Fine Beam Tracking Using Spatio–Temporal Interpolation in Wireless Power Transfer Systems

The use of transmit beamforming in a radio-frequency wireless power transfer (WPT) system can improve the energy transmission efficiency by concentrating the transmit power in the angle of departure (AoD). Optimal beamforming requires information regarding the AoD or the channel between the transmitter and receiver. A previously proposed WPT system continually estimates the optimal AoD using custom-designed pilot signals, which are known as beam-steered pilot signals and are transmitted periodically at every frame. Herein, we propose a spatio–temporal interpolation technique to improve the accuracy of beam tracking for such a WPT system. Specifically, thin-plate spline interpolation is adopted in the space domain to overcome the limited angular resolution of the beam-steered pilot signals in a three-dimensional channel. In the time domain, polynomial interpolation is performed to track the changing AoD during consecutive frames. Numerical results show that the proposed technique can improve the WPT efficiency as well as the mean square error performance of the beamforming weight vector.


I. INTRODUCTION
Recently, massive Internet of Things (IoT), which is one of the target scenarios of the fifth generation mobile communication, has garnered increasing attention. Massive IoT scenarios aim to support effective communication between massive number of connected devices [1], [2], [3], [4]. For massive IoT to be effectively deployed, it is important to develop a technology that supplies the power required by IoT devices, as well as an efficient multiple access technology that supports massive devices [5], [6], [7]. To solve the power supply problem in IoT devices, one can adopt wireless power transfer (WPT) technology, which is classified into magnetic induction, magnetic resonance, and radio-frequency (RF) radiation methods. In the magnetic induction method, miniaturization is difficult to accomplish owing to the short-range power transmission that is within 1 cm, as well as heat issues.
The associate editor coordinating the review of this manuscript and approving it for publication was Claudio Curcio .
In the magnetic resonance method, the charging distance is severely limited, and coupling and stability problems occur when many devices are used in a large space [8], [9], [10], [11]. Therefore, in this study, we investigated an RF radiation method that enables relatively long-distance WPT to mobile receivers [12], [13]. The RF radiation method can also be advanced to simultaneous wireless information and power transfer (SWIPT) systems [14], which incorporate WPT into wireless communications.
In the RF radiative WPT system, transmit beamforming is an effective method for improving energy transmission efficiency by concentrating the transmit power in the direction of the receiving device. In particular, a phased-array antenna can steer a beam toward a desired angle of departure (AoD) by adjusting the phases of the individual antenna elements. Therefore, the phased-array antenna can be effectively adopted for WPT, wireless communications, SWIPT, and radar [15], [16], [17], [18]. Optimal beamforming that conveys maximum energy to the receiver adjusts the phases of the individual antenna elements such that the signal with the same phase reaches the receiver from every individual antenna. To achieve optimal beamforming, the transmitter must be aware of the AoD [19], or equivalently, the channel state information between the transmitter and receiver [20]. The transmitter typically transmits pilot signals known by the receiver to obtain information pertaining to the AoD or channel state [21].
Recently, a simple but effective beam scanning method was proposed to realize beamforming in a WPT system [22]. The beam space was partitioned into several subspaces, and the transmitter transmitted a pilot signal steered in the AoD corresponding to each subspace. Each beam subspace was scanned by changing the AoD of the steered pilot signals. By measuring the received signal strengths corresponding to the pilot signals, the receiver can identify the optimal beam subspace, i.e., the AoD corresponding to the maximum energy reception. Subsequently, the receiver provides the optimal AoD information to the transmitter via a dedicated feedback channel. The accuracy of the beam-scanning scheme is determined by the resolution of the pilot signals in the space and time domains, as the pilot signals are intermittently transmitted for a finite number of quantized beam subspaces [22]. A direct solution for improving the accuracy in the space domain is to increase the number of beam subspaces. However, an increase in the number of beam subspaces will increase the feedback overhead because more bits are required to represent the increased number of beam subspaces. Besides, the receiver mobility deteriorates the accuracy of beam scanning in the time domain. The effect of mobility can be mitigated by reducing the interval between the pilot signals. However, this may undermine the time duration for WPT, thereby decreasing the overall power transfer efficiency.
Herein, we propose a spatio-temporal interpolation technique to improve the beamforming accuracy in both the space and time domains in a WPT system based on beam scanning. The proposed technique estimates the optimal AoD via the thin-plate spline (TPS) interpolation of the received signal strengths, which are associated with a finite number of beam subspaces. The interpolation enhances the resolution of the pilot signals, thereby improving the AoD estimation performance of the transmitter in a three-dimensional (3D) space. Furthermore, polynomial interpolation is employed to track the moving path of the receiver during consecutive time frames, which enables more frequent updates of the steering AoD. Hence, the proposed technique is expected to improve the accuracy of beamforming in both space and time domains. Numerical results are presented to demonstrate the The remainder of this paper is organized as follows: Section II describes the signal model for a WPT system and briefly reviews the beam scanning method proposed in [22]. In Section III, the spatio-temporal interpolation technique proposed herein for improving the accuracy of beam scanning in space and time domains is described. The numerical results and conclusions are presented in Sections IV and V, respectively.

