# Energy Harvesting for Internet of Things with Heterogeneous Users

Authors: Desheng Wang; Haizhen Liu; Xiaoqiang Ma; Jun Wang; Yanrong Peng; Yanyan Wu
Journal: Wireless Communications and Mobile Computing
 (2017)

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## Abstract

We study the energy harvesting problem in the Internet of Things with heterogeneous users, where there are three types of single-antenna users:ID users that only receive information,EH users that can only receive energy, andID/EH users that receive information and energy simultaneously from a multiantenna base station via power splitting. We aim to maximize the minimum signal-to-interference-plus-noise ratio (SINR) of the ID users and ID/EH users by jointly designing the power allocation at the transmitter and the power splitting strategy at the ID/EH receivers under the maximum transmit power and the minimum energy harvesting constraints. Specifically, we first apply the semidefinite relaxation (SDR), zero-forcing (ZF), and maximum ratio transmission (MRT) techniques to solve the nonconvex problems. We then apply the zero-forcing dirty paper coding (ZF-DPC) technique to eliminate the multiuser interference and derive the closed-form optimal solution. Numerical results show that ZF-DPC provides higher achievable minimum SINR than SDR and ZF in most cases.


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## Body

## 1. Introduction

Recently, the Internet of Things (IoT) [1–3] has been invading various industries. Given the fact that IoT normally consists of resource-constrained devices and relies on wireless communication for data transmission, energy efficiency is an important issue. For example, in wireless sensor networks (WSNs), a key component and enabler of IoT, the sensor nodes are normally powered by batteries that have very limited lifetime. The network lifetime can be extended by replacing or recharging the batteries, which, however, are inconvenient, costly, and environmentally unfriendly [4, 5]. As an alternative solution to prolonging the network lifetime, energy harvesting [6] has become an appealing solution that potentially provides unlimited power supply to wireless networks by scavenging energy from the environment. The radio frequency (RF) signals emitted by ambient transmitters are a viable source for wireless energy harvesting (EH). As an emerging energy harvesting technique, simultaneous wireless information and power transfer (SWIPT) [7] has drawn an upsurge of interests, where RF signals are used to charge users’ devices wirelessly. Through power splitting (PS) [8, 9], users are provided with information decoding (ID) and energy harvesting simultaneously, which brings great convenience.How to get the trade-off between the achievable information and energy harvesting is still a challenging problem, and various techniques have been developed, for example, zero-forcing (ZF), maximum ratio transmission (MRT), and semidefinite relaxation (SDR) [10–14]. It is worth noting that most existing studies only considerID/EH users that can receive information and energy simultaneously. Yet, in practice, there are still manyID users that can only transmit/receive information and do not have the ability to harvest energy as well asEH users that only harvest energy from RF signals, which should be also considered in a unified framework. Hence, in this paper, we, for the first time, consider the coexistence of all the three types of users under multiuser MISO broadcast channels. Our aim is to maximize the minimum SINR of all users except for the EH users by jointly designing the power allocations at the transmitter and the PS ratios for ID/EH users who need to harvest energy under both the maximum transmit power and the minimum EH constraints.In the remainder of this paper, we first present the system model and problem formulation and prove the feasibility of the problem. We then apply three traditional techniques, namely, semidefinite relaxation (SDR), zero-forcing (ZF), and maximum ratio transmission (MRT), to solve the problem. Moreover, it is well known that dirty paper coding (DPC), a nonlinear precoding scheme, can precancel noncausal interference without loss of information and also achieve the capacity region for MIMO broadcast channels (BCs) [15], which, however, has a relatively high computational complexity. To this end, we develop a suboptimal yet efficient solution based on zero-forcing dirty paper coding (ZF-DPC) [16]. We compare the three schemes through extensive simulations, and the results show that ZF-DPC provides better performance than SDR and ZF in most cases.


