# The Precoder Design with Covariance Feedback for Simultaneous Information and Energy Transmission Systems

Authors: Wen Zhou; Dan Deng; Junjuan Xia; Ziyun Shao
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

We consider the optimal precoder design with the assumption that the transmitter only has channel covariance information, for the multi-input multi-output (MIMO) information and energy transmission system. The objective of the system design is to maximize the average system information rate, meanwhile meeting the minimum energy requirement of the energy receiver. Following this objective, we formulate the problem as a semidefinite programming (SDP) and further transform it into a dual problem. Two methods are proposed to solve this problem: the first method decomposes the transmission covariance as a product of precoders so that the constrained optimization becomes an unconstrained one, whereas the second method derives the structure of the optimal transmission covariance analytically. Both methods are proved to be convergent and their overheads and complexity are also analyzed. The achievable rate-energy (R-E) regions for the proposed methods are presented in the simulation. Under various system settings, the superiority of the proposed methods is shown by comparing with a few existing transmission schemes.


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## Body

## 1. Introduction

The rapidly development of wireless sensor networks has created many applications such as environmental monitoring, telemedicine system, and intelligent house system [1, 2]. However, the main bottleneck of these applications is the energy limitation of mobile equipment [3–5]. Energy harvesting is a promising technique to deal with this problem, prolonging the lifetime of the network, and hence has attracted a great deal of attention [6–9].Nowadays, the techniques for wireless power transfer (WPT) can be divided into three categories [7]: (1) near-field power transfer; (2) far-field directive power beaming; (3) far-field power transfer based on radio frequency (RF). Note that low-power RF signals are ambient and can be harvested by receivers from remote transmitters such as base stations and free WiFi hotspots. Further, since RF signals carrying energy can also be used for information transmission, a novel research direction, simultaneous wireless information and power transfer (WIPT), appears. It is clear that, compared with the former two techniques, RF-based WPT is more suitable for simultaneous wireless information and power transfer (WIPT). In fact, most researches on WIPT are based on the third kind. Therefore, this article also applies RF signals in energy scavenging.In general, the researches on WIPT were gone from simpleness to complication and from single-input single-output (SISO) systems to multiple-antenna systems [10–34]. For the SISO WIPT system, the problems on the trade-off between the rate and power, the circuit design, the transmission power allocation, and so on, were thoroughly investigated [10–23].The technical papers [10, 11] are two pioneer researches on how to transmit energy and information jointly. In [10], the concept of the achievable rate-energy (R-E) region was first proposed and a few R-E expressions were derived under some discrete channels. In [11], the authors considered the WIPT over a frequency-selective channel with Gaussian noise, providing the solutions of power allocation to the discrete and continuous versions, respectively. The authors in [12] studied the designs of both the information and energy receiver over time-varying fading channels. They presented several practical circuits with different multiplexing methods. In [15], the authors studied an amplify-and-forward (AF) relaying system, in which the energy of the received RF signal is harvested by the relay and then used for the information exchange between the source and destination. Two relay protocols were proposed to facilitate the WIPT of the relay. There were also a few researches that considered the multiuser access problem, wireless resource management, or other issues [18–20].For the WIPT system with multiple antennas, the issues on the design of transmission covariance, the achievable R-E region, the system performance, and others, were addressed in [21–34]. For instance, in [21], a robust beamforming scheme was designed for the multiple-input single-output (MISO) WIPT system. This article makes an assumption that channel state information (CSI) obtained at the transmitter is not perfect and the design is based on the criterion in which the worst-case harvested energy for the energy receiver is maximized while guaranteeing that the minimum information rate for the information receiver. Utilizing a similar system model, in [24], aiming at maximizing the harvesting energy, the authors proposed an adaptive energy beamforming method according to the instantaneous CSI. In addition, the relationship between transmit power, transfer duration, and limited feedback amount was also presented. In [26], a unified study on MIMO WIPT systems was conducted, in which two practical designs were proposed in the case of the colocated receiver and their respective achievable R-E regions were also characterized. In [27], the joint two sources and relay beamforming design problem was studied for orthogonal space-time block code (OSTBC) based AF relay systems. In [29], a low-complexity technique of antenna switching for WIPT MIMO relay systems was proposed to minimize the system outage probability.Usually, in the above researches, channel state information (CSI) at the transmitter is indispensable for the design of transmission scheme, power control, and others issues. Some of them assumed that the transmitter is able to know complete instantaneous CSI. With such an assumption, the authors in [11, 26] presented the achievable R-E regions for SISO systems and MIMO systems, respectively. Whereas there were also a few studies that considered that, in practice, it is quite difficult to obtain perfect CSI, due to Doppler shift, channel noise, or other factors, and hence assumed that only incomplete instantaneous CSI can be accessed by the transmitter. For instance, the authors in [21] designed a robust beamformer for the MIMO WIPT broadcasting system, with the assumption that the transmitter only has the incomplete CSI. In the multiantenna WIPT system, the authors in [24] analyzed the performance of energy beamforming to maximize the energy efficiency with the limited CSI feedback. Besides, a robust beamformer was designed by the use of the hybrid CSI, including the statistical CSI and instantaneous CSI [31].It is clear that most existing researches use the instantaneous CSI at the transmitter to improve the system performance, including the cases of perfect CSI and imperfect CSI. It is suitable for the feedback of instantaneous CSI in slow fading channels. However, in the case of fast fading channel, frequent CSI exchange will inevitably bring much burden on the communication system. Under such a scenario, the scheme of feeding back the instantaneous CSI is no longer appropriate. On the other front, in general, the channel statistical information does not change or changes very slowly, feeding back which can reduce the amount of CSI exchange significantly. It is no doubt that studying the precoder design with channel statistical information feedback makes sense. In fact, the optimal transmission covariance matrices with only statistical CSI feedback were thoroughly investigated in point-to-point MIMO systems or interference systems [32–37]. However, these cannot be applied to MIMO WIPT systems directly. To the best of our knowledge, the precoder with only statistical CSI feedback in WIPT systems has not been studied yet.Therefore, this article focuses on the scenario in which the transmitter can only have the statistical CSI. Instead of using the instantaneous information rate as the optimization objective, the objective function becomes the expected system information rate. Consequently, the optimal precoder is designed under two constraints which include maximizing the expected information rate and meeting the minimum energy requirement. The contributions of this article are briefly summarized as follows:(i)
Unlike most existing works, we design the optimal precoder by only use of statistical CSI, channel covariance feedback. Two precoding methods are proposed: one is a numerical method by employing the classical gradient-decent algorithm; the other one is an analytical method by use of the analytical expression of the optimal precoder. In the analytical method, we present the close-form expression of the optimal transmission covariance matrix via a proposition, as a function of the number of transmitter antennas, channel covariance matrices, and so on.(ii)
We analyze the complexity and overheads of the proposed methods. A few classical existing works are also analyzed and compared, including time-switching method [26], isotropic transmission, and the precoder with hybrid CSI [31]. To the best of our knowledge, such work has not been done in existing literature. Furthermore, under various system settings, we compare the system performance of the proposed methods with that of these existing methods.(iii)
Besides, the boundary of R-E region is characterized, analytically resenting two important points of the boundary.The rest of this paper is organized as follows. Section2 describes the WIPT MIMO system model. Section 3 formulates the problem and proposes two precoding methods. Section 4 analyzes the complexity and overheads of the proposed methods. Simulation results are presented in Section 5, with concluding remarks in Section 6.Notations. Vectors are denoted by boldface lowercase letters and matrices are denoted by boldface uppercase letters. E(·) stands for the statistical expectation; Cm×n stands for the sets of m×n complex matrices. For a vector x, x~CN(μ,R) means that x follows a complex Gaussian distribution with mean μ and covariance matrix R; for a matrix X, the notations X1/2, tr(X), XH, and X∗ denote its square root, trace, Hermitian transpose, and conjugate, respectively; we write X⪰0 to mean that X is positive semidefinite; besides, Im is an m×m identity matrix.


