I Introduction and Background
Wireless Energy Harvesting (EH) has been recently considered as a key technological concept for the energy sustainability of the Internet of Things (IoT) [2, 3]. An efficient technology belonging into this concept is the Simultaneous Wireless Information and Power Transfer (SWIPT) that targets at realizing perpetual operation of low power and data hungry network nodes [4, 5]. However, to achieve the goal of energy sustainable IoT via SWIPT, the fundamental bottlenecks of the practically available EH circuits need to be effectively handled. Among these bottlenecks are the low rectification efficiency in Radio Frequency (RF) to Direct Current (DC) conversion and the relatively low receive energy sensitivity ; the latter depends strongly on the distance between the Transmitter (TX) power node and a Receiver (RX) EH node. Recent advances in multiantenna signal processing techniques for SWIPT [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] have revealed that the effective exploitation of the spatial dimension has the potential to overcome EH bottlenecks both in point-to-point systems and in multipoint communication like IoT. Thus, SWIPT from a multiantenna TX has increased potential in providing continuous replenishment of the drained energy. However, novel low complexity designs are needed to optimize the harvested power fairness among IoT nodes, while meeting their Quality of Service (QoS) demands [19, 20].
In the seminal work  focusing on the efficiency optimization of point-to-point multiantenna SWIPT systems, the trade off between achievable rate and received power for EH (also known as rate-energy trade off) was investigated for practical RX architectures. Power Splitting (PS), Time Switching (TS), and Antenna Switching (AS) architectures were proposed with the latter two being special cases of the former. Further, spatial switching architecture was recently investigated for QoS-aware harvested power maximization in MIMO SWIPT system . Based on these architectures, a lot of recent developments have lately appeared intending at enhancing the rate-energy performance of multiuser Multiple Input Single Output (MISO) SWIPT systems [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22] . These works mainly target at the optimization of TX precoding and PS operation, and can be classified into the following two categories. The first category is based on whether RXs are required to perform both Information Decoding (ID) and EH (co-located ID and EH)
. These works mainly target at the optimization of TX precoding and PS operation, and can be classified into the following two categories. The first category is based on whether RXs are required to perform both Information Decoding (ID) and EH (co-located ID and EH)[7, 8, 13, 10, 9, 12, 11, 14] or just act as ID or EH RXs (separated ID and EH) [15, 16, 17, 18]. The second category includes performance objectives like the minimization of TX power required for meeting Quality of Service (QoS) and EH constraints [7, 8, 10, 9, 11], and throughput [13, 15, 12, 14] or EH [17, 16, 18] maximization for a given TX power budget and QoS constraints. Recently in , the impact of the density of small-cell base stations together with their transmit power and the time allocation factor between EH and information transfer was analytically investigated for -tier heterogeneous cellular networks capable of SWIPT via TS. However, the jointly globally optimal TX precoding and RX PS operation for energy sustainable multiuser MISO SWIPT incorporating realistic nonlinear RF EH modeling is still unknown. Although the recent works [11, 14] considered nonlinear RF EH modeling for studying multiantenna SWIPT systems, analytical investigations on the joint designs and their efficient algorithmic implementation were not provided. More specifically,  considered that the harvested DC power is a known requirement at each EH-enabled RX, whereas  focused on a point-to-point multiantenna scenario.
I-B Paper Organization and Notations
Section II outlines the motivation and key contributions of this work. The considered system model description is presented in Section III, while Section IV details the proposed joint TX precoding and IoT PS optimization framework. Section V discusses the optimal TX precoding for the considered energy sustainable IoT problem formulation, and the Global Optimization Algorithm (GOA) along with analytical bounds for the unique global optimum are presented in Section VI. Tight closed form approximations for the optimal TX Power Allocation (PA) and RX PS ratios are presented in Section VII. A detailed numerical investigation of the proposed joint design along with the extensive performance comparisons against the relevant techniques is carried out in Section VIII. The concluding remarks are mentioned in Section IX.
Vectors and matrices are denoted by boldface lowercase and capital letters, respectively. The Hermitian transpose and trace of are denoted by and , respectively, and represents the identity matrix (). and denote the inverse and square root, respectively, of a square matrix , whereas means that is positive semi-definite. and are respectively used to represent the Euclidean norm of a complex vector and the absolute value of a complex scalar. and represent the complex and real number sets, respectively, and denotes the smallest integer larger than or equal to .
