MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer111This paper has been presented in part at IEEE Global Communications Conference (Globecom), December 5-9, 2011, Houston, USA. 222R. 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. 333C. K. Ho is with the Institute for Infocomm Research, A*STAR, Singapore (e-mail:firstname.lastname@example.org).
Wireless power transfer (WPT) is a promising new solution to provide convenient and perpetual energy supplies to wireless networks. In practice, WPT is implementable by various technologies such as inductive coupling, magnetic resonate coupling, and electromagnetic (EM) radiation, for short-/mid-/long-range applications, respectively. In this paper, we consider the EM or radio signal enabled WPT in particular. Since radio signals can carry energy as well as information at the same time, a unified study on simultaneous wireless information and power transfer (SWIPT) is pursued. Specifically, this paper studies a multiple-input multiple-output (MIMO) wireless broadcast system consisting of three nodes, where one receiver harvests energy and another receiver decodes information separately from the signals sent by a common transmitter, and all the transmitter and receivers may be equipped with multiple antennas. Two scenarios are examined, in which the information receiver and energy receiver are separated and see different MIMO channels from the transmitter, or co-located and see the identical MIMO channel from the transmitter. For the case of separated receivers, we derive the optimal transmission strategy to achieve different tradeoffs for maximal information rate versus energy transfer, which are characterized by the boundary of a so-called rate-energy (R-E) region. For the case of co-located receivers, we show an outer bound for the achievable R-E region due to the potential limitation that practical energy harvesting receivers are not yet able to decode information directly. Under this constraint, we investigate two practical designs for the co-located receiver case, namely time switching and power splitting, and characterize their achievable R-E regions in comparison to the outer bound.
MIMO system, broadcast channel, precoding, wireless power, simultaneous wireless information and power transfer (SWIPT), rate-energy tradeoff, energy harvesting.
Energy-constrained wireless networks, such as sensor networks, are typically powered by batteries that have limited operation time. Although replacing or recharging the batteries can prolong the lifetime of the network to a certain extent, it usually incurs high costs and is inconvenient, hazardous (say, in toxic environments), or even impossible (e.g., for sensors embedded in building structures or inside human bodies). A more convenient, safer, as well as “greener” alternative is thus to harvest energy from the environment, which virtually provides perpetual energy supplies to wireless devices. In addition to other commonly used energy sources such as solar and wind, ambient radio-frequency (RF) signals can be a viable new source for energy scavenging. It is worth noting that RF-based energy harvesting is typically suitable for low-power applications (e.g., sensor networks), but also can be applied for scenarios with more substantial power consumptions if dedicated wireless power transmission is implemented.444Interested readers may visit the company website of Powercast at http://www.powercastco.com/ for more information on recent applications of dedicated RF-based power transfer.
On the other hand, since RF signals that carry energy can at the same time be used as a vehicle for transporting information, simultaneous wireless information and power transfer (SWIPT) becomes an interesting new area of research that attracts increasing attention. Although a unified study on this topic is still in the infancy stage, there have been notable results reported in the literature [1, 2]. In , Varshney first proposed a capacity-energy function to characterize the fundamental tradeoffs in simultaneous information and energy transfer. For the single-antenna or SISO (single-input single-output) AWGN (additive white Gaussian noise) channel with amplitude-constrained inputs, it was shown in  that there exist nontrivial tradeoffs in maximizing information rate versus (vs.) power transfer by optimizing the input distribution. However, if the average transmit-power constraint is considered instead, the above two goals can be shown to be aligned for the SISO AWGN channel with Gaussian input signals, and thus there is no nontrivial tradeoff. In , Grover and Sahai extended  to frequency-selective single-antenna AWGN channels with the average power constraint, by showing that a non-trivial tradeoff exists in frequency-domain power allocation for maximal information vs. energy transfer.
As a matter of fact, wireless power transfer (WPT) or in short wireless power, which generally refers to the transmissions of electrical energy from a power source to one or more electrical loads without any interconnecting wires, has been investigated and implemented with a long history. Generally speaking, WPT is carried out using either the “near-field” electromagnetic (EM) induction (e.g., inductive coupling, capacitive coupling) for short-distance (say, less than a meter) applications such as passive radio-frequency identification (RFID) , or the “far-field” EM radiation in the form of microwaves or lasers for long-range (up to a few kilometers) applications such as the transmissions of energy from orbiting solar power satellites to Earth or spacecrafts . However, prior research on EM radiation based WPT, in particular over the RF band, has been pursued independently from that on wireless information transfer (WIT) or radio communication. This is non-surprising since these two lines of work in general have very different research goals: WIT is to maximize the information transmission capacity of wireless channels subject to channel impairments such as the fading and receiver noise, while WPT is to maximize the energy transmission efficiency (defined as the ratio of the energy harvested and stored at the receiver to that consumed by the transmitter) over a wireless medium. Nevertheless, it is worth noting that the design objectives for WPT and WIT systems can be aligned, since given a transmitter energy budget, maximizing the signal power received (for WPT) is also beneficial in maximizing the channel capacity (for WIT) against the receiver noise.
