Multi-Antenna Wireless Powered Communication with Energy Beamforming
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The newly emerging wireless powered communication networks (WPCNs) have recently drawn significant attention, where radio signals are used to power wireless terminals for information transmission. In this paper, we study a WPCN where one multi-antenna access point (AP) coordinates energy transfer and information transfer to/from a set of single-antenna users. A harvest-then-transmit protocol is assumed where the AP first broadcasts wireless power to all users via energy beamforming in the downlink (DL), and then the users send their independent information to the AP simultaneously in the uplink (UL) using their harvested energy. To optimize the users’ throughput and yet guarantee their rate fairness, we maximize the minimum throughput among all users by a joint design of the DL-UL time allocation, the DL energy beamforming, and the UL transmit power allocation plus receive beamforming. We solve this non-convex problem optimally by two steps. First, we fix the DL-UL time allocation and obtain the optimal DL energy beamforming, UL power allocation and receive beamforming to maximize the minimum signal-to-interference-plus-noise ratio (SINR) of all users. This problem is shown to be in general non-convex; however, we convert it equivalently to a spectral radius minimization problem, which can be solved efficiently by applying the alternating optimization based on the non-negative matrix theory. Then, the optimal time allocation is found by a one-dimension search to maximize the minimum rate of all users. Furthermore, two suboptimal designs of lower complexity are proposed, and their throughput performance is compared against that of the optimal solution.
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Recently, energy harvesting has become an appealing solution to prolong the lifetime of energy constrained wireless networks such as device centric or sensor based wireless networks. In particular, radio frequency (RF) signals radiated by ambient transmitters is a viable new source for wireless energy harvesting. As a result, the wireless powered communication network (WPCN) has drawn an upsurge of interests, where RF signals are used to wirelessly power user terminals for communication. A typical WPCN model is shown in Figure 1 , where an access point (AP) with constant power supply coordinates the downlink (DL) wireless information and energy transfer to a set of distributed user terminals that do not have embedded energy sources, as well as the wireless powered information transmission from the users in the uplink (UL).
It is worth noting that the DL simultaneous wireless information and power transfer (SWIPT) in WPCNs has been recently studied in the literature (see e.g. -), where the achievable information versus energy transmission trade-offs were characterized under different channel setups. However, the above works have not addressed the joint design of DL energy transfer and UL information transmission in WPCNs, which is another interesting problem to investigate even by ignoring the DL information transmission for the purpose of exposition. In , a WPCN with single-antenna AP and users has been studied for joint DL energy transfer and UL information transmission. A “harvest-then-transmit” protocol was proposed in  where the users first harvest energy from the signals broadcast by the AP in the DL, and then use their harvested energy to send independent information to the AP in the UL based on time-division-multiple-access (TDMA). The orthogonal time allocations for the DL energy transfer and UL information transmissions of all users are jointly optimized to maximize the network throughput. Furthermore, an interesting “doubly near-far” phenomenon was revealed in , where a far user from the AP, which receives less power than a near user in the DL energy transfer, also suffers from more signal power attenuation in the UL information transmission due to pass loss.
In this paper, we extend the study of  to WPCNs with the multi-antenna AP, as shown in Figure 2. When the AP is equipped with multiple antennas, the amount of energy transferred to different users in the DL can be controlled by designing different energy beamforming weights at the AP, while in the UL all users can transmit information to the AP simultaneously via space-division-multiple-access (SDMA), which thus has higher spectrum efficiency than orthogonal user transmissions in TDMA as considered in . To overcome the doubly near-far problem, similar to , we maximize the minimum UL throughput among all users by a joint optimization of the DL-UL time allocation, the DL energy beamforming, and the UL transmit power allocation plus receive beamforming. First, we assume that the optimal linear minimum-mean-square-error (MMSE) based receiver is employed at the AP for UL information transmission, which results in a non-convex problem. We solve this problem optimally by two steps: First, we fix the DL-UL time allocation and obtain the corresponding optimal DL energy beamforming, UL power allocation and receive beamforming solution; then, the problem is solved by a one-dimension search over the optimal time allocation. Particularly, for the joint DL energy beamforming and UL power allocation plus receive beamforming optimization, it is shown that this problem is in general non-convex. However, we establish its equivalence to a spectral radius minimization problem, which is then solved globally optimally by applying the alternating optimization technique  based on the non-negative matrix theory , . Notice that the non-negative matrix theory has been applied in the literature to the UL multiuser information transmission with transmit power control and receive beamforming (see e.g. , ,  and the references therein). Therefore, our proposed algorithm in this case can be viewed as an extension of the above works to the case with jointly optimizing the DL energy beamforming for wireless power transfer. It is also worth pointing out that in conventional multi-antenna wireless networks with both the UL and DL information transmissions, a useful tool that has been successfully applied to solve many non-convex design problems is the so-called UL-DL duality , -. Different from this conventional setup, in this paper we explore another interesting new relationship between the DL and UL transmissions in a WPCN with coupled DL energy transfer and UL information transmission optimization. Finally, to reduce the complexity of the optimal solution, we propose two suboptimal designs employing the zero-forcing (ZF) based receive beamforming in the UL information transmission.
