Magnetic MIMO Signal Processing and Optimization for Wireless Power Transfer
Abstract
In magnetic resonant coupling (MRC) enabled multipleinput multipleoutput (MIMO) wireless power transfer (WPT) systems, multiple transmitters (TXs) each with one single coil are used to enhance the efficiency of simultaneous power transfer to multiple singlecoil receivers (RXs) by constructively combining their induced magnetic fields at the RXs, a technique termed “magnetic beamforming”. In this paper, we study the optimal magnetic beamforming design in a multiuser MIMO MRCWPT system. We introduce the multiuser power region that constitutes all the achievable power tuples for all RXs, subject to the given total power constraint over all TXs as well as their individual peak voltage and current constraints. We characterize each boundary point of the power region by maximizing the sumpower deliverable to all RXs subject to their minimum harvested power constraints, which are proportionally set based on a given powerprofile vector to ensure fairness. For the special case without the TX peak voltage and current constraints, we derive the optimal TX current allocation for the singleRX setup in closedform as well as that for the multiRX setup by applying the techniques of semidefinite relaxation (SDR) and timesharing. In general, the problem is a nonconvex quadratically constrained quadratic programming (QCQP), which is difficult to solve. For the case of one single RX, we show that the SDR of the problem is tight, and thus the problem can be efficiently solved. For the general case with multiple RXs, based on SDR we obtain two approximate solutions by applying the techniques of timesharing and randomization, respectively. Moreover, for practical implementation of magnetic beamforming, we propose a novel signal processing method to estimate the magnetic MIMO channel due to the mutual inductances between TXs and RXs. Numerical results show that our proposed magnetic channel estimation and adaptive beamforming schemes are practically effective, and can significantly improve the power transfer efficiency and multiuser performance tradeoff in MIMO MRCWPT systems compared to the benchmark scheme of uncoordinated WPT with fixed identical TX current.
Wireless power transfer, magnetic resonant coupling, magnetic MIMO, magnetic beamforming, magnetic channel estimation, multiuser power region, timesharing, semidefinite relaxation.
I Introduction
\PARstartNearfield wireless power transfer (WPT) has drawn significant interests recently due to its high efficiency for delivering power to electric loads without the need of any wire. Nearfield WPT can be realized by inductive coupling (IC) for shortrange applications within centimeters, or magnetic resonant coupling (MRC) for midrange applications up to a couple of meters. Although shortrange WPT has been in widely commercial use (e.g., electric toothbrushes), midrange WPT is still largely under research and prototype. In , a milestone experiment has successfully demonstrated that based on strongly coupled magnetic resonance, a single transmitter (TX) is able to transfer watts of power wirelessly with – efficiency to a single receiver (RX) at a distance about meters. Motivated by this landmark experimental result, the research in MRC enabled WPT (MRCWPT) has grown fast and substantially (see e.g., [1] and the references therein).
MRCWPT with generally multiple TXs and/or multiple RXs has been studied in the literature [2, 3, 4, 5, 6, 7]. Under the multipleinput singleoutput (MISO) setup, [2] has studied an MRCWPT system with two TXs and one single RX, while the analytical results proposed in this paper cannot be directly extended to the case with more than two TXs. In [3], a convex optimization problem has been formulated to maximize the efficiency of MISO MRCWPT by jointly optimizing all TX currents together with the RX impedance. However, the study in [3] has not considered the practical circuit constraints at individual TXs, such as peak voltage and current constraints, and also its solution cannot be applied to the mutiRX setup. Recently, [4] has reported a wireless charger with an array of TX coils which can efficiently charge a mobile phone cm away from the charging unit, regardless of the phone’s orientation. On the other hand, under the singleinput multipleoutput (SIMO) setup, an MRCWPT system with one single TX and multiple RXs has been studied in [5], in which the load resistances of all RXs are jointly optimized to minimize the total transmit power drawn while achieving fair power delivery to the loads at different RXs, even subject to their nearfar distances to the TX. For the general multipleinput multipleoutput (MIMO) setup, in [6] it has been experimentally demonstrated that employing multiple TX coils can enhance the power delivery to multiple RXs simultaneously, in terms of both efficiency and deliverable power. However, this work has not addressed how to design the system parameters to achieve optimal performance.