Hence, in this paper we attempt to pursue a unified study on WIT and WPT for emerging wireless applications with such a dual usage. An example of such wireless dual networks is envisaged in Fig. 1, where a fixed access point (AP) coordinates the two-way communications to/from a set of distributed user terminals (UTs). However, unlike the conventional wireless network in which both the AP and UTs draw energy from constant power supplies (by e.g. connecting to the grid or a battery), in our model, only the AP is assumed to have a constant power source, while all UTs need to replenish energy from the received signals sent by the AP via the far-field RF-based WPT. Consequently, the AP needs to coordinate the wireless information and energy transfer to UTs in the downlink, in addition to the information transfer from UTs in the uplink. Wireless networks with such a dual information and power transfer feature have not yet been studied in the literature to our best knowledge, although some of their interesting applications have already appeared in, e.g., the body sensor networks  with the out-body local processing units (LPUs) powered by battery communicating and at the same time sending wireless power to in-body sensors that have no embedded power supplies. However, how to characterize the fundamental information-energy transmission tradeoff in such dual networks is still an open problem.
In this paper, we focus our study on the downlink case with simultaneous WIT and WPT from the AP to UTs. In the generic system model depicted in Fig. 1, each UT can in general harvest energy and decode information at the same time (by e.g. applying the power splitting scheme introduced later in this paper). However, from an implementation viewpoint, one particular design whereby each UT operates as either an information receiver or an energy receiver at any given time may be desirable, which is referred to as time switching. This scheme is practically appealing since state-of-the-art wireless information and energy receivers are typically designed to operate separately with very different power sensitivities (e.g., dBm for information receivers vs. dBm for energy receivers). As a result, if time switching is employed at each UT jointly with the “near-far” based transmission scheduling at the AP, i.e., UTs that are close to the AP and thus receive high power from the AP are scheduled for WET, whereas those that are more distant from the AP and thus receive lower power are scheduled for WIT, then SWIPT systems can be efficiently implemented with existing information and energy receivers and the additional time-switching device at each receiver.
For an initial study on SWIPT, this paper considers the simplified scenarios with only one or two active UTs in the network at any given time. For the case of two UTs, we assume time switching, i.e., the two UTs take turns to receive energy or (independent) information from the AP over different time blocks. As a result, when one UT receives information from the AP, the other UT can opportunistically harvest energy from the same signal broadcast by the AP, and vice versa. Hence, at each block, one UT operates as an information decoding (ID) receiver, and the other UT as an energy harvesting (EH) receiver. We thus refer to this case as separated EH and ID receivers. On the other hand, for the case with only one single UT to be active at one time (while all other UTs are assumed to be in the off/sleep mode), the active UT needs to harvest energy as well as decode information from the same signal sent by the AP, i.e., the same set of receiving antennas are shared by both EH and ID receivers residing in the same UT. Thus, this case is referred to as co-located EH and ID receivers. Surprisingly, as we will show later in this paper, the optimal information-energy tradeoff for the case of co-located receivers is more challenging to characterize than that for the case of separated receivers, due to a potential limitation that practical EH receiver circuits are not yet able to decode the information directly and vice versa. Note that similar to the case of separated receives, time switching can also be applied in the case of co-located receivers to orthogonalize the information and energy transmissions at each receiving antenna; however, this scheme is in general suboptimal for the achievable rate-energy tradeoffs in the case of co-located receivers, as will be shown later in this paper.
Some further assumptions are made in this paper for the purpose of exposition. Firstly, this paper considers a quasi-static fading environment where the wireless channel between the AP and each UT is assumed to be constant over a sufficiently long period of time during which the number of transmitted symbols can be approximately regarded as being infinitely large. Under this assumption, we further assume that it is feasible for each UT to estimate the downlink channel from the AP and then send it back to the AP via the uplink, since the time overhead for such channel estimation and feedback is a negligible portion of the total transmission time due to quasi-static fading. We will address the more general case of fading channels with imperfect/partial channel knowledge at the transmitter in our future work. Secondly, we assume that the system under our study typically operates at the high signal-to-noise ratio (SNR) regime for the ID receiver in the case of co-located receivers. This is to be compatible with the high-power operating requirement for the EH receiver of practical interest as previously mentioned. Thirdly, without loss of generality, we assume a multi-antenna or MIMO (multiple-input multiple-output) system, in which the AP is equipped with multiple antennas, and each UT is equipped with one or more antennas, for enabling both the high-performance wireless energy and information transmissions (as it is well known that for WIT only, MIMO systems can achieve folded array/capacity gains over SISO systems by spatial beamforming/multiplexing ).