Ii Motivation and Key Contributions
In this paper we are interested in the energy sustainability of IoT systems comprising of low power and data hungry network nodes capable of EH functionality. Since the lifetime of an EH IoT system  depends on the time elapsed until the first EH network node runs out of energy, maximizing the minimum (max-min) energy that can be harvested among the nodes is critical. Focusing on a MISO SWIPT multicasting IoT system where a multiantenna TX is responsible for simultaneously transferring information and power to low power and data hungry EH PS RXs, we study the EH fairness maximization problem. Our proposed design aims at confronting the short wireless energy transfer range [2, 3] of the considered multicasting system by efficient utilization of the multiple TX antennas, thus, increasing the lifetime of RF EH IoT with individual QoS constraints. In our optimization formulation, we consider a generic RF EH model for the IoT nodes that captures the nonlinear relationship between the harvested DC power and the received RF power for any practically available RF EH circuit [24, 25, 26, 27]. Moreover, we consider the general case of individual QoS requirements for the IoT nodes, which are represented by respective Signal-to-Interference-plus-Noise Ratio (SINR) constraints. Our goal is to jointly design TX precoding and RX PS in order to maximize the minimum harvested energy among RXs, while satisfying their individual QoS constraints.
The EH fairness problem has been recently investigated for secure MISO SWIPT systems . However, the existing TX precoding designs [6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18] adopted an oversimplified linear RF EH model which has been lately shown [26, 11, 14] to be incapable of capturing the operational characteristics of the available RF EH circuits. In addition, the designs in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] are based either on numerical solutions or iterative algorithms. Proofs of global optimality or analytical solutions shedding insights on the interplay between different system parameters are in general missing. Motivated by these observations, in this paper we present an efficient algorithm for obtaining the jointly globally optimal TX precoding and IoT PS design for the considered optimization objective, and provide explicit analytical insights on the presented design parameters. The key distinctions of this work compared with the state-of-the-art are: the design objective that incorporates practical energy fairness IoT demands, a novel solution methodology taking into account the nonlinearity of RF-to-DC rectification operation, and the nontrivial analytical insights on the joint solution that eventually result in efficient low complexity sub-optimal designs.
Next we summarize the novel contributions of this work.
We first present our novel EH fairness maximization problem for energy sustainable MISO SWIPT multicasting IoT systems, while incorporating the practical nonlinear RF-to-DC rectification process. This nonconvex optimization problem is then transformed to an equivalent Semi-Definite Relaxation (SDR) formulation and we prove that it possesses a unique global optimum.
We derive analytical tight upper and lower bounds for the global optimal value of the considered optimization problem. Capitalizing on these bounds, we then present an iterative GOA for the computation of the jointly globally optimal TX precoding and IoT PS ratios design. The fast convergence of the proposed algorithm to the global optimum of the targeted problem has been both analytically described and numerically validated.
We present analytical insights for the optimal TX beamforming directions by investigating the interplay between the directions for either solely optimizing EH performance or the ID one. Tight analytical approximations for the optimal TX PA and uniform PS ratio at each IoT node for a given TX precoding design are also derived. The latter insights and approximations have been used for designing two low complexity sub-optimal algorithms. These low complexity designs are suitable for low power IoT nodes and exhibit performance sufficiently close to the optimum one for certain cases of practical interest.
Our numerical results gain insights on the impact of key system parameters on the trade-off between optimized received RF power for EH at each node and their individual QoS requirements. We also carry out extensive comparative numerical investigations between our presented designs and relevant benchmark schemes which corroborate the utility of our optimization framework.
The key challenges addressed in this paper for the considered MISO SWIPT multicasting IoT systems are: (a) incorporation of nonlinear RF-to-DC rectification operation in the proposed jointly globally optimal TX precoding and IoT PS ratio design; and (b) derivation of efficient (sufficiently close to the optimum for certain cases of practical interest) low complexity joint designs addressing the limited computational capability and energy constraints of low power IoT nodes.
Iii System Description
In this section we first present the considered MISO SWIPT multicasting IoT system together with the adopted channel model and underlying signal model. Then, we introduce the generic RF EH model under consideration that is capable of capturing the rectification operation of realistic RF EH circuits.