## 2. Related Work

Initial research works in the field of IoT mainly focus on the building management [17, 18] and the related security as well as energy issues [19]. Building management includes many aspects, for example, traffic control, surveillance, energy management, and indoor environmental and air quality (IEAQ) control. The authors in [3] propose an energy-efficient large-scale and diffusive object monitoring mechanism to reduce communication costs and thus to reduce the energy consumption. In this paper, we propose using SWIPT for energy harvesting in IoT, which is convenient, costless, and environmentally friendly.SWIPT has drawn an upsurge of interests. The authors in [20] study a multiuser multiple-input single-output (MISO) broadcast SWIPT system, where there exist two kinds of users, namely, ID users and EH users; the aim is to maximize the weighted sum power transferred to all EH receivers subject to given minimum SINR constraints at different ID users. The authors in [21] consider a point-to-point wireless link over the narrow band flat-fading channel subject to time-varying cochannel interference and then propose two different schemes, namely, time switching (TS) and power splitting (PS), for distributed information and energy receivers to evaluate the performance of the system. The recent research mainly focuses on the users who can receive information and harvest energy simultaneously. The authors in [22] consider SWIPT in the multiuser single-input single-output (SISO) interference channels and maximize the minimum SINR of users under the maximum transmit power and the minimum EH constraints; then the authors propose two algorithms: a centralized algorithm to optimally solve the nonconvex problem and a distributed algorithm designed to update its transmit power and PS ratio through an iterative process. The authors in [10] utilize the ZF and MRT schemes to minimize the total transmit power at the BS under the given SINR and the minimum EH constraints. The authors in [11–13] apply the SDR technique [14] to solve the nonconvex problem.Different from most of the existing studies that only consider ID/EH users, the novelty of our work is to investigate the coexistence of ID users, ID/EH users, and EH users under the multiuser MISO broadcast channels, as shown in Figure1. Our objective is to maximize the minimum SINR of the ID and ID/EH users by jointly designing the power allocations at the transmitter and the PS ratios. Specifically, we discuss how to adapt the traditional schemes in the literature such as SDR, ZF, and MRT to the new scenario and examine their effectiveness. We further propose a new nonlinear ZF-DPC scheme for the new scenario and compare it with the traditional ones.Figure 1
A general model for the coexistence of heterogeneous users.


## 3. System Model and Problem Formulation

We consider a multiuser MISO broadcast system consisting of one BS andK users. The system model is depicted in Figure 2, where the BS is equipped with Nt>1 antennas, and each user is equipped with one antenna. We summarized the notations in Notations. The first M users can only receive information (namely, ID users), while the next N users can receive information and harvest energy through a power splitter simultaneously (namely, ID/EH users), and the other K-M-N users can only harvest energy (namely, EH users). We assume linear transmit precoding at the BS, where each user is assigned one dedicated information beam [14]. The complex baseband transmitted signal at the BS is(1)x=∑k=1Kvksk,where sk denotes the transmitted data symbol for user k and vk is the corresponding transmit beamforming vector. It is assumed that sk,k=1,…,K, are independent and identically distributed (i.i.d.) cyclic symmetric complex Gaussian (CSCG) random variables with zero mean and unit variance, denoted by sk~CN0,1.Figure 2
A multiuser MISO broadcast system.We assume quasi-static flat-fading channel for all users and for convenience denotehk as the conjugated complex channel vector from BS to user k. Then the received signal at user k is given by(2)yk=hkH∑j=1Kvjsj+nk,k=1,…,K,where nk~CN0,σk2 denotes the antenna noise at user k. Therefore, the SINR of the first M ID users is(3)SINRk=hkHvk2∑j≠khkHvj2+σk2,k=1,…,M.As for the nextN ID/EH users, we assume that each applies the PS scheme [8] to coordinate the process of information decoding and energy harvesting from the received signal, which allocates a ρk0≤ρk≤1 portion of the signal power to ID and the remaining 1-ρk portion to EH. As a result, the signal split to user k’s ID part is(4)ykID=ρkhkH∑j=1Kvjsj+nk+zk,k=M+1,…,M+N,where zk~CN0,σz2 is the additional noise introduced by user k’s ID part.Accordingly, the SINR at userk’s ID part is given by(5)SINRk=ρkhkHvk2ρk∑j≠khkHvj2+ρkσk2+σz2,k=M+1,…,M+N.On the other hand, the signal split to userk’s EH part is(6)ykEH=1-ρkhkH∑j=1Kvjsj+nk,k=M+1,…,M+N.Then, the harvested power of userk’s EH part is given by(7)Ek=ζk1-ρk∑j=1KhkHvj2+σk2,k=M+1,…,M+N,where ζk∈0,1 denotes the energy conversion efficiency of user k.For the remainingK-M-N EH users, we have(8)Ek=ζk∑j=1KhkHvj2+σk2,k=M+N+1,…,K.Assume that each EH userk (k=M+1,…,K) has a minimum harvest energy demand ek and that the BS has a maximum transmit power constraint Pmax. Our aim is to maximize the minimum SINR by jointly designing the transmit beamforming vectors vk,k=1,…,K, and the PS ratios ρk,k=M+1,…,M+N, under these constraints, which is formulated as follows:(9)maxv,ρmink=1,…,M+N⁡SINRks.t.∑k=1Kvk2≤Pmax,Ek≥ek,k=M+1,…,K,0≤ρk≤1,k=M+1,…,M+N,where(10)SINRk=hkHvk2∑j≠khkHvj2+σk2,k=1,…,M,SINRk=ρkhkHvk2ρk∑j≠khkHvj2+ρkσk2+σz2,k=M+1,…,M+N,Ek=ζk1-ρk∑j=1KhkHvj2+σk2,k=M+1,…,M+N,Ek=ζk∑j=1KhkHvj2+σk2,k=M+N+1,…,K.By introducing an auxiliary variableτ=mink=1,…,M+NSINRk, problem (9) can be reformulated as(11)maxv,ρ,ττs.t.SINRk≥τ,k=1,…,M+N,∑k=1Kvk2≤Pmax,Ek≥ek,k=M+1,…,K,0≤ρk≤1,k=M+1,…,M+N.It can be shown that problem (11) becomes a feasibility problem for any given τ (please refer to Appendix A for details). Therefore, by applying the bisection search method on τ over a specific interval and solving the feasibility problem at each step with the associated τ, the optimal solution to problem (11) is also that to problem (9).


