arXiv:1705.05212v4 [cs.IT] 1 Dec 2017 emi:[email protected]). (e-mail: uiainEgneig nvriyo oaua r Lanka Sri Moratuwa, of University [email protected]). Engineering, munication nvriyo ehooyadDsg,Snaoe(e-mail: Singapore Design, and Technology of { Conf Communications University Global au IEEE Corresponding - the [1] in ( (61750110529). at Abeywickrama.) 2016. 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ER broadcasts a single bit to the ET indicating whether the RSSI current received energy level is higher or lower than the previous, and the ET makes phase perturbations based on the feedback of the ER to obtain a satisfactory beamforming vector for the WPB stage. This means that by utilizing the ... feedback bits, the ET fine tunes its transmit beamforming ER vector, and obtains a more refined estimate of the channel. [13] .. could be considered to be the most related work to our work and it proposes the following methodology. In the training stage, firstly, each antenna is individually activated, and then, antennas are pairwise activated. The respective RSSI value for Energy each activation is fed back by the ER to the ET. Next, they Trans m itter utilize the gathered RSSI values to estimate the channel. Our proposed scheme is significantly different to [6–10, 13], Fig. 1. System model. and our contributions and the paper organization can be summarized as follows. We focus on a system consisting of picking the candidate that provides the best energy transfer. K antennas at the ET, and a single antenna at the ER. We In this paper, we propose a method, that allows us to resolve start the analysis by assuming K =2. Under this assumption, the ambiguity without ascertaining any further RSSI feedback the proposed training stage consists of N time slots. In each from the receiver. time slot, the ET will transmit using a beamforming vector In section V, we show how our results can be extended from a pre-defined codebook of size N. The ER feeds back for a single-user WET system consisting of K > 2 antennas the analog RSSI value corresponding to each beamforming at the ET. Then, we focus on selecting N. Although larger vector, i.e., the ET will receive N RSSI feedback values at N yields a higher channel estimation precision, for a given the end of the training stage. These N feedback values are time period T , a larger N will consume a larger portion of utilized to set the beamforming vector for the WPB stage. T , which will reduce the time for WPB. Therefore, larger More precisely, the feedback values are utilized to estimate N may lead to a reduction in the total transferred energy. the phase difference of the two channels between the ET and In Section VI, we present bounds for the optimal value of N ER, and this estimate is utilized in the WPB stage. To this end, that maximizes the system performance in terms of the energy the ET equally splits the power among the transmit antennas, transfer during the WPB stage. In Section VII, we validate our and pre-compensates channel phase shifts such that the signals analytical results numerically, while providing useful insights are coherently added up at the ER regardless of the channel into the system performance. Furthermore, Section VIII shows magnitudes. These ideas are introduced in Section II. that the proposed methodology can be in fact implemented In Section III, we focus on defining the aforementioned on hardware, and the experimental setup is used to further pre-defined codebook. To this end, we employ a Cramer- validate our results. Experimental validation is not common Rao lower bound (CRLB) analysis, and define the codebook in the related works, and can be highlighted as another major such that the estimator of the phase difference between the contribution of this paper. Both Section VII and Section two channels of interest achieves the CRLB, which is the VIII show that our proposed method will achieve impressive best performance that an unbiased estimator can achieve. On results, and will provide performance improvements compared top of providing a solid theoretical basis for the selection of to directly related works in the literature. It should be also the beamforming vectors for the training stage, this approach noted that the proposed methodology can be used for any also allows us to simplify derived results significantly, and application of beamforming in which processing capabilities most importantly, it leads to achieving impressive results of the receiver are limited. Section IX concludes the paper. in the WET. The defined codebook gives the ET sufficient information to obtain the N RSSI feedback values. In Section II. SYSTEM MODEL AND PROBLEM SETUP IV, we discuss how the feedback values can be utilized to We consider a MISO channel for WET. An ET consisting of set the beamforming vector for the WPB stage, through a K 2 antennas delivers energy to an ER consisting of a single maximum likelihood analysis. Our analysis takes the effect antenna≥ over a wireless medium, see Fig. 1. The transmit of noise on the measurements into account unlike [13]. The signal at the ET is given by x = ws, where w CK×1 results that we obtain are remarkably simple, requires minimal denotes the complex K-by-1 beamforming vector and∈ s de- processing, and can be easily implemented at the ET. Also, the notes the transmit symbol, which is independent of w, and results are general such that they will hold for all well known has zero-mean and unit variance (i.e., E( s 2)=1). We have fading models. However, it should be noted that the estimate, dropped the time index for notational simplicity.| | The transmit † † which is a phase value, has an ambiguity due to the use of covariance matrix is given by Cxx = E(xx ) = E(ww ), −1 tan , and hence can take two values. In [13], a similar phase where denotes the conjugate transpose. Cxx is positive semi- ambiguity is resolved by ascertaining further RSSI feedback definite,† thus the number of energy beams d can be obtained (four values) for the candidate phase values from the ER, and from the rank of Cxx [15], i.e., d = rank(Cxx) . It is assumed that the maximum transmit sum-power constraint at the ET is τ T-τ 2 P > 0. Therefore, we have E( x ) = tr(Cxx) P , where tr( ) denotes the trace of a squarek k matrix, and ≤denotes the 1 n N Wireless Power Beamforming (WPB) · k·k Euclidean norm. b b b ⊤ 1 } n N jδ1 jδK Let h = h e ,..., hK e represent the complex | 1| | | MISO channel vector between the ET and the ER. Further, Codebook B we consider a quasi-static block-fading channel model and = 2 a block-based energy transmission, where it is assumed that Fig. 2. The two-phase transmission protocol when K . the wireless channel remains constant over each transmission block. The transmission block has a length T > 0 (in practice, can be set without any loss of optimality if full channel state T is upper bounded by the channel coherence time). The information (CSI) is available at the ET. In practice, full CSI received energy (or RSSI) at the ER can be written as at the ET can be achieved by estimating the channel at the † R= ξ(h Cxxh), (1) ER, and feeding back the channel information to the ET. However, we are particularly focusing on applications with where ξ denotes the conversion efficiency of the energy tight energy constraints at the ER. Thus, such an estimation harvester [15]. process may become infeasible as channel estimation involves Our main focus is to design a single energy beam to analog to digital conversion and baseline processing, which maximize the received energy at the ER, so that the harvested require significant energy. Therefore, we focus on introducing energy is maximized at the ER. To this end, we focus on the a more energy friendly method of selecting the beamforming following optimization problem: vector, by only considering RSSI values that are fed back from † the ER to the ET. It should be noted that the feedback takes maximize ξ(h Cxxh) Cxx0 the form of real values, and RSSI values are readily available subject to tr(Cxx) P, rank(Cxx)=1. in most receiver circuits. We will first present the proposed ≤ scheme for the special case of K =2 to draw useful insights, Since rank(C ) = 1, d = 1. The solution for this xx and then, in Section V, we will extend the proposed scheme optimization problem is C⋆ = P vv†, where v denotes xx to the general case of . the dominant eigenvector of the normalized MISO channel K > 2 H H hh† Under the assumption of K = 2, the proposed scheme covariance matrix [15], i.e., = † , and F denotes khh kF k·k is as follows. The scheme consists of a training stage and the Frobenius norm. a wireless power beamforming (WPB) stage. As we have We employ equal gain transmit (EGT) beamforming for the depicted in Fig. 2, the training stage is further divided into ⋆ WET. Thus, the optimal transmit signal can be written as x = N mini slots. We define a codebook B = [b1 ... bN ] that ⋆ √P vs, which implies an optimal beamforming vector w = includes N beamforming vectors to be used in each mini slot √P v. To this end, in the training stage. Let = 1,...,N . In each mini slot N { } 1 −jφ2 −jφ ⊤ n , the ET simultaneously activates both its antennas and v = 1,e ,...,e K , (2) ∈ N B √K transmits using beamforming vector bn . This means, the B ∈   n-th element of is used in n-th mini slot. Let Rn denote the where φk = δk δ , k 2,...,K . In practice, each − 1 ∈ { } RSSI value at the ER during mini slot n . The ER will transmit antenna has its own power amplifier, which operates N feedback = R to the ET, which∈ means, N at the end properly only when the transmit power is below a pre-designed n n=1 of the trainingR { period} τ, the ET will have N RSSI feedback threshold. Therefore, there are practical difficulties in imple- values corresponding to each element in B. menting maximum ratio transmit (MRT) beamforming, where Moreover, consider the nth element of B to take the form the transmit power in some antennas may theoretically exceed ⊤ of b = P 1 ejθn , where θ is the n-th element of these threshold values. Because of this reason, although MRT n 2 n Θ. Θ is a set that includes phase values between 0 and is superior, still, EGT beamforming, where the ET equally q   splits the power among all transmit antennas, is a preferred 2π. For implementation convenience, Θ is predetermined and method in practice [16]. In this paper, we assume that the pre- does not depend on the feedback values. Further, we shall designed transmit power threshold is equal among antennas, employ estimation theory and the concept of the CRLB in and we transmit at that power. It should be noted that our order to define Θ. At the end of the training stage, the ET results can be easily extended to a case where these threshold will determine the beamforming vector wWPB to be used for values are not equal among antennas as well. More specifically, the WPB stage. The ER does not feed back in the WPB stage, the results can be extended to general sum-power or per- and typically, this stage is longer than the training stage to antenna power constraints, but the power allocation among reduce the overhead incurred in the WPB stage. From (2), it antennas will be static, and not dynamic as in a case where is not hard to see that the optimal beamforming vector should P −jφ2 ⊤ the ET employs MRT beamforming. take the form of wWPB = 2 1 e . Our challenge is From (2), we can see that the optimality of the wireless to estimate φ2 by only utilizingq . K R  energy transfer depends only on φk , and these values We denote the RSSI value at the ER during mini slot n { }k=2 ∈ N by Rn, and it is written as Gaussian assumption made on the random variable in (3), we are minimizing the largest or the worst case CRLB. R = ξ(h†C h)+ z . (3) n xx n Using (3), the N-by-1 vector representing N RSSI obser- Note that due to noise, the RSSI value will change from one vations can be written as mini slot to the other. We use random variable zn to capture the R effect of noise on Rn. More specifically, zn captures the effect = xϕ + z, (5) of all noise related to the measurement process such as noise in the channel, circuit, antenna matching network and rectifier. where xϕ is a N-by-1 vector of which the nth element takes We assume that the channel is slowly varying so that during the form of α + β cos(θn + φ2). Since xϕ is independent of the training stage and the subsequent beamforming, h can be z, R in (5) is distributed according to a multivariate Gaussian R C C 2I considered to be unknown, but non varying (fixed). Therefore, distribution, i.e., (xϕ, zz), where zz = σ N, and ∼ N IN the randomness in (3) is caused only by zn. For tractability, is the N-by-N identity matrix. We will specifically focus ⊤ and without loss of generality, we assume z = [z1,...,zN ] on φ2, which is the main parameter of interest, and derive to be an i.i.d. Gaussian random vector with zero mean and the CRLB of its estimator. Then, we will focus on finding the N variance 2. set of values θn that will minimize the derived CRLB. σ { }n=1 Under the above assumptions, for K = 2, i.e., a channel We will start by deriving the FIM of ϕ, which is formally jδ1 jδ2 ⊤ presented in the following Lemma. vector h = h1 e h2 e . we have | | | | Lemma 1: The FIM of ϕ is given by jθn  C 1 1 e xx = jθ . 2 e− n 1 N N A N D   n=1 n n=1 n Thus, (3) can be simplified as 1  N PN 2 PN  FIMϕ(R)= A A A D , σ2 n=1 n n=1 n n=1 n n ξP 2 2 Rn = h1 + h2 +2 h1 h2 cos(θn + δ2 δ1) + zn   PN PN P N  4 | | | | | || | −  D A D D2     n=1 n n=1 n n n=1 n  = α + β cos(θn + φ2)+ zn, (4)   P P P ξP 2 2 ξP where An = cos(θn + φ2) and Dn = β sin(θn + φ2). where α = ( h1 + h2 ), β = h1 h2 , and φ2 = δ2 − 4 | | | | 2 | || | − Proof: See Appendix A. δ1. Our goal is to estimate φ2. It can be seen from (4) that Rn We will first use the FIM to obtain some useful insights depends on three unknown parameters α, β, and φ2. Hence, ⊤ on the selection of . These insights can be drawn from the the parameter vector can be written as ϕ = [α β φ2] . N To implement the proposed method in this paper, we should determinant of the FIM. To this end, for N = 1 and N = R R first define Θ. In the next section, we define Θ by performing 2, det(FIMϕ( )) = 0, which implies that FIMϕ( ) is not a CRLB analysis on the parameter vector. Then, Θ will be invertible for these two cases. Since the CRLB of φ2 is the 3rd R used to define the codebook B, and in Section IV, we discuss diagonal element of the inverse of FIMϕ( ), we can conclude how the RSSI feedback values associated to the beamforming that the CRLB is unbounded when N < 3. Therefore, the estimation variance of φ is unbounded when N < 3, implying vectors in B can be used to estimate φ2 through a maximum 2 likelihood analysis. that we need at least 3 RSSI values fed back to the ET to make the proposed scheme work. On the other hand, when N 3, ≥ III. CRAMER-RAO LOWER BOUND ANALYSIS we have