Iii-a System and Channel Models
We consider a MISO SWIPT multicasting IoT system comprising of single-antenna IoT nodes and one sink node equipped with antennas that is responsible for simultaneously transferring information and power to the IoT nodes. Hereinafter, each -th IoT node is denoted by RX and the sink node is termed for simplicity TX. The multiantenna TX adopts Space Division Multiple Access (SDMA) with linear precoding according to which each RX is assigned a dedicated precoding vector (or beam) for SWIPT. We denote by the unit power data symbol at TX, which is chosen from a discrete modulation set and intended for RX. These data symbols are transmitted simultaneously through spatial separation with the aid of the linear precoding vectors . As such, the complex baseband transmitted signal from the multiantenna TX is given by . For each precoding vector associated with the data symbol , we distinguish its following two components: i) The phase part given by the normalized beamforming direction ; and ii) The amplitude part representing the power allocated to , ie, . Combining the latter two components of each yields . For the transmitted signal , we assume that there exists a total power budget , hence, it must hold .
A frequency flat MISO fading channel is assumed for each of the wireless links that remains constant during one transmission time slot and changes independently from one slot to the next. We represent by the channel vector between the -antenna TX and the single-antenna RX. The entries of each that depends on the propagation losses of the TX to RX transmission. The baseband received signal at RX can be mathematically expressed as
where represents the zero-mean Additive White Gaussian Noise (AWGN) with variance . Assuming the availability of perfect Channel State Information (CSI) at both TX and RXs, we consider PS receptions  according to which each RX splits its received RF signal with the help of a power splitter. Particularly, an fraction of the received RF power at RX is used for ID and the remaining fraction is dedicated for RF EH. Using this definition in (1), the received signal available for ID at RX is given by
where is a ZMCSCG distributed random variable with variance representing the additional noise introduced during ID at RX. The resulting SINR for at each RX can be derived as
where Similarly, the corresponding received signal available for RF EH at each RX is given by
Using the latter expression, the total received RF power at each RX that is available for EH is defined as
Iii-B RF Energy Harvesting Model
The harvested DC power at each RX after RF-to-DC rectification of the received signal is given using (5) by
where represents the received RF power at RX and denotes the RF-to-DC rectification efficiency function of the RF EH circuitry used at each of the RXs. In general, is a positive nonlinear function of the received RF power available for RF EH [24, 25, 26, 27]. This function is plotted in Fig 1(a) for two real-world RF EH circuits, namely, the commercially available Powercast P1110 Evaluation Board (EVB)  and the circuit designed in  for low power far field RF EH. It is obvious that the widely considered [6, 7, 8, 9, 13, 15, 17, 16, 18, 28] trivial linear RF EH model cannot efficiently describe practical rectification functionality, hence very recently, nonlinear models have been proposed [26, 27]. Despite the nonlinear relationship between the rectification efficiency and , we note that due to the law of energy conservation holds that at each RX is monotonically increasing with , as shown in Fig 1(b). Hence, although the form of differs for different RF EH circuits, the non-decreasing nature of with is valid for all practical RF EH circuits . In other words, the relationship between the harvested DC power and received RF power can be defined as , where represents a nonlinear non-decreasing function. We will exploit this feature in the following section including our proposed joint TX precoding and IoT PS optimization formulation.
Iv Optimization Problem Formulation
We first present our energy sustainable IoT problem formulation and describe its key mathematical properties. Then, we present an equivalent SDR formulation for this problem and prove that it possesses a unique global optimum. We finally discuss the problem’s feasibility conditions.
Iv-a Problem Definition
We are interested in the joint design of TX precoding vectors and RXs’ PS ratios that maximizes the minimum of among the RF EH RXs, while satisfying all the underlying minimum SINR requirements of all RXs. By using (3), (6), and the total TX power , the proposed optimization problem for the considered MISO SWIPT multicasting IoT system is formulated as
where constraints and represent the minimum SINR requirements and maximum TX power budget, respectively. In addition, constraint includes the boundary conditions for ’s. is a nonlinear nonconvex combinatorial optimization problem including the nonlinear function
is a nonlinear nonconvex combinatorial optimization problem including the nonlinear functionin the objective along with the coupled vectors and ratios in both the objective and constraints. Specifically, quadratic terms of appear in both the objective and constraints. To resolve these non tractable mathematical issues, we next present an equivalent SDR formulation for that can be solved optimally. Also, since RXs in the considered system are energy constrained, we assume that is solved at TX using the foreknown SINR demands along with the CSI knowledge of all involved links. After computing the optimal PS ratios, they are communicated to the corresponding RXs via appropriately designed control signals.
