I Introduction
Recently, to sustainably power wireless devices (WDs) in lowerpower Internet of Things (IoT) systems, e.g., wireless sensor networks [1], radio frequency (RF)based energy harvesting (EH), capable of harvesting energy from stable and controllable RF signals, is envisioned as a promising solution. One of the most popular application paradigms of RF EH is wireless powered communication networks (WPCNs) [2], with which WDs firstly harvest energy from RF signals and then fulfill the communication and computing operations with the harvested energy. Since RF signals carry both information and energy, simultaneous wireless information and power transfer (SWIPT) was proposed as another popular RF EH application paradigm. It was reported that by integrating beamforming technology, heterogeneous devices (e.g., information decoding (ID) and energy harvesting (EH) devices) could be served by SWIPT to meet their different requirements [3].
On the other hand, to enhance the computing capacity of IoT systems, fog computing (or mobile edge computing (MEC) as the alternative^{1}^{1}1Since some works considered mobile secarios, e.g., [4].) was proposed, which is able to reduce the long transmission delay by pushing computing, network control and storage functionalities to the network edge [5]. With fog computing, IoT WDs may offload part or all of the computation tasks to the fog server located at the network edge. There are two modes for IoT WDs to fulfill computation offloading, i.e., partial offloading and binary offloading. The partial offloading mode suits for the divisible computation tasks, which allows the computation task to be divided into two parts and among them one part is offloaded to the fog server. The binary offloading mode suits for the indivisible computation tasks, which requires the whole computation task to be either offloaded or locally computed.
To inherent the benefits of both RF EH and fog computing, some recent works have started to integrate them in a single IoT systems, see e.g., [6][9]. However, these works only considered WPCN systems with fog computing, and no SWIPT was involved. Since SWIPT realizes wireless power transfer (WPT) and wireless information transfer (WIT) at the same time, which is more suitable for latencysensitive applications. A few recent works began to study SWIPTaware fog computing systems. In [10] and [11], the energy consumption was minimized by adopting time switching (TS) mode and power splitting (PS) mode, respectively, where a user was powered by an energy access point and then the tasks were offloaded to a fog server. In [12], the energy consumption was minimized by optimizing power, time and data allocation, where multiple users were considered.
In this paper, we also focus on the optimal design of the SWIPTaware fog computing system, where the fog function integrated hybrid access point (FHAP) first transmits energy and information to EH and ID devices with SWIPT. Then EH devices complete the computation tasks with the harvested energy and fog computing paradigm. Compared with existing works, the following differences should be emphasized.
Firstly, in existing works, see e.g. [10][11], only single type of users or single user were studied. That is, only EH users, PS users or TS users were considered in their works. In view of that in practice, it is very common to deploy various types of WDs with different EH or ID requirements in a single system, we consider heterogeneous users in our work, where both EH and ID devices are investigated.
Secondly
, in existing works, although beamforming vector was optimally designed to enhance transmission efficiency for SWIPTenable fog computing networks, see e.g.
[11], energy and information signals were transmitted with the same beam vector. Considering that information and energy transmissions are with very different physical features and receiving sensitivities (e.g., dBm for ID receivers and dBm for EH receivers), we design different beamforming vectors and matrix for them. By doing so, more flexibility is achieved, which therefore is able to yield a better system performance.Thirdly, in existing works, only a part of the following system resources and configurations, i.e., transmit beamforming vectors and matrix, bandwidth allocation, time assignment, and computation offloading distribution, were jointly optimized, see e.g., [10][12]. In our work, all the configurations and resources mentioned above are jointly optimized in order to achieve the higher system performances.
The contributions of our work are summarized as follows.

An optimization problem is formulated to minimize the FHAP’s required energy, where two optimal designs, i.e., the fixed offloading time (FOT) and the optimized offloading time (OOT) designs, are presented. In the first one, energy and information beamforming designs, bandwidth allocation and computation offloading distribution, are jointly optimized. In the second one, the offloading time assignment is also jointly optimized with the system configurations and resources mentioned in the FOT design.