## 2. System Model

With reference to Figure1, this paper considers a three-node MIMO WIPT system, which includes a transmitter TX, an energy receiver RXE, and an information receiver RXD. The numbers of antennas for TX, RXE, and RXD are NT, NE, and ND, respectively. There are two kinds of links: one is the broadcast link for data or power transmission; the other is the feedback link for statistical CSI feedback [38–41]. The signal x∈CNT×1, sent from the transmitter, has NT independent data streams with its autocorrelation matrix E(xxH)=INT. Before transmission, we multiply the signal x by a precoder matrix W∈CNT×NT, and after passing through flat-fading channels, the received signals of the information receiver and energy receiver are given by (1)yD=HDWx+nD,(2)yE=HEWx+nE,where HD∈CND×NT and HE∈CNE×NT are flatting MIMO channels from the transmitter to the information receiver and energy receiver, respectively; nD∈CND×1 and nE∈CNE×1 arei.i.d Gaussian noise vectors following CN(0,σD2IND) and CN(0,σE2INE), respectively. In the followings, for the sake of convenience, we refer to HD and HE as the information channel and energy channel, respectively.Figure 1
A three-node MIMO WIPT system model.The channel matricesHD and HE can be further modeled as(3)HD=θRD1/2Hω,DθTD1/2,(4)HE=θRE1/2Hω,EθTE1/2,in which θTD and θRD are channel covariance matrices for the information channel at the transmitter end and receiver end, respectively; θTE and θRE are channel covariance matrices for the energy channel at the transmitter end and receiver end, respectively. Hω,D and Hω,E are random matrices havingi.i.d. zero-mean and unit variance complex Gaussian random variables, representing the uncorrelated scattering.After measuring channels for a period of time, the transmitter acquires channel statistical information, includingθRD, θRE, θTD, and θTE. Using the information, the optimal precoder W is designed to maximize the system information transmission rate, meanwhile satisfying the minimum energy requirement. Clearly, the scheme with statistical CSI feedback can reduce the overheads, compared to the scheme with instantaneous CSI feedback. However, it may bring the loss of information rate or energy.Finally, for the sake of performance evaluation, we define the system signal-to-noise-ratio (SNR). Since the autocorrelation of the user signalx is an identity matrix, the transmission power is expressed as(5)EWxF2=trWWH≤PT,where PT is the maximum transmission power. Obviously, WWH is the transmission covariance matrix after precoding. With (5), the system SNR is defined as(6)SNR=PTσD2,where σD2 is the noise power of the information channel HD. Without loss of generality, we assume that σD2=1.


## 3. The Proposed Precoding Methods with Covariance Feedback

In this section, we first formulate the precoding problem as a SDP and then transform it into a dual problem by introducing some auxiliary variables and constructing a dual function. Based on these, two methods are proposed to solve the dual problem. Their differences mainly lie on how to solve the dual function: one does it by the classical numerical searching, whereas the other does it by exploiting the structure of the covariance matrix.


### 3.1. Problem Formulation

First, with (1), the information transmission rate is given by(7)J=Elog⁡det⁡IND+HDQHDH,where the matrix Q∈CNT×NT, defined by WWH, is the covariance matrix of the transmitted signal.Then, with (2), the received energy per unit time can be written as(8)εng=ηEHEWxHHEWx=ηEtrHEQHEH.With (4), (8) can be further expressed as [31](9)εng=ηtrQθTEtrθRE,where the notation η is the energy transfer efficiency. Without loss of generality, we set it to 1.Finally, we try to find the optimal Q that maximizes the expected information rate, meanwhile meeting the minimum energy requirement of the energy receiver, εng. The problem, termed as P1, can be summarized as follows:(10)P1:maxQJ=Elog⁡det⁡IND+HDQHDHs.t.Q⪰0trQ≤PTtrQθTEtrθRE≥εng.Notice that unreasonable εng may make the problem P1 not feasible, and hence we present the following lemma about the range of εng (see Appendix A).Lemma 1.
Whenεng>PTλmaxθTEtrθRE, the problem P1 is not feasible, where λmaxθTE is the maximum eigenvalue of θTE; when εng<trθRE, the energy constraint of P1 is inactive, where QOPT′ is the solution of the following problem P1′:(11)P1′:maxQElog⁡det⁡IND+HDQHDHs.t.Q⪰0;trQ≤PT.
Note thatP1′ is a transmission optimization problem for the transmitter only has channel covariance information. Similar problems were well investigated and solved in the past years [33–37]. For instance, in [33], the optimal transmission covariance matrices for two cases were presented, including channel covariance and mean feedback of rank one. In [34], the precoding methods for MIMO interference channels with covariance feedback were presented. In fact, P1′ can be solved by the proposed method in [33]. In addition, similar to [26], we also use the rate-energy (R-E) region to describe all achievable rate and energy pairs, which is defined as follows:(12)ΩPT=J,εng:Elog⁡det⁡IND+HDQHDH≥J,trQθTEtrθRE≥εng,trQ≤PT,Q⪰0Clearly, given εng, the rate solution of P1 is the vertical ordinate of some boundary point of the R-E region.Remark. Recently, there are practical nonlinear models being proposed [42]. As pointed in [42], for low power, the harvested RF power increases with increasing input power; however, there are limitations on the maximum possible harvested energy. In fact, from Figure 2 of [42], we find that the nonlinearity mainly occurs in high input power region and the use of the linear energy harvesting (EH) model in low-power region is still suitable. If there is no light-of-sight (LOS) path or the distance is long enough between the transmitter and the receiver, the assumption that the received power is in low region is likely to be reasonable. Hence, assuming the received power is low, the conventional linear EH model is adopted in this article.


### 3.2. The Proposed First Precoding Method

In this section, a numerical algorithm is proposed to solve P1 and the optimalQ can be obtained subsequently. Obviously, once acquiring the optimal Q, the optimal precoder W can be readily solved by taking the square root of Q.First, the Lagrangian of P1 is given by(13)LQ,λ,u=Elog⁡det⁡IND+HDQHDH+λtrQθTEtrθRE-εng-utrQ-PTwhere λ≥0 and u≥0 are two introduced auxiliary variables. Observe that the Lagrangian combines the objective function with the transmission power and energy constraints of P1. By doing so, the number of the constraints on the covariance matrix Q is reduced. The aim of introducing the Lagrangian is to transform the original problem P1 into an easier problem. With (13), adual functionis defined as(14)gλ,u=maxQ⪰0LQ,λ,u.Consequently, the variable Q does not exist in this function.With (14), the dual problem of P1 can be formulated as (P2)(15)P2:Jdu=minλ,u≥0gλ,uClearly, P2 only involves two introduced variables λ and u and it is much simpler than P1. It is easy to verify theSlater’s condition is satisfied (see Appendix B). Therefore, there is no dual gap between P1 and P2. In other words, the two problems are equivalent.Next, we consider how to solve P2. The problem-solving trait is composed of two steps: (1) given (λ,u), obtain the optimalQand g(λ,u); (2) search over the two-dimension space of (λ,u) and find the minimum value of g(λ,u). To do those, the objective function g(λ,u) needs further arrangement. With (13) and (14), we have [20](16)gλ,u=maxQ⪰0Elog⁡det⁡IND+HDQHDH-trAQ+Tconst,where A=uINT-λtrθREθTE, Tconst=uPT-λεng. Note that the matrix A must be semidefinite, otherwise gλ,u will go to infinity. This implies that u>λtrθREλmaxθTE, where λmaxθTE is the largest eigenvalue of θTE.SinceQ is semidefinite and hence can be decomposed as Q=WWH, where W∈CNT×NT is the precoder. Replacing Q with W, the semidefinite constraint of Q will disappear. Hence, (16) can be further written as(17)gλ,u=maxWLW,λ,u=maxWElog⁡det⁡IND+HDWWHHDH-trAWWH+Tconst.Differentiating LW,λ,u with respect to W∗, we have(18)DW≜∂LW,λ,u∂W∗=EINT+HDHHDWWH-1HDHHDW-AW.Observe that solving maxWLW,λ,u is an unconstrained optimization problem. Given (λ,u), we propose the following subalgorithm to solve g(λ,u) and obtain W and Q afterwards.Given(λ,u), the proposed subalgorithm (Algorithm 1) will output the optimal W and corresponding g(λ,u). Based on Subalgorithm 1 (see Algorithm 1), a decent algorithm to solve the dual problem P2 is proposed. We term it as Main Algorithm 1 and summarize it in Algorithm 2.Algorithm 1:Subalgorithm 1: solve g(λ,u).
(1) Initialization:W0, the iteration index i≔0, and the searching step t>0. (2) Repeat At thei-th iteration, compute D(W(i)) according to (18); updateW according to Wi+1=Wi+tDWi; i≔i+1. (3) UntilLWi,λ,u-LWi-1,λ,u<ξ, where ξ is a pre- scribed threshold.Algorithm 2:Main Algorithm 1: solve the dual problem P2.
(1) Initialization: the iteration index i≔0, λ(0)≥0, u(0)≥0, u0>λ0trθREλmaxθTE, the searching step s>0. Computeg(λ0,u(0)) and W(0) according to Sub-algorithm 1 (see Algorithm 1). (2) Repeat Updateλi+1,ui+1 according to λ i + 1 = m a x 0 , λ i + s Δ λ i, (∗a) u i + 1 = m a x 0 , u i + s Δ u i, (∗b) whereΔλi and Δui are the partial derivatives of gλ,u with respect toλ and u at λ(i) and u(i), respectively. Computeg(λi+1,u(i+1)) and W(i+1) via Sub-algorithm 1 (see Algorithm 1); i≔i+1. (3) Untilg(λ,u) converges to the predetermined accuracy.It is worth noting that, in Main Algorithm 1 (see Algorithm2), Δλi and Δuiare derived as(19a)Δλi=∂gλ,u∂λλ=λi=trWiWiHθTEtrθRE-εng(19b)Δui=∂gλ,u∂uu=ui=-trWiWiH+PTThe flow chart of Main Algorithm 1 (see Algorithm2) is depicted in Figure 2. With the proposed Main Algorithm 1 (see Algorithm 2), we can obtain the optimal W, denoted as WOPT, and the minimum value of g(λ,u). According to the duality principle, the optimal precoder of the original problem P1 is WOPT, and the maximum transmission rate of P1 is g(λ,u)’s minimum value.Figure 2
The flow chart of Main Algorithm 1 (see Algorithm2).Remark 2.
In Subalgorithm 1 (see Algorithm1), the initial W can be set to an arbitrary matrix except a zero one. Since L(W,λ,u) is not a convex function of W, we had better run this algorithm several times and try different initial points to find a best value.Remark 3.
In the simulation, we compute the expected information rate in (7) by the following two steps. (1) With (3), generate some information channel samples, denoted as HD(n), where n is the index of the samples. (2) Compute the following formula to approximate the expected rate in (7): (20)∑n=0Nsam-1log⁡det⁡IND+HDnQHDHnNsam,where Nsam is the number of samples.