The CRLB is directly related to the accuracy of an estima- N−2 N−1 N 2 tion process. More precisely, the CRLB gives a lower bound det(FIMϕ(R)) = β ∆i,j,k, on the variance of an unbiased estimator. To this end, suppose i=1 j=i+1 k=j+1 ⊤ X X X we wish to estimate the parameter vector ϕ = [α β φ2] . ⊤ The unbiased estimator of ϕ is denoted by ϕˆ = [ˆα βˆ φˆ2] , where where E ϕˆ = ϕ. The variance of the unbiased estimator 2 { } θi θj θj θk θk θi var(ϕˆ ) is lower-bounded by the CRLB of ϕ, which is denoted ∆i,j,k = 4 sin − sin − sin − , by CRLB , i.e., var(ϕˆ) > CRLB . Moreover, CRLB can 2 2 2 ϕ ϕ ϕ h      i be obtained by the inverse of FIMϕ, which is the Fisher which will be non zero if Θ consists of N distinct phase information matrix (FIM) of ϕ. Since no other unbiased values1. Thus, if Θ is selected accordingly, the CRLB will estimator of ϕ can achieve a variance smaller than the exist for N 3. Along these ideas, we will use FIMϕ(R) to CRLB, the CRLB is the best performance that an unbiased derive the CRLB≥ of φ, and it is formally presented through estimator can achieve. Hence, our motivation is to select Θ the following lemma. in a manner that the estimator achieves the CRLB, and its variance is minimized. Also note that as discussed in Appendix 1Obtaining this expression analytically is straightforwardly done by com- A, the Gaussian distribution leads to the worst-case CRLB puting the determinant of a 3-by-3 matrix. However, due to being tedious, it performance for our estimation problem. Therefore, due to the is omitted to avoid any deviation from the main focus of the paper. Lemma 2: For N 3, if Θ consists of N distinct phase 2.5 ≥ Θ values, the CRLB of parameter φ2 exists, and it is given by CRLB, and φ2 randomly generated Θ fi N N MCRLB, set according to De nition 1 −1 2 2 2 σ cos(θi + φ ) cos(θj + φ ) 2 − 2 i=1 j=i+1 CRLBφ = X X h i .