II. SYSTEM MODEL
We consider a radiative WPT system to IoT devices with a planar array antenna (PAA) installed at the transmitter, as illustrated in Fig. 1. In a PAA composed of N t (= MN ) antennas, M antennas along the x-axis and N antennas along the z-axis are uniformly placed with a spacing d of one-half the wavelength, i.e., d = λ/2. This antenna configuration allows the PAA to form a directive beam with a narrow main beam in a 3D space. The receiver is assumed to be equipped with a single antenna. In the following subsections, we present the signal model for WPT and provide a brief review of the beam scanning scheme proposed in [22].
where (·) T denotes the transpose operation. Then, the N t × 1 beamformed signal vector x(t) at the transmitter can be expressed as x(t) = ws(t), where s(t) denotes the transmit energy signal. Correspondingly, the signal y(t) received at the receiver can be expressed as where P is the average received power and h is an equivalent N t × 1 channel vector constructed by stacking the column of the M × N channel matrix. n(t) is the additive white Gaussian noise process with a variance of σ 2 , and (·) H the Hermitian transpose. As the energy is typically conveyed on a narrowband tone signal, the channel vector h can be written as where θ mn and φ mn represent the elevation and azimuth angles, respectively, between the (m, n) antenna element of the PAA and the receive antenna; r mn denotes the distance between the (m, n) antenna element and the receive antenna; γ mn represents the corresponding path loss; E(θ, φ) denotes the field pattern of each antenna element, which is assumed to be a dipole antenna with directional linear pattern, i.e., B. BEAM SCANNING METHOD PROPOSED IN [22] Fig. 2 shows a transmission frame structure for WPT. The frame duration T frame is segmented into a section T pilot , in which a pilot signal is transmitted, and another section T wpt , in which WPT is performed. For beam-steered pilot transmission, the entire beam space is segmented into K subspaces, each of which corresponds to a steered AoD. The beam-steered pilot signals are transmitted toward the K predetermined AoDs, and the signal for each AoD persists for T s . The beamforming AoD for power transfer is determined based on feedback received from the receiver in the previous frame.
It is noteworthy that in (4), if the distance r 0 between the transmitter and receiver is sufficient to satisfy the far-field approximation, then one can assume that the azimuth and elevation angles between every antenna element of the PAA and the receive antenna are equal, i.e., θ mn = θ 0 , φ mn = φ 0 . Therefore, the optimal beamforming AoD is determined solely by θ 0 and φ 0 . Both the azimuth steering and elevation steering angles for K steered AoDs as well as the corresponding beamforming weight vectors are stored in a codebook. Codebook C, which is a set of beamforming weight vectors corresponding to quantized AoDs of pilot signals, is defined as The transmitter sequentially transmits the steered beam toward the beam subspace. If the beam space scanned by the beam-steered pilot signals is uniformly quantized, then the angles θ i and φ j corresponding to the beam subspace (i, j) can be expressed as where θ min and φ min represent the minimum values of the elevation and azimuth angles, respectively; θ and φ represent the step sizes between adjacent elevation angles and azimuth angles, respectively; K e and K a represent the total numbers of elevation and azimuth angles, respectively. The total number of codewords constituting the codebook, i.e., the number of beam subspaces, is K (= K e K a = 2 b e +b a = 2 b ), where b e and b a denote the numbers of quantization bits for the elevation and azimuth angles, respectively. The receiver measures the average received signal strength of each beam-steered pilot signal using a rectifier. The received signal strength p ij for the pilot signal corresponding to the AoD (θ i , φ j ) can be expressed as θ andφ, which are associated with the maximum values among the measured K average power values, are identified as being the AoD for the receiver, and the corresponding index are fed back to the transmitter. Note that the index can be represented using b (= b e + b a ) bits (b e bits for elevation angle and b a bits for azimuth angle), and it must be sent every frame. Hence, the feedback requires a dedicated communication link with minimum data rate of b/T frame [bps], which is defined as the feedback overhead.