For the FOT design, since the primal problem is nonconvex, it is relaxed by adopting semidefinite relaxation (SDR) and solved by convex optimization problem solution methods, and then, we theoretically prove that the rankone constraints are always satisfied. So, the global optimal solution is achieved. In order to provide more insights, a semiclosed optimal solution is also presented by using the dual decomposition method.

For the OOT design, since the problem is more complex and cannot be solved by using the solution method for the FOT design, we therefore present a penalty dual decomposition (PDD)based algorithm, which is able to find the suboptimal solution.

Based on the theoretical analysis and simulation results, we also discuss the performance of both proposed methods to solve both designs in terms of required energy and system computing capability (computational complexity). It suggests that if the system is with strong computing capability, the OOT design is suggested to achieve lower required energy; Otherwise, the FOT design is preferred to achieve a relatively low computation complexity.
The rest of this paper is organized as follows. Section II describes the system model. Section III and Section IV present two optimal designs, respectively. Section V analyzes the computational complexity. Section VI shows some numerical results and Section VII concludes this paper.
Ii System Model
Consider a SWIPTaware fog computing system with a antenna FHAP and heterogeneous singleantenna IoT devices as shown in Fig. 1, where the FHAP is deployed to transmit energy and information, as well as providing computation services. Both EH and ID devices exist in the system, where the set of EH devices is denoted as and that of ID devices is denoted as . ID devices desire to receive information from the FHAP, while EH devices desire to harvest energy from it. To enhance WPT and WIT efficiency, beamforming technology is employed at the FHAP, and the energy and information are transmitted simultaneously via different beam vectors. Denote and as the channel coefficients between node and node associated with the EH and ID devices, respectively^{2}^{2}2We assume that node represents the FHAP.. The channel coefficients remain constants over time since the block fading channel model is assumed.
Iia DL Model
In the DL, the FHAP transmits energy and information to EH and ID devices at the same time, where the transmitted symbol is expressed by is the energybearing signal for the
th EH device with Gaussian distribution, i.e.,
. is the information beamforming vector associated with the th ID device. with and with represent the energy and the information signals for the th EH device and the th ID device, respectively. For the th EH device, the harvested energy can be given by(1) 
where is the energy conversion efficiency and , denoting the energy transmit covariance matrix. Let be the desired information rate of the th ID device. The achievable information rate must exceed the desired information rate, which is given by
(2) 
where is the system bandwidth, is the noise power spectral density, , and .
IiB UL Model
In the UL, EH devices offload partial data to the FHAP with frequency division multiple access (FDMA). Assume each computation task is data partitioned, so it can be divided into two independent parts. Let denote the computation tasks data size of the th EH device and be the part for fog computing. Thus, the rest data with size of is for local computing. To complete fog computing, should be offloaded to the FHAP, so
(3) 
where is the bandwidth allocation factor with , is the offloading time for EH devices, and is the transmit power for offloading. Following (3), can be expressed by , where . To complete local computing, the required energy is given by
where describes the effective capacitance coefficient that depends on the chip architecture. is the local central processing unit (CPU) frequency represented by [8] with being the required CPU cycle numbers for computing per bit at the th device (in cycles/bit). Since the available energy for computation offloading and local computing is limited by the harvested energy in the DL, it satisfies that
(4) 
After offloading, the computation tasks are processed at the FHAP with delay , which satisfies that
(5) 
where is computation capacity of FHAP. For notation simplification, we define in the sequel.
IiC System Design Target
For such a SWIPTaware fog computing system, we desire to minimize the required energy at the FHAP by jointly optimizing energy and information beamforming designs at the FHAP in the DL, the bandwidth allocation and the computation offloading distribution in the UL. Firstly, the FOT design is studied in order to find some basic insights for the system in Section III. Then, the OOT design is studied for achieving a better system performance in Section IV.
Iii The Optimal FOT Design
Let being the total information beamforming vector, being bandwidth allocation vector, and being computation offloading distribution vector. The energy minimization problem for the optimal FOT design is given by
s.t.  