## 4. Three Traditional Schemes: SDR, ZF, and MRT

In this section, we introduce three traditional schemes, namely, SDR, ZF, and MRT, to solve the nonconvex problem (11) in a multiuser MISO broadcast system.


### 4.1. SDR Scheme

DefineXk=vkvkH, ∀k; then rankXk=1, ∀k. By ignoring the rank-one constraint for all Xk, the SDR of problem (11) is given by(12)maxX,ρ,ττs.t.∑k=1KtraceXk≤Pmax,hkHXkhk∑j≠khkHXjhk+σk2≥τ,k=1,…,M,ρkhkHXkhkρk∑j≠khkHXjhk+ρkσk2+σz2≥τ,k=M+1,…,M+N,ζk1-ρk∑j=1KhkHXjhk+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζk∑j=1KhkHXjhk+σk2≥ek,k=M+N+1,…,K,Xk⪰0,∀k.Problem (12) is still nonconvex, since both the SINR and harvested power constraints involve coupled Xk and ρk. This problem can be reformulated as the following problem:(13)maxX,ρ,ττs.t.∑k=1KtraceXk≤Pmax,1τhkHXkhk-∑j≠khkHXjhk≥σk2,k=1,…,M,1τhkHXkhk-∑j≠khkHXjhk≥σk2+σz2ρk,k=M+1,…,M+N,∑j=1KhkHXjhk≥ekζk1-ρk-σk2,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,∑j=1KhkHXjhk≥ekζk-σk2,k=M+1,…,M+N,Xk⪰0,∀k.Note that problem (13) is convex; therefore, the optimal solution can be obtained by applying the bisection search method on τ. Let Xk∗ and ρk∗ denote the optimal solution to problem (13). If Xk∗ satisfies rankXk∗=1, ∀k, then the optimal beamforming solution to problem (11) can be obtained by doing eigenvalue decomposition (EVD) of Xk∗=v¯kv¯kH, ∀k; otherwise we need to do Gaussian randomization [23] to choose the best solution.In AppendixB, we have proven that Xk∗ satisfies rankXk∗=1 when there are no EH users, where K=M+N, and from the numerical analysis in Section 6, we could verify that no matter whether there exist EH users, Xk∗ for the ID users and the ID/EH users always satisfy the rank-one conditions, but it is not suitable for the EH users, and thus we need to do Gaussian randomization to choose the best beamforming vectors vk for them. However, we just need to know the achievable minimum SINR which is obtained directly from Xk∗ rather than vk in this paper, and thus we can ignore the process of Gaussian randomization.The optimal solution to problem (11) can be obtained via solving problem (13) by the interior-point algorithm [24] using standard solvers, for example, CVX [25], and the complexity of the interior-point algorithm is OK+Nt23.5log1/ε [26], where ε is the desired numerical accuracy.


