Multiantenna Clustering Collaboration for WPCNs Based on Nonlinear EH

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Introduction
Wireless energy-constrained networks usually refer to the network whose nodes can only use storage energy such as small-capacity batteries for power supply due to the limitations of their size, location, and environment, such as wireless-powered communication networks (WPCNs) [1][2][3][4], wireless-powered sensor network (WPSN) [5,6], wireless body area network [7], and a new type of wireless Internet of things network, for instance, smart transportation, smart medical treatment, and smart agriculture.In these networks, most wireless users are battery-powered.However, the limited capacity of batteries makes it difficult for them to work continuously for a long time, which has become the main bottleneck restricting their large-scale deployment.For example, in WPCN and WPSN, most sensors are deployed in unattended areas with harsh environ-ment, which makes it difficult and costly to replace batteries.The use of disposable battery implantation has severely restricted the life cycle of wireless networks.In wireless body area network, wearable devices, especially the sensor devices implanted in the human body, have extremely limited power storage capacity due to the volume limitation, so it is unrealistic to extend the network life cycle by regularly replacing the device battery through frequent surgery.
In [8], nonorthogonal multiple access-(NOMA-) assisted federated learning is studied, in which a group of terminal devices forms a NOMA cluster and sends their locally trained models to a cellular base station for model aggregation.Thus, how to improve the performance of network system while ensuring the energy supply of network nodes and extending the network life cycle has become a significant problem to be solved urgently in the design of wireless energy-constrained networks.In addition, how to provide sustainable and steady power for wireless energy limited networks has become a new demand and challenge in the era of intelligent connection.
In order to work out the above-mentioned issues, academia and industry, on the one hand, lower the nodes' energy consumption (EC) and enhance the network energy efficiency and spectrum efficiency through the continuous development and evolution of advanced wireless communication technology (such as multiantenna technology and cooperative communication technology).On the other hand, the technology of energy harvesting (EH) in view of radio frequency (RF) signal is introduced to collect renewable energy or reusable energy from the surrounding environment to provide continuous and stable energy for network nodes.Further, combining wireless communication technology and EH technology based on RF signal, new technologies that are efficient and meet the needs of information and energy transmission have been proposed and studied successively.
Before 2015, the research on wireless EH only considered the linear EH model's application.Not until 2015 the nonlinear logic EH model was proposed, and both linear and nonlinear EH models began to be considered for exploration.For energy transformation efficiency, in the linear EH pattern, it is usually expressed by a constant.Therefore, [9] in 2016 pointed out that the energy transformation efficiency was a nonlinear variable, which was expressed by a polynomial function, and a heuristic EH model was proposed.However, the mathematical expressions of these two nonlinear EH models are complicated and difficult to analyze in many research problems.It is hard to obtain the studied problems' closed-form analytical results.In addition, in order to simplify the model, facilitate theoretical analysis, and depict the EH's nonlinear characteristics, the approximate nonlinear piecewise EH pattern was proposed in [10] in the same year.To describe the actual EH circuit's sensitive property, namely, the starting voltage, the modified logic EH model was proposed in 2017 in [11].When the EH receiver's input power does not reach input voltage requirement, output power is zero, indicating that the EH circuit is still in the nonworking state.In the same year, because logical EH pattern and heuristic EH pattern are with the respect to mathematics not suitable for the complex problems' analysis and derivation, through data fitting, [12,13], respectively, proposed quadratic polynomial EH model and fractional EH model for the problem under study.In 2018, in order to explore the limited sensitivity and nonlinear characteristics of far-field actual EH system, piecewise EH model I was proposed in [14].
Figure 1 illustrates a multiantenna cluster-based collaboration for WPCNs in view of nonlinear EH.The hybrid access point (HAP) with multiple antennas in WPCNs adopts wireless energy transfer to provide a cluster of distant WDs for energy and accept their data transfer.This might correspond to a real-world setting in WPSNs, in which the moving HAP is paused in its course to provide energy to a densely deployed cluster of transducers supervising a specific location.As with traditional WSNs, one of the WDs is designated as a cluster head (CH), which forwards other cluster members' (CMs') information to the HAP.The article's specific contribution is shown below.
(i) A multiantenna clustering collaboration approach is proposed in WPCNs, where one WD is specified as a CH to relay the other sensors' message.In view of the CH's high energy consumption in traditional cluster-based collaborative solutions, EB technology is applied in the HAP to equilibrate the WDs' diverse EC rates (ii) Using the proposed collaborative approach, a cooptimization issue for the design of EB, the emission time distribution between HAP and WDs, and the CH's emission power distribution is developed for max-min throughput, which can be obtained between all WDs to enhance user fairness.A highefficiency optimum solution algorithm is put forward for figuring out the nonconvex optimization issues (iii) A numerical calculus is performed to investigate the effectual various system settings on the proposed approach's performance.Compared with other benchmark approaches, we find that the collaboration scheme can validly enhance throughput performance.In addition, when the cluster nearest to the cluster center is chosen as a CH, while the cluster location is close, the channel within the cluster is strong, and the amount number of cluster collaboration is medium, the proposed coordination mode is the most effective The organization of the rest part in the article is as follows.Wireless EH models are introduced in Section 2. Section 3 illustrates the system pattern and proposes the multiantenna clustering collaboration approach based on nonlinear EH.Section 4 analyzes the per-WD throughput performance.The maximum-minimum throughput optimization issue is formulated and an optimal band-aid arithmetic is put forward in Section 5. Section 6 evaluates the proposed collaboration's performance employing simulations.In the end, Section 7 summarizes the full text.

