of the channel state information (CSI) of the eavesdropper. We minimize the total consumed power while guaranteeing the quality of service (QoS) at the information receivers (IRs) in the WPCN. The optimal solution of the proposed problem is obtained via a two-dimensional search and semidefinite programming (SDP) relaxation. The proposed solution unveils that information beamforming from the access point in the direction of the IRs is optimal and that the SDP relaxation is tight. Besides, a low-computational complexity iterative suboptimal scheme is proposed to obtain a suboptimal solution. Numerical results demonstrate that the proposed schemes can significantly reduce the power consumption in the considered WPCN compared to two baseline schemes.

## Ii System Model

In this section, we first present some notations and the considered system model. Then, we discuss the energy harvesting and hardware impairment models adopted for power-efficient resource allocation algorithm design.

### Ii-a Notation

We use boldface capital and lower case letters to denote matrices and vectors, respectively. , , , , , and represent the Hermitian transpose, trace, determinant, inverse, rank, and maximum eigenvalue of matrix , respectively; indicates that is a positive semidefinite matrix; denotes the identity matrix. denotes the space of all matrices with complex entries. represents the set of all -by- complex Hermitian matrices. and represent the absolute value of a complex scalar and the Frobenius norm, respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) vector with mean vector and covariance matrix is denoted by , and means “distributed as”. denotes statistical expectation. stands for . represents the partial derivative of function with respect to the elements of vector . Furthermore, is a diagonal matrix with the elements of on the main diagonal. returns the element in the -th row and -th column of square matrix . is a square matrix with all entries equal to except for the -th diagonal element which is equal to .

### Ii-B System Model

We focus on a WPCN which consists of a power station^{1}^{1}1In this work, we assume that the PS is connected to the main grid with a continuous and stable energy supply. (PS), an access point (AP), IRs, and one eavesdropper (Eve), cf. Figure 1. We assume that the PS, the AP, and Eve are equipped with , , and antennas^{2}^{2}2We note that an eavesdropper equipped with antennas is equivalent to multiple eavesdroppers with a total of antennas which are connected to a joint processing unit performing cooperative eavesdropping. Besides, we assume to enable secure communication., respectively. The IRs are single-antenna devices for hardware simplicity. The communication in the WPCN comprises two transmission phases as shown in Figure 1. We assume that the fading channels in both phases are frequency flat and slowly time-varying. In particular, Phase I, with a time duration of , is reserved for wireless charging, where the PS transmits a dedicated energy beam to the energy-constrained AP.

The instantaneous received signal at the AP during Phase I is given by

(1) |

where is the energy signal vector adopted in Phase I for wireless charging with covariance matrix . The channel matrix between the PS and the AP is denoted by and captures the joint effect of path loss and multipath fading. Vector represents the additive white Gaussian noise (AWGN) at the AP where denotes the noise variance at each antenna of the AP. In (1), and represent the random residual HWIs after compensation introduced at the transmitter and receiver during Phase I, respectively. The model adopted for the residual HWIs will be presented later in the next section.

In Phase II, for a time duration of , the AP transmits independent signals to the IRs simultaneously. Because of the broadcast nature of wireless channels, there is a security threat due to potential eavesdropping. To circumvent this threat, both the AP and the PS deliberately emit artificial noise to degrade the channel quality of the eavesdropper [14]. Therefore, the instantaneous received signal at IR in Phase II is given by

(2) |

where and are the information symbol for IR and the corresponding beamforming vector, respectively. Without loss of generality, we assume that . is the channel vector between the AP and IR , while denotes the channel vector between the PS and IR . Furthermore, and are the Gaussian pseudo-random energy signal sequences broadcasted, i.e., the artificial noise, by the AP and the PS, respectively, where and , denote the corresponding covariance matrices, respectively. These two noise processes are exploited by the AP and the PS to degrade the channel quality of the eavesdropper via jamming. and represent the random residual transmitter HWIs after compensation introduced by the AP and the PS in Phase II, respectively, while represents the residual receiver HWIs introduced by IR . is the AWGN at IR with noise power

The instantaneous received signal at Eve in Phase II is given by

(3) |

where and denote the channel matrices of the AP-to-Eve links and the PS-to-Eve links, respectively. is the AWGN vector at the eavesdropper with noise power . In this work, we assume that ideal hardware is available at the eavesdropper, i.e., there are no HWIs at the eavesdropper, which constitutes the worst case for communication security.

### Ii-C Hardware Impairment Model

In this paper, we adopt the general HWI model proposed in [13, Chapter 4], [24, Chapter 7]. In particular, the residual distortion caused by the aggregate effect of different HWIs, such as I/Q imbalance, phase noise, and power amplifier non-linearities is modeled as a Gaussian random variable whose variance scales with the power of the signals transmitted and received at the transmitter and the receiver, respectively. This model has been widely used in the literature to study the impact of transceiver HWIs on the performance of communication systems [10], [12], [21]. Besides, the authors of [12] showed that this model accurately captures the residual distortions caused by the joint effect of various HWIs in practical multiple-antenna systems.