### 4.2. ZF Scheme

In ZF, the conditionNt≥K must be satisfied [10], and hk are not linearly dependent. The ZF beamforming scheme can then be used to eliminate the multiuser interference by restricting vk to satisfy hiHvk=0,∀i≠k. More specifically, the ZF weight vk is provided by the solution to the following problem:(14)minvkhkHvk2s.t.HkHvk=0K-1×1,where Hk≜h1,…,hk-1,hk+1,…,hK∈CNt×K-1. Assume that vk=pkwk, where wk22=1 and the solution is given by [27](15)wkZF=INt-FhkINt-Fhk,where F=Hk†HkH and Hk†=HkHkHHk-1 is the Moore-Penrose inverse of Hk.LetGi,j=hiHwjZF2 denote the link gain, and we have Gi,j=0, i≠j. Hence, problem (11) is equivalent to the following problem:(16)maxpk,ρ,ττs.t.∑k=1Kpk≤Pmax,Gk,kpk≥τσk2,k=1,…,M,Gk,kpk≥τσk2+τσz2ρk,k=M+1,…,M+N,ζk1-ρkGk,kpk+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζkGk,kpk+σk2≥ek,k=M+N+1,…,K,pk≥0,∀k.Let xk=Gk,kpk+σk2,α=1+τσk2,β=τσz2, and γ=ek/ζk, and the closed-form solution is given by(17)pk∗=τσk2Gk,k,k=1,…,M,pk∗=1Gk,kx∗-σk2,ρk∗=1-γx∗,k=M+1,…,M+N,pk∗=γ-σk2Gk,k,k=M+N+1,…,K,where x∗=1/2α+β+γ+α+β+γ2-4αγ. Please refer to Appendix C for details.Clearly, the complexity of the ZF scheme is dominated by theK times of SVD operations. Since the complexity of each SVD operation is O(KNt2+K2Nt+K3) [28], the overall complexity of the ZF scheme is O(K2Nt2+K3Nt+K4).


### 4.3. MRT Scheme

The MRT beamforming maximizes the SNR at each receiver(hkHvk2/σk2,∀k), and it only requires the knowledge of the direct links hk; thus, it is of relatively low complexity to obtain the beamforming vectors. Assuming that vk=pkwk, where wk22=1 and pk is the power allocated to user k, the MRT beamforming [29] can be expressed as(18)wkMRT=hkhk.It is worth noting that MRT does not take into account the signals transmitted to other users and therefore it results in a strong cross-interference. Although this cross-interference is a bottleneck for conventional MISO systems, it could be beneficial for scenarios with EH constraints. Then, letGi,j=hiHwjMRT2 denote the link gain; problem (11) can be cast into a second-order cone programming (SOCP) [30] formulation as follows:(19)maxpk,ρ,ττs.t.∑k=1Kpk≤Pmax,1τGk,kpk-∑j≠kGk,jpj≥σk2,k=1,…,M,1τGk,kpk-∑j≠kGk,jpj≥σk2+σz2ρk,k=M+1,…,M+N,ζk1-ρk∑j=1KGk,jpj+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζk∑j=1KGk,jpj+σk2≥ek,k=M+N+1,…,K,pk≥0,∀k.The optimal solution to problem (19) can be obtained by using standard solvers, for example, CVX. The complexity of the MRT scheme is dominated by the K times of computing the beamforming vector wkMRT with O1 complexity each time. The SOCP algorithm for solving problem (19) is OK+N3.5log⁡1/ε [26], where ε is the preset search precision. Thus, the complexity of MRT scheme is OK+N3.5log⁡1/ε.


