## I Introduction

As a new paradigm in communications, internet of things (IoT) can provide intelligent control and smart solutions for various tasks in our lives by connecting a large variety of devices [1, 2, 3]. It has been gradually applied in various applications from smart homes, healthcare to structural and environmental monitoring, and disaster warning and so on [4, 5, 6, 7, 8]. Usually IoT networks involve a large number of sensor nodes to collect and exchange information with data center and can also be regarded as wireless sensor networks (WSNs). The success of IoT networks heavily relies on the reliability and sustainability of the sensor nodes. When a large number of sensors are deployed, power supply for the sensor nodes becomes a challenging issue to be solved. Currently, there are several available solutions to power up the sensors. The first one is to wire the sensor to a fixed power supply through cables. The installation is time consuming and location dependent, and the wire connection also limits the mobility of sensors. The second one is to power sensors by batteries. However, batteries usually have short lifetime and their maintenance and replacement are costly and difficult, especially when sensors are deployed in harsh environment or remote locations. It is even impossible when the sensors are deployed inside the building structures or human bodies[4, 5]. The third one is to self-harvest energy from natural energy sources, such as solar and wind. But the amount of harvested energy is unstable and affected by uncontrollable nature factors. Lately, a new solution “wireless power transfer (WPT)” was proposed [6, 7, 8]. It leverages the fact that energy could be transferred wirelessly through radio frequency (RF) signals. Compared to other natural based energy harvesting, the RF oriented energy harvesting is generally ubiquitous, predictable and steady with low cost[9, 10]. As reported in [9], the energy harvesters operating at 915MHz and using Dipole antennas can collect about 3.7mW and 1uW of wireless power from RF signals at distances of 0.6m and 6m, respectively. Meanwhile, advanced antenna and transceiver designs for realizing high RF energy harvesting efficiency have also been reported [10]. With high feasibility and a wide range of applications in IoT, wireless powered sensor networks (WPSNs) have thus gained considerable research interest recently.

Generally in WPSNs, wireless powered sensors firstly harvest energy from the downlink RF signal transmitted by a power source or a hybrid access point (H-AP) which serves dually as a power source and a data center, and then utilize the harvested energy for uplink information transmission [13, 14, 15]. With multiple sensors, the uplink transmission can be supported following spatial division multiple access (SDMA) or time division multiple access (TDMA) schemes. To achieve various objectives, optimal designs for SDMA-enabled and TDMA-enabled WPSNs are necessary and have been investigated in the literature [15, 16, 17, 18, 19, 20, 21]. Specifically, for SDMA-enabled WPSNs, the optimal H-AP beamforming was proposed for maximizing the uplink sum rate and maximizing the minimum uplink rate among multiple wireless powered sensors, respectively in [15] and [16]. In [17], uplink sum throughput maximization under various cooperation protocols in SDMA-enabled cognitive WPSNs was studied. Moreover, fairness-based uplink throughput maximization was discussed for multiple-input-multiple-output (MIMO) WPSNs in [18]. With respect to the TDMA-enabled WPSNs, [19, 20] studied the optimal time allocation for multiple energy harvesters to maximize uplink sum throughput for single-input-single-ouput (SISO) WPSNs. When multiple-input-single-output (MISO) TDMA-enabled WPSNs were considered, joint beamforming design and time allocation for uplink sum throughput maximization were investigated in [21]. It was found that the downlink energy beamforming design for the H-AP was similar to that in the SDMA-enabled WPSNs. For TDMA-enabled WPSNs with separate multiple-antenna power source and single-antenna data center, sum throughput maximization through optimal beamforming and time allocation was investigated in [7]. Optimal solutions were proposed for two different scenarios, i.e., the power source and the sensor nodes belong to the same or different service operator(s).