Under the above assumptions, a three-node MIMO broadcast system is considered in this paper, as shown in Fig. 2, wherein the EH and ID receivers harvest energy and decode information separately from the signal sent by a common transmitter. Note that this system model refers to the case of separated EH and ID receivers in general, but includes the co-located receivers as a special case when the MIMO channels from the transmitter to both receivers become identical. Assuming this model, the main results of this paper are summarized as follows:
For the case of separated EH and ID receivers, we design the optimal transmission strategy to achieve different tradeoffs between maximal information rate vs. energy transfer, which are characterized by the boundary of a so-called rate-energy (R-E) region. We derive a semi-closed-form expression for the optimal transmit covariance matrix (for the joint precoding and power allocation) to achieve different rate-energy pairs on the boundary of the R-E region. Note that the R-E region is a multiuser extension of the single-user capacity-energy function in . Also note that the multi-antenna broadcast channel (BC) has been investigated in e.g. – for information transfer solely by unicasting or multicasting. However, MIMO-BC for SWIPT as considered in this paper is new and has not yet been studied by any prior work.
For the case of co-located EH and ID receivers, we show that the proposed solution for the case of separated receivers is also applicable with the identical MIMO channel from the transmitter to both ID and EH receivers. Furthermore, we consider a potential practical constraint that EH receiver circuits cannot directly decode the information (i.e., any information embedded in received signals sent to the EH receiver is lost during the EH process). Under this constraint, we show that the R-E region with the optimal transmit covariance (obtained without such a constraint) in general only serves as a performance outer bound for the co-located receiver case.
Hence, we investigate two practical receiver designs, namely time switching and power splitting, for the case of co-located receivers. As shown in Fig. 3, for time switching, each receiving antenna periodically switches between the EH receiver and ID receiver, whereas for power splitting, the received signal at each antenna is split into two separate signal streams with different power levels, one sent to the EH receiver and the other to the ID receiver. Note that time switching has also been proposed in  for the SISO AWGN channel. Furthermore, note that the antenna switching scheme whereby the receiving antennas are divided into two groups with one group switched to information decoding and the other group to energy harvesting can be regarded as a special case of power splitting with only binary splitting power ratios at each receiving antenna. For these practical receiver designs, we derive their achievable R-E regions as compared to the R-E region outer bound, and characterize the conditions under which their performance gaps can be closed. For example, we show that the power splitting scheme approaches the tradeoff upper bound asymptotically when the RF-band antenna noise at the receiver becomes more dominant over the baseband processing noise (more details are given in Section IV-C).
The rest of this paper is organized as follows: Section II presents the system model, characterizes the rate-energy region, and formulates the problem for finding the optimal transmit covariance matrix. Section III presents the optimal transmit covariance solution for the case of separated receivers. Section IV extends the solution to the case of co-located receivers to obtain a performance upper bound, proposes practical receiver designs, and analyzes their performance limits as compared to the performance upper bound. Finally, Section V concludes the paper and provides some promising directions for future work.
Notation: For a square matrix , , , , and denote its trace, determinant, inverse, and square-root, respectively, while and mean that is positive semi-definite and positive definite, respectively. For an arbitrary-size matrix , and denote the conjugate transpose and transpose of , respectively. denotes an diagonal matrix with being the diagonal elements. and denote an identity matrix and an all-zero vector, respectively, with appropriate dimensions. denotes the statistical expectation. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean and covariance matrix is denoted by , and stands for “distributed as”. denotes the space of matrices with complex entries. is the Euclidean norm of a complex vector , and is the absolute value of a complex scalar . and denote the maximum and minimum between two real numbers, and , respectively, and . All the functions have base-2 by default.
Ii System Model And Problem Formulation
As shown in Fig. 2, this paper considers a wireless broadcast system consisting of one transmitter, one EH receiver, and one ID receiver. It is assumed that the transmitter is equipped with transmitting antennas, and the EH receiver and the ID receiver are equipped with and receiving antennas, respectively. In addition, it is assumed that the transmitter and both receivers operate over the same frequency band. Assuming a narrow-band transmission over quasi-static fading channels, the baseband equivalent channels from the transmitter to the EH receiver and ID receiver can be modeled by matrices and , respectively. It is assumed that at each fading state, and are both known at the transmitter, and separately known at the corresponding receiver. Note that for the case of co-located EH and ID receivers, is identical to and thus .