Currently, there are two main industrial organizations on standardizing wireless charging, namely, the Wireless Power Consortium (WPC) which developed the “Qi” standard based on magnetic induction, and the Alliance for Wireless Power (A4WP) which developed the “Rezence” specification based on magnetic resonance. The Rezence specification advocates a superior charging range, the capability to charge multiple devices concurrently, and the use of twoway Bluetooth communication between the charger and devices for realtime charging control. These features make Rezence a promising technology for highperformance wireless charging in future. However, in the current Rezence specification, one single TX coil is used in the power transmitting unit, i.e., only the SIMO MRCWPT is considered. Generally, deploying multiple TXs can help focusing their generated magnetic fields more efficiently toward one or more RXs simultaneously [4], thus achieving a magnetic beamforming gain, in a manner analogous to multiantenna beamforming in the farfield wireless information and/or power transfer based on electromagnetic (EM) wave radiation [8, 9, 10, 11]. It is worth noting that applying signal processing and optimization techniques for improving the efficiency of farfield WPT systems has recently drawn significant interests (see, e.g., the work on transmit beamforming design [12, 13], channel acquisition method [14, 15], waveform optimization [16], and power scheduling policy for WPT networks [17]). However, to our best knowledge, there has been no prior work on magnetic beamforming optimization under practical TX circuit constraints, for a MIMO MRCWPT system with arbitrary numbers of TXs and RXs, which motivates our work. The results of this paper can be potentially applied in e.g., the Rezence specification for the support of multiTX WPT for performance enhancement.
In this paper, as shown in Fig. 1, we consider a general MIMO MRCWPT system with multiple RXs and multiple TXs where the TXs’ source currents (or equivalently voltages) can be adjusted such that their induced magnetic fields are optimally combined at each of the RXs, to maximize the power delivered. We introduce the multiuser power region to characterize the optimal performance tradeoffs among the RXs, which constitutes all the achievable power tuples deliverable to all RXs subject to the given total consumed power constraint over all TXs as well as practical peak voltage and current constraints at individual TXs.
The main contributions of this paper are summarized as follows.

In order to characterize the optimal performance tradeoffs among all RXs by finding all the boundary points of the multiuser power region, we apply the technique of power profile. Specifically, we obtain each boundary point by maximizing the sumpower deliverable to all RXs subject to the minimum harvested power constraints at different RX loads which are proportionally set based on a given powerprofile vector. We propose an iterative algorithm to solve this problem, which requires to solve a TX sumpower minimization problem at each iteration to optimally allocate the TX currents.

For the special case of one single RX, identical TX resistances and without the TX peak voltage and current constraints, we show that the optimal current at each TX should be proportional to the mutual inductance between its TX coil and the RX coil. This optimal magnetic beamforming design for MISO MRCWPT system is analogous to the maximalratiotransmission (MRT) based beamforming in the farfield radiationbased WPT [9].

In general, the TX sumpower minimization problem is a nonconvex quadratically constrained quadratic programming (QCQP). For the case of one single RX, with arbitrary TX resistances and the peak voltage and current constraints at individual TXs applied, we show that the semidefinite relaxation (SDR) of the problem is tight, and thus the problem can be efficiently solved via the semidefinite programming (SDP) by using existing optimization software such as CVX [18]. For the general case with multiple RXs, based on SDR, we obtain two approximate solutions by applying the techniques of timesharing and randomization, respectively. In particular, for the special case without the TX peak and voltage constraints, the timesharing based solution is shown to be optimal.

For practical implementation of magnetic bemaforming, it is essential to obtain the magnetic channel knowledge on the mutual inductance between each pair of TX coil and RX coil. To this end, we propose a novel magnetic MIMO channel estimation scheme, which is shown to be efficient and accurate by simulations. The channel estimation and feedback design for MIMO or multiantenna based wireless communication systems has been extensively studied in the literature (see. e.g., [19] and the references therein). However, it is shown in this paper that the magnetic MIMO channel estimation problem in MRCWPT has a different structure, which cannot be directly solved by existing methods in wireless communication.