### 3.3. The Proposed Second Precoding Method

This subsection presents an indirect method for solving P2. To find the optimal covariance matrix, we show its structure by a proposition and then propose an iterative algorithm to solve an unknown diagonal matrix, which is similar to the so-calledpower-allocation matrix.First, with (16), we have the following proposition.Proposition 4.
The optimal covariance matrixQOPT has the following form:(21)QOPT=A-1/2VBΛVBHA-1/2,where A=uINT-λtrθREθTE, the columns of V are taken from the right-singular vectors of B≜θTD1/2A-1/2, and Λ=diagp1p2⋯pNT with p1≥⋯≥pNT≥0.Proof.
See AppendixC.From the above proposition, givenλ,u, the unknown parameter to be determined is the diagonal matrix Λ.Then, we propose an iterative algorithm, like the so-calledpower-allocation algorithm, to solve Λ. With a few manipulations (Appendix C), g(λ,u) in (14) can be expressed as(22)gλ,u=maxQ~⪰0Elog⁡det⁡IND+UθRDH~DVBHQ~VBH~DHUθRDH-trQ~+Tconst,where H~D=ΛθRD1/2UθRDHHω,DUBΛB, the definitions of UB and ΛB are shown in (C.3), and Tconst≜uPT-λεng.With (21) and noticing Q~=A1/2QA1/2, gλ,u can be further expressed as(23)gλ,u=maxΛ⪰0Elog⁡det⁡IND+UθRDH~DΛH~DHUθRDH-trΛ+Tconst=maxΛ⪰0Elog⁡det⁡IND+H~DΛH~DHUθRDHUθRD-trΛ+Tconst=maxΛ⪰0Elog⁡det⁡IND+H~DΛH~DH-trΛ+Tconst.To solve Λ, introducing auxiliary variables ui′≥0, i=1,…,NT, we construct the following Lagrangian function:(24)Lgp1,…,pNT,u1′,…,uNT′=Elog⁡det⁡IND+H~DΛH~DH-∑i=1NTpi+∑i=1NTpiui′+Tconst=Elog⁡det⁡IND+∑i=1NTpih~D,ih~D,iH-∑i=1NTpi+∑i=1NTpiui′+Tconst,in which the vector h~D,i is the i-th column of H~D. Consequently, the KKT (Karush-Kuhn-Tucker) optimality condition is given by(25)∂Lgp1,…,pNT,u1′,…,uNT′∂pl=EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH-1+ul′=0.(26)ul′pl=0,ul′≥0,pl≥0,l=1,…,NT.After a few manipulations, the optimality condition above can be expressed as(27)EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH=1,forpl>0;EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH≤1,forpl=0.Notice that (27) is similar, but not identical to [32, Eq. (5)], because the problem of solving gλ,u and that in [32] are somewhat different. Furthermore, as in Appendix D, (27) can be written in a simpler form,(28)Eh~D,lHCl-1h~D,l1+plh~D,lHCl-1h~D,l=1,forpi>0;Eh~D,lHCl-1h~D,l1+plh~D,lHCl-1h~D,l≤1,forpi=0,where Cl≜IND+∑i≠lpih~D,ih~D,iH. With (28), it is derived that (also see Appendix D)(29)pl=0,forEh~D,lHCl-1h~D,l≤1;pl>0,pl=1-E11+plh~D,lHCl-1h~D,l,otherwise.Similar to the processing technique introduced in [32], we present the iterative algorithm (see Algorithm 3) to solve {pl}, resulting from (29).Algorithm 3:Subalgorithm 2.1: solve {pl}.
(1) Initialization: the iteration index k≔0, pl(0)>0, where l∈Φ=1,…,NT. (2) Repeat Updatepl(k+1) according to pl(k+1)=1-E1/(1+pl(k)h~D,lH(Cl-1)kh~D,l), l∈Φ; k≔k+1. (3) Until all pl’s converge to the predetermined accuracy. (4) If there exists some l such that Eh~D,lHCl-1h~D,l≤1 but pl does not converge to 0, findlmin=argminl∈ΦE(h~D,lHCl-1h~D,l), set k≔0 andplmin=0, update Φ≔Φ-{lmin}, and go to Step (2); otherwise end this procedure.Consequently, given(λ,u), the algorithm of solving g(λ,u) based on Proposition 4 is proposed and summarized in Algorithm 4.Algorithm 4:Subalgorithm 2.2: solve g(λ,u).
(1) Obtain the matrix Λ=diag(p1p2⋯pNT), according to Sub-algorithm 2.1 (see Algorithm3). (2) Calculate the matrix Qaccording to (21). (3) Calculateg(λ,u) by substituting Q into (16).Finally, with Subalgorithms 2.2 (see Algorithm 4), the second algorithm of solving P2 is proposed as Algorithm 5.Algorithm 5:Mainalgorithm 2: solve the dual problem P2.
(1) Initialization: the iteration index i≔0, λ0≥0,u0≥0, u0>λ0trθREλmaxθTE, and s>0. Compute g(λ0,u(0)) andQ(0) according to Sub-algorithm 2.2 (see Algorithm 4). (2) Repeat Update(λi+1,u(i+1)) according to (∗a) and (∗b) in Algorithm 2, respectively; computeg(λi+1,u(i+1)) and Q(i+1), according to Sub-algorithm 2.2 (see Algorithm4); increase i≔i+1. (3) Untilg(λ,u) converges to the prescribed accuracy.Note that, for (∗a) and (∗b) (see Algorithm 2) mentioned in Main Algorithm 2 (see Algorithm 5), the partial derivatives of g(λ,u) should be modified as(30)Δλi=∂gλ,u∂λλ=λi=trQiθTEtrθRE-εng,Δui=∂gλ,u∂uu=ui=-trQi+PT.Similar to the conclusion of the previous subsection, the proposed Main Algorithm 2 (see Algorithm 5) will output the optimal covariance QOPT and the minimum g(λ,u). According to the duality principle, the maximum transmission rate for P1 is the minimum g(λ,u) and the optimal precoder WOPT can be readily acquired by taking the square root of QOPT.Remark 1.
In Subalgorithm 2.1 (see Algorithm3), the initial power stream pl0 should not be set to zero; otherwise it will be unchanged afterwards. To prove the convergence of this algorithm analytically is quite difficult; however, we show that it does converge rapidly by extensive simulations and similar conclusions have also been drawn in [32].Remark 2.
Since the dual functiong(λ,u) is convex [43] and the two constraints λ≥0 and u≥0 are linear, the problem P2 is convex and its solution is unique. Searching from an arbitrary feasible initial point is able to reach its optimal value after a few iterations. On the other hand, it is easy to find that given λ and u, g(λ,u)>–∞, that is, g(λ,u) has a lower bound. Furthermore, the objective function g(λ,u) in either Main Algorithm 1 or 2 (see Algorithm 2 or Algorithm 5) is nonincreasing because of the gradient-decent property of the proposed algorithm. Therefore, the proposed two algorithms are convergent.


### 3.4. Discussion

This subsection discusses two issues, including the difference between the proposed first method and second method and the difference between the proposed methods and the method in [32].First, the two proposed methods solve the same optimization problem, i.e., solving g(λ,u) in two different ways. The first one is anumerical method, which transforms a constrained optimization problem into an unconstrained one, and hence can adopt the classical gradient-decent method to solve the problem, whereas the second one is ananalytical method, which presents the structure of the optimal transmission covariance matrix in Proposition 4. Therefore, the second algorithm exhibits a deeper understanding of the original problem, by revealing the structure of the optimal precoder.Second, we study the precoder design or covariance matrix in three-node WIPT systems, as compared with that in point-to-point systems [32]. Clearly, the structure of the optimal covariance matrix (21) is quite different from that in [32, Theorem 1]. The power-allocation algorithm is only a part of the structure of the covariance matrix. We draw on the trait of solving the power-allocation problem in [32] and the two problems do look similar to some degree. However, there are also a few differences between them. In addition, solving the power-allocation problem also yields new contributions.(i)
The two problems are different in form: one is given by (23) and the other is shown in [32, Eq. (16)]. The derivation methods of solving the two problems are somewhat different. The authors in [32] derives the optimal matrix from the first principle [44, Section 15.2], a derivative version of the KKT conditions. However, the principle cannot be applied to the problem in this paper. Hence, we perform the derivations via the classical KKT conditions.(ii)
When solving the power-allocation problem, we believe that the derivation from [32, Eq. (5)] to [32, Eqs. (10,11)] is not such straightforward. The two formulas (27) and (29) in this paper are the counterparts of [32, Eq. (5)] and [32, Eqs. (10,11)], respectively. We give the details on how (29) is derived from (27) in Appendix D. Using the techniques in Appendix D, one can derive [32, Eqs. (10,11)] from [32, Eq. (5)].