Hence, the distortion noises caused by the transmitter HWIs at the PS in Phase I and Phase II are modeled as and , respectively. and are diagonal covariance matrices which contain on their main diagonal the distortion noise variances at each antenna of the PS in Phase I and Phase II, respectively, and are given by

(4) | |||||

(5) |

In (4) and (5), and , , respectively, are the average powers of the transmit signal at the -th antenna of the PS in Phase I and Phase II, respectively. Also,
, is a convex, continuous, and monotonically increasing distortion function which quantifies the impact of the HWIs for a given average power of the transmit signal at the -th antenna^{3}^{3}3Note that the model considered here directly maps the signal power to the distortion power while the one proposed in [13, Chapter 4] maps the signal magnitude to the distortion magnitude. , i.e., the function maps the average power of the signal to a specific distortion value. For example, the transmitter distortion function can be modeled by the following convex increasing function:

(6) |

where [Watt] is the average transmit power at antenna . Constants and are model parameters which are chosen such that they fit the measurements of the associated practical systems^{4}^{4}4In practice, the value of the distortion function of the transmitter HWIs usually grows at least linearly with respect to the transmit power which leads to ..
In Figure 2, we illustrate that the proposed model for the transmitter distortion function in (6) closely matches the experimental results in [25].

Similarly, the transmitter HWIs at the AP in Phase II are modeled as with covariance matrix

(7) |

where , . On the other hand, the received signal is affected mostly by phase noises and I/Q imbalances [24, Chapter 7]. In the considered WPCN, these residual HWIs are modeled by the receiver distortion noise at the AP, , cf. (1), where . Moreover, , where is a convex, continuous, and monotonically increasing function that models the receiver impairment characteristic. Additionally, the receiver HWIs at each IR during Phase II are given by , with , where and are introduced for notational simplicity. According to [10], a suitable choice of the receiver distortion function is [Watt], where is the average power of the received signal and is a constant model parameter with a typical range of .

### Ii-D Energy Harvesting Model

In the considered WPCN, we exploit the energy and artificial noise signals transmitted by the PS in Phase I and Phase II to charge the AP and to facilitate secure information transfer, respectively. In this paper, we adopt the practical non-linear RF-based EH model proposed in [26] to characterize the end-to-end WPT at the AP. The total energy harvested by the AP in Phase I is given by

(8) | |||||

where represents the received RF power at the AP. The parameters , , and in (8) capture the joint effects of various non-linear phenomena caused by hardware limitations in practical EH circuits. More specifically, represents the maximum power that can be harvested by the EH circuit, as the circuit becomes saturated for exceedingly large received RF powers. Moreover, and depend on several physical hardware phenomena, such as circuit sensitivity and potential current leakage. In fact, the adopted non-linear EH model was shown to accurately characterize the behavior of various practical EH circuits [26, 27]. In contrast, the conventional linear EH model, which is widely used in the literature [16]–[19, 21], may lead to performance degradation due to a severe model mismatch for resource allocation algorithm design.

###### Remark 1

In practice, the EH hardware circuit of the AP is fixed and parameters and of the non-linear model in (8) can be determined by a standard curve fitting tool.

### Ii-E Channel State Information

In practice, handshaking is performed between the legitimate PS, the AP, and the IRs. As a result, accurate CSI can be obtained by exploiting the pilot sequences embedded in the handshaking signals. In this paper, we assume that the CSI of , , and is available for resource allocation algorithm design. In contrast, the potential eavesdropper may not directly interact with the transmitters and it is difficult to obtain perfect CSI for the corresponding links. To capture the impact of imperfect CSI knowledge of the eavesdropper’s channels on the system performance, we adopt the deterministic model from [27]–[30]. To this end, the CSI of the relevant communication links is modeled as

(9) | |||||

(10) |

respectively, where and are the estimates of channel matrices and , respectively, for resource allocation. Matrices and represent the channel uncertainty which captures the joint effects of channel estimation errors and the time varying nature of the associated channels. In particular, the continuous sets and in (9) and (10), respectively, define the continuous spaces spanned by all possible channel uncertainties with respect to the associated channels. Constants and , denote the maximum values of the norms of the CSI estimation error matrices and , respectively. In practice, the values of and depend on the adopted channel estimation algorithms and the coherence times of the associated channels.

###### Remark 2

Although the eavesdropper may be passive and remain silent, its CSI can be estimated based on the power leakage of the local oscillator of its receiver RF front-end [31].