By extensive numerical results, we show that our proposed magnetic beamforming designs are practically effective, and can significantly enhance the energy efficiency as well as the multiuser performance tradeoff in MIMO MRCWPT, as compared to the benchmark scheme of uncoordinated WPT with fixed identical current at all TXs.
Notation  Meaning 

Number of TXs and RXs, respectively  
Index for TXs and RXs, respectively  
Operating angular frequency  
Phasor representation for complex voltage of TX  
Phasor representation for complex current, realpart and imaginarypart of current of TX , respectively  
, ,  TX current vector , its realpart and imaginarypart, respectively 
Phasor representation for complex current, realpart and imaginarypart of current of RX , respectively  
,  Selfinductance and capacitance of the th TX coil, respectively 
,  Selfinductance and capacitance of the th RX coil, respectively 
Total source resistance of the th TX  
Diagonal resistance matrix  
, ,  Parasitic resistance, load resistance and total resistance of RX , respectively 
,  Mutual inductance between TX and RX / TX with , respectively 
Vector of mutual inductance between RX and all TXs  
Rankone matrix for RX  
Impedance matrix, its realpart and imaginarypart, respectively  
Rankone matrix , with denoting the th column of  
Total power drawn from all TXs  
Power delivered to the load of RX  
Maximum total power drawn by all TXs  
,  Maximum amplitude of voltage and current of TX , respectively 
Powerprofile vector  
Rankone matrix with the th diagonal element being one and others zero  
Sumpower delivered to all RXs  
Rankone matrix  
Rank of optimal SDR solution  
Singular matrix of ,  
order diagonal matrix with diagonal elements given by eigenvalues of  
Transmission time of the th WPT slot in timesharing based solution  
Error of the th RX’s current in the th channeltraining slot 
The rest of this paper is organized as follows. Section II introduces the system model for MIMO MRCWPT. Section III presents the problem formulation to characterize the boundary points of the multiuser power region. Section IV presents the optimal and approximate solutions for the formulated problem under various setups. Section V presents the algorithms for magnetic MIMO channel estimation. Section VI provides the numerical results. Section VII concludes the paper.
The notations for main variables used in this paper are listed in Table I for the ease of reading. Moreover, we use the following math notations in this paper. means the operation of taking the absolute value. means that the matrix is positive semidefinite (PSD). means the operation of taking the real part. means the trace operation. is the union operation of sets. denotes the statistical expectation. means that the random vector follows the circularly symmetric complex Gaussian (CSCG) distribution with mean vector and covariance matrix . The and represent the transpose, conjugate, and conjugate transpose operations, respectively.
Ii System Model
As shown in Fig. 2, we consider a MIMO MRCWPT system with TXs each equipped with a single coil, and singlecoil RXs. We assume that the RXs are all legitimate users for wireless charging. Each TX , , is connected to a stable power source supplying sinusoidal voltage over time given by , with denoting the complex voltage and denoting the operating angular frequency. Let denote the steadystate current flowing through TX , with the complex current . The current produces a timevarying magnetic flux in the th TX coil, which passes through the coils of all RXs and induces timevarying currents in them. Let denote the steadystate current in the th RX coil, , with the complex current .
Let and denote the mutual inductance between the th TX coil and the th RX coil, and the mutual inductance between the th TX coil and the th TX coil with , respectively. The mutual inductance is a real number, either positive or negative, which depends on the physical characteristics of each pair of TX and RX coils such as their relative distance, orientations, etc. [4].^{1}^{1}1In this paper, the values of mutual inductances (i.e., magnetic channels) are assumed to be purely real, since our considered MRCWPT system operates under the nearfield condition for which EM wave radiation is negligible and hence the imaginarypart of each inductance value can be set as zero. Specifically, the negative sign of mutual inductance () indicates that the current induced at the coil of RX (TX ) due to the current flowing at the coil of TX is in the opposite of the reference direction assumed (as shown in Fig. 2, the reference current direction at each TX/RX is set to be clockwise in this paper for convenience). In this paper, we assume that the mutual coupling between any pair of RX coils is negligible, as shown in Table II later for our considered numerical example, due to their small sizes in practice and the assumption that they are well separated from each other.
