Magnetic MIMO Signal Processing and Optimization for Wireless Power Transfer

Magnetic MIMO Signal Processing and Optimization for Wireless Power Transfer

Gang Yang, , Mohammad R. Vedady Moghadam, , and Rui Zhang, Manuscript received August 21, 2016; revised December 27, 2016 and February 1, 2017; accepted February 9, 2017. The work of G. Yang was supported in part by the National Natural Science Foundation of China under Project 61601100. This work was presented in part at the IEEE International Conference on Acoustics, Speech, and Signal Processing, Shanghai, China, March 20-25, 2016.G. Yang was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583. He is now with the National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: yanggang@uestc.edu.cn).M. R. V. Moghadam is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (email: elemrvm@nus.edu.sg).R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, and also with the Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore 138632 (e-mail: elezhang@nus.edu.sg).
Abstract

In magnetic resonant coupling (MRC) enabled multiple-input multiple-output (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 single-coil 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 multi-user MIMO MRC-WPT system. We introduce the multi-user 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 sum-power deliverable to all RXs subject to their minimum harvested power constraints, which are proportionally set based on a given power-profile 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 single-RX setup in closed-form as well as that for the multi-RX setup by applying the techniques of semidefinite relaxation (SDR) and time-sharing. In general, the problem is a non-convex 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 time-sharing 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 multi-user performance trade-off in MIMO MRC-WPT systems compared to the benchmark scheme of uncoordinated WPT with fixed identical TX current.

{keywords}

Wireless power transfer, magnetic resonant coupling, magnetic MIMO, magnetic beamforming, magnetic channel estimation, multi-user power region, time-sharing, semidefinite relaxation.

I Introduction

\PARstart

Near-field 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. Near-field WPT can be realized by inductive coupling (IC) for short-range applications within centimeters, or magnetic resonant coupling (MRC) for mid-range applications up to a couple of meters. Although short-range WPT has been in widely commercial use (e.g., electric toothbrushes), mid-range 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 (MRC-WPT) has grown fast and substantially (see e.g., [1] and the references therein).

MRC-WPT with generally multiple TXs and/or multiple RXs has been studied in the literature [2, 3, 4, 5, 6, 7]. Under the multiple-input single-output (MISO) setup, [2] has studied an MRC-WPT 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 MRC-WPT 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 muti-RX 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 single-input multiple-output (SIMO) setup, an MRC-WPT 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 near-far distances to the TX. For the general multiple-input multiple-output (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 two-way Bluetooth communication between the charger and devices for real-time charging control. These features make Rezence a promising technology for high-performance 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 MRC-WPT 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 multi-antenna beamforming in the far-field 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 far-field 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 MRC-WPT 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 multi-TX WPT for performance enhancement.

In this paper, as shown in Fig. 1, we consider a general MIMO MRC-WPT 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 multi-user power region to characterize the optimal performance trade-offs 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.

Fig. 1: Example setup of our considered MIMO MRC-WPT system: a rectangular table with five built-in wireless chargers attached below its surface and four receivers randomly placed on it for wireless charging.

The main contributions of this paper are summarized as follows.

  • In order to characterize the optimal performance trade-offs among all RXs by finding all the boundary points of the multi-user power region, we apply the technique of power profile. Specifically, we obtain each boundary point by maximizing the sum-power deliverable to all RXs subject to the minimum harvested power constraints at different RX loads which are proportionally set based on a given power-profile vector. We propose an iterative algorithm to solve this problem, which requires to solve a TX sum-power 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 MRC-WPT system is analogous to the maximal-ratio-transmission (MRT) based beamforming in the far-field radiation-based WPT [9].

  • In general, the TX sum-power minimization problem is a non-convex 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 time-sharing and randomization, respectively. In particular, for the special case without the TX peak and voltage constraints, the time-sharing 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 multi-antenna 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 MRC-WPT 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 multi-user performance trade-off in MIMO MRC-WPT, 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, real-part and imaginary-part of current of TX , respectively
, , TX current vector , its real-part and imaginary-part, respectively
Phasor representation for complex current, real-part and imaginary-part of current of RX , respectively
, Self-inductance and capacitance of the -th TX coil, respectively
, Self-inductance 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
Rank-one matrix for RX
Impedance matrix, its real-part and imaginary-part, respectively
Rank-one 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
Power-profile vector
Rank-one matrix with the -th diagonal element being one and others zero
Sum-power delivered to all RXs
Rank-one 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 time-sharing based solution
Error of the -th RX’s current in the -th channel-training slot
TABLE I: List of main variable notations and their meanings

The rest of this paper is organized as follows. Section II introduces the system model for MIMO MRC-WPT. Section III presents the problem formulation to characterize the boundary points of the multi-user 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 MRC-WPT system with TXs each equipped with a single coil, and single-coil 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 steady-state current flowing through TX , with the complex current . The current produces a time-varying magnetic flux in the -th TX coil, which passes through the coils of all RXs and induces time-varying currents in them. Let denote the steady-state 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].111In this paper, the values of mutual inductances (i.e., magnetic channels) are assumed to be purely real, since our considered MRC-WPT system operates under the near-field condition for which EM wave radiation is negligible and hence the imaginary-part 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.

