Beamforming Optimization for MIMO Wireless Power Transfer with Nonlinear Energy Harvesting: RF Combining versus DC Combining

THE Internet of Things (IoT) connects sensors, actuators, machines, and other objects to Internet so that processes and services such as manufacturing, monitoring, transportation, and healthcare can be enhanced [1]. Wireless Sensor Networks (WSN) and radio frequency identification (RFID) network can be loosely viewed as possible parts of the IoT. However, the devices of IoT might be deployed in unreachable or hazard environment such that battery replacement or recharging becomes inconvenient. Besides, battery replacement or recharging becomes prohibitive and unsustainable for a large number of IoT devices. Therefore, powering the devices of IoT in a reliable, controllable, user-friendly, and cost-effective manner remains a challenging issue.

Far-field wireless power transfer (WPT) via radio-frequency has a long history and nowadays it attracts more and more attention as a promising technology for overcoming this issue. WPT utilizes a dedicated source to radiate electromagnetic energy through a wireless channel and a rectifying antenna (rectenna) at the receiver to receive and convert this energy into DC power. The major challenge of far-field WPT is to increase the DC power level at the output of the rectenna without increasing the transmit power, and to power devices located tens to hundreds of meters away from the transmitter. To overcome this challenge, the vast majority of the technical efforts in the literature have been devoted to the design of efficient rectenna [2]-[4].

Another promising approach to increase the output DC power level is to design efficient WPT signals [5]. Interestingly, it was observed through RF measurements in the RF literature that the RF-to-DC conversion efficiency is a function of the input waveforms. In [6], [7], a multisine signal excitation is shown through analysis, simulations and measurements to enhance the DC power and RF-to-DC conversion efficiency over a single sinewave signal. In [8], various input waveforms (OFDM, white noise, chaotic) are considered and experiments show that waveforms with high peak to average power ratio (PAPR) increase RF-to-DC conversion efficiency. However, the main limitation of those methods is not only the lack of a systematic approach to design waveforms, but also the fact that they operate without Channel State Information (CSI) at the Transmitter (CSIT) and Receiver (CSIR). Inspired by communications, CSI is also very helpful to WPT to adjust dynamically the transmit signal as a function of the channel state and the multipath fading. The first systematic analysis, design and optimization of waveforms for WPT was conducted in [9]. Those waveforms are adaptive to the CSI and jointly exploit a beamforming gain, the frequency-selectivity of the channel and the rectenna nonlinearity so as to maximize the amount of harvested DC power. Since then, further enhancements have been made to waveform optimization adaptive to CSI with the objective to reduce the complexity of the design and extend to large scale multi-antenna multi-sine WPT [10]-[12], to account for limited feedback [13], to energize multiple devices (multi-user setting) [10], [14], to transfer information and power simultaneously [15]-[17], and enable efficient wireless powered communications [18], [19].

In addition to optimizing the rectenna circuit and waveform, the use of multiple rectennas, also known as multiport rectennas, at the receiver can increase the output DC power level. In [20]-[24], multiport rectennas have been designed for ambient RF energy harvesting, which is similar to WPT but does not have a controllable and dedicated transmitter. It was shown that using multiport rectennas can linearly increase the output DC power level with the number of rectennas at the receiver while keeping a compact multiport rectenna size as single rectenna. Two combinings, DC and RF combinings, for the multiple rectennas at the receiver has been investigated in [25]. However, the investigation is at the level of RF circuit design and does not consider the impact on communication and signal designs, including CSI acquisition, as well as adaptive waveform and beamforming optimization.

In this paper, we utilize multiple rectennas at the receiver for WPT to increase the output DC power level. Together with the multiple antennas available at the transmitter, a multiple-input and multiple-output (MIMO) WPT system is formed. The contributions of the paper are summarized as follow.

First, we analyze the model of MIMO WPT system with the rectenna nonlinearity. Two combining schemes for multiple rectenna at the receiver, DC and RF combinings, are modeled and analyzed. This contrasts with prior works [9]-[19] that assumed a single rectenna per device.