## Iii Resource Allocation Problem Formulation

In this section, we first define the system performance metrics and then we formulate the resource allocation algorithm design as an optimization problem.

### Iii-a Achievable Data Rate and Secrecy Rate

Given perfect CSI at the receiver, the achievable data rate of IR in Phase II is given by

(11) | |||||

is the received signal-to-interference-plus-noise ratio (SINR) at IR . We note that since the pseudo-random artificial noise signals are known to all legitimate transceivers, the impact of the artificial noise signals on the desired signal can be removed at IR via interference cancellation, i.e., and have been removed^{5}^{5}5We note that the interference caused by the transmitter HWIs cannot be removed at IR as it is an unknown random process. in (11).

On the other hand, as the computational capability of the eavesdropper is not known, we focus on the worst-case scenario for facilitating secrecy provisioning. In particular, we assume that the eavesdropper is equipped with a noiseless receiver and is able to remove all multiuser interference via successive interference cancellation before attempting to decode the information of IR . As a result, the maximum rate at which the eavesdropper can decode the information intended for IR is given by

(12) | |||||

(13) |

is the interference covariance matrix of the eavesdropper. The achievable secrecy rate between the AP and IR is given by

(14) |

As can be seen from (12) and (14), in principle, both the artificial noise signals and the transmitter HWI signals can enhance communication secrecy by degrading the capacity of the channel of the eavesdropper.

### Iii-B Total Power Consumption

In this section, we study the power consumption in both transmission phases. Due to the residual HWIs at the transmitter of the PS, a portion of the transmitted power is wasted during Phase I. More specifically, the total power consumption in Phase I is given by

(15) |

where accounts for the constant circuit power consumption at the PS. Besides, to capture the power inefficiency of practical power amplifiers, we introduce a linear multiplicative constant for the power radiated by the PS [17, 32]. For example, if , then for every Watt of power radiated in the RF, the PS consumes Watt of power which leads to a power amplifier efficiency of . In Phase II, the PS transmits artificial noise signals to degrade the channel quality of the eavesdropper and the associated power consumption is

(16) |

On the other hand, in Phase II, the AP transmits independent information signals to the IRs concurrently. Besides, artificial noise is also emitted by the AP to degrade the channel quality of the eavesdropper. Because of the residual transmitter HWIs, a portion of the power is also wasted at the AP. Hence, the total power consumption at the AP is given by

(17) |

Here, is the constant circuit power consumption and denotes the power amplifier inefficiency at the AP.

### Iii-C Optimization Problem Formulation

In the following, we formulate an optimization problem for the minimization of the total power consumption in both transmission phases of the considered WPCN while guaranteeing secure communication in the presence of residual HWIs, imperfect CSI, and a non-linear EH model. The considered optimization problem is given by:

(18) | |||||

The objective function in (18) takes into account the total power consumption in Phases I and II at the PS and the AP^{6}^{6}6The power consumptions are optimized in both phases, even though the AP is wirelessly charged by the PS only in Phase I. In fact, besides the energy harvested in Phase I, the AP can also use residual energy, , remaining from previous transmission phases for transmission in the current Phase II, cf. constraint C4. Hence, the power consumption of Phase II should also be minimized as well., cf. (15)–(17). Constraint is imposed such that the achievable data rate of IR in Phase II satisfies a minimum required data rate . On the other hand, taking into account the impact of CSI imperfectness, i.e., sets and , constant in limits the maximum tolerable capacity achieved by Eve in attempting to decode the message of IR . In practice, is set by the system operator to ensure secure communication^{7}^{7}7We note that the solution of (18) guarantees a minimum secrecy rate of for IR [8].. in constraint specifies the total time available for both phases. is a constraint on the overall energy consumption at the AP during Phase II. The total available energy at the AP comprises the energy harvested from the dedicated energy signal transmitted by the PS in Phase I and a constant energy . In practice, may represent the residual energy at the AP from previous transmissions or energy obtained from other sources. Furthermore, in is an auxiliary optimization variable which represents the received RF power at the AP. In particular, ensures that is always smaller or equal to the minimum harvested power. is a non-negativity constraint on the durations of Phase I and Phase II, respectively. in constraints and limit the maximum transmit power of the PS in Phase I and Phase II, respectively. Similarly, in constraint specifics the maximum transmit power allowance of the AP in Phase II. Constraint , , and constrain matrices , and to be positive semidefinite Hermitian matrices such that they are valid covariance matrices.

## Iv Resource Allocation Algorithm Design

The resource allocation problem in (18) is a non-convex optimization problem. In fact, the right-hand side of constraint is a quasi-concave function with respect to and . Besides, the optimization variables are coupled in the objective function and in constraint . Also, constraint involves infinitely many possibilities due to the uncertainties of the channel estimation errors. Furthermore, the log-det function in constraint is generally intractable. In this section, we first study the design of a globally optimal resource allocation scheme. The performance of this scheme serves as a performance upper bound for any suboptimal scheme. Then, we derive a computationally efficient suboptimal resource allocation algorithm.