## 3.1. Problem Formulation

First, with (1), the information transmission rate is given by(7)J=Elog⁡det⁡IND+HDQHDH,where the matrix Q∈CNT×NT, defined by WWH, is the covariance matrix of the transmitted signal.Then, with (2), the received energy per unit time can be written as(8)εng=ηEHEWxHHEWx=ηEtrHEQHEH.With (4), (8) can be further expressed as [31](9)εng=ηtrQθTEtrθRE,where the notation η is the energy transfer efficiency. Without loss of generality, we set it to 1.Finally, we try to find the optimal Q that maximizes the expected information rate, meanwhile meeting the minimum energy requirement of the energy receiver, εng. The problem, termed as P1, can be summarized as follows:(10)P1:maxQJ=Elog⁡det⁡IND+HDQHDHs.t.Q⪰0trQ≤PTtrQθTEtrθRE≥εng.Notice that unreasonable εng may make the problem P1 not feasible, and hence we present the following lemma about the range of εng (see Appendix A).Lemma 1.
Whenεng>PTλmaxθTEtrθRE, the problem P1 is not feasible, where λmaxθTE is the maximum eigenvalue of θTE; when εng<trθRE, the energy constraint of P1 is inactive, where QOPT′ is the solution of the following problem P1′:(11)P1′:maxQElog⁡det⁡IND+HDQHDHs.t.Q⪰0;trQ≤PT.
Note thatP1′ is a transmission optimization problem for the transmitter only has channel covariance information. Similar problems were well investigated and solved in the past years [33–37]. For instance, in [33], the optimal transmission covariance matrices for two cases were presented, including channel covariance and mean feedback of rank one. In [34], the precoding methods for MIMO interference channels with covariance feedback were presented. In fact, P1′ can be solved by the proposed method in [33]. In addition, similar to [26], we also use the rate-energy (R-E) region to describe all achievable rate and energy pairs, which is defined as follows:(12)ΩPT=J,εng:Elog⁡det⁡IND+HDQHDH≥J,trQθTEtrθRE≥εng,trQ≤PT,Q⪰0Clearly, given εng, the rate solution of P1 is the vertical ordinate of some boundary point of the R-E region.Remark. Recently, there are practical nonlinear models being proposed [42]. As pointed in [42], for low power, the harvested RF power increases with increasing input power; however, there are limitations on the maximum possible harvested energy. In fact, from Figure 2 of [42], we find that the nonlinearity mainly occurs in high input power region and the use of the linear energy harvesting (EH) model in low-power region is still suitable. If there is no light-of-sight (LOS) path or the distance is long enough between the transmitter and the receiver, the assumption that the received power is in low region is likely to be reasonable. Hence, assuming the received power is low, the conventional linear EH model is adopted in this article.


## 3.2. The Proposed First Precoding Method

In this section, a numerical algorithm is proposed to solve P1 and the optimalQ can be obtained subsequently. Obviously, once acquiring the optimal Q, the optimal precoder W can be readily solved by taking the square root of Q.First, the Lagrangian of P1 is given by(13)LQ,λ,u=Elog⁡det⁡IND+HDQHDH+λtrQθTEtrθRE-εng-utrQ-PTwhere λ≥0 and u≥0 are two introduced auxiliary variables. Observe that the Lagrangian combines the objective function with the transmission power and energy constraints of P1. By doing so, the number of the constraints on the covariance matrix Q is reduced. The aim of introducing the Lagrangian is to transform the original problem P1 into an easier problem. With (13), adual functionis defined as(14)gλ,u=maxQ⪰0LQ,λ,u.Consequently, the variable Q does not exist in this function.With (14), the dual problem of P1 can be formulated as (P2)(15)P2:Jdu=minλ,u≥0gλ,uClearly, P2 only involves two introduced variables λ and u and it is much simpler than P1. It is easy to verify theSlater’s condition is satisfied (see Appendix B). Therefore, there is no dual gap between P1 and P2. In other words, the two problems are equivalent.Next, we consider how to solve P2. The problem-solving trait is composed of two steps: (1) given (λ,u), obtain the optimalQand g(λ,u); (2) search over the two-dimension space of (λ,u) and find the minimum value of g(λ,u). To do those, the objective function g(λ,u) needs further arrangement. With (13) and (14), we have [20](16)gλ,u=maxQ⪰0Elog⁡det⁡IND+HDQHDH-trAQ+Tconst,where A=uINT-λtrθREθTE, Tconst=uPT-λεng. Note that the matrix A must be semidefinite, otherwise gλ,u will go to infinity. This implies that u>λtrθREλmaxθTE, where λmaxθTE is the largest eigenvalue of θTE.SinceQ is semidefinite and hence can be decomposed as Q=WWH, where W∈CNT×NT is the precoder. Replacing Q with W, the semidefinite constraint of Q will disappear. Hence, (16) can be further written as(17)gλ,u=maxWLW,λ,u=maxWElog⁡det⁡IND+HDWWHHDH-trAWWH+Tconst.Differentiating LW,λ,u with respect to W∗, we have(18)DW≜∂LW,λ,u∂W∗=EINT+HDHHDWWH-1HDHHDW-AW.Observe that solving maxWLW,λ,u is an unconstrained optimization problem. Given (λ,u), we propose the following subalgorithm to solve g(λ,u) and obtain W and Q afterwards.Given(λ,u), the proposed subalgorithm (Algorithm 1) will output the optimal W and corresponding g(λ,u). Based on Subalgorithm 1 (see Algorithm 1), a decent algorithm to solve the dual problem P2 is proposed. We term it as Main Algorithm 1 and summarize it in Algorithm 2.Algorithm 1:Subalgorithm 1: solve g(λ,u).
(1) Initialization:W0, the iteration index i≔0, and the searching step t>0. (2) Repeat At thei-th iteration, compute D(W(i)) according to (18); updateW according to Wi+1=Wi+tDWi; i≔i+1. (3) UntilLWi,λ,u-LWi-1,λ,u<ξ, where ξ is a pre- scribed threshold.Algorithm 2:Main Algorithm 1: solve the dual problem P2.
(1) Initialization: the iteration index i≔0, λ(0)≥0, u(0)≥0, u0>λ0trθREλmaxθTE, the searching step s>0. Computeg(λ0,u(0)) and W(0) according to Sub-algorithm 1 (see Algorithm 1). (2) Repeat Updateλi+1,ui+1 according to λ i + 1 = m a x 0 , λ i + s Δ λ i, (∗a) u i + 1 = m a x 0 , u i + s Δ u i, (∗b) whereΔλi and Δui are the partial derivatives of gλ,u with respect toλ and u at λ(i) and u(i), respectively. Computeg(λi+1,u(i+1)) and W(i+1) via Sub-algorithm 1 (see Algorithm 1); i≔i+1. (3) Untilg(λ,u) converges to the predetermined accuracy.It is worth noting that, in Main Algorithm 1 (see Algorithm2), Δλi and Δuiare derived as(19a)Δλi=∂gλ,u∂λλ=λi=trWiWiHθTEtrθRE-εng(19b)Δui=∂gλ,u∂uu=ui=-trWiWiH+PTThe flow chart of Main Algorithm 1 (see Algorithm2) is depicted in Figure 2. With the proposed Main Algorithm 1 (see Algorithm 2), we can obtain the optimal W, denoted as WOPT, and the minimum value of g(λ,u). According to the duality principle, the optimal precoder of the original problem P1 is WOPT, and the maximum transmission rate of P1 is g(λ,u)’s minimum value.Figure 2
The flow chart of Main Algorithm 1 (see Algorithm2).Remark 2.
In Subalgorithm 1 (see Algorithm1), the initial W can be set to an arbitrary matrix except a zero one. Since L(W,λ,u) is not a convex function of W, we had better run this algorithm several times and try different initial points to find a best value.Remark 3.
In the simulation, we compute the expected information rate in (7) by the following two steps. (1) With (3), generate some information channel samples, denoted as HD(n), where n is the index of the samples. (2) Compute the following formula to approximate the expected rate in (7): (20)∑n=0Nsam-1log⁡det⁡IND+HDnQHDHnNsam,where Nsam is the number of samples.