The rest of this paper is organized as follows. Section 2 presents the multi-antenna WPCN model with the harvest-then-transmit protocol. Section 3 formulates the minimum throughput maximization problem. Section 4 presents the optimal solution for this problem based on non-negative matrix theory. Section 5 presents two suboptimal designs with lower complexity. Section 6 provides numerical results to compare the performances of proposed solutions. Finally, Section 7 concludes the paper.
Notation: Scalars are denoted by lower-case letters, vectors by bold-face lower-case letters, and matrices by bold-face upper-case letters. I and 0 denote an identity matrix and an all-zero matrix, respectively, with appropriate dimensions. For a square matrix S , S denotes the trace of S ; S 0 ( S 0 ) means that S is positive (negative) semi-definite. For a matrix M of arbitrary size, M and M denote the conjugate transpose and rank of M , respectively. denotes the statistical expectation. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean x and covariance matrix is denoted by x ; and stands for “distributed as”. denotes the space of complex matrices. x denotes the Euclidean norm of a complex vector x . For two real vectors x and y , x y means that x is greater than or equal to y in a component-wise manner.
Consider a WPCN consisting of one AP and users, denoted by , , as shown in Figure 2. It is assumed that the AP is equipped with antennas, while each is equipped with one antenna. The conjugated complex DL channel vector from the AP to and the reversed UL channel vector are denoted by g and h , respectively. We assume that all channels follow independent quasi-static flat fading, where g ’s and h ’s remain constant during one block transmission time, denoted by , but in general can vary from block to block.
In this paper, we assume that all ’s have no conventional energy supplies (e.g. fixed batteries) available and thus need to replenish energy from the signals sent by the AP in the DL. However, we assume that an energy storage device (ESD) in the form of rechargeable battery or super-capacitor is still equipped at each user terminal to store the energy harvested from received RF signals for future use. In particular, we adopt the “harvest-then-transmit” protocol proposed in , as shown in Figure 3, which is described as follows. In each block, during the first () amount of time, the AP broadcasts energy signals in the DL to transfer energy to all ’s simultaneously, while in the remaining amount of time of the block, all ’s transmit their independent information to the AP simultaneously in the UL by SDMA using their harvested energy from the DL. For convenience, we normalize in the rest of this paper without loss of generality.
More specifically, during the DL phase, the AP transmits with energy beams to broadcast energy to all ’s, as shown in Figure 2(a), where can be an arbitrary integer that is no larger than . The baseband transmit signal x is thus expressed as
where v denotes the th energy beam, and is its energy-carrying signal. It is assumed that ’s are independent and identically distributed (i.i.d.) random variables (RVs) with zero mean and unit variance. Then the transmit power of the AP in the DL can be expressed as x v . Suppose that the AP has a transmit sum-power constraint ; thus, we have v . The received signal in the DL at is then expressed as (by ignoring the receiver noise that is in practice negligible for energy receivers)
Due to the broadcast nature of wireless channels, the energy carried by all energy beams, i.e., v ’s (), can be harvested at each . As a result, the harvested energy of in the DL can be expressed as
where denotes the energy harvesting efficiency at the receiver. Define V v v . Then, the average transmit power available for in the subsequent UL phase of information transmission is given by
where denotes the circuit energy consumption at which is assumed to be constant over blocks. For convenience, we assume , , in the sequel to focus on transmit power for UL information transmission. Notice that thanks to multiple antennas equipped at the AP, we can schedule the UL transmit power at each via a proper selection of the DL energy beams in V , which is not possible in a single-input single-output (SISO) WPCN with single-antenna AP as considered in .