We denote the selfinductance and the capacitance of the th TX coil (th RX coil) by () and (), respectively. The capacitance values are set as and , such that all TXs and RXs have the same resonant angular frequency, . Let denote the total source resistance of the th TX. Define the diagonal resistance matrix as . The resistance of each RX , denoted by , consists of the parasitic resistance and the load resistance , i.e., . The load is assumed to be purely resistive. It is also assumed that the load resistance is sufficiently larger than the parasitic resistance at each RX such that . This is practically required to ensure that most of the energy harvested by the coil at each RX can be delivered to its load.
In our considered MRCWPT system, we assume that there is a controller installed which can communicate with all TXs and RXs (e.g., using Bluetooth as in the Rezence specification) such that it can collect the information of all system parameters (e.g., RX loads and currents) required to design and implement magnetic beamforming. We also assume that the RXs all have sufficient initial energy stored in their batteries, which enables them to conduct the necessary current measurement and send relevant information to the central controller to implement magnetic beamforming. However, for simplicity, we ignore the energy consumed for such operations at RXs. Last, for convenience, we treat the complex TX currents ’s as design variables,^{2}^{2}2In practice, it may be more convenient to use voltage source instead of current source. Therefore, after designing the TX currents ’s, the corresponding voltages ’s can be computed and set by the controller accordingly (see (3) and (8)). Moreover, in the case of adjustable voltage sources, impedance matching can be conducted in series with the sources, each of which can be adjusted in real time to match the current flowing in its corresponding TX to the optimal value obtained by magnetic beamforming design. which can be adjusted by the controller in real time to realize adaptive magnetic beamforming.
By applying Kirchhoff’s circuit law to the th RX, we obtain its current as
(1) 
Denote the vector of all TX currents as . Moreover, denote the vector of mutual inductances between the th RX coil and all TX coils as , and define the rankone matrix . From (1), the power delivered to the load of the th RX is
(2) 
Similarly, by applying Kirchhoff’s circuit law to each TX , we obtain its source voltage as
(3) 
Next, we derive the total power drawn from all TXs in terms of the vector of TX currents . Let us define an impedance matrix as
(4) 
where the elements in and are respectively given by
(5)  
(6) 
Note that the matrices and are all symmetric, since . Denote the th column of the matrices by , respectively. We also define the rankone matrices . It can be shown that both and ’s are PSD matrices. The matrix can be also rewritten as
(7) 
Accordingly, the source voltage of each TX given in (3) can be equivalently reexpressed as
(8) 
From (4) and (8), the total power drawn from all TXs is given by
(9) 
Iii Problem Formulation
In this section, we first introduce the multiuser power region to characterize the optimal performance tradeoffs among all RXs in a MIMO MRCWPT system introduced in Section IIIA. Then, we formulate an optimization problem to find each boundary point of the power region corresponding to a given “powerprofile” vector.
Iiia Multiuser Power Region
In this subsection, we define the multiuser power region under practical circuit constraints at TXs. In particular, the power region consists of all the achievable power tuples that can be received by all RXs subject to the following constraints: the total power drawn by all TXs needs to be no larger than a given maximum power , i.e., ; the peak amplitude of the voltage (current ) at each TX needs to be no larger than a given threshold (), i.e., . In this case, it can be easily verified that the maximum transmit power at each TX is indeed capped by . Accordingly, to avoid the trivial case that the constraint is never active, we consider that holds in this paper. The power region is thus formally defined as
(10) 
where are given in (2), (8), and (9), respectively. Note that the union operation in (10) has considered the possibility that some power tuples may be achievable only through “timesharing (TS)” of a certain set of achievable power tuples each corresponding to a different set of feasible ’s and ’s.