It is worth noting that the EH receiver does not need to convert the received signal from the RF band to the baseband in order to harvest the carried energy. Nevertheless, thanks to the law of energy conservation, it can be assumed that the total harvested RF-band power (energy normalized by the baseband symbol period), denoted by , from all receiving antennas at the EH receiver is proportional to that of the received baseband signal, i.e.,
where is a constant that accounts for the loss in the energy transducer for converting the harvested energy to electrical energy to be stored; for the convenience of analysis, it is assumed that in this paper unless stated otherwise. We use to denote the baseband signal broadcast by the transmitter at the th symbol interval, which is assumed to be random over , without loss of generality. The expectation in (1) is thus used to compute the average power harvested by the EH receiver at each fading state. Note that for simplicity, we assumed in (1) that the harvested energy due to the background noise at the EH receiver is negligible and thus can be ignored.555The results of this paper are readily extendible to study the impacts of non-negligible background noise and/or co-channel interference on the SWIPT system performance.
On the other hand, the baseband transmission from the transmitter to the ID receiver can be modeled by
where denotes the received signal at the th symbol interval, and denotes the receiver noise vector. It is assumed that ’s are independent over and . Under the assumption that is random over , we use to denote the covariance matrix of . In addition, we assume that there is an average power constraint at the transmitter across all transmitting antennas denoted by . In the following, we examine the optimal transmit covariance to maximize the transported energy efficiency and information rate to the EH and ID receivers, respectively.
Consider first the MIMO link from the transmitter to the EH receiver when the ID receiver is not present. In this case, the design objective for is to maximize the power received at the EH receiver. Since from (1) it follows that with , the aforementioned design problem can be formulated as
Let and the (reduced) singular value decomposition (SVD) of be denoted by , where and , each of which consists of orthogonal columns with unit norm, and with . Furthermore, let denote the first column of . Then, we have the following proposition.
The optimal solution to (P1) is .
See Appendix A.
Given , it follows that the maximum harvested power at the EH receiver is given by . It is worth noting that since is a rank-one matrix, the maximum harvested power is achieved by beamforming at the transmitter, which aligns with the strongest eigenmode of the matrix , i.e., the transmitted signal can be written as , where is an arbitrary random signal over with zero mean and unit variance, and is the transmit beamforming vector. For convenience, we name the above transmit beamforming scheme to maximize the efficiency of WPT as “energy beamforming”.
Next, consider the MIMO link from the transmitter to the ID receiver without the presence of any EH receiver. Assuming the optimal Gaussian codebook at the transmitter, i.e., , the transmit covariance to maximize the transmission rate over this MIMO channel can be obtained by solving the following problem :
The optimal solution to the above problem is known to have the following form : , where is obtained from the (reduced) SVD of expressed by , with , , , , and with the diagonal elements obtained from the standard “water-filling (WF)” power allocation solution :
with being the so-called (constant) water-level that makes . The corresponding maximum transmission rate is then given by . The maximum rate is achieved in general by spatial multiplexing  over up to spatially decoupled AWGN channels, together with the Gaussian codebook, i.e., the transmitted signal can be expressed as , where is a Gaussian random vector , and denote the precoding matrix and the (diagonal) power allocation matrix, respectively.
It is worth noting that in Problem (P1), it is assumed that the transmitter sends to the EH receiver continuously. Now suppose that the transmitter only transmits a fraction of the total time denoted by with . Furthermore, assume that the transmit power level can be adjusted flexibly provided that the consumed average power is bounded by , i.e., or . In this case, it can be easily shown that the transmit covariance also achieves the maximum harvested power for any , which suggests that the maximum power delivered is independent of transmission time. However, unlike the case of maximum power transfer, the maximum information rate reliably transmitted to the ID receiver requires that the transmitter send signals continuously, i.e., , as assumed in Problem (P2). This can be easily verified by observing that for any and , where the equality holds only when , since is a nonlinear concave function of . Thus, to maximize both power and rate transfer at the same time, the transmitter should broadcast to the EH and ID receivers all the time. Furthermore, note that the assumed Gaussian distribution for transmitted signals is necessary for achieving the maximum rate transfer, but not necessary for the maximum power transfer. In fact, for any arbitrary complex number that satisfies , even a deterministic transmitted signal , achieves the maximum transferred power in Problem (P1). However, to maximize simultaneous power and information transfer with the same transmitted signal, the Gaussian input distribution is sufficient as well as necessary.
