The expected massive number of “Internet of Things” (IoT) devices coming online over the next decade is conditioned on first solving critical challenges, specially in terms of efficient massive access, light communication protocols, and sustainable powering mechanisms. As for the latter, existing solutions that rely on wired powering are usually cost-prohibitive or infeasible for ubiquitous deployment. Meanwhile battery-powered solutions face other drawbacks : i) limited lifetime, influenced by activity/usage, ii) frequent maintenance as different devices, and corresponding lifetimes, may coexist in the same environment, and iii) the battery waste processing problem. Therefore, alternative powering mechanisms and related technologies need to be developed in the coming years to realize the vision of a data-driven sustainable society, enabled by near-instant, secure, unlimited and green massive connectivity [1, 2].
Radio frequency (RF) wireless energy transfer (WET) (hereinafter just referred to as WET) constitutes an appealing technology to be further researched, developed and exploited for powering IoT deployments [1, 2, 3, 4]. Different from energy harvesting (EH) solutions relying on other energy sources, WET allows: i) small-form factor, ii) native multi-user support, and iii) relatively long range energy coverage. However, the ultra-low end-to-end power transfer efficiency (PTE) of WET systems, together with strict regulations on the electromagnetic field (EMF) radiation, are critical factors that may limit WET feasibility in practice. By addressing these challenges, while exploiting further advancements on ultra-low-power integrated circuits, WET may emerge as a revolutionary technology that will finally cut the “last wires” (i.e., cables for energy recharging ) for a truly wireless and autonomous connectivity. In fact, IoT industry is already strongly betting on this promising technology, proof of which is the variety of emerging enterprises with a large portfolio of WET solutions, for example Powercast, TransferFi and Ossia111See https://www.powercastco.com, https://www.transferﬁ.com and https://www.ossia.com..
Recent works on WET have mainly targeted extremely low-power/cost IoT applications, e.g., wireless sensor networks and RFIDs, because of the ultra-low PTE (see  and references therein). However, several promising technologies such as i) energy beamforming and waveform optimization [4, 5, 6, 7], ii) distributed [8, 9] and massive antenna [10, 11, 12, 13, 14, 15, 9] systems, iii) smart reflect arrays and reconfigurable metasurfaces , iv) motor-equipped power beacons (PBs) , flying PBs , PBs with rotary antennas ), and v) mobile computation offloading and crowd sensing , may broaden WET applicability, and turn it plausible for powering more energy-hungry IoT devices, e.g., smartphones, game console controllers, electronic toys. In this work, we leverage some of above enablers to investigate WET’s feasibility for power-hungry indoor charging. Specifically, optimal and sub-optimal energy beamforming schemes are proposed and discussed for a cell-free (distributed) massive multiple-input multiple-output (mMIMO) radio stripes system deployed in an indoor small-area environment (thus with limited serving distances) for multi-user RF wireless charging as illustrated in Fig. 1.
I-a Related Work
The max-min throughput optimization problem for a mMIMO wireless powered communication network (WPCN) is addressed in . Therein, authors show that the asymptotically, in the number of transmit antennas, optimal energy beamformer is akin to maximum ratio transmission (MRT) in MIMO communications. Taking advantage of the latter results, authors in  investigate the overall PTE and the energy efficiency (EE) of a wireless powered communication network (WPCN), where a mMIMO base station (BS) uses WET to charge single-antenna EH users in the downlink. A piece-wise linear EH model just considering sensitivity and saturation impairments is adopted, and it is shown that increasing the transmit power improves the EE as the number of antennas becomes large. However, authors in [10, 11] assume that all the harvested energy is used for uplink information transmission, ignoring other important energy consumption sources. Meanwhile, a low-complexity mMIMO WET scheme based on the retrodirective beamforming technique, where all EH devices send a common beacon simultaneously to the PB in the uplink and the PB simply conjugates and amplifies its received sum-signal for downlink WET, is investigated in  relying on the asymptotically optimum MRT-like precoding . The performance in terms of received net energy (harvested energy minus energy consumed during training) of a multi-user mMIMO system with Rician fading channels is analyzed in . Two different schemes are considered: i) training-based WET, and ii) line-of-sight (LOS) beamforming, and authors derive a user-specific path loss threshold for switching between them. The appeal of LOS-based beamforming is also illustrated in , where an MRT-based energy beamforming algorithm with low complexity for mMIMO is proposed. However, the system performance is analyzed only in the ultra-low EH regime in [13, 7], while the channel assumption in  is also very restrictive. Finally, readers may refer to [14, 15] where authors investigate the performance of mMIMO-enabled simultaneous wireless information and power transfer (SWIPT) systems.