## 3.3. The Proposed Second Precoding Method

This subsection presents an indirect method for solving P2. To find the optimal covariance matrix, we show its structure by a proposition and then propose an iterative algorithm to solve an unknown diagonal matrix, which is similar to the so-calledpower-allocation matrix.First, with (16), we have the following proposition.Proposition 4.
The optimal covariance matrixQOPT has the following form:(21)QOPT=A-1/2VBΛVBHA-1/2,where A=uINT-λtrθREθTE, the columns of V are taken from the right-singular vectors of B≜θTD1/2A-1/2, and Λ=diagp1p2⋯pNT with p1≥⋯≥pNT≥0.Proof.
See AppendixC.From the above proposition, givenλ,u, the unknown parameter to be determined is the diagonal matrix Λ.Then, we propose an iterative algorithm, like the so-calledpower-allocation algorithm, to solve Λ. With a few manipulations (Appendix C), g(λ,u) in (14) can be expressed as(22)gλ,u=maxQ~⪰0Elog⁡det⁡IND+UθRDH~DVBHQ~VBH~DHUθRDH-trQ~+Tconst,where H~D=ΛθRD1/2UθRDHHω,DUBΛB, the definitions of UB and ΛB are shown in (C.3), and Tconst≜uPT-λεng.With (21) and noticing Q~=A1/2QA1/2, gλ,u can be further expressed as(23)gλ,u=maxΛ⪰0Elog⁡det⁡IND+UθRDH~DΛH~DHUθRDH-trΛ+Tconst=maxΛ⪰0Elog⁡det⁡IND+H~DΛH~DHUθRDHUθRD-trΛ+Tconst=maxΛ⪰0Elog⁡det⁡IND+H~DΛH~DH-trΛ+Tconst.To solve Λ, introducing auxiliary variables ui′≥0, i=1,…,NT, we construct the following Lagrangian function:(24)Lgp1,…,pNT,u1′,…,uNT′=Elog⁡det⁡IND+H~DΛH~DH-∑i=1NTpi+∑i=1NTpiui′+Tconst=Elog⁡det⁡IND+∑i=1NTpih~D,ih~D,iH-∑i=1NTpi+∑i=1NTpiui′+Tconst,in which the vector h~D,i is the i-th column of H~D. Consequently, the KKT (Karush-Kuhn-Tucker) optimality condition is given by(25)∂Lgp1,…,pNT,u1′,…,uNT′∂pl=EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH-1+ul′=0.(26)ul′pl=0,ul′≥0,pl≥0,l=1,…,NT.After a few manipulations, the optimality condition above can be expressed as(27)EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH=1,forpl>0;EtrIND+∑i=1NTpih~D,ih~D,iH-1h~D,lh~D,lH≤1,forpl=0.Notice that (27) is similar, but not identical to [32, Eq. (5)], because the problem of solving gλ,u and that in [32] are somewhat different. Furthermore, as in Appendix D, (27) can be written in a simpler form,(28)Eh~D,lHCl-1h~D,l1+plh~D,lHCl-1h~D,l=1,forpi>0;Eh~D,lHCl-1h~D,l1+plh~D,lHCl-1h~D,l≤1,forpi=0,where Cl≜IND+∑i≠lpih~D,ih~D,iH. With (28), it is derived that (also see Appendix D)(29)pl=0,forEh~D,lHCl-1h~D,l≤1;pl>0,pl=1-E11+plh~D,lHCl-1h~D,l,otherwise.Similar to the processing technique introduced in [32], we present the iterative algorithm (see Algorithm 3) to solve {pl}, resulting from (29).Algorithm 3:Subalgorithm 2.1: solve {pl}.
(1) Initialization: the iteration index k≔0, pl(0)>0, where l∈Φ=1,…,NT. (2) Repeat Updatepl(k+1) according to pl(k+1)=1-E1/(1+pl(k)h~D,lH(Cl-1)kh~D,l), l∈Φ; k≔k+1. (3) Until all pl’s converge to the predetermined accuracy. (4) If there exists some l such that Eh~D,lHCl-1h~D,l≤1 but pl does not converge to 0, findlmin=argminl∈ΦE(h~D,lHCl-1h~D,l), set k≔0 andplmin=0, update Φ≔Φ-{lmin}, and go to Step (2); otherwise end this procedure.Consequently, given(λ,u), the algorithm of solving g(λ,u) based on Proposition 4 is proposed and summarized in Algorithm 4.Algorithm 4:Subalgorithm 2.2: solve g(λ,u).
(1) Obtain the matrix Λ=diag(p1p2⋯pNT), according to Sub-algorithm 2.1 (see Algorithm3). (2) Calculate the matrix Qaccording to (21). (3) Calculateg(λ,u) by substituting Q into (16).Finally, with Subalgorithms 2.2 (see Algorithm 4), the second algorithm of solving P2 is proposed as Algorithm 5.Algorithm 5:Mainalgorithm 2: solve the dual problem P2.
(1) Initialization: the iteration index i≔0, λ0≥0,u0≥0, u0>λ0trθREλmaxθTE, and s>0. Compute g(λ0,u(0)) andQ(0) according to Sub-algorithm 2.2 (see Algorithm 4). (2) Repeat Update(λi+1,u(i+1)) according to (∗a) and (∗b) in Algorithm 2, respectively; computeg(λi+1,u(i+1)) and Q(i+1), according to Sub-algorithm 2.2 (see Algorithm4); increase i≔i+1. (3) Untilg(λ,u) converges to the prescribed accuracy.Note that, for (∗a) and (∗b) (see Algorithm 2) mentioned in Main Algorithm 2 (see Algorithm 5), the partial derivatives of g(λ,u) should be modified as(30)Δλi=∂gλ,u∂λλ=λi=trQiθTEtrθRE-εng,Δui=∂gλ,u∂uu=ui=-trQi+PT.Similar to the conclusion of the previous subsection, the proposed Main Algorithm 2 (see Algorithm 5) will output the optimal covariance QOPT and the minimum g(λ,u). According to the duality principle, the maximum transmission rate for P1 is the minimum g(λ,u) and the optimal precoder WOPT can be readily acquired by taking the square root of QOPT.Remark 1.
In Subalgorithm 2.1 (see Algorithm3), the initial power stream pl0 should not be set to zero; otherwise it will be unchanged afterwards. To prove the convergence of this algorithm analytically is quite difficult; however, we show that it does converge rapidly by extensive simulations and similar conclusions have also been drawn in [32].Remark 2.
Since the dual functiong(λ,u) is convex [43] and the two constraints λ≥0 and u≥0 are linear, the problem P2 is convex and its solution is unique. Searching from an arbitrary feasible initial point is able to reach its optimal value after a few iterations. On the other hand, it is easy to find that given λ and u, g(λ,u)>–∞, that is, g(λ,u) has a lower bound. Furthermore, the objective function g(λ,u) in either Main Algorithm 1 or 2 (see Algorithm 2 or Algorithm 5) is nonincreasing because of the gradient-decent property of the proposed algorithm. Therefore, the proposed two algorithms are convergent.


## 3.4. Discussion

This subsection discusses two issues, including the difference between the proposed first method and second method and the difference between the proposed methods and the method in [32].First, the two proposed methods solve the same optimization problem, i.e., solving g(λ,u) in two different ways. The first one is anumerical method, which transforms a constrained optimization problem into an unconstrained one, and hence can adopt the classical gradient-decent method to solve the problem, whereas the second one is ananalytical method, which presents the structure of the optimal transmission covariance matrix in Proposition 4. Therefore, the second algorithm exhibits a deeper understanding of the original problem, by revealing the structure of the optimal precoder.Second, we study the precoder design or covariance matrix in three-node WIPT systems, as compared with that in point-to-point systems [32]. Clearly, the structure of the optimal covariance matrix (21) is quite different from that in [32, Theorem 1]. The power-allocation algorithm is only a part of the structure of the covariance matrix. We draw on the trait of solving the power-allocation problem in [32] and the two problems do look similar to some degree. However, there are also a few differences between them. In addition, solving the power-allocation problem also yields new contributions.(i)
The two problems are different in form: one is given by (23) and the other is shown in [32, Eq. (16)]. The derivation methods of solving the two problems are somewhat different. The authors in [32] derives the optimal matrix from the first principle [44, Section 15.2], a derivative version of the KKT conditions. However, the principle cannot be applied to the problem in this paper. Hence, we perform the derivations via the classical KKT conditions.(ii)
When solving the power-allocation problem, we believe that the derivation from [32, Eq. (5)] to [32, Eqs. (10,11)] is not such straightforward. The two formulas (27) and (29) in this paper are the counterparts of [32, Eq. (5)] and [32, Eqs. (10,11)], respectively. We give the details on how (29) is derived from (27) in Appendix D. Using the techniques in Appendix D, one can derive [32, Eqs. (10,11)] from [32, Eq. (5)].


## 4. Overhead and Complexity Issues

This section analyzes the overheads and computational complexity of the proposed precoding methods and a few other existing methods.