## 4.1. SDR Scheme

DefineXk=vkvkH, ∀k; then rankXk=1, ∀k. By ignoring the rank-one constraint for all Xk, the SDR of problem (11) is given by(12)maxX,ρ,ττs.t.∑k=1KtraceXk≤Pmax,hkHXkhk∑j≠khkHXjhk+σk2≥τ,k=1,…,M,ρkhkHXkhkρk∑j≠khkHXjhk+ρkσk2+σz2≥τ,k=M+1,…,M+N,ζk1-ρk∑j=1KhkHXjhk+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζk∑j=1KhkHXjhk+σk2≥ek,k=M+N+1,…,K,Xk⪰0,∀k.Problem (12) is still nonconvex, since both the SINR and harvested power constraints involve coupled Xk and ρk. This problem can be reformulated as the following problem:(13)maxX,ρ,ττs.t.∑k=1KtraceXk≤Pmax,1τhkHXkhk-∑j≠khkHXjhk≥σk2,k=1,…,M,1τhkHXkhk-∑j≠khkHXjhk≥σk2+σz2ρk,k=M+1,…,M+N,∑j=1KhkHXjhk≥ekζk1-ρk-σk2,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,∑j=1KhkHXjhk≥ekζk-σk2,k=M+1,…,M+N,Xk⪰0,∀k.Note that problem (13) is convex; therefore, the optimal solution can be obtained by applying the bisection search method on τ. Let Xk∗ and ρk∗ denote the optimal solution to problem (13). If Xk∗ satisfies rankXk∗=1, ∀k, then the optimal beamforming solution to problem (11) can be obtained by doing eigenvalue decomposition (EVD) of Xk∗=v¯kv¯kH, ∀k; otherwise we need to do Gaussian randomization [23] to choose the best solution.In AppendixB, we have proven that Xk∗ satisfies rankXk∗=1 when there are no EH users, where K=M+N, and from the numerical analysis in Section 6, we could verify that no matter whether there exist EH users, Xk∗ for the ID users and the ID/EH users always satisfy the rank-one conditions, but it is not suitable for the EH users, and thus we need to do Gaussian randomization to choose the best beamforming vectors vk for them. However, we just need to know the achievable minimum SINR which is obtained directly from Xk∗ rather than vk in this paper, and thus we can ignore the process of Gaussian randomization.The optimal solution to problem (11) can be obtained via solving problem (13) by the interior-point algorithm [24] using standard solvers, for example, CVX [25], and the complexity of the interior-point algorithm is OK+Nt23.5log1/ε [26], where ε is the desired numerical accuracy.


## 4.2. ZF Scheme

In ZF, the conditionNt≥K must be satisfied [10], and hk are not linearly dependent. The ZF beamforming scheme can then be used to eliminate the multiuser interference by restricting vk to satisfy hiHvk=0,∀i≠k. More specifically, the ZF weight vk is provided by the solution to the following problem:(14)minvkhkHvk2s.t.HkHvk=0K-1×1,where Hk≜h1,…,hk-1,hk+1,…,hK∈CNt×K-1. Assume that vk=pkwk, where wk22=1 and the solution is given by [27](15)wkZF=INt-FhkINt-Fhk,where F=Hk†HkH and Hk†=HkHkHHk-1 is the Moore-Penrose inverse of Hk.LetGi,j=hiHwjZF2 denote the link gain, and we have Gi,j=0, i≠j. Hence, problem (11) is equivalent to the following problem:(16)maxpk,ρ,ττs.t.∑k=1Kpk≤Pmax,Gk,kpk≥τσk2,k=1,…,M,Gk,kpk≥τσk2+τσz2ρk,k=M+1,…,M+N,ζk1-ρkGk,kpk+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζkGk,kpk+σk2≥ek,k=M+N+1,…,K,pk≥0,∀k.Let xk=Gk,kpk+σk2,α=1+τσk2,β=τσz2, and γ=ek/ζk, and the closed-form solution is given by(17)pk∗=τσk2Gk,k,k=1,…,M,pk∗=1Gk,kx∗-σk2,ρk∗=1-γx∗,k=M+1,…,M+N,pk∗=γ-σk2Gk,k,k=M+N+1,…,K,where x∗=1/2α+β+γ+α+β+γ2-4αγ. Please refer to Appendix C for details.Clearly, the complexity of the ZF scheme is dominated by theK times of SVD operations. Since the complexity of each SVD operation is O(KNt2+K2Nt+K3) [28], the overall complexity of the ZF scheme is O(K2Nt2+K3Nt+K4).