Second, for DC combining, assuming perfect CSIT can be attained and making use of the rectenna model, we optimize the beamforming for multiple antennas at the transmitter in MIMO WPT system. We formulate an optimization problem to adaptively change the beamforming as a function of the CSIT so as to maximize the total DC power of all rectenna outputs. By solving a non-convex posynomial maximization problem with semi-definite relaxation (SDR), we optimize the beamforming and we also numerically show that the proposed algorithm finds a stationary point of the problem for the tested channel realizations.

Third, for RF combining, assuming perfect CSIT and CSIR and making use of the rectenna model, we optimize the beamformers at both the transmitter and the receiver in the MIMO WPT system. The global optimal beamforming weights at both the transmitter and the receiver are obtained in closed form. Additionally, a practical RF combining circuit using RF phase shifter and RF power combiner is proposed for the MIMO WPT system. Assuming perfect CSIT and CSIR and making use of the rectenna model, the analog receive beamforming is optimized by solving a non-convex optimization problem with SDR. We numerically show that SDR is very tight and the proposed algorithm can find nearly the global optimal solution for the tested channel realizations.

Fourth, scaling laws of the output DC power for multiple-input single-output (MISO) WPT system and single-input multiple-output (SIMO) with DC and RF combinings are analytically derived as a function of the number of transmit antennas $M$ and the number of receive antennas $Q$. Those scaling laws confirm the benefits of using multiple antennas at the transmitter or receiver and show that different combining schemes leads to different output DC power. They also highlight that RF combining significantly outperforms DC combining since the receive beamforming in RF combining leverages the rectenna nonlinearity more efficiently.

Fifth, the beamforming for DC and RF combinings, adaptive to the CSI and accounting for the rectenna nonlinearity, are shown through realistic circuit evaluations to boost the output DC power level. Moreover, the circuit simulations show that RF combining outperforms DC combining in terms of the output DC power level since the receive beamforming in RF combining leverages the rectenna nonlinearity more efficiently.

Sixth, the impact of DC combining versus RF combining on the WPT system design is also discussed and a comprehensive comparison of DC and RF combinings is provided.

It is also worth contrasting our contributions with the recent MIMO WPT systems proposed in [26], [27]. Our work is different from [26] in several aspects: 1) the nonlinearity of the rectenna is not considered in [26] while this work shows that exploiting such nonlinearity is important to boost the output DC power, and 2) only DC combining is considered in [26] while this work considers both DC and RF combinings and shows that RF combining can boost the output DC power by leveraging the nonlinearity of the rectenna. On the other hand, our work is different from [27] in several aspects: 1) this work focuses on the low power (e.g. below mW input power) WPT scenario so that a nonlinear rectenna model based on Taylor expansion of the diode I-V characteristics is used while [27] focuses on medium and large power WPT scenario so that a sigmoidal function-based rectenna model which reflects the property of saturated output dc power is used, 2) the generic architecture in [27] uses many power splitters and power combiners, which in reality causes high insertion loss and is not suitable for low power WPT scenario, so this work focuses on the two practical combining schemes with low complexity for the low power WPT MIMO system, and 3) this work proposes a more practical RF combining circuit consisting of phase shifters and RF power combiner and the optimization for the phase shifts in the practical RF combining is also provided.

In addition, we would like to clarify the differences between the beamforming design in this work and the waveform design in [9]. Both of them are effective approaches to increase the output DC power in WPT system. However, waveform design focuses on how to allocate power (with adaptive magnitude and phase) at different frequency tones while beamforming design focuses on how to allocate power (with adaptive magnitude and phase) at different transmit antennas and how to combine power from different receive antennas in RF combining. In other words, waveform design is considered from the perspective of frequency domain while beamforming design is from spatial domain. Besides, waveform design is performed at the transmitter. However, in MIMO WPT system, beamforming design is not only performed at the transmitter but can be also performed at the receiver (in the case of RF combining). Interestingly, it remains a future work to jointly optimize the waveform design and the beamforming design to further improve the output DC power.

As a takeaway message, this paper again shows the crucial role played by the rectenna nonlinearity. It is well understood from [9], [15], [28] that nonlinearity favors a different waveform, modulation, input distribution and transceiver architecture as well as a different use of the RF spectrum in WPT, and wireless information and power transfer (WIPT). What this paper further highlights is that nonlinearity also changes how to make use of multiple receive antennas in WPT, and therefore how to design the corresponding beamformers. As shown in this paper, if we ignore the nonlinearity of the rectenna and assume the (inaccurate) linear model of the rectenna [26], [29], DC combining and RF combining would lead to the same output DC power.