As evinced above, single-transmitter mMIMO technology for WET has been considerably investigated. Surprisingly, much less efforts have been put to investigate distributed mMIMO technology, which makes WET more feasible by shortening the channel distances . To the best of our knowledge, there is no prior work addressing cell-free mMIMO with radio stripes  for WET, in which actual PBs consist of antenna elements and circuit-mounted chips inside the protective casing of a cable/stripe, allowing imperceptible installation and alleviating the problem of deployment permissions . We show later that it is indeed possible to deliver significant amounts of energy over practical third generation partnership project (3GPP) channels to multiple EH receivers by using this technology in an indoor small-area environment.
On the other hand, WET systems are subject to strict transmit constraints that limit their performance in practice. Specifically, the EMF caused by energy radiation must be strictly limited in the presence of humans (and other living species) to reduce the risks of potential biological effects (e.g., tissue heating) [20, 21]. Concerns about adverse consequences of EMF exposure have resulted in the establishment of exposure limits222EMF limits are set by the International Commission on Non-Ionizing Radiation Protection (ICNIRP) in most of Europe . such as the maximum permissible exposure (MPE), also called power density, measured in . Although several works have considered such important constraints in WET systems, e.g., [22, 23], to the best of our knowledge none has yet addressed them in the context of mMIMO WET, let alone in a distributed cell-free mMIMO setup for WET.
This article considers an indoor cell-free mMIMO system with radio stripes for multi-user WET under EMF-related constraints. Our main contributions are:
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We introduce an indoor cell-free mMIMO system with radio stripes for wirelessly charging user equipments (UEs). The radio stripes system is composed of multi-antenna PBs subject to maximum transmit power constraints, which coherently beamform the energy signals towards the devices using channel state information (CSI) obtained from uplink training, and PTE information of the EH circuits at the UEs.
We formulate the semi-definite programming (SDP)-based precoding design that minimizes the transmit power subject to stringent EH requirements per user. Moreover, we apply the successive convex approximation (SCA) technique to reduce complexity, while a sub-optimal design based on MRT precoding is also proposed. For the special case of a single UE scenario, the MRT precoding is reformulated, and we show that it requires at most low complexity iterative operations.
We propose an analytical framework to evaluate the RF power density in the proximity of the UEs, and that caused at a random point in the network. We formulate these figures in terms of EMF constraints and incorporate them to the original optimization problem accordingly.
We show that the spatial transmit gains from incorporating more PBs to the radio stripes at high frequencies do not compensate the path loss, demanding more transmit energy resources. However, the EMF radiation exposure can be more easily controlled at higher frequencies.
We provide numerical evidence that training is not a very critical (limiting) factor for the considered indoor (short-range) system as just little energy is required to attain a performance similar to that of a system with ideal CSI. Moreover, we show that the system incurs in a greater transmit power consumption once the per-PB power and EMF-related constraints are considered.
We discuss key trade-offs between spatial division multiple access (SDMA) and time division multiple access (TDMA). We show that TDMA is more efficient when serving a small number of UEs, while SDMA may be preferable when the number of UEs is large.
The remainder of this paper is organized as follows. Section II introduces the system model and problem formulation. Sections III and IV present the global optimization framework, and alternative low-complexity solutions, respectively, while Section V addresses the single UE case. Section VI introduces the analytical framework for
EMF-related metrics. Finally, Section VII presents numerical results, and Section VIII concludes the article.