## 4.3. MRT Scheme

The MRT beamforming maximizes the SNR at each receiver(hkHvk2/σk2,∀k), and it only requires the knowledge of the direct links hk; thus, it is of relatively low complexity to obtain the beamforming vectors. Assuming that vk=pkwk, where wk22=1 and pk is the power allocated to user k, the MRT beamforming [29] can be expressed as(18)wkMRT=hkhk.It is worth noting that MRT does not take into account the signals transmitted to other users and therefore it results in a strong cross-interference. Although this cross-interference is a bottleneck for conventional MISO systems, it could be beneficial for scenarios with EH constraints. Then, letGi,j=hiHwjMRT2 denote the link gain; problem (11) can be cast into a second-order cone programming (SOCP) [30] formulation as follows:(19)maxpk,ρ,ττs.t.∑k=1Kpk≤Pmax,1τGk,kpk-∑j≠kGk,jpj≥σk2,k=1,…,M,1τGk,kpk-∑j≠kGk,jpj≥σk2+σz2ρk,k=M+1,…,M+N,ζk1-ρk∑j=1KGk,jpj+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζk∑j=1KGk,jpj+σk2≥ek,k=M+N+1,…,K,pk≥0,∀k.The optimal solution to problem (19) can be obtained by using standard solvers, for example, CVX. The complexity of the MRT scheme is dominated by the K times of computing the beamforming vector wkMRT with O1 complexity each time. The SOCP algorithm for solving problem (19) is OK+N3.5log⁡1/ε [26], where ε is the preset search precision. Thus, the complexity of MRT scheme is OK+N3.5log⁡1/ε.


## 5. Nonlinear Precoding Scheme: ZF-DPC

In this section, we present the ZF-DPC scheme that applies ZF with QR decomposition to eliminate the causal interference and then uses DPC to eliminate the noncausal interference.Denote the MISO channels byH≜h1,h2,…,hKH∈CK×Nt. Then the QR decomposition [31] obtained by applying Gram-Schmidt orthogonalization to the rows of HH is HH=RQH; then H can be expressed as(20)H=QRH.Letm=rankH≤min⁡K,Nt. Then Q is a K×m lower triangular matrix (i.e., it has zeros above its main diagonal), and denote qi,j as the i,jth element of Q. R is an Nt×m subunitary matrix with m orthonormal columns. By letting the beamforming matrix BZF=R, the transmit signal of ZF-DPC is given by (21)xZF=BZFuZF,where uZF=u1,u2,…,umT and it satisfies the power constraints uZF2≤Pmax. Then the received signal of user k is given by the set of interference channels:(22)yk=qk,kuk+∑j<kqk,juj+nk,k=1,…,m,while no information is sent to users m+1,…,K. In this case, there is no point in maximizing the minimum SINR, and we assume that hk are not linearly dependent and the condition Nt≥K must be satisfied, and we have m=K.Based on (20)–(22), the received signals y=y1,y2,…,yKT can be written as(23)y=HBZFuZF+n=QuZF+n,where n=n1,n2,…,nKT. According to the QR decomposition of the channel matrix, different ordering among users will lead to different channel gain. For the sake of convenience, we assume that the coding order is 1,2,…,K. Note that the interuser interference from the off-diagonal entries of Q can be cancelled by successive DPC by the precoding matrix W=Q-1diagq1,1,q2,2,…,qK,K and uZF=Ws¯ZF, where s¯ZF is the transmitted signal. Then, the received signals can be written as y=diagq1,1,q2,2,…,qK,Ks¯ZF+n. Thus the BC is transformed into K independent channels with qk,k2,k=1,…,K, being the channel gain. Denote pk as the power allocated to user k, and the SINR of user k is given by(24)SINRk=qk,k2pkσk2,k=1,…,M,SINRk=ρkqk,k2pkρkσk2+σz2,k=M+1,…,M+N.The EH of user k is(25)Ek=ζk1-ρkqk,k2pk+σk2,k=M+1,…,M+N,Ek=ζkqk,k2pk+σk2,k=M+N+1,…,K.Now problem (11) can be reformulated as follows:(26)maxpk,ρ,ττs.t.∑k=1Kpk≤Pmax,qk,k2pk≥τσk2,k=1,…,M,qk,k2pk≥τσk2+τσz2ρk,k=M+1,…,M+N,ζk1-ρkqk,k2pk+σk2≥ek,k=M+1,…,M+N,0≤ρk≤1,k=M+1,…,M+N,ζkqk,k2pk+σk2≥ek,k=M+1,…,M+N,pk≥0,∀k.Similar to the ZF scheme, letxk=qk,k2pk+σk2,α=1+τσk2,β=τσz2, and γ=ek/ζk, and then the solution is(27)pk∗=τσk2qk,k2,k=1,…,M,pk∗=1qk,k2x∗-σk2,ρk∗=1-γx∗,k=M+1,…,M+N,pk∗=γ-σk2qk,k2,k=M+N+1,…,K,where x∗=1/2α+β+γ+α+β+γ2-4αγ.The complexity of the ZF-DPC scheme is divided into two parts. The first part is to perform QR decomposition, the complexity of which isOK2Nt; the other part is to perform pseudoinverse and matrix multiplication (multiply a diagonal matrix) for both K×K matrices, and the complexity is OK3 [32] and OK, respectively. Thus the overall complexity of the ZF-DPC scheme is OK2Nt+K3.