Next, we apply the technique of powerprofile vector [5] to characterize all the boundary points of the power region, where each boundary power tuple corresponds to a Paretooptimal performance tradeoff among the RXs. Let denote the sumpower delivered to all RXs, i.e., . Accordingly, we set , where the coefficients ’s are subject to and . The vector is a given powerprofile vector that specifies the proportion of the sumpower delivered to each RX . With each given , the maximum achievable sumpower thus corresponds to a boundary point of the power region; Fig. 3 illustrates the characterization of the power region boundary via the power profile technique for the case of RXs.
IiiB Optimization Problem
In this subsection, we formulate an optimization problem to find different boundary points of the power region. Denote the dimensional complex space by , and let denote the rankone matrix with the th diagonal element being one and all other elements being zero.
From the definition in (10), each boundary point of the power region can be obtained by solving the following RX sumpower maximization problem with a given powerprofile vector (for the case when TS is not required to achieve the boundary point of the multiuser power region corresponding to the given power profile ; see Proposition 2 in Section IV for the case when TS is required),
(11a)  
s.t.  (11b)  
(11c)  
(11d)  
(11e) 
where the inequalities (11b), (11c) and (11e) are due to (2), (8), and (9), respectively. Given a powerprofile vector , can be solved by a bisection search over , where in each search iteration, it suffices to solve a feasibility problem that checks whether all constraints of can be satisfied for some given . The converged optimal value of is denoted by .
The feasibility problem can be equivalently solved by first obtaining the minimum sumpower drawn from all TXs by solving the following problem, denoted by , and then comparing it with the given total power constraint for all TXs, . Specifically, the TX sumpower minimization problem is given by
(12a)  
s.t.  (12b)  
(12c)  
(12d) 
To summarize, the overall algorithm for solving (P0) is given in Algorithm 1. Note that in the rest of this paper, we focus on solving problem . However, is in general a nonconvex QCQP problem [20] due to the constraints in (12b). Although solving nonconvex QCQPs is difficult in general [21], we study the optimal and approximate solutions to under various setups in Section IV. Notice that for solving , it is essential for the controller to have the knowledge of the mutual inductance values between any pair of TX coils as well as any pair of TX and RX coils. In practice, the TXTX mutual inductance is constant with fixed TX positions and thus can be measured offline and stored in the controller. However, due to the mobility of RXs (such as phones, tablets), the TXRX mutual inductance is timevarying in general and thus needs to be estimated periodically. The magnetic channel estimation problem will be addressed later in Section V.
Last, note that an alternative approach to characterize the boundary of the multiuser power region is to solve a sequence of weighted sumpower maximization (WSPMax) problems for the RXs. Compared to the TX sumpower minimization problem with the given RX minimum load power constraints, the WSPMax problem with the given maximum total TX power can be considered as its “dual” problem. In practice, how to select weights in WSPMax so as to satisfy the minimum load power requirement at each RX is challenging. Hence, in this paper, we study due to its practical usefulness in satisfying any given RX load power requirements.
Iv Solutions to Problem ()
In this section, we first present the optimal solution to () for the special case without TX peak voltage and current constraints (12c) and (12d), and then study the solution to () for the general case with all constraints.
Iva Optimal Solution to () without Peak Voltage and Current Constraints
In this subsection, we consider () for the ideal case without the TX peak voltage and current constraints given in (12c) and (12d), respectively, to obtain useful insights and the performance limit of magnetic beamforming.
Denote the dimensional real space by . Let , where . It is then observed that the realpart and the imaginarypart contribute in the same way to the total TX power in (12a) as well as the delivered load power in (12b), since both and ’s are symmetric matrices. As a result, we can set without loss of generality and adjust only, i.e., we need to solve
(13a)  
s.t.  (13b) 
Denote the space of order real matrices by . Let . The SDR of is thus given by
(14a)  
s.t.  
(14b)  
(14c) 
In general, is a convex relaxation of by dropping the rankone constraint on . This relaxation is tight, if and only if the solution obtained for , denoted by , is of rank one. In the following, we discuss the solutions to as well as that for for the two cases with one single RX and multiple RXs, respectively.