Next, in the UL phase, each utilizes its harvested energy in the previous DL phase to transmit information to the AP, as shown in Figure 2(b). The transmit signal of in the UL is then expressed as
where ’s denote the information-carrying signals of ’s, which are assumed to be i.i.d. circularly symmetric complex Gaussian (CSCG) RVs with zero mean and unit variance, denoted by , , and denotes the transmit power of . Note that V , . The received signal at the AP in the UL is then expressed as
where n denotes the receiver additive white Gaussian noise (AWGN). It is assumed that n 0 I . In this paper, we assume that the AP employs linear receivers to decode ’s in the UL. Specifically, let w denote the receive beamforming vector for decoding , . Define p and W w w . Then, the signal-to-interference-plus-noise ratio (SINR) for decoding ’s signal is expressed as
Thus, the achievable rate (in bps/Hz) for the UL information transmission of can be expressed as
Notice that there exists a non-trivial trade-off in determining the optimal DL-UL time allocation to maximize since to increase the transmit power , more time should be allocated to DL energy transfer according to (Equation 4), while this will reduce the UL information transmission time from (Equation 6).
In this paper, we are interested in maximizing the minimum (max-min) throughput of all ’s in each block by jointly optimizing the time allocation , the DL energy beams V , the UL transmit power allocation p and receive beamforming vectors W , i.e.,
It is worth noting that the number of energy beams, i.e., , is a design variable in problem (Equation 7). After the DL energy beamforming solution V is obtained, we can set the optimal value of as the number of columns in V .
Problem (Equation 7) is non-convex due to the coupled design variables in the objective function as well as the UL transmit power constraints. Note that if we fix and V V , then problem (Equation 7) reduces to the following UL SINR balancing problem with the users’ individual power constraints.
The above problem has been solved in the literature. For example, in  problem (Equation 8) was decoupled into subproblems, each with one individual user power constraint and thus solvable by the non-negative matrix theory based algorithm proposed in . In the following two sections, we propose both optimal and suboptimal algorithms to solve problem (Equation 7), respectively.
To summarize, problem (Equation 7) can be solved in the following two steps: First, given any , we solve problem (Equation 9) to find ; then, we solve problem (Equation 10) to find the optimal by a simple one-dimension search over . In the rest of this section, we thus focus on solving problem (Equation 9) with given . It is worth noting that as will be shown later in the numerical results in Section 6, with the optimal solution to problem (Equation 9) for certain , denoted by p W V , the users’ individual power constraints in (Equation 9) are not necessarily all tight, i.e., there may exist some ’s such that V . This indicates that power control is in general needed in the UL information transmission since the optimal strategy for each user is not to always transmit with its maximum available power using the harvested energy from the DL power transfer.
By introducing a common SINR requirement for all ’s, problem (Equation 9) can be reformulated as the following problem.
Note that even if we fix V V in problem (Equation 11), which reduces to the well-known SINR balancing problem given in (Equation 8), this problem in general is still non-convex over p , W and , and as a result its optimal solution cannot be obtained by convex optimization techniques . However, the non-negative matrix theory ,  has been used in e.g., , , and  to obtain the optimal solution to problem (Equation 8). By extending the results in , , and , in the following we present an efficient algorithm to solve problem (Equation 11) with the joint DL energy beamforming optimization based on the non-negative matrix theory.
First, we transform the SINR balancing problem given in (Equation 11) into an equivalent spectral radius minimization problem, where the spectral radius of a matrix B , denoted by B , is defined as its maximum eigenvalue in absolute value , . Define D W w h w h , W w w , and the non-negative matrix W as
where W denotes the entry on the th row and th column of W . Furthermore, define
where e denotes a vector with its th component being , and all other components being . Then we have the following theorem.