Organization: Section II introduces the MIMO WPT system model and Section III briefly revisits the rectenna models. Section IV and Section V tackle the beamforming optimization for DC and RF combinings, respectively. Section VI demonstrates the beneficial role of the rectenna nonlinearity. Section VII analytically derives the scaling laws for the MIMO WPT system. Section VIII evaluates the performance and Section IX discusses the impact of DC and RF combinings on the WPT system design. Section X concludes the work.

Notations: Bold lower and upper case letters stand for vectors and matrices, respectively. A symbol not in bold font represents a scalar. $\mathscr{\mathcal{E}}\left\{\cdot\right\}$ refers to the expectation/averaging operator. $\Re\left\{x\right\}$, $\Im\left\{x\right\}$, and $\left|x\right|$ refer to the real part, imaginary part, and modulus of a complex number $x$. $\left\|\mathbf{x}\right\|$ and $\left\|\mathbf{x}\right\|_{1}$ refer to the $l_{2}$-norm and $l_{1}$-norm of a vector $\mathbf{x}$, respectively. $\arg\left(\mathbf{x}\right)$ refers to a vector with each element being the phase of the corresponding element in a vector $\mathbf{x}$. $\mathbf{X}^{T}$, $\mathbf{X}^{H}$, $\left[\mathbf{X}\right]_{ij}$, $\mathrm{Tr}\left(\mathbf{X}\right)$, and $\mathrm{rank}\left(\mathbf{X}\right)$ refer to the transpose, conjugate transpose, $\left(i,j\right)$th element, trace, and rank of a matrix $\mathbf{X}$, respectively. $\mathbf{X}\succeq 0$ means that $\mathbf{X}$ is positive semi-definite. $\chi_{k}^{2}$ denotes the chi-square distribution with $k$ degrees of freedom. $\mathcal{CN}\left(\mathbf{0},\mathbf{\Sigma}\right)$ denotes the distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean vector $\mathbf{0}$ and covariance matrix $\mathbf{\Sigma}$ and $\sim$ stands for “distributed as”. $\mathbf{I}$ and $\mathbf{0}$ denote an identity matrix and an all-zero vector, respectively. log is in base $e$.

We consider a point-to-point MIMO WPT system. There are $M$ antennas at the transmitter and $Q$ antennas at the receiver. The transmitted signal at time $t$ on the $m$th transmit antenna can be expressed by

where $\omega_{c}$ denotes the center frequency, $w_{T,m}=s_{T,m}e^{j\phi_{T,m}}$ denotes the complex weight with $s_{T,m}$ and $\phi_{T,m}$ referring to the amplitude and phase of the signal on the $m$th transmit antenna, and we take the real part of $w_{T,m}e^{j\omega_{c}t}$ to convert a phasor into a sinusoidal function of time $t$. Stacking up all transmit signals, we can write the transmit signal vector as

where the complex weights are collected into a vector $\mathbf{w}_{T}=\left[w_{T,1},w_{T,2},...,w_{T,M}\right]^{T}$. The transmitter is subject to a transmit power constraint given by

where $P$ denotes the transmit power and the scaler $\frac{1}{2}$ is due to that the average power of the sinusoidal function $x_{m}\left(t\right)$ is $\frac{1}{2}\left|w_{T,m}\right|^{2}$.

The signals transmitted by the multiple transmit antennas propagate through a wireless channel. The received signal at the $q$-th receive antenna can be expressed as

where $\mathbf{h}_{q}=\left[h_{q1},h_{q2},...,h_{qM}\right]$ denotes the channel vector (a row vector) for the $q$-th receive antenna with $h_{qm}$ referring to the complex channel gain between the $m$th transmit antenna and the $q$th receive antenna. We collect all $\mathbf{h}_{q}$ into a matrix $\mathbf{H}=\left[\mathbf{h}_{1}^{T},\mathbf{h}_{2}^{T},...,\mathbf{h}_{Q}^{T}\right]^{T}$ where $\mathbf{H}$ represents the $Q\times M$ channel matrix of the MIMO WPT system. We assume that the channel matrix $\mathbf{H}$ is perfectly known to the transmitter.