IvA1 SingleRX Case
Let denote the order identity matrix. For the case of single RX (i.e., RX with ), the optimal solution to is obtained in closedform as follows.
Theorem 1.
For the case of , the optimal solution to is , where is a constant such that the constraint (13b) holds with equality, and is the eigenvector associated with the minimum eigenvalue, denoted by , of the matrix
(15) 
where is chosen such that . Particularly, for the case of identical TX resistances, i.e., with , the optimal solution to () is simplified to
(16) 
Proof.
Please refer to Appendix A. ∎
Theorem 1 implies that for the case of single RX and identical TX resistances, the optimal current of each TX is proportional to the mutual inductance between the TX and RX . This is analogous to the maximalratiotransmission (MRT) based beamforming in the farfield wireless communication [8]. However, magnetic beamforming operates in the nearfield and thus the phase of each TX current only needs to take the value of or , i.e., the current is a positive or negative real number depending on its positive or negative mutual inductance with the RX, while in wireless communication beamforming operates over the farfield, and as a result, the beamforming weight at each transmit antenna needs to be of the opposite phase of that of the wireless channel, which can be an arbitrary value within and .
IvA2 MultipleRX Case
For the general case of multiple RXs, is a separable SDP with constraints. We directly obtain the following result from [22, Thm. 3.2].
Proposition 1.
For the case of , the rank of the optimal solution to is upperbounded by
(17) 
From Lemma 1, we have the following corollary.
Corollary 1.
For , the SDR in is tight, i.e., the optimal solution to is always rankone, which is given by . The optimal solution to is thus .
Note that for , the optimal solution of to may have a rank higher than 1, which is thus not feasible to . In general, can be efficiently solved by existing software such as CVX [18].
In the following, we propose a timesharing (TS) based scheme to achieve the same optimal value of problem . Let be the rank of the obtained solution for , i.e., , with . Denote the singularvaluedecomposition (SVD) of by , where is an matrix with and is an order diagonal matrix with the diagonal elements given by .
To perform magnetic beamforming in a TS manner, we divide WPT into orthogonal time slots, indexed by , where slot takes a portion of the total transmission time given by , with and . In particular, we set
(18) 
In the th slot, the TX current vector is then given by
(19) 
We have the following result on the TS scheme.
Proposition 2.
Proof.
With the TS scheme, the total delivered power to each RX over time slots is
(20) 
and the total transmit power is given by
(21) 
Clearly, by using the above TS scheme, the delivered power and the total transmit power are the same as those by using the solution to . Hence, the proof is completed. ∎
In general, since the optimal value of is a lower bound of that of , the above TS scheme thus achieves a TX sumpower that is no larger than the the optimal value of . Thus, the resulting solution can be considered to be optimal for (P2) if TS is allowed. Notice that in such cases, TS is required to achieve the boundary point of the multiuser power region with the given power profile vector . In summary, the aforementioned procedure to solve is given in Algorithm 2.
IvB Solution to () with All Constraints
In this subsection, we consider () with all the constraints. Denote the space of order complex matrices by . Let . The SDR of is given by
(22a)  
s.t.  
(22b)  
(22c)  
(22d)  
(22e) 
Like , is also convex. By exploiting its structure, we obtain the following result on the rank of the optimal solution to .
Theorem 2.
The rank of the optimal solution to is upperbounded by
(23) 
Proof.
Please refer to Appendix B. ∎
The optimal solution to can be efficiently obtained by CVX [18]. Moreover, from Theorem 2, we directly obtain the following corollary.
Corollary 2.
For in the case of , the SDR in is tight, i.e., the optimal solution to is always of rankone with , where is thus the optimal solution to .
For the general case of , if the solution to is of rankone with , then is the optimal solution to ; however, for the case of , in the following we propose two approximate solutions for based on TS and randomization, respectively.