Wireless EH Models
Wireless EH technology adopts radio frequency (RF) signals as EH sources.Compared with traditional natural EH sources (such as wind, solar, and lightning), wireless EH technology has been widely concerned by industry and academia because of its better controllability and stability.The existing wireless EH models for energy collectors (ECs) can be roughly separated into two categories: linear EH pattern and nonlinear EH pattern.
2.1.Linear EH Pattern.In the Linear EH model, the EC's output and input energy increase linearly, which is International Journal of Distributed Sensor Networks where η and x are the EC's transformation efficiency and input power, respectively.It can be seen from Equation (1) that η is independent of x.However, the actual EH circuit has nonlinear characteristics in the peer-to-peer energy transformation.To be specific, in the realistic EH circuit, while the input power arrives a certain level, the collected energy will reach saturation value and no longer increase linearly; that is, there is a maximum value of EH.Therefore, the application of linear models to practical EH circuits may lead to uncoordinated resource allocation.On the basis of linear EH models, researchers have proposed nonlinear EH models that can be roughly separated to five categories.

Nonlinear EH Pattern
2.2.1.The First Kind of Nonlinear EH Model (Case I).On the basis of the linear EH pattern, [14] proposes a new inputoutput relationship for the EC after testing dozens of practical ECs.Be associated with the linear EH pattern, case I is written as where η x is the EC's transformation efficiency and x is the input power from the EC.Different from the linear EH pattern, case I's energy transformation efficiency is not independent of the EC's input power constant.Here, the EC's transformation efficiency is given as the input power's function, which is where p 0 , p 1 , p 2 , q 0 , q 1 , q 2 , and q 3 are obtained through curve fitting in [10][11][12][13].For the EC of [10][11][12][13][14][15], the energy input and output relation of EC can be expressed as As can be seen from Equation ( 4), within the input power's certain range, the EC's output power advances with input power.While the input power arrives a fixed value, the EC's input power reaches saturation state.According to Equation (4), while the input power advances, the nonlinear EH model's output power is virtually close to p 2 /p 1 .In addition, the linear EH pattern f x = ηx is regarded as obtained through disposing of Equation ( 4)'s highest terms in the numerator and denominator while x is tiny.This is the circumstance while the input power is tiny.

The Second Kind of Nonlinear EH Model (Case II).
The actual parametric nonlinear EH model proposed by Boshkovska et al. [15] can describe the nonlinear characteristics of the actual EH circuit's peer-to-peer power conversion, as shown below:

Energy flow
Information flow where where the parameters a, b, and c are invariable regulated through fitting the standard curve.The term b/c is summed to satisfy the case that the output power is zero while the EC's input power is small.In addition, similar to case I and case II, the EH model described in Equation ( 6) is more able to describe the nonlinearity and saturation of EH by the EC than the linear EH model.However, compared with case I and case II, the nonlinear EH model expressed in Equation (6) (case III) is simpler to operate, and it is easier to deal with the mean derivation.