## 6. Numerical Analysis

In this section, we numerically evaluate the performance of the proposed schemes. For simplicity, we assume thatζk=ζ, ek=e,k=M+1,…,K, and σc2=σz2=10-4,∀k; the BS is equipped with Nt=8 antennas, and the transmit power is no more than Pmax.We have the following initial observations. First, we can see from Figure3 that the system’s performance in terms of the achievable minimum SINR under different ζ is almost the same when ζ exceeds a certain value, say 0.6, in our simulation. We use ζ=1 in the next simulations without loss of generality.Figure 3
Achievable minimum SINR versus transmit power under differentζ.Second, we can see from Figure4 that the performance of the MRT scheme remains almost unchanged when the transmit power increases, due to the strong cross-interference from other users, and the ZF scheme preforms better than the MRT scheme except when K=8, M=4, and N=2 and the transmit power is relatively small, which also indicates that interference is beneficial to EH. A simple method to ensure that the direct links are much stronger than the interference links is to strengthen the direct channel links by a multiplicative constant δ [33]; that is, δhi,i→hi,i, which is, however, infeasible in our scenario with only one BS. We do not consider the MRT scheme in the remaining part, since it is inferior to ZF in most cases.Figure 4
Comparison of ZF and MRT schemes.


### 6.1. SINR versus Transmit Power

In this part, we investigate the achievable minimum SINR of the proposed schemes under different transmit power of the BS whene=0 dBm.From Figures5 and 6, we can see that as the transmit power Pmax increases from 0 dBm to 30 dBm, the system’s performance of all three schemes increases substantially, and the performance of ZF is always the worst. When the number of users is relatively small, the performances of the three schemes are almost the same, especially when K=2,M=1, and N=1 due to less interference from other users. Yet, as shown in Figure 5, with the increasing number of users (e.g., K=4 and K=8), the performances of different schemes differ significantly.Figure 5
Achievable minimum SINR versus transmit power with different numbers of users.Figure 6
Achievable minimum SINR versus transmit power with fixed number of EH users.Specifically, when the transmit power is relatively small, SDR has better performance than ZF-DPC, while with higher transmit power, the performance of ZF can approach SDR, and ZF-DPC has a gain of about2dB and about 10dB over SDR and ZF, respectively, when K=8,M=4, and N=2. We can also see from Figure 6 that when the number of total users K and the number of EH users are fixed, with the increasing number of ID users M (the number of ID/EH users decreasing correspondingly), the performances of the three schemes all improve about 3dB. The reason is that, with fewer energy harvesters, the BS can allocate more power to the information receivers.Figure7 shows that our proposed ZF-DPC technique provides higher achievable minimum SINR than SDR and ZF even for the traditional scenario with ID/EH users only.Figure 7
Achievable minimum SINR versus transmit power with ID/EH users only.