III. PROPOSED SPATIO-TEMPORAL INTERPOLATION TECHNIQUE
In the conventional beam scanning scheme, the beam weight vector is updated in each frame. Increasing the beam-scanning resolution increases the pilot overhead as the number of feedback bits increases. In addition, an increase in the pilot interval T pilot within the frame reduces the power transfer interval T wpt , degrading the WPT efficiency, which is defined as the ratio of the received signal power to the transmit signal power. Meanwhile, in the time domain, the capability of the transmitter in reflecting the mobility of the receiver across frames is limited because the beam weight vector, i.e., the AoD, is updated only once every frame. If the frame duration is decreased to achieve more frequent updates, then the ratio of T wpt to T frame will be reduced because T pilot cannot be changed to preserve the accuracy in the space domain. This may decrease the overall efficiency of the WPT. In this section, we propose a technique to improve the accuracy of the conventional beam-scanning scheme presented in Section II-B that does not involve altering the beam scanning resolution and frame duration.

A. SPATIAL INTERPOLATION
We used a TPS interpolation algorithm [24] to identify a more accurate AoD than that estimated using the conventional beam scanning algorithm. TPS interpolation utilizes the idea of an elastic shell and can adjust the smoothing level to establish a model for arbitrarily scattered data. A more accurate beam-steering AoD can be estimated by applying TPS interpolation to the beam scanning results.
The measured power z ij corresponding to the AoD (θ i , φ j ) estimated through beam scanning and a thin-plate physical model f (θ i , φ j ) are correlated as follows: z ij = f (θ i , φ j ), i = 1, 2, . . . , K e , j = 1, 2, . . . , K a . (8) Note that the function f is initially unknown and obtained as the outcome of the TPS interpolation. In other words, the purpose of the TPS interpolation is to find a function f that well describes the variation of the received power with θ and φ. In particular, the TPS model can be extracted by solving a mixed target minimization problem: where the first term denotes the external energy that reflects the sum of squared errors between the fitted surface and the data, and E b (f ) is the bending energy that acts as roughness penalty. 0 ≤ α ≤ 1 is a parameter that controls the smoothing level, i.e., smoothness of the allowable deformation by changing the weight for the bending energy [25]. The higher α is, the less exact the fit to data becomes; If α = 0, the data will be fit exactly, whereas, if α is close to 1, the solution tends to approximate the least-squares one. The bending energy E b (f ) in (9) can be expressed as [26] E The solution to the problem expressed in (9) can be obtained by solving a linear system of equations. Specifically, the TPS model f can be represented as where θ and φ denote the elevation and azimuth angles, respectively, and o ij ≜ (θ − θ i ) 2 + (φ − φ j ) 2 , i = 1, 2, . . . , K e , j = 1, 2, . . . , K a . Subsequently, the function is determined using a set of K + 3 parameters: ρ ij (i = 1, 2, . . . , K e , j = 1, 2, . . . , K a ), d 0 , d 1 , and d 2 . These parameters can be computed by applying block Gaussian elimination to the following linear system of equations [27]: where g ij ≜ o 2 ij ln o ij and i = 1, 2, . . . , K e , j = 1, 2, . . . , K a . Using the estimated TPS model f , the elevation and azimuth angles of the beam steering can be expressed as whereθ andφ denote the steering AoD corresponding to the maximum average received signal strengths among the K candidate values obtained at the receiver. The problem in (13) is to find the optimal angles of θ and φ that maximize the energy transferred to the receiver, and it can be solved using an exhaustive search over a search space of two quantized AoDs centered atθ andφ, as the optimal angles will be near the quantized anglesθ andφ. As a result, TPS interpolation identifies a more appropriate AoD (θ * , φ * ) surrounding the initially estimated AoD obtained via beam scanning.