(6a)  
(6b)  
(6c) 
where describes the energy consumption per bit at the FHAP (in joule/bit). In the objective function of Problem , both transmitting energy and computation energy are considered [8]. Problem is nonconvex due to the nonconvex constraints (2) and (4). To handle the nonconvexity, new matrix variables denoting with are defined. Then, (2) is rewritten by
(7) 
where and . Moreover, (4) is reexpressed as
(8)  
where . Since , which is convex for joint and , both (7) and (8) are convex now. By introducing , we adopt SDR to relax Problem to be Problem [13], which is given by
s.t.  
(9a) 
where . Problem is convex, which can be solved by known methods, e.g., interior point method.
Proposition 1.
The optimal solution to Problem always satisfies .
Proof:
The proof is shown in Appendix A. ∎
Proposition 1 indicates that the optimal , to Problem
can be obtained by eigenvalue decomposition of
.In order to provide more insights of computation offloading, we present the semiclosed forms as shown in Proposition 2 by using the dual decomposition method.
Proposition 2.
For any given dual variables , the optimal solution and to problem satisfies that
1) When , none of EH devices offloads the tasks to the FHAP, i.e., ;
2) When , for the th EH device with , it has
and where is expressed by (20), and for , it has and .
Proof:
The proof is shown in Appendix B. ∎
Iv The Optimal OOT Design
Considering that the offloading time also have influence on the system performances, we jointly optimize with the variables of Problem in this section. When becomes a variable, becomes nonconvex w.r.t. in (8). To deal with the nonconvex constraint, firstly, we introduce the auxiliary variable with . Then, the energy minimization problem of the OOT design can be given by
s.t.  
(10a)  
(10b)  
(10c) 
With the nonconvex equality constraints in Problem , the solution method proposed in Section III cannot be applied anymore, so a PDDbased method is designed to find the suboptimal solution to Problem [14], which consists of two layers, where the inner layer solves the augmented Lagrangian (AL) problem and the outer layer updates the penalty parameter or the dual variables.
To apply the PDDbased method to solve Problem with the coupling equalities in (10c), Problem is firstly transformed into the AL problem form, i.e.,
s.t. 
where is the penalty parameter and is the dual variable associated with constraint (10c). is defined as a vector to collect all , i.e., . Note that when , Problem is equivalent with Problem [14]. Then, we find the minimum of Problem () by an iteration process. One can observe that Problem () is with a nonconvex objective function and a group of convex constraints. The convex constraints can be divided into two independent sets, i.e, set and set , with separated variables. That is including (6a)(6b) is only associated with and including (5), (6c), (7), (9a), (10a)(10b) is associated with . Since and are independent, Problem () becomes convex when is fixed; on the other hand, when is fixed, Problem () becomes convex. Thus, with the dual variables and the penalty parameter at the th iteration denoted as , the block coordinate descent (BCD) method [15] with two independent blocks, i.e., and , is used to solve the primal variables in Problem (). Then, the th dual variables and the penalty parameter can be updated by the th ones, which are given by
(11) 
and
(12) 
where is the iteration step size. By defining
(13) 
as the stopping criterion, the proposed PDD algorithm is summarized as Algorithm 1. Following [14], the proposed PDDbased algorithm converges to a KarushKuhnTucker (KKT) solution to Problem ().
V Computational Complexity Analysis
According to [16], the computational complexity can be analyzed by discussing the number of constraints and the scale of variables. Follow it, we analyze the computational complexity of FOT design and OOT design in this section.
Firstly, for the FOT design, Problem is with the scale of variables , which is on the order of . Since there are linear matrix inequality (LMI) constraints with the size of and LMI constraints with the size of , to solve Problem , the computational complexity is .
For the OOT design, to solve Problem , there are two layers, where the inner layer is with two blocks. So, by denoting and as the computational complexity of the two blocks, respectively, the computational complexity to solve Problem is , where and are the number of iterations for the inner layer and the outer layer, respectively. For block one with , the scale of variables is on the order of , and then . For block two with , the scale of variables is on the order of , and then , where .
Without loss of generality, we suppose and , where and are constants. Then, can be approximated to and can be approximated to . As a result, and are further approximated to and , respectively.