Fig. 2: System model of MIMO MRC-WPT.

We denote the self-inductance 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 MRC-WPT 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,222In 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 rank-one 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 rank-one 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 re-expressed as

(8)

From (4) and (8), the total power drawn from all TXs is given by

(9)

Note that from (5), it follows that in (9) in general depends on the mutual inductances ’s between all TXs and RXs, but does not depend on the mutual inductances ’s among the TXs.

Iii Problem Formulation

In this section, we first introduce the multi-user power region to characterize the optimal performance trade-offs among all RXs in a MIMO MRC-WPT system introduced in Section III-A. Then, we formulate an optimization problem to find each boundary point of the power region corresponding to a given “power-profile” vector.

Iii-a Multi-user Power Region

In this subsection, we define the multi-user 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 “time-sharing (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 power-profile vector [5] to characterize all the boundary points of the power region, where each boundary power tuple corresponds to a Pareto-optimal performance trade-off among the RXs. Let denote the sum-power delivered to all RXs, i.e., . Accordingly, we set , where the coefficients ’s are subject to and . The vector is a given power-profile vector that specifies the proportion of the sum-power delivered to each RX . With each given , the maximum achievable sum-power 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.

Iii-B 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 rank-one matrix with the -th diagonal element being one and all other elements being zero.

Fig. 3: Illustration of characterization of power region boundary via the technique of power profile in a two-user case.

From the definition in (10), each boundary point of the power region can be obtained by solving the following RX sum-power maximization problem with a given power-profile vector (for the case when TS is not required to achieve the boundary point of the multi-user 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 power-profile 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 sum-power 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 sum-power 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 non-convex QCQP problem [20] due to the constraints in (12b). Although solving non-convex 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 TX-TX 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 TX-RX mutual inductance is time-varying in general and thus needs to be estimated periodically. The magnetic channel estimation problem will be addressed later in Section V.

1:  Initialization: , , and a small positive number ( is set in our simulations).
2:  while  do
3:     .
4:     if  is not feasible then
5:         Go to step 8.
6:     else if  then
7:         Obtain the optimal solutions as .
8:         .
9:     else
10:         Obtain the optimal solutions as .
11:         .
12:     end if
13:  end while
14:  return  the optimal value and solution of as and , respectively.
Algorithm 1 : Algorithm for

Last, note that an alternative approach to characterize the boundary of the multi-user power region is to solve a sequence of weighted sum-power maximization (WSPMax) problems for the RXs. Compared to the TX sum-power 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.

Iv-a 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 real-part and the imaginary-part 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 rank-one 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.

Iv-A1 Single-RX Case

Let denote the -order identity matrix. For the case of single RX (i.e., RX with ), the optimal solution to is obtained in closed-form 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 maximal-ratio-transmission (MRT) based beamforming in the far-field wireless communication [8]. However, magnetic beamforming operates in the near-field 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 far-field, 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 .

Iv-A2 Multiple-RX 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 upper-bounded 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 rank-one, 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 time-sharing (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 singular-value-decomposition (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.

For the case without peak voltage and current constraints, the TS scheme given in (18) and (19) achieves the same optimal value of .

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 sum-power 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 multi-user power region with the given power profile vector . In summary, the aforementioned procedure to solve is given in Algorithm 2.

1:  Input parameters: , for
2:  Solve , obtain its solution as .
3:  if  then
4:     return  . (TS is not applied)
5:  else
6:     return  , and , for . (TS is applied)
7:  end if
Algorithm 2 : Algorithm for with TS

Iv-B 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 upper-bounded 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 rank-one with , where is thus the optimal solution to .

For the general case of , if the solution to is of rank-one 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.

Iv-B1 TS-based Solution

We note that the TS scheme proposed in Section IV-A2 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 TS-based 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 linear-programming (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 TS-based solution for ; otherwise is regarded as infeasible, which implies that the RX sum-power needs to be decreased in the next bisection search iteration in Algorithm 1.

Iv-B2 Randomization-based Solution

The randomization technique is a well-known method applied to extract a feasible approximate QCQP solution from its SDR solution. Before presenting the proposed randomization-based 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 all-zero 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 randomization-based 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 quasi-static, 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.

1:  Initialization: the solution to , a large positive integer (set as in our simulations), set .
2:  Compute the SVD of as .
3:  for  do
4:     Generate a random vector , where .
5:     if the problem (28) is feasible,  then
6:        Obtain .
7:        .
8:     end if
9:  end for
10:  return   if ; otherwise, declare is infeasible.
Algorithm 3 : Randomization-based Solution for

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 TX-TX 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 matrix-form

(31)

Let . The voltage matrix in (31) can be rewritten as

(32)

With known and , the matrix is known by the central controller.

V-a Channel Estimation with Perfect RX-Current Knowledge

For the case with perfect RX-current knowledge of at the central controller, it suffices to use