We briefly revisit two simple and tractable models of the rectenna circuit derived in the past literatures [9], [10]. The goal of describing two different models is to emphasize the rectenna nonlinearity in WPT systems. Those two models account for the rectenna nonlinearity through the higher order terms in the Taylor expansion of the diode I-V characteristics while having a simple and tractable expression. Interestingly, those two models are shown to be equivalent in terms of optimization even though they rely on different physical assumptions [10].

Consider a rectifier with input impedance $R_{\mathrm{in}}$ connected to a receive antenna as shown in Fig. 1. The signal $y\left(t\right)$ impinging on the antenna has an average power $P_{\mathrm{av}}=\mathscr{\mathcal{E}}\left\{y\left(t\right)^{2}\right\}$. The receive antenna is assumed lossless and modeled as an equivalent voltage source $v_{s}\left(t\right)$ in series with an impedance $R_{\mathrm{ant}}=50\,\Omega$ as shown in Fig. 1. With perfect matching ($R_{\mathrm{in}}=R_{\mathrm{ant}}$), the input voltage of the rectifier $v_{\mathrm{in}}\left(t\right)$ can be related to the received signal $y\left(t\right)$ by $v_{\mathrm{in}}\left(t\right)=y\left(t\right)\sqrt{R_{\mathrm{ant}}}$.

A rectifier is always made of a nonlinear rectifying component such as diode followed by a low pass filter with load [2], [3], [21]-[24] as shown in Fig. 1. The current $i_{d}\left(t\right)$ flowing through an ideal diode (neglecting its series resistance) relates to the voltage drop across the diode $v_{d}\left(t\right)$ = $v_{\mathrm{in}}\left(t\right)$ − $v_{\mathrm{out}}\left(t\right)$ as $i_{d}\left(t\right)=i_{s}\left(e^{\frac{v_{d}\left(t\right)}{nv_{t}}}-1\right)$ where $i_{s}$ is the reverse bias saturation current, $v_{t}$ is the thermal voltage, $n$ is the ideality factor (assumed equal to 1.05). Based on the Taylor expansion of the diode I-V characteristics and some physical assumptions, the analysis of the rectifier circuit is simplified and therefore two simple and tractable models are provided as shown in the following subsections.

Consider the DC combining scheme for the multiple receive antenna system as shown in Fig. 2. Each receive antenna is connected to a rectifier so that the RF signal received by each antenna is individually rectified. Using the aforementioned voltage model of the nonlinear rectenna, the output DC voltage of the $q$th rectifier (connected to the $q$th receive antenna) is given by

where $\mathscr{\mathcal{E}}\left\{y_{q}\left(t\right)^{i}\right\}$ is given by

with $\zeta_{i}=\frac{1}{2\pi}\int_{0}^{2\pi}\sin^{i}t\>\textrm{d}t$ and specifically $\zeta_{2}=\frac{1}{2}$, $\zeta_{4}=\frac{3}{8}$, and $\zeta_{6}=\frac{5}{16}$.

The output DC power of all rectifiers are combined together by a DC combining circuit such as multiple-input and multiple-output (MIMO) switching DC-DC converter [32]-[34] as shown in Fig. 2. The total output DC power is then given by $P_{\mathrm{out}}=\sum_{q=1}^{Q}v_{\mathrm{out},q}^{2}/R_{L}$ where we assume each rectifier has the same load $R_{L}$. Therefore, we aim to maximize the total output DC power subject to the transmit power constraint, which can be formulated as

Observing the expression of the objective function $P_{\mathrm{out}}$, we find that it is hard to determine whether $P_{\mathrm{out}}$ monotonically increases/decreases with $\Re\left\{w_{T,m}\right\}$ or $\Im\left\{w_{T,m}\right\}$. Hence, we introduce the auxiliary variables $r_{q}=\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$ where $r_{q}>0$ and we equivalently rewrite the total output DC power as