Iv-B Semi-Definite Relaxation (SDR) Transformation
Using the definition in and ignoring the rank- constraint for each , an equivalent formulation can be obtained after applying some algebraic rearrangements to the constraints and objective of , as follows:
Constraints and represent the equivalent transformations for and , respectively. We have particularly replaced the TX precoding vectors with their respective matrix definition . An additional variable has been also included to reformulate the max-min problem to the simpler maximization problem having additional constraints, as represented by the new constraint . We have also replaced the harvested power maximization problem in with the corresponding received RF power for EH maximization in by using the two key results as discussed next in Lemmas 1 and 2.
The power at each RX is jointly pseudoconcave in and .
The max-min problem of among the RXs is equivalent to the problem of maximizing the corresponding minimum received RF powers .
From the discussion in Sec III-B it follows that each harvested DC power is a non-decreasing function of the corresponding . It also holds that the non-decreasing transformation of the pseudoconcave function is pseudoconcave [29, 30]. Using these properties together with Lemma 1, we conclude that, since is jointly pseudoconcave in and , the same holds for . In addition, it is known that a pseudoconcave function has a unique global maximum [31, Chap 3.5.9]. Hence, maximizing the minimum among is equivalent to maximizing the minimum among , and function defines the mathematical formula connecting their globally optimal solutions.
Using the latter two lemmas, we next prove the generalized convexity of along with its equivalence to .
having the unique globally optimal solution is an equivalent formulation for .
We first show that belongs to the special class of generalized convex problems [31, Chapter 4.3] that possess the unique global optimality property. Actually, and in are linear (ie, convex) constraints. Due to the linearity of the expression and the convexity of in and , is jointly quasiconvex. In addition, is jointly pseudoconcave from Lemma 1. Combining these properties of constraints along with the linearity of objective and result in [31, Theorem 4.3.8], yields that the Karush Kuhn Tucker (KKT) point of is its globally optimal solution.
It follows from Lemma 2 that the harvested DC power max-min problem is equivalent to maximizing the minimum among the received RF powers. Using this result together with the epigraph transformation [30, Chap 4.2.4] of , we obtain with an implicit rank- constraint to be satisfied by the globally optimal TX precoding matrix . As it will be proven in the following lemma, this condition is always implicitly met. Hence, and are equivalent and the globally optimal solution of can be obtained from the globally optimal solution of , where the optimal TX precoding vector for each RX is derived from the EigenValue Decomposition (EVD) of
is derived from the EigenValue Decomposition (EVD) of.
The optimal solution of implicitly satisfies the rank- condition for .
Keeping constraints and in implicit and associating the Lagrange multipliers , , and , respectively, with the constraints , , and , the Lagrangian function of is defined as
with and . As and , it holds . Using this function, the dual function of is given by
Since has a unique globally optimal solution (cf Theorem 1), it holds from the strong duality principle [31, Section 6.2] that its solution can be also obtained from the solution of the following dual problem:
Denoting the optimal solution of as , the optimal power and that maximize the Lagrangian in (IV-B) is the optimal solution of . Since the variables are decoupled from the remaining variables and as shown in (IV-B), we can compute by solving the following equivalent problem (note that constant terms have been discarded in this equivalent formulation):
In (9), and are obtained by substituting the optimal solutions of into and , respectively. Here we have implicitly used and , where the latter is imposed in order to have a bounded solution for . Using these properties along with the results proved in [8, Prop 1] (or [17, Th 1]), the rank- property of the optimal solution of (9) can be shown by contradiction. Hence, each has to be a rank- matrix like .
The outcomes of our energy sustainable IoT problem formulation that focuses on the practical QoS-aware harvested power fairness maximization are different from the objectives in existing multiuser SWIPT works [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. This will be shown analytically in Section V and through numerical validations in Section VIII. The same holds for the joint TX precoding and IoT PS design of the proposed optimization problem that will be presented in the following sections.
Iv-C Feasibility Conditions
The feasibility of depends on the underlying SINR constraints of all RXs that need to be simultaneously met for a given total TX power budget . To check whether can be satisfied, we solve the following problem:
, which does not consider EH (ie, for each RX), has been widely studied and its globally optimal solution denoted by is given by [32, eq (10)]. If , then both and are feasible, otherwise they are not. Also, to ensure in , needs to satisfy , where is the receive energy sensitivity of the RF EH circuit [3, 25].
V Optimal TX Precoding Design
Here we provide insights on the optimal TX precoding design for our energy sustainable IoT problem. These insights will be used later for implementing efficient algorithms for the jointly global optimal TX precoding and IoT PS design.