Now, consider the case where both the EH and ID receivers are present. From the above results, it is seen that the optimal transmission strategies for maximal power transfer and information transfer are in general different, which are energy beamforming and information spatial multiplexing, respectively. It thus motivates our investigation of the following question: What is the optimal broadcasting strategy for simultaneous wireless power and information transfer? To answer this question, we propose to use the Rate-Energy (R-E) region (defined below) to characterize all the achievable rate (in bits/sec/Hz or bps for information transfer) and energy (in joule/sec or watt for power transfer) pairs under a given transmit power constraint. Without loss of generality, assuming that the transmitter sends Gaussian signals continuously (cf. Remark II.1), the R-E region is defined as
In Fig. 4, an example of the above defined R-E region (see Section III for the algorithm to compute the boundary of this region) is shown for a practical MIMO broadcast system with separated EH and ID receivers (i.e., ). It is assumed that . The transmitter power is assumed to be watt(W) or dBm. The distances from the transmitter to the EH and ID receivers are assumed to be meter and meters, respectively; thus, we can exploit the near-far based energy and information transmission scheduling, which may correspond to, e.g., a dedicated energy transfer system (to “near” users) with opportunistic information transmission (to “far” users), or vice versa. Assuming a carrier frequency of MHz and the power pathloss exponent to be 4, the distance-dependent signal attenuation from the AP to EH/ID receiver can be estimated as 40dB and 80dB, respectively. Accordingly, the average signal power at the EH/ID receiver is thus 30dBmdBdBm and 30dBmdBdBm, respectively. It is further assumed that in addition to signal pathloss, Rayleigh fading is present, as such each element of channel matrices and is independently drawn from the CSCG distribution with zero mean and variance dBm (for EH receiver) and dBm for (for ID receiver), respectively (to be consistent with the signal pathloss previously assumed). Furthermore, the bandwidth of the transmitted signal is assumed to be 10MHz, while the receiver noise is assumed to be white Gaussian with power spectral density dBm/Hz (which is dominated by the receiver processing noise rather than the background thermal noise) or average power dBm over the bandwidth of 10MHz. As a result, considering all of transmit power, signal attenuation, fading and receiver noise, the per-antenna average SNR at the ID receiver is equal to dB, which corresponds to in the equivalent signal model for the ID receiver given in (2) with unit-norm noise. In addition, we assume that for the EH receiver, the energy conversion efficiency is 50%. Considering this together with transmit power and signal attenuation, the average per-antenna signal power at the EH receiver is thus (30dBmdB) W.
From Fig. 4, it is observed that with energy beamforming, the maximum harvested energy rate for the EH receiver is around mW, while with spatial multiplexing, the maximum information rate for the ID receiver is around Mbps. It is easy to identify two boundary points of this R-E region denoted by and , respectively. For the former boundary point, the transmit covariance is , which corresponds to transmit beamforming and achieves the maximum transferred power to the EH receiver, while the resulting information rate for the ID receiver is given by . On the other hand, for the latter boundary point, the transmit covariance is , which corresponds to transmit spatial multiplexing and achieves the maximum information rate transferred to the ID receiver , while the resulting power transferred to the EH receiver is given by .
Since the optimal tradeoff between the maximum energy and information transfer rates is characterized by the boundary of the R-E region, it is important to characterize all the boundary rate-power pairs of for any . From Fig. 4, it is easy to observe that if , the maximum harvested power is achievable with the same transmit covariance that achieves the rate-power pair ; similarly, the maximum information rate is achievable provided that . Thus, the remaining boundary of yet to be characterized is over the intervals: , . We thus consider the following optimization problem:
Note that if takes values from , the corresponding optimal rate solutions of the above problems are the boundary rate points of the R-E region over . Notice that the transmit covariance solutions to the above problems in general yield larger rate-power pairs than those by simply “time-sharing” the optimal transmit covariance matrices and for EH and ID receivers separately (see the dashed line in Fig. 4).666By time-sharing, we mean that the AP transmits simultaneously to both EH and ID receivers with the energy-maximizing transmit covariance (i.e. energy beamforming) for portion of each block time, and the information-rate-maximizing transmit covariance (i.e. spatial multiplexing) for the remaining portion of each block time, with .
Problem (P3) is a convex optimization problem, since its objective function is concave over and its constraints specify a convex set of . Note that (P3) resembles a similar problem formulated in ,  (see also  and references therein) under the cognitive radio (CR) setup, where the rate of a secondary MIMO link is maximized subject to a set of so-called interference power constraints to protect the co-channel primary receivers. However, there is a key difference between (P3) and the problem in : the harvested power constraint in (P3) has the reversed inequality of that of the interference power constraint in , since in our case it is desirable for the EH receiver to harvest more power from the transmitter, as opposed to that in  the interference power at the primary receiver should be minimized. As such, it is not immediately clear whether the solution in  can be directly applied for solving (P3) with the reversed power inequality. In the following, we will examine the solutions to Problem (P3) for the two cases with arbitrary and (the case of separated receivers) and (the case of co-located receivers), respectively.
Iii Separated Receivers
Consider the case where the EH receiver and ID receiver are spatially separated and thus in general have different channels from the transmitter. In this section, we first solve Problem (P3) with arbitrary and and derive a semi-closed-form expression for the optimal transmit covariance. Then, we examine the optimal solution for the special case of MISO channels from the transmitter to ID and/or EH receivers.