which is a posynomial so that we express it in a compact form that $P_{\mathrm{out}}=\sum_{k=1}^{K}g_{k}\left(\mathbf{r}\right)$ where $K$ is the number of monomials in the posynomial and $g_{k}\left(\mathbf{r}\right)=\rho_{k}r_{1}^{\xi_{1k}}r_{2}^{\xi_{2k}}\cdots r_{Q}^{\xi_{Qk}}$ is the $k$th monomial with $\mathbf{r}=\left[r_{1},r_{2},...r_{Q}\right]^{T}$ where $\xi_{1k}$, …, $\xi_{Qk}$, and $\rho_{k}$ are constants and $\rho_{k}>0$. When $n_{0}=4$ (which is considered in Section VIII), we have $P_{\mathrm{out}}=\frac{1}{R_{L}}\sum_{q=1}^{Q}\left(\beta_{2}^{2}\zeta_{2}^{2}r_{q}^{2}+2\beta_{2}\beta_{4}\zeta_{2}\zeta_{4}r_{q}^{3}+\beta_{4}^{2}\zeta_{4}^{2}r_{q}^{4}\right)$ and there are $K=3Q$ monomials. Therefore, we equivalently rewrite the problem (10)-(11) as

We find that the objective function (13) monotonically increases with $r_{q}$. Hence, the constraint $r_{q}=\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$ can be replaced with $r_{q}\leq\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$ without affecting the optimal solution of the problem (13)-(15). However, $r_{q}\leq\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$ is still a non-convex constraint so we use SDR to transform it to a convex constraint [35]. By introducing an auxiliary positive semi-definite matrix variable $\mathbf{W}_{T}=\mathbf{w}_{T}\mathbf{w}_{T}^{H}$, we equivalently rewrite the problem (13)-(15) as

We use SDR to relax the rank-1 constraint (20), but the relaxed problem (16)-(19) is still a non-convex optimization problem. Therefore, we introduce an auxiliary variable $t_{0}>0$ and equivalently rewrite the relaxed problem (16)-(19) as

However, $1/\sum_{k=1}^{K}g_{k}\left(\mathbf{r}\right)$ is not a posynomial which prevents the transformation to a convex constraint. Therefore, the idea is to upper bound $1/\sum_{k=1}^{K}g_{k}\left(\mathbf{r}\right)$ by a monomial. The choice of the upper bound relies on the fact that an arithmetic mean (AM) is greater or equal to the geometric mean (GM). Hence, we have that

where $\gamma_{k}\geq 0$ and $\sum_{k=1}^{K}\gamma_{k}=1$. We replace the constraint (25) with $t_{0}\prod{}_{k=1}^{K}\left(\frac{g_{k}\left(\mathbf{r}\right)}{\gamma_{k}}\right)^{-\gamma_{k}}\leq 1$ in a conservative way. For a given choice of $\left\{\gamma_{k}\right\}$, the problem (21)-(25) is now replaced by

which looks like a GP problem but actually it is not a standard GP problem because the left sides of the constraints (28), (29), and (30) are not posynomials. To see the details, we rewrite the monomial term as $t_{0}\prod{}_{k=1}^{K}\left(\frac{g_{k}\left(\mathbf{r}\right)}{\gamma_{k}}\right)^{-\gamma_{k}}=c_{1}t_{0}r_{1}^{\alpha_{1}}r_{2}^{\alpha_{2}}\cdots r_{Q}^{\alpha_{Q}}$ where $c_{1}=\prod{}_{k=1}^{K}\left(\frac{\rho_{k}}{\gamma_{k}}\right)^{-\gamma_{k}}$ and $\alpha_{q}=-\sum{}_{k=1}^{K}\xi_{qk}\gamma_{k}$ for $q=1,...,Q$. We introduce auxiliary variables $\tilde{t}_{0}=\log t_{0}$ and $\tilde{r}_{q}=\log r_{q}$ (so that $e^{\tilde{t}_{0}}=t_{0}$ and $e^{\tilde{r}_{q}}=r_{q}$ ). We use the logarithmic transformation for the objective function (27) and the constraints (29) and (31) so we can equivalently rewrite the problem (27)-(31) as

which is a convex problem that can be solved within polynomial time by CVX [36] which adopts interior point method.