N−2 N−1 N φ 2 1.5 β ∆i,j,k i=1 j=i+1 k=j+1 X X X / MCRLB Proof: See Appendix A. φ N 1 Having derived the CRLB of φ , our goal is to find θn CRLB 2 { }n=1 that will minimize the derived CRLB for any given φ2. However, it should be noted that the CRLBφ is a function N 0.5 of φ . Therefore, the CRLB minimizing θn will be 2 { }n=1 functions of φ2 as well. This will lead to implementation difficulties as Θ is supposed to be predefined. Therefore, we 0 0 500 1000 1500 resort to averaging out the effect of φ2. To this end, we assume Realizations φ2 to be uniformly distributed in (0, 2π], and computing the expectation over φ2 leads to the modified Cramer-Rao lower Fig. 3. Comparison between CRLB and MCRLB. bound (MCRLB) [17]. The MCRLB is formally presented through the following lemma, and the proof is skipped since its trivial. Definition 1: Θ is a set of phase values between 0 and 2π, and it is defined to be Θ= θ ,...,θ , where θ = 2(n−1)π Lemma 3: The MCRLB of parameter φ2 is given by 1 N n N for n . { } E ∈ N MCRLBφ = φ[CRLBφ] The intuition behind this definition is that getting RSSI N−1 N values with the maximum spatial diversity provides us the 2 σ 1 cos(θi θj ) − − best estimate. Using the phase values in Θ, N RSSI feedback i=1 j=i+1 = X X h i . (6) values can be obtained. It should be stressed that although N−2 N−1 N our initial goal was minimizing the CRLB, we ended up 2 β ∆i,j,k minimizing the MCRLB, which is obviously not the same i=1 j=i+1 k j X X =X+1 thing. As shown in [18], sometimes, depending on the aver- After obtaining the MCRLB, our goal shifts to finding aging, minimizing the MCRLB might lead to inferior results. N Therefore, in order to check the effectiveness of the MCRLB the MCRLB minimizing θn n=1. Determining the MCRLB N { } for our application, a simple test was carried out, and the minimizing θ analytically for a general case is not n n=1 results are illustrated in Fig. 3. In this test, for a predetermined straightforward{ } due to the complexity of (6). To develop N, we randomly generated θ N and φ assuming they insights, we will first focus on the N = 3 case and derive n n=1 2 are uniformly distributed between{ } 0 and 2π, and evaluated the the MCRLB minimizing θ ,θ ,θ . To this end, without 1 2 3 CRLB in Lemma 2. As shown in the figure, this was done any loss of generality, we{ assume θ} to be zero and θ and 1 2 for 1500 realizations. Then, we evaluated the MCRLB for θ are set relative to θ . Then, we repeat the process for 3 1 the same N, and θ N selected according to Θ defined N =4. From these two derivations, we can observe a pattern n n=1 in Definition 1. The{ comparison} is presented in Fig. 3, and in the MCRLB minimizing θ values, and we define Θ by φ n it can be seen that the MCRLB with Θ defined according making use of this pattern. In Section VII, through numerical to Definition 1 is a very reasonable approximation for the evaluations, we validate the selection of Θ for arbitrary values lower bound of the CRLB. Therefore, although minimizing of N. the MCRLB instead of the CRLB is suboptimal, the loss of Lemma 4: Let θ = 0. For N = 3, Θ = 0, 2π/3, 4π/3 1 { } optimality is negligibly small. minimizes MCRLBφ, and the corresponding minimum value 2 2σ Having obtained the feedback values at the ET, the next is 3β2 . For N = 4, Θ = 0,π/2, π, 3π/2 minimizes 2 question is how these feedback values can be used to estimate { } 2σ MCRLBφ, and the corresponding minimum value is 4β2 . the phase difference between the two channels. This, question Proof: See Appendix A. is addressed in the next section. It is interesting to note that in both cases, the phase values in Θ are equally spaced over [0 2π). For an example, when IV. ESTIMATION OF THE CHANNEL PHASE DIFFERENCE φ2 N =3, θ1 θ2 = θ2 θ3 = θ3 θ1 =2π/3. When N =4, the phase| difference− | | between− | adjacent| − elements| in the set turns A. Estimating φ2 in a Noiseless Environment out to be 2π/4. Also, by observing this pattern,we can expect We will first look at a simplified scenario similar to [13] 2 2σ the minimum MCRLBφ to behave like Nβ2 with N. To this by neglecting the effect of noise. If there is no noise in the end, we will define Θ for N elements as follows. network, we have Rn = α + β cos(θn + φ ) for n . If 2 ∈ N N =3, we can simply calculate φ2 by solving three simulta- It is not hard to see that to obtain the solution of φ2, we have neous equations, after obtaining three RSSI feedback values. to first estimate α and β, and these non-essential parameters The result is formally presented in the following theorem and are referred to as nuisance parameters [19]. However, thanks this value of φ2 should intuitively give satisfactory results in to the way we have defined Θ, we can obtain an ML estimate low noise environments. The proof is skipped as it is trivial. of φ2 without estimating the nuisance parameters. These ideas Theorem 1: In a noiseless environment, for N =3, and Θ are formally presented in the following theorem. defined according to Definition 1, the estimate of the phase Theorem 2: For a sample of N i.i.d. RSSI observations, φ2 difference between the two channels between the ET and the can be estimated by ER is given by N √3λ Rn sin θn φˆ = tan−1 2,3 , (7) − 2 −1  n=1  λ2,1 + λ3,1 φˆ = tan , (10) ! 2 NX   where λi,j = Ri Rj for i, j 1,2,3 .  Rn cos θn  − ˆ ∈ { }   It should be noted that φ2 has an ambiguity due to the  n=1  −1 ˆ ˆ ˆ  X  use of tan , and φ2 can take two values φ2,1 and φ2,2, 2(n−1)π ˆ ˆ where θn = for n . such that φ2,2 = φ2,1 π. In [13], the ambiguity is resolved N ∈ N by introducing an ambiguity− resolution stage, right after the Proof: See Appendix B. training stage. In the ambiguity resolution stage, the ET We can observe that tan(φ2) is the ratio between two weighted sums of the same set of RSSI values. The i-th RSSI sequentially beamforms using each candidate value of φˆ2, and obtains the respective RSSI values from the ER through value in the denominator is weighted by the cosine of an angle, feedback. Then, the ET picks the candidate that provides the i.e., cos(θi), where as in the numerator, the same RSSI value best energy transfer to set the beamforming vector for the is weighted by the cosine of the same angle, but shifted by 90 degrees, i.e., π . Setting these angles according WPB stage. It should also be noted that [13] requires the ET cos( 2 + θi) to acquire four more feedback values to resolve the ambiguity to Definition 1 gives us the best estimate of tan(φ2), which as the phase difference is given as cos−1. In this paper, we leads to the best estimate of φ2. Note that the result in Theorem propose a more energy efficient method, that allows us to 2 is easy to calculate, requires minimal processing, and can resolve the ambiguity without ascertaining any further RSSI be easily implemented at the ET. We should stress that the feedback values. This will be discussed later in this section. simplicity of the result was mainly possible due to the CRLB analysis performed in Section III to define Θ. However, it B. Estimating φ in Noisy Environments 2 should be noted that similar to the noiseless case, φ2 has an Now, we will focus on the more general scenario. Firstly, ambiguity due to the use of tan−1, and next, we will discuss we will present the following auxiliary results, that will be how this can be resolved. directly used in the proofs of the main results. 