All the aforementioned optimal designs consider fully wireless powered sensor networks, in which all the sensor nodes are wireless powered and the downlink is dedicated for wireless power transfer only^{1}^{1}1If the H-AP has information to be transmitted to the sensor nodes, the information can be transmitted in another dedicated downlink phase.. However in practice, mixed power solution may be adopted to achieve enhanced reliability in WSNs. Particularly, most of the sensor nodes are wireless powered, but the key sensor node is powered up by traditional battery/wire power for reliable communications. This network can be regarded as a partially wireless powered sensor network. In this partially WPSN, dual functions of RF signals in wireless information and power transfer can be exploited simultaneously in the downlink to further improve the spectral efficiency. In other words, the downlink RF signals can carry not only energy to wireless powered sensors but also information to the key sensor node with traditional power supply, which results in simultaneous wireless information and power transfer (SWIPT) in the downlink[11, 12]. This partially WPSN with downlink SWIPT can achieve efficient communications with a simple mixed power supply solution and thus is attractive for practical applications[23]. Nevertheless, optimal design for such WPSNs with downlink SWIPT is rarely investigated due to the difficulty in coupled downlink and uplink design as well as mixed power and information transfer. As far as we know, there is only one available optimal design for WPSNs with downlink SWIPT reported in [23]. It considered a SDMA-enabled WPSN with multiple users. Multiple downlink SWIPT phases were introduced to sequentially transmit information to one sensor while power up the others and equal time duration was assumed for all the downlink and uplink phases. Joint design of downlink beamforming and uplink power allocation were investigated to maximize the minimum downlink and uplink signal-to-interference-and-noise ratios (SINRs).

Here we take a step forward to investigate joint design of beamforming and time allocation for partially WPSNs with downlink SWIPT. To guarantee quality of service (QoS) at the key sensor node with traditional power supply, downlink rate constraint is taken into account and uplink sum rate maximization under such downlink rate constraint is mainly concerned. The implementation of the downlink SWIPT as well as the consideration of downlink rate constraint makes the optimal design challenging and also differentiates our design from the prior ones [15, 16, 17, 18, 19, 20, 21, 23]. Both SDMA and TDMA schemes are considered for the uplink transmission. Notice that extra time allocation in the uplink is involved in the TDMA-enabled WPSNs with downlink SWIPT and its optimal design hasn’t been discussed before. The formulated uplink sum rate maximization problems for both SDMA-enabled and TDMA-enabled partially WPSNs are originally non-convex with coupled optimization variables. But they can be reformulated as concave problems through certain transformation. After analyzing the feasibility of uplink sum rate maximization problems and the influence of the downlink rate constraint, semi-closed-form optimal solutions for both SDMA-enabled and TDMA-enabled WPSNs are proposed with guaranteed global optimality. Complexity analysis is also provided to justify the advantage of our proposed solutions in low complexity. The effectiveness and optimality of our proposed optimal solutions are finally demonstrated by simulations.

The rest of the paper is organized as follows. The partially WPSNs are introduced and the optimal design problems are formulated in Section II. Feasibility analysis and problem reformulation are given in Section III. Semi-closed-form optimal solutions for SDMA-enabled and TDMA-enabled WPSNs are derived and meaningful insights in the optimal solutions are pointed out in Section IV. Simulation results are presented in Section V and finally conclusions are drawn in Section VI.

Notation

: Throughout this paper, bold-faced lowercase and uppercase letters stand for vectors and matrices, respectively. The symbols

, , , and denote transpose, conjugate, Hermitian, inverse and trace of matrix , respectively. In addition, denotes the vector formed by stacking the columns of a matrix, while represents a diagonal matrix constructed from a vector. denotes a-dimensional identity matrix. All eigenvalues and singular values in our work are arranged in a decreasing order. Finally,

indicates approaches and .## Ii system model and problem formulation

As shown in Fig. 1, we consider a partially WPSN with wireless powered sensor nodes, also called energy harvesters (ERs), , one battery/wire powered sensor node, also called information receiver (IR), and one hybrid access point (H-AP). The H-AP serves as not only power source to power up the sensor nodes ERs but also the data center to communicate with all sensor nodes. The H-AP and ERs are equipped with and antennas for effective wireless power transfer and harvesting, respectively, while the single-antenna IR is considered. This mixedly powered sensor network leverages wireless power transfer technique to solve for the challenging power supply problem for the majority of the sensor nodes, while adopts traditional battery/wire power only at the key sensor node to guarantee its communications. It can achieve efficient communications with simple power supply solution and has a wide range of practical applications.

In the partially WPSNs, two different phases are involved for power transfer and communications. In the first downlink phase, the H-AP transfers power to the ERs and sends information to the IR simultaneously. In other words, SWIPT is conducted in the downlink. Then in the second uplink phase, all the sensor nodes transmit sensing data to the H-AP. Since the ERs do not have a fixed power supply, their data transmissions are only powered by the harvested energy in the first downlink phase.