Note that the tightness of the upper bound (26) heavily depends on the choice of $\left\{\gamma_{k}\right\}$. Following [37], [38], an iterative procedure can be used to tighten the bound, where at each iteration the problem (27)-(31) is solved for an updated set of $\left\{\gamma_{k}\right\}$. Assuming a feasible $\mathbf{r}^{\left(i-1\right)}$ at iteration $i-1$, compute $\gamma_{k}=g_{k}\left(\mathbf{r}^{\left(i-1\right)}\right)/\sum_{k=1}^{K}g_{k}\left(\mathbf{r}^{\left(i-1\right)}\right)\>\forall k$ at iteration $i$ and solve problem (27)-(31) to obtain $\mathbf{r}^{\left(i\right)}$. Repeat the iterations till convergence. Note that the successive approximation method that we used is also known as a successive convex approximation or inner approximation method [39]. It cannot guarantee to converge to the global optimal solution of the problem (21)-(25), but it converges to a stationary point [39], [40] which is denoted as $\mathbf{W}_{T}^{\star}$. Due to the equivalence, $\mathbf{W}_{T}^{\star}$ is also a stationary point of the problem (16)-(19).

If $\mathrm{rank}\left(\mathbf{W}_{T}^{\star}\right)=1$, the SDR is tight so that $\mathbf{W}_{T}^{\star}$ is a stationary point of the problem (16)-(20). Using eigenvalue decomposition (EVD), we obtain $\mathbf{W}_{T}^{\star}=\mathbf{w}_{T}^{\star}\mathbf{w}_{T}^{\star H}$ and therefore $\mathbf{w}_{T}^{\star}$ is a stationary point of the problem (10)-(11). Otherwise, for the case of $\mathrm{rank}\left(\mathbf{W}_{T}^{\star}\right)>1$, we aim to extract a suboptimal rank-1 solution from $\mathbf{W}_{T}^{\star}$. A commonly adopted approach is the so-called Gaussian randomization method [35]. Particularly, we first use the EVD to decompose $\mathbf{W}_{T}^{\star}=\mathbf{U}_{T}\mathbf{\Sigma}_{T}\mathbf{U}_{T}^{H}$ where $\mathbf{U}_{T}$ and $\mathbf{\Sigma}_{T}$ are $M\times M$ unitary and diagonal matrices, respectively. Then, we generate $M$-dimensional random vectors $\mathbf{n}^{\left(l\right)}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{I}\right)$ ($l=1,\ldots,L$), multiply $\mathbf{n}^{\left(l\right)}$ by $\mathbf{U}_{T}\mathbf{\Sigma}_{T}^{1/2}$, and scale the norm of $\mathbf{U}_{T}\mathbf{\Sigma}_{T}^{1/2}\mathbf{n}^{\left(l\right)}$ to $\sqrt{2P}$ ($P$ refers to the transmit power), so that we obtain $\mathbf{w}_{T}^{\left(l\right)}=\sqrt{2P}\frac{\mathbf{U}_{T}\mathbf{\Sigma}_{T}^{1/2}\mathbf{n}^{\left(l\right)}}{\left\|\mathbf{U}_{T}\mathbf{\Sigma}_{T}^{1/2}\mathbf{n}^{\left(l\right)}\right\|}$ ($l=1,\ldots,L$). Finally, we evaluate the output DC power $P_{\mathrm{out}}$ for each $\mathbf{w}_{T}^{\left(l\right)}$ and choose the best one as the final transmit beamforming weight vector $\mathbf{w}_{T}^{\star}$, which has good performance even though it is not guaranteed to be a stationary point of the problem (10)-(11). Algorithm 1 summarizes the procedure for optimizing the beamforming with DC combining. In Section VIII, we numerically show that $\mathbf{W}_{T}^{\star}$ is rank-1 so that Algorithm 1 finds a stationary point of the problem (10)-(11) for all tested channel realizations.

Lastly, it is worth investigating the choice of the truncation order $n_{0}$. Our approach for beamforming optimization with DC combining as shown in Algorithm 1 is applicable to any truncation order $n_{0}$. However, choosing a large truncation order $n_{0}$ is not practical since the optimization complexity exponentially increases with $n_{0}$. In the simplest case $n_{0}=2$, we have that $v_{\mathrm{out},q}=\beta_{2}\mathscr{\mathcal{E}}\left\{y_{q}\left(t\right)^{2}\right\}$. Therefore, in the MISO WPT system (where $Q=1$), truncating to the second order is equivalent to the linear rectenna model [26], [29]. This has also been shown in [9]. However, when it comes to the MIMO WPT system with DC combining, truncating to the second order is not equivalent to the linear rectenna model [26], [29] due to the DC combining operation $\sum_{q=1}^{Q}v_{\mathrm{out},q}^{2}/R_{L}$. On the other hand, it is shown in [9] and [10] that $n_{0}=4$ is a good choice which effectively characterizes the rectenna nonlinearity while the complexity is not high. In Section VII, we show the numerical experimental results of the beamforming optimization with DC combining based on the rectenna model truncated to the 4th order.