B. TEMPORAL INTERPOLATION
We propose an algorithm to estimate the moving path of the mobile receiver to update the steering AoD semi-continuously within the WPT section of the frame. The proposed scheme applies n-th-order polynomial interpolation across frames to the steering AoDs obtained via beam scanning with or without TPS. Note that the frame duration T frame is fixed. Without temporal interpolation, the beamforming weight vector can be updated only once per frame. If we adopt the temporal interpolation as proposed in our paper, the moving path of the receiver is continuously estimated with a continuous function, and thus, the beamforming weight vector can be updated with infinite precision. However, too many updates will cause excessive complexity at the transmitter. We introduce U , which denotes the number VOLUME 11, 2023 of updates of the beamforming weight vector within each frame interval, as a parameter to compromise between the update rate and complexity. As a result, using the n-th-order polynomial equation resulting from the interpolation, the beam-steering AoD of the transmitter is updated U times with a constant update period T update within each frame along the estimated moving path. The parameter U determines the update frequency of the steering direction, and it is related to T wpt and T update as U = T wpt /T update . Under the assumption that the receiver moves with time t on the xz-plane in Fig. 1, the n-th-order polynomials p x (t) and p z (t) for the location of the receiver in each axis can be written as p x (t) ≜ q x0 + q x1 t + q x2 t 2 + · · · + q xn t n , p z (t) ≜ q z0 + q z1 t + q z2 t 2 + · · · + q zn t n .
To estimate the n-th-order polynomial coefficients, beam scanning results for at least n + 1 consecutive frames are necessary. Using the data, the coefficients of the polynomials can be obtained through linear regression based on the least squares criterion. Specifically, let {t 1 , t 2 , . . . , t n+1 } and {(x 1 , z 1 ), (x 2 , z 2 ), . . . , (x n+1 , z n+1 )} be the time sequence and the corresponding locations of the receiver measured at n + 1 consecutive frames. Then, the sum of squared errors, S x and S z , are defined as Correspondingly, the least squares solutions for the coefficients {q x0 , q x1 , . . . , q xn } and {q z0 , q z1 , . . . , q zn } in (14) can be obtained as which yields By estimating the coefficients q x0 , q x1 , . . . , q xn of the polynomial p x (t) and q z0 , q z1 , . . . , q zn of the polynomial p z (t), an expected moving path equation can be identified, and the position change of the receiver can be tracked continuously in the x-axis and z-axis, respectively. Accordingly, the beam-steering AoD that changes within the frame can be updated.

IV. NUMERICAL RESULTS
In this section, an evaluation of the performance of the proposed spatio-temporal interpolation technique is presented. A 7 × 7 PAA, i.e., M = N = 7, is considered, and the frequency of the transmit signal is set to 5.8 GHz. The frame duration and pilot signal duration for each steering AoD are set as T frame = 1000 ms and T s = 1 ms, respectively. The receiver is assumed to be located on a 5 m × 5 m receiving plane, located 2 m away from the transmitter. The ranges of the steering angles are assumed to be θ ∈ [−50 • , 50 • ] and φ ∈ [−50 • , 50 • ]. The angular difference between adjacent beams is denoted as θ = 100 • /2 b e , φ = 100 • /2 b a , where b e and b a are the numbers of bits for the uniform quantization of the elevation and azimuth angles, respectively. Table 1 summarizes the parameters used in the simulations.
As performance measures, we consider the mean square error (MSE) of the beamforming weight estimation and WPT efficiency. The MSE is defined as where β(θ, φ) ≜ kd cos θ sin φ +kd sin θ cos φ, and the WPT efficiency η is defined as the ratio of the received signal power to the transmit power:

A. EFFECT OF SPATIAL INTERPOLATION
We first validate the performance gain of TPS interpolation, assuming that the receiver is located at a fixed location.   a lower MSE than the Max-Pilot scheme when b ≥ 6. This implies that the number of bits for quantization should be sufficiently large to achieve an effective TPS interpolation; low-resolution beam scanning with only a few bits results in an inaccurate raw estimation, which cannot be overcome by TPS interpolation. A trend similar to that mentioned above was exhibited in the corresponding WPT efficiency, as shown in Fig. 5.