2(n−1)π C. Resolving the ambiguity of the estimate of Lemma 5: Let θn = for n . Then, φ2 N ∈ N N N N We propose a method of selecting the correct estimate of φ2 and resolving the ambiguity without ascertaining any further sin(θn+φ )= sin [2(θn + φ )] = cos(θn+φ ) 2 2 2 RSSI feedback values from the ER. This idea is formally n=1 n=1 n=1 X X N X presented through the following theorem. ˆ ˆ = cos [2(θn + φ2)]=0. Theorem 3: Let φ2,1 and φ2,2 be the possible solutions for n=1 the estimate of φ2, where φˆ2,2 = φˆ2,1 π. Then, the RSSI X ⋆ − Proof: See Appendix B. maximizing solution φ2 is given by Based on the assumption that the effect of noise is i.i.d. N Gaussian, estimating becomes a classical parameter esti- φ2 φˆ , if Rn cos(θn + φˆ , ) > 0 φˆ⋆ = 2 1 2 1 , (11) mation problem. Thus, a maximum likelihood estimate of φ2 2  n=1  X can be obtained by finding the value of φ2 that minimizes φˆ2,2 otherwise N 2 2(n−1)π E , Rn (α + β cos(θn + φ )) . (8) where θn = for n . − 2 N ∈ N n=1 Proof: See Appendix B. X h i Differentiating E with respect to φ2, and setting it to zero Again we should stress that this simplicity in the ambiguity gives us resolving process was made possible due to the methodology we have followed in defining Θ. The simplicity in our results N N can be used to further reduce the amount of feedback required Rn sin (θn + φ2)= α sin (θn + φ2) to make the proposed scheme work. This reduction will be n =1 n=1 (9) directly proportional to the resources that you have at the ER. X X N β For an example, using the expression in Theorem 2, the ER + sin [2(θn + φ2)]. 2 can calculate tan φˆ2 at the ER, and feedback this value instead n=1 X τ ()K-1 T-τ()K-1 (2), the beamforming vector for the WPB stage can be set as ⊤ P −jφˆ2 −jφˆK 1st (k -1)th (K-1 )th wWPB = K 1,e ,...,e . Wireless Power Beamforming (WPB) slot slot slot It shouldq beh noted that the methodi that we have proposed K τ for estimating φk is a heuristic scheme that activates a { }k=2 τk,n pair of antennas at a time. Therefore, by using an example, we will justify the proposed method when compared to the case K 1 n N of jointly estimating φk k=2, where the ET simultaneously activates all K antennas.{ } When K = 3, the RSSI value for b b b 1 } n N the n-th mini slot can be written as Codebook B Rn = α1 + β2 cos(θn + φ2)+ β3 cos(θn + φ3)+ Fig. 4. The two-phase transmission protocol when K > 2. β , cos(φ φ )+ zn, 2 3 2 − 3 where α , β , β , and β are the parameters that depend on of feeding back N RSSI values. Then, the ET can calculate 1 2 3 2,3 h , h , and h . Therefore, the parameter vector becomes φˆ , and request for two further feedback values from the ER 1 2 3 2 [|α | β| |β β| | φ φ ]. It can be shown by studying the to resolve the ambiguity, similar to the method suggested in 1 2 3 2,3 2 3 FIM of the parameter vector that we need at least six RSSI [13]. This method effectively reduces the amount of feedback feedback values in order to estimate φ and φ . Even with our from N to 3. Furthermore, if the ER has enough resources to 2 3 proposed pairwise antenna activation policy, we need at least calculate φˆ , as the condition obtained for ambiguity resolution 2 six RSSI feedback values when K = 3. Therefore, we may is remarkably simple as well, the ER can directly feedback φˆ⋆ 2 not achieve a significant feedback reduction. to the ET. This method will reduce the amount of feedback Having discussed on the amount of feedback, the greater from N to 1. These examples give ample evidence to highlight that the results in this paper can be applied and further concern is with the ambiguity resolution. When it comes to optimized for many different applications of beamforming. In phase estimation, ambiguity resolving is a serious practical difficulty. However, we have given a very simple ambiguity the next section, we will study the case where K > 2. resolution procedure in our proposed scheme, without re- questing further feedback from the receivers. The ambiguity V. EXTENSIONOFRESULTSFORASINGLE-USER WET resolution when φ K is estimated jointly, is not at all SYSTEM WHEN K > 2 k k=2 straightforward. Therefore,{ } the channel learning methodology K proposed for is still reasonable and justifiable. When K > 2, the ET has to estimate φk k=2 (refer K > 2 (2)), and for this, we propose a pair wise transmit{ } antenna activation policy. To this end, when a pair of antennas is VI. THE SELECTION OF N activated, the phase difference of the channels between the According to the CRLB analysis in Section III, we can 2 activated antennas and the ER can be estimated by using 2σ expect the minimum variance of φˆ to scale like 2 with the same method that we have proposed for K = 2. This k Nβ N. This means, larger N values yield a higher channel pairwise activation is repeated for different antenna pairs until K estimation precision. However, larger N will increase the time we have estimated φk k=2. It is not hard to see that the most { } K spent in training, which will eventually reduce the time for straightforward way of selecting the best through φk k=2 WPB. This may lead to a reduction in the total transferred pairwise activation is by doing an exhaustive{ search} after the energy. Therefore, it is not hard to see that N affects the activation of all possible antenna pairs, and selecting the WET system performance greatly, and we will focus on setting this maximizing K . However, this approach is too complex φk k=2 important parameter in this section. to be feasible{ in practice.} Therefore, in this paper, we propose a We will first derive an expression to approximate the suboptimal method of estimating φ K , that still guarantees k k=2 received signal strength in the WPB stage, which we denote satisfactory results. { } as R . When K(> 2), from (1), we have The proposed extension is as follows. The training stage is WPB further divided into K 1 time slots, such that each time slot K K−1 K − ˆ consists of N mini-slots, see Fig. 4. This means, there will be RWPB = α1 + βi cos ∆φi + βi,j i=2 i=2 j=i+1 (K 1) N mini-slots in total in the training stage. When X   X X K =− 2,× we had only one time slot, and N mini-slots. Let cos ∆φˆi ∆φˆj , (12) = 2,...,K . In the (k 1)th time slot, where k , − Kthe ET{ simultaneously} activates− the -th antenna and the∈ Kst   k 1 where ∆φˆ and ∆φˆ denote the error in estimating φ and φ , antenna, and transmits using each element in B. This allows i j i j respectively, for i, j 2,...,K and i = j. α1, βi, and βi,j us to estimate φk using the same method proposed for K =2 are the parameters that∈{ depend on} channel6 magnitudes between as only a pair of antennas is activated. This also allows us to K the ET and the ER similar to the ones defined in Section V. obtain φˆk at the end of the training stage. Then, from We assume the estimation errors to be small and approximately k=2 n o 2 12