Consider the RF combining scheme for the multiple antennas as shown in Fig. 3. All receive antennas are connected to an RF combining circuit. The RF combining circuit includes fixed RF power combiner such as T-junction and reconfigurable power combiner with variable power ratio and phase shifts [41] to adapt to the channel. The received signal at all receive antennas are combined together so that the RF combined signal $\widetilde{y}\left(t\right)$ can be expressed as

where $\mathbf{w}_{R}$ denotes the receive beamforming weight vector. Since the RF combining circuit is passive, the output power of the RF combining circuit should be equal or less than the input power, which results in a constraint of $\left\|\mathbf{w}_{R}\right\|^{2}\leq 1$. The RF combined signal $\widetilde{y}\left(t\right)$ is then rectified by the only one rectifier in the RF combining scheme as shown in Fig. 3. Using the aforementioned voltage model of the nonlinear rectenna, the output DC voltage of the one rectifier is given by

where $\mathscr{\mathcal{E}}\left\{\widetilde{y}\left(t\right)^{i}\right\}$ is given by

with $\zeta_{i}=\frac{1}{2\pi}\int_{0}^{2\pi}\sin^{i}t\>\textrm{d}t$ (same as (9)).

We aim to maximize the output DC power $P_{\mathrm{out}}=v_{\mathrm{out}}^{2}/R_{L}$ subject to the transmit power constraint. Noticing that $P_{\mathrm{out}}$ monotonically increases with $\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}$ irrespectively of the truncation order $n_{0}$, we have the following equivalent problem

Therefore, the choice of the truncation order $n_{0}$ has no effect on the beamforming optimization with RF combining in the MIMO WPT system, which is different from the case of DC combining.

In this section, we provide some insights into the role of nonlinearity. This is crucial to understand why RF combining outperforms DC combining in the output DC power level. The voltage model $v_{\mathrm{out}}=\sum_{i\,\mathrm{even},\,i\geq 2}^{n_{0}}\beta_{i}\mathscr{\mathcal{E}}\left\{y\left(t\right)^{i}\right\}$ characterizes the rectenna nonlinearity which is crucial in WPT system. Using Jensen’s inequality, we have $\mathscr{\mathcal{E}}\left\{y\left(t\right)^{i}\right\}\geq\left(\mathscr{\mathcal{E}}\left\{y\left(t\right)^{2}\right\}\right)^{\frac{i}{2}}=\left(P_{\mathrm{RF}}\right)^{\frac{i}{2}}$ ($P_{\mathrm{RF}}$ is the input RF power) for $i$ is even and $i\geq 2$ so that $v_{\mathrm{out}}\geq\sum_{i\,\mathrm{even},\,i\geq 2}^{n_{0}}\beta_{i}\left(P_{\mathrm{RF}}\right)^{\frac{i}{2}}$, which implies that the RF-to-DC efficiency increases with the input RF power. This is consistent with the measured RF-to-DC efficiency of rectifier circuits in [21]-[24], [30].