B. EFFECT OF TEMPORAL INTERPOLATION
In this section, we consider the receiver mobility. The receiver is assumed to move within a receiving plane on the xz-plane based on a coordinated turn model. The coordinated turn model models the motion of the target when it is assumed that the target rotates at a constant velocity in two-dimensional space. Let g(k) ≜ [x(k),ẋ(k), z(k),ż(k)] T be a vector representing the movement of the receiver at the k-th time instant, where (x(k), z(k)) denotes the location of the receiver and (ẋ(k),ż(k)) denotes the movement speed along each axis [28]. Then, g(k + 1) at the (k + 1)-th time instant is determined as [28] where denotes the angular velocity, and t is the interval between adjacent time instants. It is assumed that follows a uniform distribution on [−100, 100] • /sec and updated every two frames. The time interval t was set to 100 ms. We performed a comparison on the following three cases: no interpolation (Max-Pilot), spatial interpolation only (TPS), and spatio-temporal interpolation (TPS-Poly: TPS interpolation in space and polynomial interpolation in time) in terms of the MSE and WPT efficiency. For the temporal interpolation, we assumed a second-order polynomial model, i.e., n = 2. 1 Fig. 6 shows the moving path estimated via the second-order polynomial interpolation, along with the original moving path. TPS interpolation was performed to estimate the receiver location in every frame, and the location of the receiver within each frame was updated U = 10 times based on the estimated moving path. The observations on the receiver location on the xz-plane corresponding to three preceding frames are used to estimate the polynomials for the moving path. For instance, the observations obtained by the TPS interpolation in the 2nd, 3rd, and 4th frames are (2.5000, 1.9354), (2.3282, 2.3840), and (1.8430, 2.4179), respectively, and the polynomials p x (t) = 3.590 − 0.3721t − 0.0161t 2 and p z (t) = 1.876 + 0.271t − 0.0339t 2 can be derived by solving (17). Using these polynomials, the location of the receiver and the corresponding beamforming 1 The optimal order of the polynomial model depends on the movement model of the receiver. For the coordinated turn model in (20), we empirically discovered that n = 2 provides the best performance in most cases. Hence, we set n = 2 in our study.  weight vector are updated U times until the observation in the 5th frame is available. It is seen that the estimated moving path is typically consistent with the original path. In specific outer regions, however, the location estimated via TPS deviated significantly from the original moving path owing to the limited radiation pattern of the PAA; for certain steering angles far from the bore-sight, the angles associated with the maximum directional antenna gain tends to deviate from the intended steering angles. Consequently, the estimated moving path came to deviate from the original moving path.
Figs. 7 and 8 show how the MSE and WPT efficiency vary with time as the receiver moves following the original path shown in Fig. 6. The MSE and WPT efficiency are shown to fluctuate because the beamforming weight vector is updated in discrete times while the receiver location is continuously varying. Note that the MSE and WPT efficiency were not averaged out, that is, they are instantaneous values. The steering angles of beamforming were updated only once every frame in the cases of Max-Pilot and TPS, whereas they were updated U = 10 times in the case of spatio-temporal interpolation. Owing to more frequent updates of the beam-steering angles due to the moving path estimation, the proposed spatio-temporal interpolation yielded a lower MSE and a higher WPT efficiency, particularly in time intervals across frames. It is worth mentioning that the higher update rate was implemented by temporal interpolation across frames, without changing the frame duration. The effect of the aforementioned relatively large errors in the moving path estimation during a short period on the WPT efficiency was marginal; the corresponding time sections are denoted as t A and t B in Fig. 8. Table 2 summarizes the relative WPT efficiency of the three schemes, where the relative WPT efficiency of each scheme is defined as the WPT efficiency divided by that obtained with a perfect estimation of the moving path. This reflects the WPT efficiency in (19) and the accuracy of the path estimation at the same time. The efficiency was averaged over 1,000 independent realizations of random moving path that persists for 30 seconds. The superior performance of the proposed TPS-Poly scheme is attributed to higher update rate of the beamforming weight vector. As expected, relative WPT efficiency is shown to improve as the number of updates (U ) increases. Remarkably, the proposed spatio-temporal interpolation achieves 96.43% of the ideal WPT efficiency for the case of U = 10.

V. CONCLUSION
We proposed a spatio-temporal interpolation technique to improve the beamforming accuracy in a WPT system based on beam scanning using beam-steered pilot signals. Spatial interpolation was performed to enhance the spatial resolution of beam scanning using a limited number of bits for quantization, whereas temporal interpolation was used to enhance the time resolution of beam scanning by continually tracking the moving path of the receiver. TPS interpolation and polynomial interpolation were adopted for spatial and temporal interpolations, respectively. Numerical simulations were performed to validate the performance gain of the proposed scheme in terms of the MSE of the beamforming weight vector and the WPT efficiency. It was discovered that the proposed spatio-temporal interpolation technique enhanced the beamforming accuracy without increasing the pilot overhead, resulting in significant improvements in the MSE and WPT efficiency. The fine beam steering method based on the proposed interpolation technique can be exploited in wireless communications as well as the WPT, e.g., unmanned aerial vehicle communications and inter-satellite optical communications.