### 6.2. SINR versus EH

In this part, we investigate the relationship between the achievable minimum SINR and the minimum energy harvesting of the users, with fixed transmit power of the BS (Pmax=30dBm).We can see from Figure8 that the performances of the three schemes are similar when the number of users is small and the performances of different schemes differ significantly with the increasing number of the users.Figure 8
Achievable minimum SINR versus minimum EH constraint with different number of users.From Figure9, we can see that ZF-DPC performs worse than SDR only in the very high minimum harvest energy e regions. With the increase of the ID users M (the number of ID/EH users decreasing correspondingly), the performances of the three schemes all improve about 4dB.Figure 9
Achievable minimum SINR versus minimum EH constraint with fixed number of EH users.Figure10 compares the proposed schemes when there are only ID/EH users, which indicates that our proposed ZF-DPC scheme is superior to the other schemes in most cases.Figure 10
Achievable minimum SINR versus minimum EH constraint with ID/EH users only.All the three figures again indicate the trade-off between energy harvesting and information transmission, which is a fundamental issue in the power splitting design. When the users need to harvest more energy, the achievable SINR that corresponds to the capacity of information transmission becomes lower.


## 6.1. SINR versus Transmit Power

In this part, we investigate the achievable minimum SINR of the proposed schemes under different transmit power of the BS whene=0 dBm.From Figures5 and 6, we can see that as the transmit power Pmax increases from 0 dBm to 30 dBm, the system’s performance of all three schemes increases substantially, and the performance of ZF is always the worst. When the number of users is relatively small, the performances of the three schemes are almost the same, especially when K=2,M=1, and N=1 due to less interference from other users. Yet, as shown in Figure 5, with the increasing number of users (e.g., K=4 and K=8), the performances of different schemes differ significantly.Figure 5
Achievable minimum SINR versus transmit power with different numbers of users.Figure 6
Achievable minimum SINR versus transmit power with fixed number of EH users.Specifically, when the transmit power is relatively small, SDR has better performance than ZF-DPC, while with higher transmit power, the performance of ZF can approach SDR, and ZF-DPC has a gain of about2dB and about 10dB over SDR and ZF, respectively, when K=8,M=4, and N=2. We can also see from Figure 6 that when the number of total users K and the number of EH users are fixed, with the increasing number of ID users M (the number of ID/EH users decreasing correspondingly), the performances of the three schemes all improve about 3dB. The reason is that, with fewer energy harvesters, the BS can allocate more power to the information receivers.Figure7 shows that our proposed ZF-DPC technique provides higher achievable minimum SINR than SDR and ZF even for the traditional scenario with ID/EH users only.Figure 7
Achievable minimum SINR versus transmit power with ID/EH users only.


## 6.2. SINR versus EH

In this part, we investigate the relationship between the achievable minimum SINR and the minimum energy harvesting of the users, with fixed transmit power of the BS (Pmax=30dBm).We can see from Figure8 that the performances of the three schemes are similar when the number of users is small and the performances of different schemes differ significantly with the increasing number of the users.Figure 8
Achievable minimum SINR versus minimum EH constraint with different number of users.From Figure9, we can see that ZF-DPC performs worse than SDR only in the very high minimum harvest energy e regions. With the increase of the ID users M (the number of ID/EH users decreasing correspondingly), the performances of the three schemes all improve about 4dB.Figure 9
Achievable minimum SINR versus minimum EH constraint with fixed number of EH users.Figure10 compares the proposed schemes when there are only ID/EH users, which indicates that our proposed ZF-DPC scheme is superior to the other schemes in most cases.Figure 10
Achievable minimum SINR versus minimum EH constraint with ID/EH users only.All the three figures again indicate the trade-off between energy harvesting and information transmission, which is a fundamental issue in the power splitting design. When the users need to harvest more energy, the achievable SINR that corresponds to the capacity of information transmission becomes lower.


## 7. Conclusion

In this paper, we, for the first time, studied the energy harvesting in the IoT for MISO SWIPT systems with heterogeneous users. We developed four schemes, namely, SDR, ZF, MRT, and ZF-DPC, to solve the nonconvex problem. We then compared them through simulations and provided suggestions on how to select the proper scheme in different scenarios. In the future work, we plan to solve the problems when there are multiple transmitters and the perfect CSI is not available.



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*Source: 1858532-2017-07-30.xml*