Please refer to Appendix ?.
Theorem ? implies that problem (Equation 11) is equivalent to the following spectral radius minimization problem.
Next, we propose an iterative algorithm to solve problem (Equation 13) by applying the alternating optimization technique . Specifically, by fixing the UL receive beamforming W W , we first optimize the DL energy beamforming V by solving the following DL problem:
Let V denote the optimal solution to problem (Equation 14), then by fixing V V , we optimize W by solving the following UL problem:
The above procedure is iterated until both W and V converge.
First, consider problem (Equation 14). For convenience, define X W D W W , and y W D W W W W . Furthermore, let X W denote the entry on the th row and th column of X W , and e X W denote the th entry of e X W , . Then we have the following proposition.
Please refer to Appendix ?.
Thanks to the fact that A W V ’s are all non-negative matrices, problem ( ?) is a convex optimization problem, which thus can be efficiently solved by CVX . Let S denote the optimal covariance solution to problem ( ?); then the optimal S number of DL energy beams, i.e., V v v , for problem (Equation 14) can be obtained by computing the eigenvalue decomposition (EVD) of S .
Last, by iteratively solving problems (Equation 14) and (Equation 15), we can solve problem (Equation 13), for which the overall algorithm is summarized in Table ?. Since the objective value of problem (Equation 13) is increased after each iteration, a monotonic convergence can be guaranteed for Algorithm ?. However, since problem (Equation 13) is a non-convex optimization problem, in general whether the converged solution is globally optimal to problem (Equation 13) remains unknown. In the following theorem, we show the global convergence of Algorithm ?.
Initialize a feasible V and update W via the algorithm in Table IV of . Set A W V and ;
DL: fix W W and update V by solving problem ( ?);
UL: fix V V and update W via the algorithm in Table IV of ;
Set A W V ;
until , where is a small positive number to control the algorithm accuracy.
Please refer to Appendix ?.
Due to the equivalence between problems (Equation 11) and (Equation 13) shown in Theorem ?, Theorem ? implies that we can apply Algorithm ? to obtain the optimal solution to problem (Equation 11). Let W and V denote the optimal solution to problem (Equation 13) obtained by Algorithm ?. We define A W V . Then, according to Theorem ?, the optimal value of problem (Equation 11), , is equal to A W V , and the optimal power solution p can be obtained from the dominant eigenvector of A W V , i.e., p .
In the previous section, we propose the optimal algorithm to solve problem (Equation 7) based on the techniques of alternating optimization and non-negative matrix theory. Note that the optimal algorithm requires a joint optimization of the DL energy beams V and the UL transmit power allocation p plus receive beamforming W . Moreover, the optimal time allocation for needs to be obtained by an exhaustive search. In this section, we propose two suboptimal solutions for problem (Equation 7) under the assumption that the number of users is no larger than that of antennas at the AP, i.e., ; hence, in the UL, the AP can employ the suboptimal ZF-based receivers (instead of MMSE-based receivers in the optimal algorithms) to completely eliminate the inter-user interference, which simplifies the design as shown next.
Define H h h h h , , which constitutes all the UL channels except h . Then with ZF-based receivers in the UL, we aim to solve problem (Equation 7) with the additional constraints: H w 0 , . Let the singular value decomposition (SVD) of H be denoted as
where X and Y are unitary matrices, and is a rectangular diagonal matrix. Furthermore, Y and Y consist of the first and the last right singular vectors of H , respectively. Note that Y forms an orthogonal basis for the null space of H , thus w must be in the following form: w Y w , , where w is an arbitrary complex vector of unit norm. It can be shown that to maximize the rate of , w should be aligned to the same direction as the equivalent channel Y h . Thus, we have
Note that unlike the MMSE-based receivers in Section 4, the above ZF receivers are not related to p and hence do not depend on V and .
where Y h denotes the power of the equivalent UL channel for . Based on the achievable rate expression given in (Equation 17) with ZF receive beamforming, we further propose two suboptimal solutions to obtain , p , and V for problem (Equation 7) in the following two subsections, respectively.
5.1Suboptimal Solution 1
Define , , and S v v . By introducing a common throughput requirement , problem (Equation 18) can be transformed into the following equivalent problem.
where p .