V-a Optimal TX Precoding Structure
The Lagrangian function of given by (IV-B) can be rewritten in terms of the precoding vectors as
Then, from Theorem 1, each optimal can be derived by solving (KKT condition for optimal TX precoding vector), which after few algebraic manipulations simplifies to
Since in (11) is a scalar, the optimal beamforming direction for each RX can be obtained as
The Lagrange multipliers and in (12) respectively correspond to the constraints and , and are respectively related to the EH and SINR requirements for RX. When , coincides with the optimal TX precoding for ID (ie, no EH) as given by [32, eq (10)]. Whereas, yields , which refers to Maximal Ratio Transmission (MRT) for RX. Therefore, the structure of is a modified version of the regularized Zero Forcing (ZF) beamformer  that balances the trade off between minimizing interference solely for efficient ID and maximizing the intended signal strength for efficient EH.
V-B TX Precoding Design
As noted in the above discussion, the optimal TX beamforming direction needs to balance the trade-off between the beamforming directions intended for (i) maximizing the harvested energy fairness and (ii) the one targeting efficient information transfer by meeting the SINR demands with minimum required TX power budget. Capitalizing this insight we propose the following weighted TX beamforming direction:
where represents the relative weight between the TX beamforming direction for efficient ID and the corresponding direction for efficient EH. We next derive the latter directions from their respective optimal TX precoding vectors and , respectively.
V-B1 Energy Fairness Maximization (EFM)
By setting (ie, no ID requirement at RXs) in , we focus solely on maximizing the EH fairness of the considered multicasting IoT system. For this setting, reduces to the following EH fairness optimization problem:
Since, has a linear objective and constraints, it is convex. Let denote its jointly optimal solution.
The optimal solution of implicitly satisfies the rank- condition for . Hence, for each RX is derived from the EVD of .
Keeping constraint in implicit and associating the Lagrange multipliers and with the constraints and , respectively, the Lagrangian function of is defined as
where . Following similar steps to the proof of Lemma 3, the optimal precoding can be obtained by solving the equivalent problem defined below:
where and are obtained by substituting the optimal solutions of the dual problem for into and . Lastly, using and along with the results in Lemma 3, the rank- properties of the optimal solution of (15), and thus that of in , can be shown by contradiction.
V-B2 Information Decoding (ID)
When solely targeting TX precoding for enhancing the ID performance, we consider the case (ie, no EH requirement at RXs). To derive , we focus on solving as defined in Section IV-C that seeks for the precoding design minimizing the total TX power, while meeting the individual SINR requirements. Also, is feasible only if .
In the following section we present an iterative GOA for that utilizes the optimal TX precoding vectors and .
Vi Joint TX Precoding and RX Power Splitting
Although exhibits generalized convexity as shown in Section IV-B, standard optimization tools (eg, the CVX Matlab package ) cannot be used due to the fact that constraint does not satisfy the Disciplined Convex Programming (DCP) rule set; this constraint includes the coupled term . To resolve this issue, we summarize in the sequel an iterative GOA for solving that capitalizes on our derived tight upper and lower bounds for the optimal of and uses and of Section V-B.
Vi-a Tight Analytical Bounds for the Optimal in
Vi-A1 Upper Bound on
Clearly, the optimal solution of as defined in Section V-B1, provides an upper bound for because there is no SINR constraint to be met. However, we next present a tighter upper bound that can be obtained from the solution of the following problem:
In , we seek for the minimum TX power required to meet together with the SINR demands . The objective and constraints of this problem are jointly convex in with satisfying the rank- constraint. In addition, satisfies the DCP rule set, hence, we can efficiently compute its jointly optimal solution using . The tight upper bound for can thus be obtained as
Note that, due to the presence of in and along with the fact that and , it holds that .
Vi-A2 Lower Bound on
With , is feasible and its solution can be lower bounded as
To find a tighter lower bound, we then set in and denote its jointly optimal solution by . The lower upper bound for can be then derived using the solution of as
Lastly, since and , it yields .
Note that the tightness of the presented lower and upper bounds will be later numerically validated in Section VIII.
Vi-B Global Optimization Algorithm (GOA)
The proposed GOA for efficiently solving is based on one-dimensional Golden Section Search (GSS) over the feasible range of values, as given by the previously derived bounds and . Its detailed algorithmic steps for the case where is feasible are outlined in Algorithm 1. Due to the generalized convexity of and the tightness of and , the proposed GOA converges fast to the optimal satisfying the TX power budget expressed by constraint within an acceptable tolerance .