### 4.1. The Overhead

Table1 lists the overheads of the proposed methods with other methods. We mainly consider the overhead that is required to compute the precoder.Table 1
Overhead of computing the precoders.
Precoding method Overhead the proposed two methods Tx obtains the channel covariance matricesθTE,θTD,θRE,θRD and noise power σD2, by measure- ment over a period of time.  isotropic transmission No overhead is needed.  the precoder with hybrid CSI feedback [31] Tx obtains the energy channel covariance matricesθTE,θRE and noise power σD2, by measurement over a period of time; Tx also requires the instant CSI of the information channel.  time switching method It is the same with the proposed methods.Then, we make a brief review of the other three existing methods listed in Table1. The isotropic transmission refers to the case in which the transmitter sends NT data streams with equal power, i.e., the precoder matrix W=PTINT. The time-switching scheme refers to the case in which each transmission period is divided into two time slots, one for data and the other one for energy. Let β with 0≤β≤1 be the percentage of transmitting time for the data time slot. The achievable R-E region of time-switching scheme is given by(31)Ω′PT=⋃0≤β≤1J,εng:βElog⁡det⁡IND+HDQ1HDH≥J,trQ1≤PT,1-βtrQ2θTEtrθRE≥εng,trQ2≤PT,Q1⪰0,Q2⪰0.in which Q1 and Q2 are the transmission covariance matrices for data and energy transmission phases, respectively. For the precoder with hybrid CSI feedback [31], only statistical CSI is available between the transmitter and energy receiver, while instant CSI is available between the transmitter and data receiver. What is slightly different from the precoder [31], herein we do not consider the channel estimation error brought by limited-length pilots and noises, since it is not the focus of this article.From Table1, observe that the isotropic transmission does not need the overhead; however, it has the worst performance, which will be shown in the simulations. Compared with the precoder with hybrid CSI feedback, the proposed methods only require the statistical channel information and do not need the instant CSI of the information channel, which reduces the overall overhead significantly in the long run. In addition, the proposed methods have the same overhead with the time-switching method.


### 4.2. The Computational Complexity

The number of floating-point flops is often used to measure of the complexity of an algorithm. Here, a flop is defined as one multiplication or division of two floating-point numbers, and one flop has the computational complexityO(1). The addition and subtraction operations are neglected, since they are much quicker.Note that the matrix product X1X2 requires O(mnk) flops, where X1∈Cm×n, X2∈Cn×k; for an n×n matrix, both its inverse and eigendecomposition operations require O(n3) flops [45]; for an m×n matrix, its singular value decomposition (SVD) requires O(mn2) flops [45].For the precoder with hybrid CSI feedback, the problem of solving it has the similar form with P3 [26] and hence can be solved by the proposed method in [26]. Note that the complexity for searching u and λ in this method is O(22)=O(1) [46]. For each iteration (searching u and λ), the main computations lie in [26, Eq. (5) of Theorem 3.1] and have the complexity of O(NT2ND+ND2NT). Therefore, if the precoder with hybrid CSI feedback is solved by the method [26], the complexity is O(NT2ND+ND2NT).For the time-switching method, it needs to compute two precoders. The first precoder is for energy transmission and the second is for information transmission. For the first one, it mainly involves the calculation of the eigenvectors ofθTE and hence has the complexity of O(NT3). For the second one, it can be solved by the proposed method in [27] and this method has the complexity of O(NT3+ND3). For the above, it is concluded that the time-switching method has the complexity of O(NT3+ND3).For the proposed first method, suppose that, for Main Algorithm 1 and Subalgorithm 1 (see Algorithms2 and 1), the numbers of iterations required to reach the given accuracy are T11 and T12, respectively. Observe that the main computation of this method lies in (18) and has the complexity of O(NT3+NT2ND). Therefore, the complexity of the proposed first method is T11T12O(NT3+NT2ND).For the proposed second method, suppose that the number of iterations required to reach the given accuracy for Main Algorithm 2 (see Algorithm5) is T21. Observe that there are two loops in Subalgorithm 2.1 (see Algorithm 3), termed as the outer loop and inner loop. Suppose that the number of iterations for the outer loop is T22 and clearly 1≤T22≤NT. For the inner loop, assume that the average number of iterations required to calculate the power allocated to each eigenvector is T23. Observe that the main computations lie in Step (2) of Subalgorithm 2.1 (see Algorithm 3) and (21). With the above, it is easy to find that the overall complexity of this method is T21T23NTT22-T22T22-1/2OND3+ONT3. Note that, we find T22=1~2 and T23=5~30 usually in the simulation. The complexity of the proposed methods and a few existing methods is listed in Table 2.Table 2
Computational complexity to compute the precoder.
Precoding method Complexity The proposed first method T 11 T 12 O N T 3 + N T 2 N D.  The proposed second method T 21 T 23 N T T 22 - T 22 T 22 - 1 / 2 × O N D 3 + O N T 3  isotropic transmission No need to compute  the precoder with hybrid CSI feedback [31] O N T 2 N D + N D 2 N T  time switching method O N T 3 + N D 3


## 4.1. The Overhead

Table1 lists the overheads of the proposed methods with other methods. We mainly consider the overhead that is required to compute the precoder.Table 1
Overhead of computing the precoders.
Precoding method Overhead the proposed two methods Tx obtains the channel covariance matricesθTE,θTD,θRE,θRD and noise power σD2, by measure- ment over a period of time.  isotropic transmission No overhead is needed.  the precoder with hybrid CSI feedback [31] Tx obtains the energy channel covariance matricesθTE,θRE and noise power σD2, by measurement over a period of time; Tx also requires the instant CSI of the information channel.  time switching method It is the same with the proposed methods.Then, we make a brief review of the other three existing methods listed in Table1. The isotropic transmission refers to the case in which the transmitter sends NT data streams with equal power, i.e., the precoder matrix W=PTINT. The time-switching scheme refers to the case in which each transmission period is divided into two time slots, one for data and the other one for energy. Let β with 0≤β≤1 be the percentage of transmitting time for the data time slot. The achievable R-E region of time-switching scheme is given by(31)Ω′PT=⋃0≤β≤1J,εng:βElog⁡det⁡IND+HDQ1HDH≥J,trQ1≤PT,1-βtrQ2θTEtrθRE≥εng,trQ2≤PT,Q1⪰0,Q2⪰0.in which Q1 and Q2 are the transmission covariance matrices for data and energy transmission phases, respectively. For the precoder with hybrid CSI feedback [31], only statistical CSI is available between the transmitter and energy receiver, while instant CSI is available between the transmitter and data receiver. What is slightly different from the precoder [31], herein we do not consider the channel estimation error brought by limited-length pilots and noises, since it is not the focus of this article.From Table1, observe that the isotropic transmission does not need the overhead; however, it has the worst performance, which will be shown in the simulations. Compared with the precoder with hybrid CSI feedback, the proposed methods only require the statistical channel information and do not need the instant CSI of the information channel, which reduces the overall overhead significantly in the long run. In addition, the proposed methods have the same overhead with the time-switching method.


## 4.2. The Computational Complexity

The number of floating-point flops is often used to measure of the complexity of an algorithm. Here, a flop is defined as one multiplication or division of two floating-point numbers, and one flop has the computational complexityO(1). The addition and subtraction operations are neglected, since they are much quicker.Note that the matrix product X1X2 requires O(mnk) flops, where X1∈Cm×n, X2∈Cn×k; for an n×n matrix, both its inverse and eigendecomposition operations require O(n3) flops [45]; for an m×n matrix, its singular value decomposition (SVD) requires O(mn2) flops [45].For the precoder with hybrid CSI feedback, the problem of solving it has the similar form with P3 [26] and hence can be solved by the proposed method in [26]. Note that the complexity for searching u and λ in this method is O(22)=O(1) [46]. For each iteration (searching u and λ), the main computations lie in [26, Eq. (5) of Theorem 3.1] and have the complexity of O(NT2ND+ND2NT). Therefore, if the precoder with hybrid CSI feedback is solved by the method [26], the complexity is O(NT2ND+ND2NT).For the time-switching method, it needs to compute two precoders. The first precoder is for energy transmission and the second is for information transmission. For the first one, it mainly involves the calculation of the eigenvectors ofθTE and hence has the complexity of O(NT3). For the second one, it can be solved by the proposed method in [27] and this method has the complexity of O(NT3+ND3). For the above, it is concluded that the time-switching method has the complexity of O(NT3+ND3).For the proposed first method, suppose that, for Main Algorithm 1 and Subalgorithm 1 (see Algorithms2 and 1), the numbers of iterations required to reach the given accuracy are T11 and T12, respectively. Observe that the main computation of this method lies in (18) and has the complexity of O(NT3+NT2ND). Therefore, the complexity of the proposed first method is T11T12O(NT3+NT2ND).For the proposed second method, suppose that the number of iterations required to reach the given accuracy for Main Algorithm 2 (see Algorithm5) is T21. Observe that there are two loops in Subalgorithm 2.1 (see Algorithm 3), termed as the outer loop and inner loop. Suppose that the number of iterations for the outer loop is T22 and clearly 1≤T22≤NT. For the inner loop, assume that the average number of iterations required to calculate the power allocated to each eigenvector is T23. Observe that the main computations lie in Step (2) of Subalgorithm 2.1 (see Algorithm 3) and (21). With the above, it is easy to find that the overall complexity of this method is T21T23NTT22-T22T22-1/2OND3+ONT3. Note that, we find T22=1~2 and T23=5~30 usually in the simulation. The complexity of the proposed methods and a few existing methods is listed in Table 2.Table 2
Computational complexity to compute the precoder.
Precoding method Complexity The proposed first method T 11 T 12 O N T 3 + N T 2 N D.  The proposed second method T 21 T 23 N T T 22 - T 22 T 22 - 1 / 2 × O N D 3 + O N T 3  isotropic transmission No need to compute  the precoder with hybrid CSI feedback [31] O N T 2 N D + N D 2 N T  time switching method O N T 3 + N D 3


## 5. Simulation Results

Computer simulation is employed to evaluate the performance of the proposed precoders of solving the optimal precoder and compare a few other existing methods. A three-node MIMO WIPT system is considered, where all nodes have the same number of antennas. We adopt the exponential correlation model as the channel correlation matrix, with its (i,j)-th entry being ρi–j, in which the constant ρ is the correlation coefficient. Unless specified otherwise, for θTD, θTE, θRD, and θRE, their correlation coefficients are set to 0.1, 0.7, 0.2, and 0.3, respectively.