Problem (Equation 19) can be shown to be convex, and thus it can be solved efficiently by e.g., the interior-point method . Let , p , S and denote the optimal solution to problem (Equation 19). Then the optimal power allocation solution to problem (Equation 18) can be obtained as , and the optimal S number of energy beams v ’s can be obtained by the EVD of S .
5.2Suboptimal Solution 2
Problem (Equation 18) still requires a joint optimization of V , and p . To further reduce the complexity, in this subsection we propose another suboptimal solution for problem (Equation 18) by separating the optimization of DL energy beamforming and UL power allocation. First, the DL energy beams v ’s are obtained by solving the following weighted sum-energy maximization problem.
where denotes the energy weight for . Note that intuitively, to guarantee the rate fairness among the users, in the DL we should transfer more energy to users with weaker channels (e.g., more distant from the AP) by assigning them with higher energy weights. Therefore, we propose the following energy weight assignment rule that takes the doubly near-far effect into account: g , . Let and denote the maximum eigenvalue and its corresponding unit-norm eigenvector of the matrix G , respectively. From , the optimal value of problem (Equation 20) given a set of ’s is then obtained as , which is achieved by and v , i.e., only one energy beam is used. Next, by substituting v into problem (Equation 18), the corresponding optimal time allocation and power allocation p can be obtained by solving the following problem:
It is worth noting that all ’s should transmit at full power in the UL in this case since they cause no interference to each other due to the ZF receivers used at the AP. As a result, without loss of generality we can substitute g v into problem (Equation 21) to remove the variable p , which results in the following equivalent problem:
It can be shown that g v is a concave function over , and thus problem (Equation 22) is a convex optimization problem, which can be solved efficiently by the interior-point method . Alternatively, we can apply the bisection method  to search for the optimal , while with given , the optimal time allocation can be efficiently obtained by solving a convex feasibility problem, for which the details are omitted here for brevity.
In this section, we provide numerical examples to validate our results. We consider a multi-antenna WPCN in which the AP is equipped with antennas, and there are users.
where is set to be , denotes the distance between and AP, is a reference distance set to be m, and is the path loss exponent set to be . Moreover, we assume that the channel reciprocity holds for the UL and DL channels, i.e., h g , . The channel vectors g ’s are generated from independent Rician fading. Specifically, g is expressed as
where g is the line of sight (LOS) deterministic component, g denotes the Rayleigh fading component consisting of i.i.d. CSCG RVs with zero mean and unit covariance, and is the Rician factor set to be 3. Note that for the LOS component, we use the far-field uniform linear antenna array model with g with , where is the spacing between successive antenna elements at the AP, is the carrier wavelength, and is the direction of to the AP. We set , and . The average power of g is then normalized by in (Equation 23).
In this subsection, we investigate the performance of the optimal solution proposed in Section 4. In this numerical result, we set m, m, m, and m. Specifically, the channels H h h and G g g are given by
First, we investigate the impact of on the max-min throughput among ’s. Let denote the max-min throughput achieved by MMSE receivers given the time allocation . For the purpose of comparison, we also study the max-min throughput achieved by ZF receivers, denoted by . Note that . Also note that can be obtained by solving problem (Equation 18) with . Figure 4 shows versus over . It is observed that both and are first increasing and then decreasing over . The reason is as follows. It can be observed from (Equation 6) that when is small, the available transmit power for users given in (Equation 4) is the dominant factor and thus increasing increases the DL energy transfer time and hence the UL transmit power and throughput. However, when becomes large, the UL transmission time becomes the limiting factor and as a result increasing decreases the UL transmission time and thus the throughput. It is also observed that MMSE receiver achieves higher throughput than ZF receiver for any given .