IvB1 TSbased Solution
We note that the TS scheme proposed in Section IVA2 for cannot be directly applied to due to the additional peak voltage and current constraints. This is because the current solutions given in (19) in general may not satisfy these peak constraints at all TXs over all the time slots. To tackle this problem, we treat the time allocation ’s and the current scaling factors, denoted by with , for all slots as design variables, such that all peak constraints can be satisfied over all slots. Recall ’s are subject to , and ; and with a little abuse of notations, we still use ’s to denote the singular vectors obtained from the SVD of the optimal solution to , similar to those defined for to . In the th slot, the TX current vector is then set as
(24) 
Let , and . Moreover, we denote , and nonnegative constants , , , and . We then formulate the following problem to obtain the TSbased solution for .
(25a)  
s.t.  
(25b)  
(25c)  
(25d)  
(25e)  
(25f) 
We define a set of new variables as . Problem is thus rewritten as the following linearprogramming (LP), which can be efficiently solved by e.g., CVX [18].
(26a)  
(26b)  
(26c)  
(26d)  
(26e)  
(26f) 
If the above is feasible, there is a feasible TSbased solution for ; otherwise is regarded as infeasible, which implies that the RX sumpower needs to be decreased in the next bisection search iteration in Algorithm 1.
IvB2 Randomizationbased Solution
The randomization technique is a wellknown method applied to extract a feasible approximate QCQP solution from its SDR solution. Before presenting the proposed randomizationbased solution, we first describe the steps for generating feasible random vectors from SDR solution. Recall the SVD of as . Define . A random vector is specifically generated as follows:
(27) 
where , with representing an allzero column vector of length .
To further generate a random vector that is feasible to , we scale the vector by with , i.e., . If the resulting problem shown as follows is feasible, a feasible is thus found; otherwise no feasible vector can be obtained from this .
(28a)  
s.t.  (28b)  
(28c)  
(28d) 
The proposed algorithm for obtaining the randomizationbased solution is summarized as Algorithm 3.
V Magnetic Channel Estimation
For implementation of magnetic beamforming in practice, it is necessary for the central controller at the TX side to estimate the mutual inductance between each pair of TX coil and RX coil, namely magnetic MIMO channel estimation. Note that in this paper, the mutual inductances ’s are assumed to be quasistatic, i.e., they remain constant over a certain block of time, but may change from one block to another, since the RXs are mobile devices in general. Hence, ’s need to be estimated periodically over time. For practical implementation, at the beginning of each transmission period, we treat all the magnetic channels ’s as unknown real parameters. For convenience, we denote the magnetic channel matrix by with elements given by ’s. In the next, we first consider magnetic MIMO channel estimation for the ideal case with perfect RX current knowledge and then the practical case with imperfect current knowledge.
We assume that each RX can feed back its measured current to the central controller by using existing communication module. One straightforward method to estimate is given in [5], where by switching off all the other TXs and RXs, TX can estimate with RX based on the current measured and fed back by RX . However, this method may not be efficient for estimating the magnetic MIMO channel , since it requires synchronized on/off operations of all TXs and RXs and also needs at least iterations to estimate all ’s. Alternatively, we propose more efficient methods that can simultaneously estimate the magnetic MIMO channel in () time slots. In the th slot, we apply a source voltage on TX , and the current is measured by TX . From Kirchhoff’s circuit laws, the voltage of TX is
(29) 
In practice, randomly generated voltage values are assigned over different TXs as well as over different time slots.
Define the matrices and with elements given by ’s and ’s, respectively. Moreover, define the matrix with elements given by ’s and the matrix with elements given by
(30) 
Since the fixed TXTX mutual inductance can be measured offline and the TX currents ’s as well as voltages ’s can be measured by the TXs, the matrices and are assumed to be known by the central controller perfectly. From (29), the voltages at all TXs over time slots can be written in the following matrixform
(31) 
Let . The voltage matrix in (31) can be rewritten as
(32) 
With known and , the matrix is known by the central controller.
Va Channel Estimation with Perfect RXCurrent Knowledge
For the case with perfect RXcurrent knowledge of at the central controller, it suffices to use