The Fourth Kind of Nonlinear EH Model (Case IV).
There are nonlinear components, for example, diodes in the EH circuit, so the output energy p out = Θ p in of the EC has a nonlinear relationship with the input energy p in of the EC.And this nonlinear function must meet the below characteristics: (i) While p in is less than the EC's threshold of sensitivity p 0 , define p out = Θ p in = 0 With the increase of p in , the conversion efficiency Θ p in /p in of the EC will gradually increase until it arrives the maximum value and then decrease (iv) For all p in of Θ p in ≤ P max , where P max is the maximum energy collected while the EH circuit arrives saturation Based on characteristics (i)-(iv), the curve of the relation formula should be an S shaped curve.In order to meet this feature, a logical model is put forward in [16], but [15]'s model does not meet the sensitivity feature (i).To solve this problem, [17] modifies the model proposed in [15], which can be expressed as In view of the above equation, the collected energy at the All the features for (i)-(iv) can be verified from the above equation.Also, the parameters' (τ's and v's) steepness can control functions, i.e., in equation ( 9).
2.2.5.The Fifth Kind of Nonlinear EH Model (Case V).The collected energy through the EH circuit is sculptured as Here, Function η P n R denotes the function of the input power energy conversion efficiency.Define P in Δ = P sen in , P sat in .P sen in represents the sensitivity of the EC.For every input power less than the sensitivity, the energy harvested through the EC is 0. When x ≤ P sen in , P x = 0. Let P sat in denote the saturation value of the energy collected through the EC, after which the collected energy is all constant.
(i) EC's function harvested energy P: assume The experiments prove that the above assumption is consistent with the EH curve in the EH circuit in the prior art.The energy conversion efficiency is written as where x is in units of milliwatt, W is the fitting polynomial's order in units of dBm, and ω i W i=0 is the polynomial's coefficient.For the model, the polynomial of the fitting function's order is W = 10 and W = 12, respectively.Furthermore, P in = 10 −4 25 , 10 1 6 and P in = 10 −1 2 , 10 are all in units of milliwatt.International Journal of Distributed Sensor Networks

System Model
The HAP has power supply stablility and is installed with M-antenna (N > >M in practice).Every sensor with one antenna has a built-in battery to save wireless collected energy from the HAP.To be specific, the HAP multicasts radio energy to the sensors and accepts the sensors' wireless information transfer (WIT).The HAP and each WD operate in the identical frequency band and carry out a timedivision duplex circuit to realize the separation of energy and information transfer.One of the sensors is chosen as the CH to assist transferring other CMs' WIT in this paper.Without loss of generality, make W 0 and W 1 , ⋯, W N−1 index as the CH and CMs, respectively.Suppose all channels are mutual independence, keeping to quasistatic flat-fading; in this way, the coefficients of all channels hold constant for transferring time per block, expressed as T; however, it can be different by block.Make a i ∈ C M×1 represent the channel coefficient's vector between the HAP and W i , where a i ~CN 0, σ 2 i I and σ 2 i , i = 0, 1, ⋯, N − 1, represents the average channel gain.Besides, c j CN 0, δ 2 j represents path coefficient between the CH and the j-th CM (j = 1, ⋯, N − 1).Here, h i ≜ a i 2 and g i ≜ c i 2 represent the relevant channel gains, in which • represents the 2-norm operator.

Clustering
Collaborative Protocol in View of Nonlinear EH. Figure 2 illustrates the run of our proposed clustering cooperation protocol based on nonlinear EH in a transport time block.At the start of a transfer block, it spends a constant duration τ 0 on acting channel estimation (CE).At the CE's section, the WDs alternate order to radio their pilot signals, making the HAP and the CH to know c i (i = 1, ⋯, N − 1) and a i (i = 0, 1, ⋯, N − 1), separately.After that, the CH transmits the HAP c i 's estimation, making the HAP have all the information about CSI in the mesh.
After the CE's phase, three phases are run for the system.During the first stage, time duration is τ 1 , and the HAP multicasts wireless energy with a settled transmitting power of P.During the next two stages with T − τ 0 − τ 1 , the N sensors transfer their separate nessages to the HAP adopting their acquired energy.To be specific, firstly, the N − 1 CMs transfer sequentially to the CH, in which the i-th CM transfers at the time of τ 2,i (i = 1, ⋯, N − 1).During the third phase, the CH transfers the N − 1 CMs' decoded information to the HAP in conjunction with its own information.Make τ 3,i (i = 0, 1, ⋯, N − 1) as the time spent on transferring the i-th WD's message.Obviously, time allocations meet the below inequations Note that τ 0 is a given coefficient.Without lossing generality, this article supposes T = 1.On account of the global CSI's knowledge, the HAP can figure out the optimum time allocation, which is thus broadcast to each WD so that timeswitching circuits are kept by them with synchronizing for either information and energy transmission.The next part will deduce the proposed collaborative protocol's throughput performance and formulate the optimization problem of maximum-minimum (max-min) throughput.