Next, we study the performance of the optimal solutions to problem (Equation 11) proposed in Section 4 with . Figure 5 shows the convergence performance of Algorithm ? with different initial points of V . Specifically, two initial points of V are obtained by solving problem (Equation 20) with and h g , , respectively. It is observed that Algorithm ? does converge to the optimal solution in only - iterations for both initial points. It is also observed that the initial point of V obtained by setting h g , , in problem (Equation 20) is better than that obtained by setting , , to make Algorithm ? converge faster. The reason is as follows. When we fix , , in problem (Equation 20), in the DL the users more far away from the AP tend to be allocated with less energy, i.e., incurring the doubly near-far effect in the WPCN. However, by setting h g , , the users with poorer channels are assigned with higher priority in the DL power transfer, and thus have more transmit power in the UL information transmission.
|User Index||V (mW)||(mW)|
Furthermore, to illustrate whether power control is needed in the UL information transmission, i.e., each user transmits at maximum power or not using the energy harvested from the DL power transfer, we show the values of V versus , , in Table 1, where p and V denote the optimal solution to problem (Equation 11) with . It is observed that the three users that are nearer to the AP, i.e., , and , should not transmit at maximum power, and thus in general given the optimal DL energy beams V , UL power control is needed to maximize the minimum SINR of all users in problem (Equation 11).
Last, we study the impact of the number of antennas at the AP on the max-min throughput performance. In this example, we activate one more antenna among the antennas at each time. Figure 6 shows the max-min throughput achieved by the optimal solution in Section 4 versus the number of active antennas at the AP. Note that for the case when there is only one active antenna at the AP, since spatial transmit/receive beamforming cannot be utilized, we adopt the TDMA based solution proposed in  for the SISO WPCN. It is observed from Figure 6 that the max-min throughput increases significantly with the number of active antennas at the AP.
In this subsection, we compare the max-min throughput by the optimal solution in Section 4 with MMSE receivers and the two suboptimal solutions in Section 5 with ZF receivers. In this example, it is assumed that all users are of the same distance to the AP, i.e., , . Figure 7 shows the max-min throughput over . For the purpose of comparison, we also plot the max-min throughout achieved by solving problem (Equation 22) where the energy beams V are randomly generated rather than obtained via solving problem (Equation 20). It is observed that the throughput decays drastically as increases for all optimal and suboptimal solutions. It is also observed that for all values of , the throughput by MMSE receiver outperforms those of the three suboptimal solutions by ZF receiver. However, when is small, it is observed that both Suboptimal Solutions 1 and 2 with ZF receiver achieve the throughput very close to the optimal solution with MMSE receiver. This is because in this case the available power for UL transmission is large for all ’s, and thus ZF receiver is asymptotically optimal with high signal-to-noise ratio (SNR). Furthermore, it is observed that with ZF receiver, Suboptimal Solution 2 performs very close to Suboptimal Solution 1, although it is based on separate optimizations of DL energy beamforming and UL power allocation to achieve lower complexity. However, if the energy beams are randomly generated instead of via solving problem (Equation 20), there is a significant loss in the achieved max-min throughput observed with ZF receiver.
This paper has studied a wireless powered communication network (WPCN) with multi-antenna AP and single-antenna users. Under a harvest-then-transmit protocol, the minimum throughput among all users is maximized by a joint optimization of the DL-UL time allocation, DL energy beamforming, and UL transmit power allocation plus receive beamforming. We solve this problem optimally via a two-stage algorithm. First, we fix the DL-UL time allocation and propose an efficient algorithm to obtain the corresponding optimal DL energy beamforming and UL power allocation plus receive beamforming solution based on the techniques of alternating optimization and non-negative matrix theory. Then, the problem is solved by a one-dimension search over the optimal DL-UL time allocation. Furthermore, two suboptimal solutions of lower complexity are proposed with ZF based receive beamforming, and their performances are compared to the optimal solution.
- L. Liu is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:firstname.lastname@example.org).
- R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:email@example.com). He is also with the Institute for Infocomm Research, A*STAR, Singapore.
- K. C. Chua is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:firstname.lastname@example.org).
- In practice, for the UL information transmission, the channels h ’s can be estimated by the AP based on the pilot signals sent by individual ’s, while for the DL power transfer, the channels g ’s can be obtained by the AP via, e.g., sending the pilot signal to all ’s and collecting channel estimation feedback from individual ’s. To focus on the performance upper bound, in this paper we assume that such channel knowledge is perfectly known at the AP for both DL and UL transmissions in each block.
- Note that holds in our example; thus, the two ZF receiver based solutions in Section 5 are both feasible.
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