### 5.1. The Convergence of the Proposed Methods

Figure3 shows the convergence of the proposed Subalgorithm 2.1 (see Algorithm 3). Two cases with NT=ND=NE=2 and NT=ND=NE=3 are considered. For both cases, the system SNR is 5 dB, the energy threshold εng is set to 10, and the initial power matrix Λ(0)=I. As in Figure 3(a), observe that, after 10 iterations, the power matrix converges to about diag{0.64,0.26}. Similar behavior can be found in Figure 3(b), with the corresponding power matrix converging to diag{0.92,0.49,0.33}. These results have well demonstrated the convergence of Subalgorithm 2.2 (see Algorithm 4).Figure 3
The convergence of the Subalgorithm 2.1 (see Algorithm3).
(a)
Case 1 (b)
Case 2Figure4 plots the convergence of the proposed two precoding methods which correspond to Main Algorithms 1 and 2 (see Algorithms 2 and 5). In this figure, NT=ND=NE=2, εng=10, and the initial (λ,u) pairs are (0.1, 0.8) and (0, 0.5) for the first method and the second method, respectively. Two cases with SNR = 5 dB and SNR = 6 dB are considered. Observe that at SNR = 5 dB, the dual function g(λ,u) with the proposed first method decreases from 4.02 bit to 3.66 bit after 4 iterations. Further increasing iterations only marginally decrease g(λ,u). Furthermore, the values of the dual function g(λ,u) using the proposed two methods converge to the same minimum, i.e., about 3.66 bit. Similar behavior can be found at SNR = 6 dB.Figure 4
The convergence of the proposed two precoding methods.To sum up, the above results indicate that the proposed two methods are convergent; with the same system configuration, the solutions to problem P1 using two precoding methods are identical, which also demonstrates the validity of the proposed methods.


### 5.2. The R-E Trade-Offs with Different Transmission Schemes

Figure5 is a plot of the information transmission rate versus SNR with different energy thresholds. Three cases with εng=6, εng=10, and no energy constraint are considered. For all cases, NT=ND=NE=2. The information rate is solved by the proposed first method. In fact, both methods yield the same result, as will be illustrated later. Obviously, it is seen that the case with no energy constraint is the upper bound of the other two cases. When SNR is less than a threshold, the problem P1 has no solution. This is because that even allocating the total power to θTE’s eigenvector having maximum eigenvalue cannot meet the energy requirement. For instance, the SNR threshold for εng=6 is about 2.47 dB. This value can also be calculated by the first argument of Lemma 1. On the other front, at relatively low feasible SNR, there is a certain gap between the case with energy constraint and that with no energy constraint. This is due to the fact that the energy constraint decreases the achievable information rate. However, with increasing SNR, the gap becomes smaller and smaller until zero. This is due to the fact that the energy constraint becomes inactive gradually.Figure 5
The information transmission rate versus SNR, with different energy thresholds.Figure6 presents the R-E trade-offs with the proposed two precoding methods. Two cases with SNR = 5 dB and SNR = 6 dB are considered. For both cases, we set NT=ND=NE=2 and εng=10. Observe that, for each case, the trade-off curve with either the first or second method is horizontal at small energy threshold, and at a flexion point (A1 or A2), further increase of εng will result in the decrease of transmission rate. This is because that the information rate is nearly not affected by the energy constraint when εng is small and increasing εng will result in that the energy constraint becomes more and more active. There is also a cut-off point (B1 or B2), which corresponds to the maximum power, and exceeding the maximum power will result in no solution to P1. This is because the harvested energy is limited by the maximum transmission power. The flexion point and cut-off point can also be calculated by Lemma 1. In fact, the coordinates of two points obtained by Lemma 1 are accordance with those obtained by numerical method. Moreover, with the same system configuration, the trade-off curve with the first method nearly overlaps that with the second method and hence, the proposed two methods are cross-validated.Figure 6
The R-E trade-off comparison with the proposed two precoding methods.Figure7 compares several transmission schemes, including the proposed precoder, time-switching scheme, the isotropic transmission, and the precoder with hybrid CSI feedback [31]. Since the proposed two methods yield the same precoder, we may as well use one of them, say, the second one. Two scenarios including a 2×2 MIMO WIPT system and a 3×3 MIMO WIPT system are investigated, and the corresponding results are plotted in Figures 7(a) and 7(b), respectively. Note that, the correlation coefficients for θTD, θTE, θRD, and θRE are 0.3, 0.5, 0.2, and 0.3, respectively. In Figure 7(a), observe that the achievable R-E region with the proposed precoder contains that with the isotropic transmission or time-switching scheme. The R-E region of isotropic transmission is a rectangular, whose upper boundary is very close to that of Ω(PT). This is because that the coefficient for θTD is small and hence the isotropic transmission is nearly optimal for data transmission. The boundary of Ω′(PT) is simply a line by connecting the two points (Jmax,0) and (εng,max,0). This can be derived by ranging β from 0 to 1. Also observe that the R-E region of the scheme with hybrid CSI feedback is much broader than the proposed scheme. This is because that the former utilizes the instant CSI of the information channel, whereas the latter only has the statistical CSI of the information channel. That is, the increase of average rate is an exchange of additional overhead. Similar phenomena can be found in Figure 7(b). In addition, the optimal precoders with a few cases are listed in Table 3.Table 3
The optimal covariance matrices and precoders with different cases.
Case Optimal covariance matrix Optimal Precoder 2 × 2 system,SNR = 5 dB, εng=10 1.58 1.31 1.31 1.58 1.26 1.14 1.14 1.26  2 × 2 system,SNR = 6 dB, εng=10 1.99 0.72 0.72 1.99 1.41 0.85 0.84 1.41  3 × 3 system, SNR = 5 dB, εng=15 1.050.530.360.531.070.530.360.531.05 1.02 0.73 0.60 0.73 1.03 0.73 0.60 0.73 1.02Figure 7
The R-E trade-offs with several transmission schemes.
(a)
A2×2 MIMO WIPT system (b)
A3×3 MIMO WIPT systemTo sum up, the results in Figure5 imply that the energy constraint decreases the achievable information rate at low feasible SNR evidently and it becomes gradually inactive with increasing SNR. The results in Figure 6 indicate that the proposed two methods can be cross-validated and the precalculation of the flexion point and cut-off point by Lemma 1 will help to find a rough R-E boundary curve. The results in Figure 7 demonstrate that the proposed precoder is superior to both the isotropic transmission and time-switching scheme in terms of the R-E region; however, it is inferior to the precoder with hybrid CSI feedback, since the latter utilizes more CSI.


### 5.3. The Effects of the Number of Transmitter Antennas on the R-E Trade-Off

In Figures8 and 9, NE=ND=2 and SNR = 5 dB. The figures investigate the behavior of the proposed algorithms when the number of transmit antennas is large. Observe that, in Figure 8, we find that increasing NT results in much broader R-E regions. The right boundary of the R-E region, whose horizontal coordinate is the maximum harvested energy εng,max, has a limit corresponding to the limit in Figure 9(a). In Figure 9(a), increasing NT results in an εng,max limit around 35.7 joules/s. On the other hand, the maximum achievable rate of the R-E region is characterized by Figure 9(b). In Figure 9(b), for small NT, increasing NT makes the maximum rate increase rapidly. For large NT, e.g., 64, further increase of NT can only marginal increase the maximum rate. However, we cannot conclude that the maximum rate has a limit because the transmitter utilizes the CSI of information channel.Figure 8
The effect of transmitter antenna on the rate-energy regions.Figure 9
The effect of transmitter antenna number,NT, on the maximum information rate Jmax and harvested energy εng,max.
(a)
ε n g , m a x versus NT (b)
J m a x versus NTRecall that the capacity or the maximum rate for an open-loop point-to-point MIMO system has a limit when the number of transmitter antennas is large [47]. However, it does not hold for a closed-loop MIMO system because the transmitter has the CSI.