The Analysis of Per-WD Throughput
4.1.Phase I: Energy Transmission.Note that the CH becomes the network's performance bottleneck, because the CH requires to transfer a total of N messages, which will consume obviously more energy than the other CMs.During the first stage of τ 1 , the HAP transfers w t ∈ C M×1 stochastic energy signals on the M-antenna, in which the HAP's transmitting power is influenced on where tr • , • H , and Q ≥ 0 express a matrix's trace, the complex conjugation operator, and the EB matrix, respectively.The receiver noise power is denoted by n 1 i t , and thus, the received energy signal by the i-th WD can be written as With ignoring interference power, the obtained energy through sensors is indicated as [7] E is an input power's typical Sigmoid function, Ω is used to ensure this model's zero input and zero output and keep the quantity constant, M is the circuit's maximum output power at saturation, which is also a constant, and a and b are related to the impedance, capacitance, and conduction voltage of the diode in the circuit, keeping their values constant.

4.2.
Phase II: Intracluster Transfer.Suppose in the second phase the CMs will consume the collected energy to the CH.Then, the i-th CM's transmitting power is The i-th WD's baseband signal transferred in the second stage and the receiver noise with power are denoted by s = N 0 , respectively.Then, the signal received at the CH is Thus, the i-th CM's messages are decoded through the CH with a fixed rate by In the meantime, the HAP can eavesdrop on the CMs' transfer as well, so that it receives within the i-th CM's transfer where n 2 H,i t ~CN 0, N 0 I .For the sake of simplicity, the decoding's EC is ignored, and the data transmission's EC is only considered.However, by introducing an unchanging circuit power term, our proposed approach is readily expanded when the decoding EC is nonzero.

Phase III:
Cluster-to-HAP Transfer.When completing to decode the CMs' information, the CH sends the N − 1 CMs' information to the HAP one by one, along with its own information.The CH's baseband signal and the i-th CM's reencoded baseband signal transmitted in the third stage are denoted by s 3 0 t and s 3 i t , respectively.In addition, suppose the power adopted to transfer the i-th WD's information is expressed as P CH NL 3,i and E s Then, at the HAP, the i-th WD information's received signal is The aggregated EC through the CH is upper bound on the energy it harvests E 0 , i.e., Suppose that maximal ratio combining (MRC) is adopted by the HAP to maximize the receive signal-to-noise power ratio (SNR), in which the i-th WD's combiner output SNR is The time distribution and transmitting power are denoted as τ = τ 1 , τ 2,1 ,⋯,τ 2,N−1 , τ 3,0 , τ 3,1 ,⋯,τ 3,N−1 ′ and However, the second and third stages can receive every CM's information.On this occasion, the HAP could codecode every CM's information at a given rate across two stages by

25
where (19) gives τ, Q expresses the information extracted from the received signal by the HAP, where an optimal MRC receiver is used in (18) (during the second phase) that is written as τ, P CH NL , Q expresses the transmissions' attainable rates from the CH to the HAP, which is The max-min throughput is a WPCN's significant performance index, defined as Because of user collaboration and beamforming design, τ, Q , i.e., the throughput within intracluster communication, are concave functions.Meanwhile, R 0 τ, P CH NL and V 3 i τ, P CH NL , i.e., the data rates within cluster-to-HAP communication, are also not concave functions.In addition, the (10)'s left hand side is not convex as well.Therefore, the (P2)'s present form is a nonconvex issue and lacks efficient optimization algorithm.The algorithm for optimal solution (P2) will be proposed in the next subsection.