1.8 2/N Curve Θ set according to Definition 1 10 1.6 Randomly generated Θ Method in Theorem 1: SNR=30dB Method in Theorem 2: SNR=30dB Method in Theorem 2: SNR=20dB 1.4 8 Method in Theorem 2: SNR=10dB The method in [13]: SNR=30dB 1.2 Method in Theorem 1: SNR=10dB φ

1 6 MCRLB 0.8

4 0.6 Root mean square error (RMSE) 0.4 2

0.2

0 0 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 N - Size of the Codebook N - Size of the Codebook

MCRLB = = 1 ˆ Fig. 5. The behavior of the φ with N when β σ . Fig. 6. The behavior of the root mean squared error (RMSE) of φ2 for different SNR values when β = σ = 1. equal to each other in a given transmission block, i.e., ∆φˆi ˆ ˆ ≈ where ψ = ω1ω2τk,n . We need at least 3 RSSI feedback ∆φj ∆φ. Hence, we have ω1τk,n+Ef ≈ values to estimate the channels between the ET and the ER. K K−1 K Therefore, the lower bound of N ⋆ is 3. If T is long enough to RWPB = α1 + cos ∆φˆ βi + βi,j . (13) harvest energy, Etotal is strictly positive. Therefore, we have   i=2 i=2 j=i+1 X X X ω2 Since the minimum variance of the estimates behave like 0 <ω T 3(K 1)τk,n 1 3(K 1)Ef 2 1 2σ − − − 3 − − 2 , we can write ∆φˆ = ε/√N, where ε is a constant. Nβ  9(K 1)Ef   ˆ ω2 < 3 − < 3, Also, by using the small-angle approximation cos ∆φ = ⇒ − ω1(T 3(K 1)τk,n) ˆ2 − − ∆φ ω2   1 2 , RWPB = ω1 1 N , where ω1 = α1 + because ω1, Ef , τk,n > 0, and T > 3(K 1)τk,n. Also, since − − ω1τk,n − K−1 K 2 < 1, ψ< 3. Therefore, from (15), the upper bound   K ω1τk,n+Ef K  (Pi=2 βi)ε β + β and ω = . When T i=2 i i,j 2 2ω1 of the N ⋆ is 3 , which completes the proof. i=2 j=i+1 (K−1) PK = 2, ω X= αX+ β , and when N , R turns In the next section,q we will validate our results using numerical 1 1 2 → ∞ WPB out to be ω1. evaluations. Let Ef and τk,n be the energy required and time re- VII. NUMERICAL EVALUATIONS quired to feed back a single RSSI value, respectively. Hence, In this section, we present some numerical examples to N(K 1)τ is the time taken for the training stage. If the k,n validate our proposed schemes, and to provide useful insights transmission− block length is T , the harvested energy during a single transmission block can be written as on channel learning and wireless power beamforming. As a start, in Lemma 4, we have focused on MCRLBφ, and we Etotal = T N(K 1)τk,n RWPB N(K 1)Ef have given the formal proof for the minimum MCRLBφ value, − − − − considering N = 3 and N = 4, respectively. Then, based on   ω2 = ω1 T N(K 1)τk,n 1 N(K 1)Ef . the pattern, we expected that the minimum MCRLBφ to take 2 − − − N − − 2σ    (14) the form of Nβ2 for arbitrary values of N. Validation of this result is presented in Fig. 5. For the numerical evaluations, we Using this expression, we will provide bounds for the optimal have set β = σ = 1, and we have calculated MCRLB ac- value of N through the following theorem. φ cording to Lemma 3, while setting the phase values according Theorem 4: Let N ⋆ be the optimal value of N, and T > ⋆ 3T to Θ in Definition 1. We can see that setting the phase values N(K 1)τk,n. Then, 3 N . − ≤ ≤ (K−1) according to Definition 1 allows us to achieve the minimum Proof: For positive valuesq of N, (14) is convex. By MCRLB as the values lie on the 2/N curve. The figure also differentiating E with respect to N and setting it to zero, total shows how the average MCRLBφ behaves if the phase values the optimal value of N that maximizes Etotal can be given as in Θ are chosen randomly, for a given N. It can be seen that the average MCRLB values lie above the 2/N curve, with L φ N ⋆ = ψ , (15) the gap reducing when N is increased. Due to this reason, s (K 1) − a MCRLBφ value obtained by a randomly generated Θ can 5.5 Our Proposed Method 5 1.06 Exhaustive Method

4.5 1.04 Method in Theorem 2: SNR=30dB Method in Theorem 2: SNR=20dB Method in Theorem 2: SNR=10dB 1.02 The Method in [13]: SNR=30dB Method in Theorem 1: SNR=10dB

1 Energy

2 0.98 1.5

Average loss in harvested energy (%) 1 0.96

0.5

0 0.94 3 4 5 6 7 8 9 10 0 500 1000 1500 N - Size of the Codebook No of channel realizations when N=4, K=10, and SNR=20dB

Fig. 7. The behavior of the average percentage loss in harvested energy for Fig. 9. Comparison between proposed and baseline method for single user different SNR values when β = σ = 1. channel learning and WPT when K > 2.