Since Problem (P3) is convex and satisfies the Slater’s condition , it has a zero duality gap and thus can be solved using the Lagrange duality method.777It is worth noting that Problem (P3) is convex and thus can be solved efficiently by the interior point method ; in this paper, we apply the Lagrange duality method for this problem mainly to reveal the optimal precoder structure. Thus, we introduce two non-negative dual variables, and , associated with the harvested power constraint and transmit power constraint in (P3), respectively. The optimal solution to Problem (P3) is then given by the following theorem in terms of and , which are the optimal dual solutions of Problem (P3) (see Appendix B for details). Note that for Problem (P3), given any pair of () and , there exists one unique pair of and .
The optimal solution to Problem (P3) has the following form:
where , is obtained from the (reduced) SVD of the matrix given by , with , , and , with .
See Appendix B. Note that this theorem requires that , implying that (recall that is the largest eigenvalue of matrix ), which is not present for a similar result in  under the CR setup with the reversed interference power constraint. One algorithm that can be used to solve (P3) is provided in Table I of Appendix B. From Theorem III.1, the maximum transmission rate for Problem (P3) can be shown to be , for which the proof is omitted here for brevity.
Next, we examine the optimal solution to Problem (P3) for the special case where the ID receiver has one single antenna, i.e., , and thus the MIMO channel reduces to a row vector with . Suppose that the EH receiver is still equipped with antennas, and thus the MIMO channel remains unchanged. From Theorem III.1, we obtain the following corollary.
In the case of MISO channel from the transmitter to ID receiver, i.e., , the optimal solution to Problem (P3) reduces to the following form:
where , with and denoting the optimal dual solutions of Problem (P3). Correspondingly, the optimal value of (P3) is .
See Appendix C.
From (6), it is observed that the optimal transmit covariance is a rank-one matrix, from which it follows that beamforming is the optimal transmission strategy in this case, where the transmit beamforming vector should be aligned with the vector . Moreover, consider the case where both channels from the transmitter to ID/EH receivers are MISO, i.e., , and with . From Corollary III.1, it follows immediately that the optimal covariance solution to Problem (P3) is still beamforming. In the following theorem, we show a closed-form solution of the optimal beamforming vector at the transmitter for this special case, which differs from the semi-closed-form solution (6) that was expressed in terms of dual variables.
In the case of MISO channels from transmitter to both ID and EH receivers, i.e., , and , the optimal solution to Problem (P3) can be expressed as , where the beamforming vector has a unit-norm and is given by
where , , with , and with denoting the phase of complex number . Correspondingly, the optimal value of (P3) is given by
The proof is similar to that of Theorem 2 in , and is thus omitted for brevity.
It is worth noting that in (7), if , the optimal transmit beamforming vector is based on the principle of maximal-ratio-combining (MRC) with respect to the MISO channel from the transmitter to the ID receiver, and in this case, the harvested power constraint in Problem (P3) is indeed not active; however, when , the optimal beamforming vector is a linear combination of the two vectors and , and the combining coefficients are designed such that the harvested power constraint is satisfied with equality.
In Fig. 5, we show the achievable R-E regions for the case of MISO channels from the transmitter to both EH and ID receivers. We set . For the purpose of exposition, it is assumed that and , with denoting the correlation between the two unit-norm vectors and . This channel setup may correspond to the practical scenario where the EH and ID receivers are equipped at a single device (but still physically separated), and as a result their respective MISO channels from the transmitter have the same power gain but are spatially correlated due to the insufficient spacing between two separate receiving antennas. From Theorem III.2, the R-E regions for the three cases of , and are obtained, as shown in Fig. 5. Interestingly, it is observed that increasing enlarges the achievable R-E region, which indicates that the antenna correlation between the EH and ID receivers can be a beneficial factor for simultaneous information and power transfer. Note that in this figure, we express energy and rate in terms of energy unit and bits/channel use, respectively, since their practical values can be obtained by appropriate scaling based on the realistic system parameters as for Fig. 4.
Iv Co-Located Receivers
In this section, we address the case where the EH and ID receivers are co-located, and thus possess the same channel from the transmitter, i.e., and thus . We first examine the optimal solution of Problem (P3) for this case, from which we obtain an outer bound for the achievable rate-power pairs in the R-E region. Then, we propose two practical receiver designs, namely time switching and power splitting, derive their optimal transmission strategies to maximize the achievable rate-power pairs, and finally compare the results to the R-E region outer bound.
Iv-a Performance Outer Bound
Consider Problem (P3) with . Recall that the (reduced) SVD of is given by , with , , and . From Theorem III.1, we obtain the following corollary.