To understand the beneficial role of rectenna nonlinearity, we first consider the MIMO WPT system with linear rectenna model which has a constant RF-to-DC conversion efficiency $\eta$. In DC combining, the RF power received by the $q$th antenna is $P_{\mathrm{RF},q}=\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$. Each receive antenna is connected to a rectifier so that the output DC power at the $q$th rectifier is given by $P_{\mathrm{out},q}=\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}\eta$. Therefore, the total output DC power is given by $P_{\mathrm{out}}^{\mathrm{DC\>Combining}}=\frac{1}{2}\left\|\mathbf{H}\mathbf{w}_{T}\right\|^{2}\eta$ where $\mathbf{H}=\left[\mathbf{h}_{1}^{T},\mathbf{h}_{2}^{T},...,\mathbf{h}_{Q}^{T}\right]^{T}$. On the other hand, in RF combining, the RF power of the combined RF signal is $P_{\mathrm{RF}}=\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}$. The combined RF signal is input into the single rectifier so that the output DC power is given by $P_{\mathrm{out}}^{\mathrm{RF\>Combining}}=\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}\eta$. It is obvious that the optimal receive beamforming is $\mathbf{w}_{R}^{\star}=\frac{\mathbf{H}\mathbf{w}_{T}}{\left\|\mathbf{H}\mathbf{w}_{T}\right\|}$ so that we have $P_{\mathrm{out}}^{\mathrm{RF\>Combining}}=\frac{1}{2}\left\|\mathbf{H}\mathbf{w}_{T}\right\|^{2}\eta$ which is equal to the total output DC power in DC Combining. Therefore, if we use the linear rectenna model, RF combining has the same performance as DC combining even though the optimal receive beamforming $\mathbf{w}_{R}^{\star}$ is used in RF combining.

However, rectenna is a nonlinear device and the rectenna nonlinearity cannot be ignored. The RF-to-DC conversion efficiency $\eta$ is a function of the input signal waveform and the input power level. For single-tone input signal, $\eta$ increases with the input RF power level, which is denoted as $\eta\left(P_{\mathrm{RF}}\right)$. Considering the rectenna nonlinearity, the total output DC power of DC and RF combining will be different. In DC combining, the output DC power is given by

where $\eta\left(\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}\right)$ denotes the RF-to-DC efficiency at the input RF power level of $\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}$. Therefore, nonlinearity exists in DC combining so that it can be leveraged to increase the output DC power. On the other hand, in RF combining, the output DC power is given by $P_{\mathrm{out}}^{\mathrm{RF\>Combining}}=\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}\eta\left(\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}\right)$ where $\eta\left(\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}\right)$denotes the RF-to-DC efficiency at the input RF power level of $\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}$. The optimal receive beamforming is still $\mathbf{w}_{R}^{\star}=\frac{\mathbf{H}\mathbf{w}_{T}}{\left\|\mathbf{H}\mathbf{w}_{T}\right\|}$ which simultaneously maximize the terms $\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}$ and $\eta\left(\frac{1}{2}\left|\mathbf{w}_{R}^{H}\mathbf{H}\mathbf{w}_{T}\right|^{2}\right)$. Then, the output DC power with $\mathbf{w}_{R}^{\star}$ is given by

which shows that nonlinearity also exists in RF combining so that it can be leveraged. Therefore, we can conclude that rectenna nonlinearity plays a beneficial role in both DC and RF combinings to increase the output DC power. However, the receive beamforming in RF combining can leverage the nonlinearity more efficiently than DC combining which has no receive beamforming. In DC combining, the efficiency for each rectenna is $\eta\left(\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}\right)$ while in RF combining the efficiency is $\eta\left(\frac{1}{2}\left\|\mathbf{H}\mathbf{w}_{T}\right\|^{2}\right)$. Since the efficiency increases with the input RF power, we have $\eta\left(\frac{1}{2}\left\|\mathbf{H}\mathbf{w}_{T}\right\|^{2}\right)\geq\eta\left(\frac{1}{2}\left|\mathbf{h}_{q}\mathbf{w}_{T}\right|^{2}\right)$ so that $P_{\mathrm{out}}^{\mathrm{RF\>Combining}}\geq P_{\mathrm{out}}^{\mathrm{DC\>Combining}}$. Hence, we can conclude that RF combining outperforms DC combining because the receive beamforming in RF combining can leverage the nonlinearity more efficiently than DC combining. This conclusion is also verified in the next sections of scaling laws analysis and performance evaluations.

In order to get insights into the fundamental limits of MIMO WPT system and get insights into the role of the combining strategy, we want to quantify how the output DC power $P_{\mathrm{out}}$ scales as a function of the number of transmit antennas $M$ and the number of receive antennas $Q$. In the following, we consider MISO WPT system and SIMO WPT system, respectively, and for SIMO WPT system we consider the DC combining and RF combining, respectively. We mainly consider the scaling laws with the truncation order $n_{0}=4$. We assume that the channel gains $h$ are modeled as i.i.d. CSCG random variables with zero mean and unit variance. In this Section, we also assume that $R_{L}=1\;\Omega$ for simplicity.