1 1

0.9 0.9 SNR=0dB SNR=20dB 0.8 0.8

0.7 0.7

lit 0.6 0.6

0.5 0.5 Meth o d in Theorem 2 : SNR= 20dB

0.4 Meth o d in Theorem 2 : SNR= 10dB Met ho d in[13]: SNR = 30dB 0.4

Cumulative probabi y Meth o d in Theorem 1 : SNR= 30dB Cumulative Probability 0.3 0.3

0.2 0.2

0.1 0.1

0 0.7 0.8 0.9 1 1.1 1.2 1.3 0 Energ y 2 3 4 6 8 10 12 14 16 17 Optimal N = 2 Fig. 8. Empirical CDF illustrations of energy when K . Fig. 10. The behaviour of N ⋆, when K = 2 and T is 100 times longer than τk,n. be achieved using a lower number of feedback values, if Θ RMSE of calculating phase values according to the method is defined according to Definition 1. This is vital as we are proposed in Theorem 1, where the effect of noise is neglected. dealing with a receiver having a tight energy constraint, and Furthermore, Fig. 6 illustrates that our proposed method allows we have to also minimize the time spent for the training stage. the ET to achieve significant gains when compared to other Finally, as expected, we can observe that when N increases, works in the literature, even with lower SNR values. the lower bound on the variance of φˆ2 decreases. Fig. 7 illustrates the average loss in harvested energy In Theorem 2, we have presented an ML estimate of φ2. (percentage) due to using the the proposed methodology, Fig. 6 illustrates the behavior of the root mean squared error compared to performing energy beamforming with perfect (RMSE) with N for different SNR values. Θ is defined CSI. We can see that the loss is rather acceptable given the according to Definition 1. As expected, for higher SNR values, practicality of the proposed method. Fig. 8 illustrates the we have lower RMSE values, and the RMSE values converge respective energy transfer performance of each case considered to zero with N. It is interesting to note that even when N =3, in Fig. 6, using empirical cumulative distribution functions. the phase error is not significantly large. For example, when This alternative form of representation is used for improved N =3 and SNR= 10dB, RMSE is 7.88◦. It is also interesting clarity. The important point to notice in the figure is that the to note that when N = 3, the RMSE of calculating phase variance has decreased with SNR. This is because the increase values according to Theorem 2 is approximately equal to the in SNR leads to a better estimation, and the ET can guarantee Power Receiving Antenna 350 Training stage 300 WPB stage

250

VOUT RFIN Super-capacitor Antenna ( 200 Feedback

P1110 RSSI mV) Microcontroller Transmitter 150

DSET MSP430F5529 nRF24L01 100 DOUT GND ALK 50

0 Sensors 0 1 2 3 Feedback no WPB

Fig. 12. The RSSI values corresponding to each stage when the ET and the Fig. 11. The hardware block diagram of the ER. ER are 2m apart and N = 3.