In the case of co-located EH and ID receivers with , the optimal solution to Problem (P3) has the form of , where with the diagonal elements obtained from the following modified WF power allocation:
with and denoting the optimal dual solutions of Problem (P3), . The corresponding maximum transmission rate is .
See Appendix D.
The algorithm in Table I for solving Problem (P3) with arbitrary and can be simplified to solve the special case with . Corollary IV.1 reveals that for Problem (P3) in the case of , the optimal transmission strategy is in general still spatial multiplexing over the eigenmodes of the MIMO channel as for Problem (P2), while the optimal tradeoffs between information transfer and power transfer are achieved by varying the power levels allocated into different eigenmodes, as shown in (9). It is interesting to observe that the power allocation in (9) reduces to the conventional WF solution in (3) with a constant water-level when , i.e., the harvested power constraint in Problem (P3) is inactive with the optimal power allocation. However, when and thus the harvested power constraint is active corresponding to the Pareto-optimal regime of our interest, the power allocation in (9) is observed to have a non-decreasing water-level as ’s increase. Note that this modified WF policy has also been shown in  for power allocation in frequency-selective AWGN channels.
Using Corollary IV.1, we can characterize all the boundary points of the R-E region defined in (4) for the case of co-located receivers with . For example, if the total transmit power is allocated to the channel with the largest gain , i.e., and , the maximum harvested power is achieved by transmit beamforming. On the other hand, if transmit spatial multiplexing is applied with the conventional WF power allocation given in (9) with , the corresponding becomes the maximum transmission rate, . However, unlike the case of separated EH and ID receivers in which the entire boundary of is achievable, in the case of co-located receivers, except the two boundary rate-power pairs and , all the other boundary pairs of may not be achievable in practice. Note that these boundary points are achievable if and only if (iff) the following premise is true: the power of the received signal across all antennas is totally harvested, and at the same time the carried information with a transmission rate up to the MIMO channel capacity (for a given transmit covariance) is decodable. However, existing EH circuits are not yet able to directly decode the information carried in the RF-band signal, even for the SISO channel case; as a result, how to achieve the remaining boundary rate-power pairs of in the MIMO case with the co-located EH and ID receiver remains an interesting open problem. Therefore, in the case of co-located receivers, the boundary of given by Corollary IV.1 in general only serves as an outer bound for the achievable rate-power pairs with practical receiver designs, as will be investigated in the following subsections.
Iv-B Time Switching
First, as shown in Fig. 3(a), we consider the time switching (TS) scheme, with which each transmission block is divided into two orthogonal time slots, one for transferring power and the other for transmitting data. The co-located EH and ID receiver switches its operations periodically between harvesting energy and decoding information between the two time slots. It is assumed that time synchronization has been perfectly established between the transmitter and the receiver, and thus the receiver can synchronize its function switching with the transmitter. With orthogonal transmissions, the transmitted signals for the EH receiver and ID receiver can be designed separately, but subject to a total transmit power constraint. Let with denote the percentage of transmission time allocated to the EH time slot. We then consider the following two types of power constraints at the transmitter:
Fixed power constraint: The transmitted signals to the ID and EH receivers have the same fixed power constraint given by , and , where and denote the transmit covariance matrices for the ID and EH transmission time slots, respectively.
Flexible power constraint: The transmitted signals to the ID and EH receivers can have different power constraints provided that their average consumed power is below , i.e., .
Note that the TS scheme under the fixed power constraint has been considered in  for the single-antenna AWGN channel. The achievable R-E regions for the TS scheme with the fixed (referred to as ) vs. flexible (referred to as ) power constraints are then given as follows:
It is worth noting that must be true since any pair of and that satisfy the fixed power constraint will satisfy the flexible power constraint, but not vice versa. The optimal transmit covariance matrices and to achieve the boundary of with the fixed power constraint are given in Section II (assuming ). In fact, the boundary of is simply a straight line connecting the two points and (cf. Fig. 7) by sweeping from 0 to 1.
Similarly, for the case of flexible power constraint, the transmit covariance solutions for and to achieve any boundary point of can be shown to have the same set of eigenvectors as those given in Section II (assuming ), respectively; however, the corresponding time allocation for and power allocation for and remain unknown. We thus have the following proposition.
In the case of flexible power constraint, except the two points and , all other boundary points of the region are achieved as ; accordingly, can be simplified as
See Appendix E.
The corresponding optimal power allocation for and can be easily obtained given (12) and are thus omitted for brevity. Proposition IV.1 suggests that to achieve any boundary point of with and , the portion of transmission time allocated to power transfer in each block should asymptotically go to zero when , where denotes the number of transmitted symbols in each block. For example, by allocating symbols per block for power transfer and the remaining symbols for information transmission yields as , which satisfies the optimality condition given in Proposition IV.1.