Defining the total time duration for the two phases as a unit, we assume the first slot is utilized for downlink transmission and the remaining slot is allocated for uplink transmission. In the uplink transmission, two multiple access schemes, i.e., SDMA and TDMA, are considered. Specifically, for the SDMA-enabled WPSN, the IR and all ERs simultaneously transmit information to the H-AP in the slot, while for the TDMA-enabled WPSN, each ER and the IR sequentially transmit information to the H-AP in the and slots, respectively, with . In general, SDMA outperforms TDMA in terms of uplink sum rate. However, TDMA is easy to implement with low signal detection complexity at the receiver. Both of them are widely adopted in wireless communication systems [21, 23].

### Ii-a SDMA-enabled WPSN

In the first downlink phase, the H-AP adopts the SWIPT technique to transmit energy and information to ERs and the IR simultaneously. The energy-carrying information signal is denoted as with covariance matrix . The transmission power can be written as and should usually satisfy the power constraint , where is the maximum allowable transmission power. Denoting the downlink channels from the H-AP to the IR and ER as and , respectively, the received signals at ERs and IR can be respectively written as

(1) | ||||

(2) |

where and are the additive white Gaussian noises (AWGNs) at the ER and IR, respectively. According to (1), the harvested energy at ER in this slot can be expressed as

(3) |

where denotes the energy harvesting efficiency of ER . Meanwhile, based on (2), the achievable downlink rate of the IR is expressed as

(4) |

In the second uplink stage, the IR and ER simultaneously transmit the information signals with and with covariance matrix to the H-AP, respectively. The transmission power at the IR is fixed as . Since the energy at the ER is only coming from energy harvesting in the first phase, the transmission energy at the ER should not exceed the harvested energy , i.e., . By denoting and as the uplink channels from the IR and ER to the H-AP, respectively, we have the received signal at the H-AP as

(5) |

where denotes the received AWGN noise at the H-AP. Similarly to [24], we assume that successive interference cancellation technique is adopted at the H-AP, and thus the achievable uplink sum rate of the SDMA-enabled WPSN can be formulated as

(6) |

where . In this paper, we aim at maximizing the uplink sum rate in (6) while guaranteeing the downlink communication quality-of-service (QoS), i.e., the downlink information rate in (4), by jointly optimizing the time splitting ratio , downlink energy beamforming and uplink information beamforming . Mathematically, the uplink sum rate maximization problem in the SDMA-enabled WPSN is formulated as

(7) |

Here, the constraint CR1 corresponds to the maximum transmission power constraint at the H-AP, CR2 models the downlink QoS constraint at the IR node where denotes the required minimum downlink rate, and CR3 considers the uplink transmission energy constraints at the ERs since their uplink energies are only coming from the harvested energies (3) in the downlink. Notice that our uplink sum rate maximization is different from those for fully WPSNs in [15, 16, 17, 18, 19, 20, 21], since both energy and information transfers are conducted simultaneously in the downlink and additional downlink rate constraint CR2 is considered to guarantee the QoS for the information transfer to the IR. Moreover, our rate maximization problem also differs from that in [23] with additional uplink energy constraints CR3. Clearly, the downlink rate constraint and the uplink energy constraints are practical and necessary to be considered in partially WPSNs. However, their consideration complicates the optimization problem with highly coupled variables , and the problem becomes non-convex and difficult to solve.

### Ii-B TDMA-enabled WPSN

For the TDMA-enabled WPSN, the received signals at the ER and the IR in the downlink phase are the same as that in (1) and (2), respectively. However, in the uplink phase, the IR and each ER sequentially transmit signals to the H-AP within the slots of and , respectively. Therefore, the achievable uplink sum rate is given by

(8) |

Accordingly, the uplink sum rate maximization problem in the TDMA-enabled WPSN is formulated as

(9) |

where . Similarly to CR3 in (II-A), the constraint CR4 models the uplink transmission energy constraints but with the uplink slot for the ER adjusted as . Additionally, the constraint CR5 is introduced due to the implementation of TDMA protocol. Clearly, the problem (II-B) is also nonconvex and is more challenging than the problem (II-A) since more time slots are involved in the vector for optimization. Obviously, the key challenge in the problems (II-A) and (II-B) comes from the downlink rate constraint CR2, its feasibility will be discussed first before we proceed to solve for the optimization problems.