200 RSSI a certain energy transfer with a high probability, that is, lower 180 Maximum RSSI outage. 160 RSSI in WPB, N=3 RSSI in WPB, N=4 140 In Section V, we have extended the proposed channel RSSI in WPB, N=5 learning and WPB scheme for K > 2 using a suboptimal, 120 but energy efficient method. A simple test was carried out ( 100 in order to check the performance of the proposed method. RSSI mV) 80 60 When ξ = 1, N = 4, K = 10 and SNR= 20dB, we 4 10 40 randomly generated θn and φk assuming that 20 { }n=1 { }k=2 they are uniformly distributed between 0 and 2π, and w 0 WPB 0 50 79 100 150 200 250 300 350 is calculated using the exhaustive method and the proposed θ(Degrees ) method, respectively. As shown in Fig. 9, this was done ◦ ◦ ◦ Fig. 13. The RSSI values when θi is changed from 0 to 360 with 1 for 1500 channel realizations. Although the feedback load is resolution. reduced considerably, it is not hard to see that our proposed method still exhibits impressive results. On average, the loss is only 2.2%. However, we can see that the variance has to the RSSI. As the storage device of our design, we use increased by shifting to the suboptimal method, similar to what a low leakage 0.22F super-capacitor. The output of P1110 was highlighted using Fig. 8. charges the super-capacitor and the super-capacitor powers We should note that although larger N yields a higher the microcontroller, the feedback transmitter and the sensors. channel estimation precision, this reduces the time for WPB. An Ultra-Low-Power MSP430F5529 microcontroller is used Therefore, a larger N may lead to a reduction in the total to read the RSSI values and transmit them via the feedback transferred energy. In Section VI, we have obtained bounds transmitter. When functioning, the microcontroller and the for the value of N that maximizes the energy transfer during feedback transmitter are on sleep mode, and after each 500 ms ⋆ the WPB stage, i.e. bounds on N . Fig. 10 illustrates the interval, both wake up from sleep in order to read the RSSI and ⋆ behaviour of the CDF of N , when K =2 and T = 100τk,n. transmit it to the ET. NORDIC nRF24L01 single chip 2.4GHz For these parameters, from Theorem 4, the theoretical lower transceiver has been used as the feedback transmitter. When bound and the upper bound are 3, and 17, respectively. Fig. 10 the ER operates in active mode (reading RSSI values and is consistent with these results and depict that the bounds are transmitting), it consumes only 12.8 µJ/ms and it consumes tight as well. We can observe that when the SNR increases, the negligible energy in sleep mode. The SDR used in our ET is CDF shifts to the left. This is because for better channels, we USRP B210, which has 2 2 MIMO capability. CRYSTEC RF need less feedback for an accurate estimation, and hence, the power amplifiers (CRBAMP× 100-6000) are used to amplify the optimal N will lie closer to the lower bound of the region with RF power output of the USRP B210. All the real-time signal a higher probability. Having done the numerical evaluations, processing tasks, channel phase difference (φk) estimation and we will further validate our results experimentally in the next setting beamforming vectors in both training and WPB stages section. were performed on a laptop using the GNU Radio framework. We use 915Mhz as the beamforming frequency. The same VIII. EXPERIMENTAL VALIDATION transceiver chip used in the ER, nRF24L01, is used as the In our experimental setup, the ET consists of 2 antennas and feedback receiver at the ET side. For the experiment, the ET delivers energy to an ER consisting of a single antenna. The and the ER are 2 meters apart. Using this setup, for N = 3, implementation of our ER is shown in Fig. 11. We use Pow- Fig. 12 illustrates the training stage and the WPB stage, and ercast P1110 power-harvester, which has an operating band we can see a clear gain by the proposed method. ranging from 902 to 928MHz. P1110 has an analog output Then, we focused on validating the result on phase esti- (DOUT), which provides an analog voltage level corresponding mation. For this, we changed θn from 0 to 360 degrees with R TABLE I where fR|xϕ ( , xϕ) denotes the conditional density function EXPERIMENTAL RESULTS of R given xϕ. Since xϕ and z are two independent vectors, ˆ ˆ ◦ fR (R, xϕ)= fz(R xϕ), where fz( ) denotes the density N φ2 Error |θ−79 | |xϕ − · 3 71◦ 8◦ function of z. Now, the first derivative of the log likelihood 4 77◦ 2◦ function can be written as 5 78◦ 1◦ ∂l(R, xϕ) ∂l(R xϕ) = − ∂ϕ ∂ϕ ◦ 1 resolution, and collected all respective RSSI values (see ∂xϕ ∂l(R xϕ) = − Fig. 13). Since it was not practical to collect all the 360 RSSI − ∂ϕ ∂z values using the harvested energy via the feedback transmitter, ∂x ∂l(z) = ϕ . we used a wired feedback for this experiment. Fig. 13 shows − ∂ϕ ∂z ◦ that the maximum RSSI occurs when θn = 79 . Therefore, ∂l(R, xϕ) the maximum energy transfer happens at that point. Using the FIMϕ(R) is defined as the covariance matrix of , same set of values, we estimated φˆ (Θ defined according ∂ϕ 2 i.e., to Definition 1) for N = 3, N = 4, N = 5 and N = 6, respectively. The results are tabulated in Table I. It is not ∂l(R, x ) ∂l(R, x ) T FIM (R)= E ϕ ϕ hard to see that the errors are significantly small, and they are ϕ xϕ,z ∂ϕ ∂ϕ     consistent with the numerical evaluations as well. Further, by T T ∂xϕ ∂l(z) ∂l(z) ∂xϕ using our proposed scheme, and based on the assumption that = E . xϕ,z ∂ϕ ∂z ∂z ∂ϕ the conversion efficiency of the power-harvester is fixed, we     can extend the range of the ER by 52% on average. This has Since xϕ and z are two independent vectors, been calculated based on the experimental results considering ∂x ∂l(z) ∂l(z)T ∂xT free space loss. FIM (R)= E ϕ E ϕ ϕ xϕ ∂ϕ z ∂z ∂z ∂ϕ     IX. CONCLUSIONS ∂x ∂xT = E ϕ FIM(z) ϕ , This paper has proposed a novel channel estimation method- xϕ ∂ϕ ∂ϕ ology to be used in a multiple antenna single user WET   where FIM(z) is the FIM with respect to z. Let z˜ denote system. The ET transmits using beamforming vectors from a non-Gaussian vector having same size as z. We have a codebook, which has been pre-defined using a Cramer- FIM(z˜) > FIM(z) [20]. This implies that Rao lower bound analysis. RSSI value corresponding to each beamforming vector is fed back to the ET, and these values ∂x ∂xT ∂x ∂xT E ϕ FIM(z˜) ϕ > E ϕ FIM(z) ϕ . have been used to estimate the channel through a maximum xϕ ∂ϕ ∂ϕ xϕ ∂ϕ ∂ϕ likelihood analysis. The channel estimation has then been used     Therefore, the Gaussian distribution minimizes FIM of ϕ. to set the beamforming vector for the WET. The results that Also, the Gaussian distribution maximizes the CRLB of ϕ have been obtained are simple, requires minimal processing, since the CRLB is given by the inverse of the FIM, which and can be easily implemented. The paper has also studied completes the proof. how the estimation ambiguities can be resolved in an energy efficient manner. The analytical results in the paper have been B. Proof of Lemma 1 validated numerically, as well as experimentally, while pro- We have viding interesting insights. It has been shown that the results 1 A1 D1 ∂xϕ . . . in the paper are more appealing compared to existing multiple = . . . . (16) antenna channel estimation methods in WET, especially when ∂ϕ . . .  1 A D there are tight energy constraints and hardware limitations at  N N  the ER. Also, the methods can be used for many applications By using the FIM of a Gaussian random vector in [21], and of beamforming, where processing capabilities of the receiver using the fact that Czz is independent of ϕ, the FIM of R is limited. can be written as ⊤ APPENDIX A ∂xϕ ∂xϕ FIMϕ(R)= Czz . CRAMER-RAO LOWER BOUND ANALYSIS ∂ϕ ∂ϕ     A. Worst-case CRLB Performance Substituting from (16) completes the proof. Lemma 6: The Gaussian distribution minimizes/maximizes C. Proof of Lemma 2 the FIM/CRLB of ϕ. When and has distinct elements, Proof: The log likelihood function of (5) can be written N 3 Θ N R ≥ , and R is invertible. There- as det(FIMϕ( )) = 0 FIMϕ( ) fore, computing the6 third diagonal element of the inverse of R R R l( , xϕ) = log fR|xϕ ( , xϕ), FIMϕ( ) completes the proof. D. Proof of Lemma 4 which is again independent of α and β. Now, By differentiating (6) with respect to θ and θ , respectively, N 2 3 ∂2E and by setting θ1 = 0, we obtain two expressions which are ˆ = Rn cos(θn + φˆ2,1), ∂φ2 φ2 =φ2,1 functions of θ and θ . Equating the two expressions to zero 2 n=1 2 3   X and simultaneously solving them under the constraints θ2,θ3 and (0, 2π], and θ = θ = θ , gives us θ = 2π/3 and θ =∈ 1 2 3 2 3 N 4π/3. Evaluating6 the Hessian6 matrix at the stationary point ∂2E ˆ ˆ = Rn cos(θn + φˆ2,1). (0, 2π/3, 4π/3) shows that the stationary point is a minimum. ∂φ2 φ2 =φ2,2=(φ2,1−π) −  2  n=1 Substituting (0, 2π/3, 4π/3) in (6) gives us 2σ2/3β2, which X completes the proof for N =3. Following the same lines for It is not hard to see that if (17) is positive for one possible N =4 completes the proof of the lemma. solution, then (17) is negative for the other possible solution. Moreover, as discussed in Theorem 2, φˆ2,1 and φˆ2,2 are critical APPENDIX B points of (8) (first derivative of (8) was zero at these points). ESTIMATION OF φ2 Therefore, we can claim that one of the candidate solutions A. Proof of Lemma 5 is a local minima, while the other is a local maxima. Since we want to find the local minima of E which maximizes the ˜ ˜ 2π Let θn = (n 1)θN , where θN = . Now, RSSI, the solution which satisfies the second derivative test − N ⋆ for the local minima gives us the correct estimate φ2, which N completes the proof. sin(θn + φ )= sin(φ )+ + sin((N 1)θ˜N + φ ) 2 2 ··· − 2 n=1 REFERENCES X j((N−1)θ˜N ) jφ2 = Im 1+ + e e [1] S. Abeywickrama, T. Samarasinghe, and C. K. Ho, ··· ˜ n(1 ejNθN )ejφ2  o “Wireless energy beamforming using signal strength = Im − ˜ . 