It is worth noting that the boundary of in the flexible power constraint case is achieved under the assumption that the transmitter and receiver can both operate in the regime of infinite power in the EH time slot due to , which cannot be implemented with practical power amplifiers. Hence, a more feasible region for is obtained by adding peak888Note that the peak power constraint in this context is different from the signal amplitude constraint considered in , . transmit power constraints in (IV-B) as and , with . Similar to Proposition IV.1, it can be shown that the boundary of the achievable R-E region in this case, denoted by , is achieved by . Note that we can equivalently denote the achievable R-E region defined in (IV-B) or (12) without any peak power constraint as .
Iv-C Power Splitting
Next, we propose an alternative receiver design called power splitting (PS), whereby the power and information transfer to the co-located EH and ID receivers are simultaneously achieved via a set of power splitting devices, one device for each receiving antenna, as shown in Fig. 3(b). In order to gain more insight into the PS scheme, we consider first the simple case of a single-antenna AWGN channel with co-located ID and EH receivers, which is shown in Fig. 6(a). For the ease of comparison, the case of solely information transfer with one single ID receiver is also shown in Fig. 6(b).
The receiver operations in Fig. 6(a) are explained as follows: The received signal from the antenna is first corrupted by a Gaussian noise denoted by at the RF-band, which is assumed to have zero mean and equivalent baseband power . The RF-band signal is then fed into a power splitter, which is assumed to be perfect without any noise induced. After the power splitter, the portion of signal power split to the EH receiver is denoted by , and that to the ID receiver by . The signal split to the ID receiver then goes through a sequence of standard operations (see, e.g., ) to be converted from the RF band to baseband. During this process, the signal is additionally corrupted by another noise , which is independent of and assumed to be Gaussian and have zero mean and variance . To be consistent to the case with solely the ID receiver, it is reasonable to assume that the antenna noise and processing noise have the same distributions in both Figs. 6(a) and 6(b). It is further assumed that to be consistent with the system model introduced in Section II.
For this simple SISO AWGN channel, we denote the transmit power constraint by and the channel power gain by . It is then easy to compute the R-E region outer bound for this channel with co-located ID and EH receivers, which is simply a box specified by three vertices , and , with and . Interestingly, we will show next that under certain conditions, the PS scheme can in fact achieve all the rate-energy pairs in this R-E region outer bound; without loss of generality, it suffices to show that the vertex point is achievable.
With reference to Fig. 6(a), we discuss the PS scheme in the following three regimes with different values of antenna and processing noise power.
(Case I): In this ideal case with perfect receiving antenna, the antenna noise can be ignored and thus we have and . Accordingly, it is easy to show that the SNR, denoted by , at the ID receiver in Fig. 6(a) is . The achievable R-E region in this case is then given by . This region can be shown to coincide with the R-E region for the TS scheme with the flexible power constraint given by (12) for the SISO case.
(Case II): This is the most practically valid case. Since , we can show that in this case is given by . Accordingly, the achievable R-E region in this case is given by . It is easy to show that enlarges strictly as increases from 0 to 1.
(Case III): In this ideal case with perfect RF-to-baseband signal conversion, the processing noise can be ignored and thus we have and . In this case, the SNR for the ID receiver is given by , regardless of the value of . Thus, to maximize the power transfer, ideally we should set , i.e., splitting infinitesimally small power to the ID receiver since both the signal and antenna noise are scaled identically by the power splitter and there is no additional processing noise induced after the power splitting. With , the achievable R-E region in this case is given by , which becomes identical to the R-E region outer bound (which is a box as defined earlier).
Therefore, we know from the above discussions that only for the case of noise-free RF-band to baseband processing (i.e., Case III), the PS scheme achieves the R-E region outer bound and is thus optimal. However, in practice, such a condition can never be met perfectly, and thus the R-E region outer bound is in general still non-achievable with practical PS receivers. In the following, we will study further the achievable R-E region by the PS scheme for the more general case of MIMO channels. It is not difficult to show that if each receiving antenna satisfies the condition in Case III, the R-E region outer bound defined in (4) with is achievable for the MIMO case by the PS scheme (with each receiving antenna to set ). For a more practical purpose, we consider in the rest of this section the “worst” case performance of the PS scheme (i.e., Case I in the above), when the noiseless antenna is assumed (which leads to the smallest R-E region for the SISO AWGN channel case). The obtained R-E region will thus provide the performance lower bound for the PS scheme with practical receiver circuits. In this case, since there is no antenna noise and the processing noise is added after the power splitting, it is equivalent to assume that the aggregated receiver noise power remains unchanged with a power splitter at each receiving antenna. Let with denote the portion of power split to the EH receiver at the th receiving antenna, . The achievable R-E region for the PS scheme (in the worst case) is thus given by