## 5.1. The Convergence of the Proposed Methods

Figure3 shows the convergence of the proposed Subalgorithm 2.1 (see Algorithm 3). Two cases with NT=ND=NE=2 and NT=ND=NE=3 are considered. For both cases, the system SNR is 5 dB, the energy threshold εng is set to 10, and the initial power matrix Λ(0)=I. As in Figure 3(a), observe that, after 10 iterations, the power matrix converges to about diag{0.64,0.26}. Similar behavior can be found in Figure 3(b), with the corresponding power matrix converging to diag{0.92,0.49,0.33}. These results have well demonstrated the convergence of Subalgorithm 2.2 (see Algorithm 4).Figure 3
The convergence of the Subalgorithm 2.1 (see Algorithm3).
(a)
Case 1 (b)
Case 2Figure4 plots the convergence of the proposed two precoding methods which correspond to Main Algorithms 1 and 2 (see Algorithms 2 and 5). In this figure, NT=ND=NE=2, εng=10, and the initial (λ,u) pairs are (0.1, 0.8) and (0, 0.5) for the first method and the second method, respectively. Two cases with SNR = 5 dB and SNR = 6 dB are considered. Observe that at SNR = 5 dB, the dual function g(λ,u) with the proposed first method decreases from 4.02 bit to 3.66 bit after 4 iterations. Further increasing iterations only marginally decrease g(λ,u). Furthermore, the values of the dual function g(λ,u) using the proposed two methods converge to the same minimum, i.e., about 3.66 bit. Similar behavior can be found at SNR = 6 dB.Figure 4
The convergence of the proposed two precoding methods.To sum up, the above results indicate that the proposed two methods are convergent; with the same system configuration, the solutions to problem P1 using two precoding methods are identical, which also demonstrates the validity of the proposed methods.


## 5.2. The R-E Trade-Offs with Different Transmission Schemes

Figure5 is a plot of the information transmission rate versus SNR with different energy thresholds. Three cases with εng=6, εng=10, and no energy constraint are considered. For all cases, NT=ND=NE=2. The information rate is solved by the proposed first method. In fact, both methods yield the same result, as will be illustrated later. Obviously, it is seen that the case with no energy constraint is the upper bound of the other two cases. When SNR is less than a threshold, the problem P1 has no solution. This is because that even allocating the total power to θTE’s eigenvector having maximum eigenvalue cannot meet the energy requirement. For instance, the SNR threshold for εng=6 is about 2.47 dB. This value can also be calculated by the first argument of Lemma 1. On the other front, at relatively low feasible SNR, there is a certain gap between the case with energy constraint and that with no energy constraint. This is due to the fact that the energy constraint decreases the achievable information rate. However, with increasing SNR, the gap becomes smaller and smaller until zero. This is due to the fact that the energy constraint becomes inactive gradually.Figure 5
The information transmission rate versus SNR, with different energy thresholds.Figure6 presents the R-E trade-offs with the proposed two precoding methods. Two cases with SNR = 5 dB and SNR = 6 dB are considered. For both cases, we set NT=ND=NE=2 and εng=10. Observe that, for each case, the trade-off curve with either the first or second method is horizontal at small energy threshold, and at a flexion point (A1 or A2), further increase of εng will result in the decrease of transmission rate. This is because that the information rate is nearly not affected by the energy constraint when εng is small and increasing εng will result in that the energy constraint becomes more and more active. There is also a cut-off point (B1 or B2), which corresponds to the maximum power, and exceeding the maximum power will result in no solution to P1. This is because the harvested energy is limited by the maximum transmission power. The flexion point and cut-off point can also be calculated by Lemma 1. In fact, the coordinates of two points obtained by Lemma 1 are accordance with those obtained by numerical method. Moreover, with the same system configuration, the trade-off curve with the first method nearly overlaps that with the second method and hence, the proposed two methods are cross-validated.Figure 6
The R-E trade-off comparison with the proposed two precoding methods.Figure7 compares several transmission schemes, including the proposed precoder, time-switching scheme, the isotropic transmission, and the precoder with hybrid CSI feedback [31]. Since the proposed two methods yield the same precoder, we may as well use one of them, say, the second one. Two scenarios including a 2×2 MIMO WIPT system and a 3×3 MIMO WIPT system are investigated, and the corresponding results are plotted in Figures 7(a) and 7(b), respectively. Note that, the correlation coefficients for θTD, θTE, θRD, and θRE are 0.3, 0.5, 0.2, and 0.3, respectively. In Figure 7(a), observe that the achievable R-E region with the proposed precoder contains that with the isotropic transmission or time-switching scheme. The R-E region of isotropic transmission is a rectangular, whose upper boundary is very close to that of Ω(PT). This is because that the coefficient for θTD is small and hence the isotropic transmission is nearly optimal for data transmission. The boundary of Ω′(PT) is simply a line by connecting the two points (Jmax,0) and (εng,max,0). This can be derived by ranging β from 0 to 1. Also observe that the R-E region of the scheme with hybrid CSI feedback is much broader than the proposed scheme. This is because that the former utilizes the instant CSI of the information channel, whereas the latter only has the statistical CSI of the information channel. That is, the increase of average rate is an exchange of additional overhead. Similar phenomena can be found in Figure 7(b). In addition, the optimal precoders with a few cases are listed in Table 3.Table 3
The optimal covariance matrices and precoders with different cases.
Case Optimal covariance matrix Optimal Precoder 2 × 2 system,SNR = 5 dB, εng=10 1.58 1.31 1.31 1.58 1.26 1.14 1.14 1.26  2 × 2 system,SNR = 6 dB, εng=10 1.99 0.72 0.72 1.99 1.41 0.85 0.84 1.41  3 × 3 system, SNR = 5 dB, εng=15 1.050.530.360.531.070.530.360.531.05 1.02 0.73 0.60 0.73 1.03 0.73 0.60 0.73 1.02Figure 7
The R-E trade-offs with several transmission schemes.
(a)
A2×2 MIMO WIPT system (b)
A3×3 MIMO WIPT systemTo sum up, the results in Figure5 imply that the energy constraint decreases the achievable information rate at low feasible SNR evidently and it becomes gradually inactive with increasing SNR. The results in Figure 6 indicate that the proposed two methods can be cross-validated and the precalculation of the flexion point and cut-off point by Lemma 1 will help to find a rough R-E boundary curve. The results in Figure 7 demonstrate that the proposed precoder is superior to both the isotropic transmission and time-switching scheme in terms of the R-E region; however, it is inferior to the precoder with hybrid CSI feedback, since the latter utilizes more CSI.


## 5.3. The Effects of the Number of Transmitter Antennas on the R-E Trade-Off

In Figures8 and 9, NE=ND=2 and SNR = 5 dB. The figures investigate the behavior of the proposed algorithms when the number of transmit antennas is large. Observe that, in Figure 8, we find that increasing NT results in much broader R-E regions. The right boundary of the R-E region, whose horizontal coordinate is the maximum harvested energy εng,max, has a limit corresponding to the limit in Figure 9(a). In Figure 9(a), increasing NT results in an εng,max limit around 35.7 joules/s. On the other hand, the maximum achievable rate of the R-E region is characterized by Figure 9(b). In Figure 9(b), for small NT, increasing NT makes the maximum rate increase rapidly. For large NT, e.g., 64, further increase of NT can only marginal increase the maximum rate. However, we cannot conclude that the maximum rate has a limit because the transmitter utilizes the CSI of information channel.Figure 8
The effect of transmitter antenna on the rate-energy regions.Figure 9
The effect of transmitter antenna number,NT, on the maximum information rate Jmax and harvested energy εng,max.
(a)
ε n g , m a x versus NT (b)
J m a x versus NTRecall that the capacity or the maximum rate for an open-loop point-to-point MIMO system has a limit when the number of transmitter antennas is large [47]. However, it does not hold for a closed-loop MIMO system because the transmitter has the CSI.


## 6. Conclusions

For a typical three-node wireless MIMO information and energy transmission system, the precoder design with only covariance feedback is studied. By using the convex optimization technique, and after a few transformations, we transform the precoder design problem into a dual one. The dual problem is proved to have no duality gap bySlater’s condition. Then, two algorithms have been proposed to solve this problem. The first one decomposes the transmission covariance and emphasizes numerically searching the optimal precoder, while the second one emphasizes exploiting the structure of the covariance, which is presented by Proposition 4. Their convergence, complexity, etc. have also been addressed by theoretical analysis or computer simulation.On the other hand, the R-E region is often used to measure the WIPT system performance. We have exploited the key features of the R-E boundary curve by Lemma1. In fact, the precalculation of the flexion point and cut-off point by this lemma will help to find a rough R-E boundary curve. Furthermore, we compare the proposed methods with a few existing transmission schemes. Simulation results show that the proposed precoder is superior to the isotropic transmission and time-switching scheme and it has the broadest R-E region compared with the other two regions. However, the proposed precoder is inferior to the precoder with hybrid CSI feedback, since the latter one utilizes more CSI.The proposed precoding methods and above results can be a reference for practical system implementation in the future. We will further improve the system performance by incorporating the other wireless transmission techniques [48–52].



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*Source: 8472186-2018-06-14.xml*