1 ejθN feedback,” in Proc. IEEE Global Telecommunications n − o Conference, pp. 1 – 6, Dec. 2016. jNθ˜N j2π N We have e = e =1. Hence, n=1 sin(θn + φ2)=0. [2] R. Zhang and C. K. Ho, “MIMO broadcasting for Following similar steps for the other summations of interest simultaneous wireless information and power transfer,” P completes the proof. in Proc. IEEE Global Telecommunications Conference, pp. 1 – 5, Dec. 2011. B. Proof of Theorem 2 [3] X. Chen, Z. Zhang, H. H. Chen, and H. Zhang, “Enhanc- When θn = 2(n 1)π/N, from Lemma 5, we have ing wireless information and power transfer by exploiting N − N n=1 sin(θn + φ2)= n=1 sin [2(θn + φ2)]=0. Therefore, multi-antenna techniques,” IEEE Commun. Mag., vol. 53, (9) can be simplified and written as N R sin (θ + φ )= pp. 133–141, April 2015. P P n=1 n n 2 0, which is independent of α and β. By expanding [4] J. Xu, S. Bi, and R. Zhang, “Multiuser MIMO wireless P sin (θn + φ2) we get, energy transfer with coexisting opportunistic communi- N N cation,” IEEE Wireless Commun. Letters, vol. 4, pp. 273– 276, Jun. 2015. sin φ2 Rn cos θn + cos φ2 Rn sin θn =0, n=1 n=1 [5] J. Xu and R. Zhang, “A general design framework for X X MIMO wireless energy transfer with limited feedback,” which can be directly used to obtain (10), completing the IEEE Trans. Signal Process., vol. 64, pp. 2475–2488, proof. Feb. 2016. C. Proof of Theorem 3 [6] Y. Zeng and R. Zhang, “Optimized training design for wireless energy transfer,” IEEE Trans. Commun., vol. 63, By taking the second derivative of (8) with respect to φ , 2 pp. 536–550, Feb. 2015. we have [7] G. Yang, C. K. Ho, R. Zhang, and Y. L. Guan, “Through- N N ∂2E put optimization for massive MIMO systems powered by = Rn cos(θn + φ2) α cos(θn + φ2) wireless energy transfer,” IEEE J. Sel. Areas Commun., ∂φ2 − 2 n=1 n=1 X X vol. 33, pp. 1640–1650, Aug. 2015. N [8] Y. Zeng and R. Zhang, “Optimized training for net β cos [2(θn + φ )]. − 2 energy maximization in multi-antenna wireless energy n=1 X transfer over frequency-selective channel,” IEEE Trans. When θn = 2(n 1)π/N, from the Lemma 5, we have Commun., vol. 63, pp. 2360–2373, Jun. 2015. N − N n=1 cos(θn +φ2)= n=1 cos [2(θn + φ2)]=0. Therefore, [9] G. Yang, C. K. Ho, and Y. L. Guan, “Dynamic resource allocation for multiple-antenna wireless power transfer,” N P ∂2E P IEEE Trans. Signal Process., vol. 62, pp. 3565 – 3577, = R cos(θ + φ ), (17) ∂φ2 n n 2 Jul. 2014. 2 n=1 X [10] J. Xu and R. Zhang, “Energy beamforming with one- Tharaka Samarasinghe (S’11-M’13) was born bit feedback,” IEEE Trans. Signal Process., vol. 62, in Colombo, Sri Lanka, in 1984. He received the B.Sc. degree in engineering from the Department pp. 5370–5381, Oct. 2014. of Electronic and Telecommunication Engineering, [11] A. Huhn and H. Schwetlick, “Channel estimation meth- University of Moratuwa, Sri Lanka, in 2008, where ods for diversity and MIMO in LR-WPAN systems,” he received the award for the most outstanding undergraduate upon graduation. He received the in Proc. IEEE International Conference on Consumer Ph.D. degree from the Department of Electrical and Electronics, pp. 223 – 224, Jan. 2010. Electronic Engineering, University of Melbourne, [12] Q. Wang, D. Wu, J. Eilert, and D. Liu, “Cost analysis Australia, in 2012. He was a Research Fellow at the Department of Electrical and Computer Systems of channel estimation in MIMO-OFDM for software Engineering, Monash University, Australia, from 2012 to 2014. He has defined radio,” in Proc. IEEE Wireless Communications been with the Department of Electronic and Telecommunication Engineering, and Networking Conference, pp. 935 – 939, Apr. 2008. University of Moratuwa, Sri Lanka, since January 2015, where he is currently a Senior Lecturer. His research interests are in communications theory, [13] S. Lakshmanan, K. Sundaresan, S. Rangarajan, and information theory, and wireless networks. R. Sivakumar, “Practical beamforming based on RSSI measurements using off-the-shelf wireless clients,” in Proc. Internet measurement conference, pp. 410–416, Nov. 2009. [14] M. Guillaud, D. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” in Proc. IEEE International Symposium on and Its Applications,, Chin Keong Ho (S’05-M’07) received the B.Eng. pp. 28 – 31, Aug. 2005. (First-Class Hons., minor in business admin.) and M.Eng. degrees from the Department of Electrical [15] R. Zhang and C. K. Ho, “MIMO broadcasting for simul- Engineering, National University of Singapore, in taneous wireless information and power transfer,” IEEE 1999 and 2001, respectively. He received the Ph.D. Trans. Wireless Commun., vol. 12, pp. 1989–2001, May degree at the Eindhoven University of Technology, The Netherlands, where he concurrently conducted 2013. research work in Philips Research. Since August [16] S.-H. Tsai, “Equal gain transmission with antenna selec- 2000, he has been with Institute for Infocomm Re- 2 tion in MIMO communications,” IEEE Wireless Com- search (I R) , ASTAR, Singapore. He is Lab Head of Energy-Aware Communications Lab, Department of mun., vol. 10, pp. 1470–1479, May 2011. Advanced Communication Technology, in I2R. His research interest includes [17] T. Pany, Navigation signal processing for GNSS software green wireless communications with focus on energy-efficient solutions and receivers. Artech House, 1 ed., 2010. with energy harvesting constraints; cooperative and adaptive wireless com- munications; and implementation aspects of multi-carrier and multi-antenna [18] F. Gini, R. Reggiannini, and U. Mengali, “The modified communications. His work in unified study of wireless power and wireless Cramer-Rao bound in vector parameter estimation,” IEEE communications received the IEEE Marconi Prize Paper Award in 2015. Trans. Commun., vol. 46, pp. 52–60, Jan. 1998. [19] J. C. Spall and D. C. Chin, “On system identification with nuisance parameters,” in Proc. American Control Conference, pp. 530 – 531, July 1994. [20] O. Rioul, “Information theoretic proofs of entropy power inequalities,” IEEE Trans. Inf. Theory, vol. 57, pp. 33 – 55, Jan. 2011. [21] S. Kay, Fundamentals of Statistical Signal Processing: Chau Yuen (S’04-M’06-SM’12) received the BEng and PhD degree from Nanyang Technological Uni- Estimation Theory. Prentice-Hall, 1993. versity (NTU), Singapore, in 2000 and 2004 respec- tively. Dr Yuen was a Post Doc Fellow in Lucent Technologies Bell Labs, Murray Hill during 2005. During the period of 2006 2010, he worked at the Institute for Infocomm Research (I2R, Singapore) as a Senior Research Engineer, where he was involved in an industrial project on developing an 802.11n Samith Abeywickrama (S’16) received the B. Sc. Wireless LAN system, and participated actively in degree in engineering from the Department of Elec- o Long Term Evolution (LTE) and LTEAdvanced tronic and Telecommunication Engineering, Univer- (LTEA) standardization. He joined the Singapore University of Technology sity of Moratuwa, Sri Lanka, in 2016. He then joined and Design from June 2010, and received IEEE Asia-Pacific Outstanding with Singapore University of Technology and De- Young Researcher Award on 2012. Dr Yuen serves as an Editor for IEEE sign, as a researcher. His research interests include Transactions on Communications and IEEE Transactions on Vehicular Tech- proof-of-concept and taking advanced theoretical nology. He has 2 US patents and published over 300 research papers at ideas all the way to practice using software-defined international journals or conferences. (SDR), wireless power transfer, wireless spec- trum sensing, and angle of arrival (AOA) estimation.