Boldface lowercase/uppercase letters denote column vectors/matrices;is a vector of ones; while
is the identity matrix. Superscripts, and are the complex conjugate, transpose and Hermitian operations, respectively. and are the Euclidean norm of a vector and trace of a matrix, respectively. The curled inequality symbol denotes generalized inequality: between vectors, it represents component-wise inequality; between symmetric matrices, it represents matrix inequality. Meanwhile, is the set of non-negative real numbers, is the set of Hermitian matrices of dimensions , is the set of complex numbers, respectively, and is the imaginary unit. Additionally, , and are the minimum, fractional part, and floor functions, respectively, while is the absolute (or cardinality for sets) operation. denotes statistical expectation, outputs the real part of the argument, and is partial differentiation. Finally, is the big-O notation, and is a circularly-symmetric complex Gaussian random vector with mean and covariance matrix .
Ii Radio Stripes System Model
As in Fig. 1, we consider a cell-free mMIMO radio stripes network comprising of a set of PBs333Although not relevant for the scenario discussed here, we may assume that front-haul connections goes from central processing unit (CPU)., each with antennas, wirelessly powering single antenna UEs in the downlink. In practice, should hold, and might be required so that the radio stripes system can energize relatively power hungry devices such as mobile phones, console controllers, etc. Such UEs may have been scheduled in advance for the system to charge, thus may be part of a greater set of UEs with charging demands.
The PBs are equally placed on a radio stripe of length (m) which is wrapped around a square perimeter of the same length, e.g., a room or office, at a height from the floor level. Then, the number of PBs that can be deployed is upper bounded by
by considering half-wavelength spaced antenna elements, where is the system operation wavelength. Moreover, we consider quasi-static block Rician fading, such that the channel between and is distributed as
where is the LOS component and is the positive semi-definite covariance matrix. However, the methods and analysis presented in Sections III-V hold independently of the channel fading distribution. Further, the coherence block consists of
channel uses. Prior to the downlink WET, there is an uplink channel estimation phase consisting ofchannel uses for pilots transmissions from the UEs. Therefore, the downlink WET occurs over the remaining channel uses. Both phases are illustrated in Fig. 1 and described in the following.
Ii-a Uplink Channel Estimation
We assume there are mutually orthogonal length pilot vector signals , with , which are used by the UEs for channel estimation. We assume that and the pilot sequence assigned to is , thus, each UE owns a unique pilot sequence. Then, the pilot signal received at in the CSI acquisition phase is
where is the fixed transmit power of , and is the noise at the receiver modeled with independent entries distributed as .
Herein, we assume the least square (LS) channel estimate
where the channel error estimate is distributed as
thus, . Note that there is no need of using more complex methods such as the minimum mean square error (MMSE) estimator for this kind of short-distance setup. This is because pilots are orthogonal, and , thus, channel estimates are already very precise when using LS. Moreover, the MMSE method would require prior knowledge of and , which are subject to estimation errors that propagate to the actual estimate of .
Ii-B Downlink WET
At each channel use of the downlink WET phase, each transmits energy symbols such that the RF signal received by is given by
where is the precoding vector associated to . Note that, in general, the number of powering signals, thus, the number of precoders, does not necessarily need to match the number of UEs. However, as the energy requirements per UE become stringent, a dedicated beam per UE may be required. Moreover, noise impact is ignored since it is practically null for EH purposes, and it is assumed that symbols
are independent random variables, i.e.,, , normalized so as to satisfy a unit average power constraint, i.e., . Then, the incident average RF power available at each per channel use of the WET phase is given by
where comes from using (6), follows after re-arranging terms, and is immediately attained from leveraging the assumption of independent and power-normalized signals. Meanwhile, the energy harvested by in a block (comprising both phases) converges to
for sufficiently large , e.g., , and considering unit time blocks without loss of generality. Note that is a non-decreasing function modeling the relation between the incident RF power and harvested direct current (DC) power at , and and are the DC sensitivity and saturation levels, respectively. Although, in general, also depends on the modulation and incoming waveform , we may ignore such effects if no optimization and/or different modulation schemes are utilized as is the case here.
Ii-C Problem Formulation
The goal is to minimize the power consumption of the radio stripes system by properly designing/setting the precoding vectors .