(P2) Optimization
Algorithm.Above all, define W ≜ τ 1 Q ≥ 0. With the total transmitted power constraint is at (14), which we have Therefore, the variables are changed as 19) and (26) are represented as the functions of τ and z = z 1 ,⋯,z N−1 :

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in which ρ i ≜ g i /N 0 and ρ i ≜ h i /N 0 are parameters.

Simulation Result Analysis
This part assesses our proposed collaborative approach's performance.In the simulations below, suppose that the HAP's transmitted power P = 3 watts (W), energy collecting efficiency η = 0 51 (please see the detailed product specifications on the website of Powercast Co. (http://www .powercastco.com)), the quantity of HAP antennas M = 5, the antenna gain G A = 2, the carrier frequency f c = 915 MHz, the path-loss factor α = 3, and the interference power N 0 = 10 −10 W for all receivers' bandwidth.The average channel gain between every two sensors (HAP or one WD) keeps to a path-loss pattern.For example, make d H,i as the distance between the HAP and the i-th WD, and then, the mean channel gain δ 2 i = G A 3 × 10 8 /4πd H,i f c α .In addition, 15 WDs are uniform distribution in a circle of radius r meters (m), and the circle's center is d m away from the HAP.Every point in the graphs is an individual WD location with an average of 20 [1].International Journal of Distributed Sensor Networks Figure 3 examines cluster-to-HAP information links' effect the total throughput's performance through setting r = 3 and changing d.We can observe from Figure 3 that the proposed collaborative approach gains significant performance advantage over the benchmark method.However, when d is further increased, very low data rates of both schemes are achieved because the energy signal dramatically attenuates over distance.The simulation results demonstrate that the collaborative approach's effective working scope under consideration is essentially confined by the relatively low energy transfer efficiency.As a matter of fact, WPC is only available if the power does not travel very far, which allows the WDs to gather enough power to transmit information.In a practical applications, the performance can be enhanced in several ways, for example, augmenting the number of HAP antennas, optimizing the routing for moving HAP, or adding the HAP's transmitting power.
In Figure 4, throughput performance's stability is evaluated as the number of WDs N is increased from 5 to 50 by setting d = 6 m and r = 3 m. Figure 4(a) shows that the  8 International Journal of Distributed Sensor Networks maximum-minimum throughput of both approaches lessens with N. The worst-performing WD lessens throughput due to the shorter average transfer time per sensor.Specially, as N increases from 35 to 50, the lessen in maximumminimum throughput is modest.However, Figure 4(b) shows that the total throughput adds with N, although the throughput per WD may lower.This manifests that there is a trade-off between the throughput per WD and the total network.Even so, we can still become conscious of the proposed approach's important performance improvement over the benchmark method, while the network is big (for example, N equals 30).

Conclusions
In this article, a clustering collaborative approach is proposed for WPCNs based on nonlinear EH, in which one sensor is specified as a CH to help other WDs' propagation.An efficient algorithm is put forward to reach the optimal maximum-minimum throughput among the sensors, through cooptimizing the design of EB, the emission time distribution among the HAP and sensors, and the CH's emission power distribution.A large number of simulations in real mesh settings demonstrated that our suggested approach might remarkably advance user fairness and spectral efficiency, compared with nontrivial benchmark approaches.

Figure 1 :
Figure 1: A diagram of the considered clustering collaboration for WPCNs, in which W 0 is the CH and the remaining N − 1 WDs are CMs.

3. 1 .
Channel Model.A WPCN is considered as demonstrated in Figure1, which is composed of one HAP and N WDs.

Figure 3 :Figure 4 :
Figure 3: The effect of CH selection on max-min throughput when d = 6 m.EB technology is used in HAP in (a), but not in (b).
[14] is the conventional logic function relative to the RF power received by the user SU n .The actual nonlinear EH model can describe the nonlinearity caused by hardware constraints by adjusting the parameters a n , b n , and M n .In particular, M n represents the collected maximum power at the EH user SU n while the EH circuit is satiation.anandbn are concerned with a minute EH circuit specification.These three parameters can be obtained through the given standard curve in view of a fixed EH circuit's measured data.2.2.3.The Third Kind of Nonlinear EH Model (Case III).Case I and case II can describe the saturation of nonlinear models, but it is difficult to derive the mean value, the collected energy's probability density function, and the cumulative distribution function.On this basis,[14]proposed a simpler nonlinear EH model, which is written as