### Iv-a Optimal Resource Allocation

To obtain a globally optimal resource allocation scheme, we first perform the following transformation steps. In particular, we introduce auxiliary optimization matrices, , , and , which account for the HWIs at the PS and the AP, respectively. Additionally, we introduce auxiliary optimization variables, , which represent the distortion noise terms caused by the receiver HWIs at the IRs. Then, we transform problem (18) into the following equivalent rank-constrained SDP optimization problem^{8}^{8}8In this paper, equivalent means that the transformed problem and the original problem share the same optimal solution. with optimization variable set :

(19) | |||||

where and are the equivalent required and the maximum tolerable SINRs at IR and the eavesdropper, respectively. The power consumption in Phase I and Phase II in the objective function can be rewritten as

(20) | |||||

(21) | |||||

(22) |

Furthermore, constraint in (18) is replaced by constraint in (19). These two constraints are equivalent when and , cf. [28]. Constraints , , and are equivalent to the original constraints , , and , respectively, as the new convex constraints are satisfied with equality at the optimal solution.

Next, we handle the coupling of the optimization variables in the objective function and constraint . When both and are fixed, we can solve (19) for the remaining optimization variables. In fact, for a fixed , the quasi-concavity of the right-hand side of constraint is also resolved. Besides, the right-hand side of constraint is concave with respect to . As a result, we study the optimal resource allocation by assuming that the optimal values of and satisfying constraints and are found by a two-dimensional grid search.

Hence, we recast the original problem as an equivalent rank-constrained SDP optimization problem. To this end, we define and rewrite the problem in (19) as

(23) | |||||

where is a set of auxiliary optimization variables. We note that the new sets of constraint pairs , , , and are equivalent to the original constraints , , , and , respectively, as the new constraint pairs are satisfied with equality for the optimal solution. Besides, constraints and are imposed to guarantee that holds for the optimal solution.

Next, we handle the infinitely many possibilities in . First, by introducing an auxiliary optimization matrix , constraint can be equivalently written as:

(24) | |||

(25) |

Then, we introduce a lemma to overcome the infinitely many inequalities in and .

###### Lemma 1

[Robust Quadratic Matrix Inequalities [33]] Let a quadratic matrix function be defined as

(26) |

where , and are arbitrary matrices with appropriate dimensions. Then, the following two statements are equivalent:

(27) |

for matrix and is an auxiliary constant.

Then, constraint can be equivalently transformed into:

(28) |

Similarly, we can transform constraint into its equivalent form:

(29) |

Next, we relax the non-convex constraint in by removing it from the problem formulation. Therefore, for a given and , the equivalent SDP relaxed formulation of (19) is given by:

(30) | |||||

where constraint is due to the use of Lemma 1 for handling the infinitely many constraints associated with the channel estimation errors.

The optimization problem in (30) is a standard convex optimization problem and can be solved efficiently by numerical convex program solvers such as CVX [34]. However, by solving (30) numerically, in general, there is no guarantee that the optimal solution of (30) satisfies of the original problem formulation in (18), i.e., . Hence, in the following, we study the structure of the solution of the SDP relaxed problem in (30).

###### Theorem 1

Proof: Please refer to the Appendix.

Theorem 1 states that the globally optimal solution of (30) can be obtained by information beamforming for each IR, despite the HWIs at the transceivers. Moreover, beamforming is also optimal for wireless charging and jamming in Phase I and Phase II, respectively, even when the non-linearity of the EH circuits, the residual HWIs, and the imperfect CSI are taken into account. In summary, we first discretize the continuous optimization variables . Then, we solve (30) for each pair of satisfying . Finally, we obtain the globally optimal solution^{9}^{9}9Note that the optimality of this method depends on the resolution of the discretization. of (18) by employing a two-dimensional search over all combinations of to find the minimum objective value.

### Iv-B Suboptimal Solution

Although the method proposed in the last section achieves the globally optimal solution of (18), it requires a two-dimensional search with respect to optimization variables and the number of SDPs to be solved increase quadratically with the resolution of the search grid. To reduce the computational complexity, we propose in the following a suboptimal resource allocation algorithm. In fact, (18) is jointly convex with respect to and when the other optimization variables are fixed. As a result, an iterative alternating optimization method [35] is proposed to obtain a locally optimal solution of (18) and the algorithm is summarized in Table I. The algorithm is implemented by a repeated loop. In line 2, we first set the iteration index to zero and initialize the resource allocation policy. Variables and denote the time allocation policy in the -th iteration. Then, in each iteration, for a given intermediate beamforming policy , we solve