## Iii Feasibility Analysis

### Iii-a Feasibility of downlink rate constraint

Different from most of the prior designs for WPSNs, downlink rate constraint CR2 is considered here due to the adoption of simultaneous wireless power and information transfer in the downlink. Clearly from (II-A) and (II-B), the feasible downlink rate threshold is limited by the achievable downlink rate in (4) which depends on the downlink beamforming , while is also constrained by the maximum downlink transmission power in CR1. The feasible downlink rate threshold is thus upper bounded by

(10) |

It is easily observed that the optimal for the problem (III-A) is , and then the problem (III-A) reduces to a conventional rate maximization problem for a MISO system and has been solved in [25]. More specifically, the optimal solution of for the problem (III-A) is , which means that the downlink beamforming is aligned for information transmission only. The upper bound of feasible downlink rate threshold is correspondingly given as [25]. Since both SDMA-enabled and TDMA-enabled WPSNs have the same downlink process, this upper bound is applicable for both problems (II-A) and (II-B). In other words, we can conclude that when , the problems (II-A) and (II-B) are feasible.

### Iii-B Tightness of downlink rate constraint

In the feasible region , the tightness of downlink rate constraint CR2 depends on the actual rate threshold and will heavily affect the optimal solutions for the problems (II-A) and (II-B). Now we take the problem (II-A) for tightness investigation first. Specifically, if neglecting the downlink rate constraint , the problem (II-A) reduces to the sum rate maximization problem for fully WPSNs in [17], and the corresponding optimal downlink beamforming is given as , where

is the dominated eigenvector of a certain linear combination of the covariance matrices for all ERs’ downlink channels, i.e.,

[17]. This solution means that the downlink beamforming is aligned for energy transfer only.Given this downlink beamforming, the achievable downlink rate can be expressed as , and the corresponding optimal achievable uplink sum rate is in fact an upper bound of that of the problem (II-A) with the constraint . If the downlink rate threshold is less than the rate , i.e., , the downlink rate constraint can be automatically satisfied with the inequality strictly holding, i.e., . In other words, when , the constraint is inactive and can be ignored in the problem (II-A). However, when , the downlink rate constraint cannot be ignored and we have the following result.

###### Lemma 1.

When , the downlink rate constraint in the problem (II-A) is tight, i.e., the optimal solution will exist at the boundary with satisfied.

###### Proof.

Please see Appendix A. ∎

With respect to the problem (II-B), by neglecting the constraint , although the corresponding optimal downlink beamforming and the achievable downlink rate ( and ) cannot be directly obtained based on the results in [17] due to the extra time allocations and , they can be derived using the joint concavity of the problem (II-B) proved in Section III. C. Then the above tightness result also holds for the problem (II-B) in the TDMA-enabled WPSNs. Since the tightness proof is similar to that in Appendix A, it is omitted here for conciseness.

### Iii-C Problem reformulation

In order to solve the uplink sum rate maximization problem (II-A) for the SDMA-enabled WPSNs effectively, we define two new variables and . Then the problem (II-A) can be reformulated as

(11) |

Clearly, the objective function of the problem (III-C) is the perspective of the concave function . According to [26, p. 39], the concavity is preserved by the perspective operation. Therefore, the objective function is strictly and jointly concave with respect to (w.r.t.) . In addition, all constraints in (III-C) are convex. We then can conclude that the problem (III-C) is jointly concave w.r.t. . Similarly, by redefining , the uplink sum rate maximization problem (II-B) for TDMA-enabled WPSNs can also be reformulated as

(12) |

Following a similar logic of proving the concavity of the problem (III-C), the problem (III-C) is also jointly concave w.r.t. Both problems (III-C) and (III-C) can be numerically solved by standard convex optimization technique [26], and the globally optimal beamforming and and the time slots can then be obtained with simple variable substitution. However, the numerical solution not only has high computational complexity but also provides little insight. In the following, we will propose insightful semi-closed-form optimal solutions for the jointly concave problems with low complexity.

## Iv Semi-closed-form optimal solutions for SDMA-enabled and TDMA-enabled WPSNs

It is noticed that when the time splitting ratio is given, the problems (III-C) and (III-C) are still jointly concave w.r.t. the other design variables . It inspires us to solve for the optimal solution for the other design variables by fixing , and then find the optimal afterwards.

### Iv-a SDMA-enabled uplink sum rate maximization

When is given, the SDMA-enabled uplink sum rate maximization problem (III-C) can be rewritten as

(13) |

and its corresponding Lagrangian function is given by

(14) |

where and

(15) |

Here, , are the non-negative lagrangian multipliers corresponding to constraints , and in the problem (IV-A), respectively. While an are the lagrangian multipliers corresponding to and , respectively. denotes the Hermitian square root of the positive definite matrix . Since the problem (IV-A) is concave, its Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for the optimal solution. Based on the KKT conditions and the definitions and , the optimal beamforming and lagrangian multipliers should follow the structure shown in the following theorem.