Vi Numerical Results
In the simulations, the following system parameters are set according to [8, 18, 17], where , , J/bit, MHz, dBm, GHz, , s, J, and . For EH devices, Kbits, , and cycles/bit . For ID devices, dB, . The distance between the FHAP and the th EH device is randomly selected with m and the Rician channel model is adopted with Rician factor being . The distance between the FHAP and the th ID device is randomly selected with m and Rayleigh channel model is considered. Moreover, the pathloss exponent is assumed to be .
For the FOT design, we set . Fig. 2 and Fig. 3 compare the partial offloading mode and two benchmark modes, i.e., “local computing only” mode and “offloading only” mode, versus and , respectively. Fig. 2 shows the minimal required energy at the FHAP versus . It is observed that the partial offloading mode is superior to the two benchmark modes. Moreover, the “offloading only” mode is worse than the “local computing only” mode with the increment of . It means that more sufficient energy supply for EH devices motivates local computing rather than offloading, because local computing requires less energy than computation offloading. Fig. 3 depicts the minimal required energy at the FHAP versus , where the partial offloading mode shows the best performance. One can also see that the “offloading only” mode is better than the “local computing only” mode with a relatively small while the “offloading only” mode is worse than the “local computing only” mode with a relatively large .
For the OOT design, Fig. 4 plots the convergency of the PDDbased method with dB. It is shown that the minimal required energy at the FHAP converge within 6 iterations for outer iterations of Algorithm 1, and the constraint violation reduces to the threshold also in a few iterations for outer iterations of Algorithm 1.
Fig. 5 compares the minimal required energy at the FHAP versus with two fixed and the optimized obtained by the PDDbased method. It is seen that by optimizing , the required energy can be greatly reduced. That is the OOT design achieves much better performance than the FOT design. However, as shown in Fig. 6, to achieve such a performance gain, the running time associated with two designs are different. The running time of the OOT design is much higher than that of the FOT design, and the former one increases faster than the latter one with the increment of user numbers.
Vii Conclusions
This paper studied a SWIPTaware fog computing network consisting of a FHAP and multiple heterogeneous IoT devices. An energy consumption minimization problem was formulated by jointly optimizing energy and information beamforming designs at the FHAP, bandwidth allocation and computation offloading distribution with two designs. For the FOT design, the SDR was adopted and it was proved that the rankone constraints were always satisfied, the global optimal solution was guaranteed. For the OOT design, since the nonconvexity, a PDDbased algorithm was proposed to achieve a suboptimal solution. Simulation results suggest that if the system is with strong enough computing capability, the OOT design is suggested to achieve lower required energy; Otherwise, the FOT design is preferred to achieve a relatively low computation complexity.
Acknowledgements
This work is supported in part by National Key R&D Program of China (no. 2016YFE0200900), in part by the General Program of the National Natural Science Foundation of China (NSFC)(no. 61671051), in part by the Fundamental Research Funds for the Central Universities (no. 2017YJS046).
(14a)  
(14b) 
Appendix A: Proof for Proposition 1
In order to analyze the optimal , the Lagrangian function of Problem is rewritten as (14), where we only consider the the part about . Hence, according to term in (14b), a new problem is constructed as
where and is the dual variable associated with (9a). Since the minimal value of cannot be unbounded and ,
should be a positive definite matrix with probability one, i.e.,
. Then, the KKT conditions of (Appendix A: Proof for Proposition 1) associated with are given by(15a)  
(15b)  
(15c) 
From (15c), we have
(16) 
Moreover, from a basic inequality for the rank of matrices and (15b), we have that
(17)  
Therefore, based on (16) and (17), it implies that . According to (2), , so , which implies that .
Appendix B: Proof for Proposition 2
Following (14b), for any given , , we have that
(18) 
where is the (c) term in (14b). When , becomes . Hence, the optimal solution to problem (18) satisfies . However, when , the KKT conditions are listed as
(19a)  
(19b)  
(19c)  
(19d)  
(19e) 
where represent the optimal dual variables and is the firstorder derivative of . According to (19a)(19c), it is derived that
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