###### Theorem 1.

For any given time splitting ratio , the optimal lagrangian multipliers , the optimal downlink beamforming and the optimal uplink beamforming to the problem (IV-A) are expressed as

(16a) | |||

(16b) | |||

(16c) | |||

(16d) | |||

(16e) | |||

(16f) |

where and are the maximum eigenvalue and the corresponding dominated eigenvector of , respectively, is defined as the -dimensional right singular matrix of

based on the singular value decomposition (SVD)

, and the diagonal matrix consists of singular values of . In addition, denotes the achievable downlink rate function as .###### Proof.

Please see Appendix B. ∎

Theorem 1 provides the semi-closed-form optimal solutions for the problem (IV-A). By iteratively solving from the lagrangian multipliers as well as the downlink and uplink beamforming based on (16a)-(16f), the optimal solution can be obtained. The convergence of the iterative calculation and the global optimality of the obtained solution are also guaranteed since the problem (IV-A) is jointly concave. Moreover, from Theorem 1, we have the following insightful observations.

1) The optimal downlink beamforming

is a rank-1 matrix, whose eigenspace is uniquely determined by the dominant eigenvector in the joint eigenspace spanned by

ERs’ downlink channels and the IR’s downlink channel . Since SWIPT is conducted in the downlink, the downlink beamforming should provide a good balance between energy transfer and information transmission. Whether the optimal downlink beamforming aligns toward the space of the ERs’ channels for energy transfer or that of the IR’s channel for information transmission is controlled by the lagrangian multipliers and depends on the downlink rate constraint. Specifically, when the downlink rate constraint is not high, i.e., , holds and the optimal downlink beamforming is fully aligned with the eigenspace of the ERs’ channels. In other words, the optimal downlink beamforming is designed only aiming at energy transfer while ignoring the need of information transmission since the required information transmission can be automatically satisfied.However, when the downlink rate constraint is high, i.e., , we have , the need for information transmission cannot be ignored and the optimal downlink beamforming shifts from the space of the ERs’ channels toward that of the IR’s channel. When the downlink rate constraint is as high as the maximum rate , and the downlink beamforming should be fully aligned with the IR’s channel to meet the strict information transmission requirement. To some extent, the optimal lagrangian multiplier therefore can be regarded as an indicator of the relativity between the downlink beamforming and the IR’s downlink channel. High means high relativity between the optimal downlink beamforming and the IR’s downlink channel.

2) It is seen from (16b)(16e) that the optimal uplink beamforming depends on the downlink beamforming through the lagrangian multiplier . This is due to the fact that the uplink transmission energy in each ER is constrained by its energy harvested in the downlink, i.e., the constraint . Moreover, the optimal uplink beamforming for the th ER is related to other ERs’ uplink beamforming, i.e., , through the matrix . To solve for the coupled uplink beamforming in (16b), the iterative water-filling procedure [27] can be applied. With the concavity of the problem (IV-A), the iterative water-filling procedure is guaranteed to converge to the globally optimal . Interested readers can refer to [27] for the detailed iterative process.

3) When , the optimal lagrangian multiplier is determined by the nonlinear function . After analysis, we find the function has the monotonically increasing property as follows.

###### Lemma 2.

Given the downlink beamforming structure in (16a), the function is monotonically increasing w.r.t. and converges to when .

###### Proof.

Please see Appendix C. ∎

Based on Lemma 2, the optimal satisfying can be uniquely determined by the bisection search. Now with the semi-closed-form optimal solution in Theorem 1 for the problem (IV-A), our remaining task is to find the optimal time splitting ratio to achieve the maximum uplink sum rate. Mathematically, it is to solve the problem . Since the objective function is concave w.r.t. , the Golden section search can be utilized to find the globally optimal [29].

### Iv-B TDMA-enabled uplink sum rate maximization

Due to the involvement of additional uplink time allocation vector, the TDMA-enabled uplink sum rate maximization in (III-C) is more challenging than the problem (III-C). As far as we know, the joint design of beamforming and time allocation vector for TDMA-enabled WPSNs is rarely discussed in the literature. Here we will follow a similar approach as that for SDMA-enabled WPSNs to solve this challenging problem. To be specific, by fixing the time splitting ratio , the problem (III-C) can be rewritten as

(17) |

where denotes the uplink time allocation. Clearly, the problem (IV-B) is jointly concave w.r.t. , and the corresponding Lagrangian function is expressed as

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