The system is subject to per-PB power and per-UE EH constraints at each WET phase. This is, each PB circuit is subject to a total power constraint , and each has a target energy requirement per block. With a proper knowledge of devices EH hardware characteristics, i.e., , such constraint can be transformed to
where the hat accent denotes estimation since the true channel realizations are never known with perfection, thus and can be evaluated using (8) and (7), respectively, but substituting by . In practice, a constraint in the form of (9) does not prevent that the true harvested energy may fall below the threshold for poorly estimated channel realizations. However, as we shall see later in Section VII, channel estimates are indeed very accurate in the considered setup. Moreover, small fluctuations around average out when considering consecutive coherence time intervals and will not impact the WET performance in practice.
Since is a non-decreasing function, it is invertible as long as .444Obviously is not practically viable; while in case , we can set to be . Therefore, the optimization problem can be formulated as
which is non-convex due to (10b). In Sections III-V we discuss different optimization approaches to solve . Meanwhile, note that the radio stripes deployment topology enables a distributed 3D beamforming (avoiding the generation of strong beams since transmit antenna elements are spread over a square perimeter) pointing to the locations of the UEs, where the EMF exposure levels are expected to be the greatest. However, , in its current form, does not prevent increased EMF levels at other spatial points as well. A EMF-aware optimization, under which EMF-related constraints are incorporated into , is addressed later in Section VI.
Iii Global Optimization Framework
Let us proceed by defining
Then, departing from (7) we have that
where and . Now, can be optimally found by solving the following (convex) SDP
where is a sensing matrix designed to collect the total transmit power of each PB, thus, it has unit entries from the th until the th element of each row , and zero in the remaining entries. Notice that (14a), (14b) and (14c) are equivalent to (10a), (10b) and (10c), respectively. After solving using standard convex optimization frameworks, e.g., CVX , the composite precoding vectors , with equal to the rank of
, can be obtained as the eigenvectors of.
Interestingly, the precoding optimization for information multicasting, where the same information-bearing signal is simultaneously transmitted to all users, usually relies on a semi-definite relaxation with a structure similar to that of (14) . Therefore, after solving the SDP, it requires a last step to force rank-1 multicast beamforming, e.g., via Gaussian randomization, thus leading to sub-optimal solutions. Meanwhile, the optimum energy precoding here is not rank constrained since energy can be collected from any arbitrary superposition of energy carrying symbols, thus, directly obtained from solving .
Iii-a Problem Feasibility
Note that by removing the per-PB power constraint (14c) from , the resulting relaxed problem is always feasible. This is because for any precoding phase design, the corresponding transmit power could boundlessly increase until constraints in (14b) are met. Therefore, the problem feasibility can be verified by i) finding the minimum that allows satisfying (14b)(14d), and ii) verifying that it is indeed not greater than the physical per-PB power constraint. Note that the feasibility may be controlled by the charging scheduling decision, which establishes the set of UEs to be served, thus regulating the influence of constraint (14b).
Iii-B Complexity Analysis
Interior point methods are usually adopted to efficiently find the optimal solution of SDP formulations. In this case, it can be shown that solving requires around iterations, with each iteration requiring arithmetic operations , and where represents the solution accuracy attained when the algorithm ends. Consequently, both, determining the problem feasibility and finding the optimum solution, become computationally costly as the total number of antenna elements of the radio stripes system increases. To achieve complexity reduction, a low-complexity optimization approach is proposed next as an efficient alternative to attain near-optimum performance.
Iv Low-Complexity Optimization
In this section, we elaborate on finding a low-complexity solution for the original non-convex problem . We adopt the SCA technique [29, 30], wherein the non-convex problem is recast as a sequence of convex subproblems, and then iteratively solved until convergence.
Iv-a Optimal Precoder Design
First, let us rewrite as
which comes from a procedure similar to that leading to (13). Now, observe that constraint (10b) can be written as , where is quadratic, thus a convex function [31, Ch. 3]. Therefore, the rewritten constraint constitutes a difference of convex functions, thus its best convex approximation can be obtained by replacing with its first-order approximation . The first-order Taylor approximation of around a fixed operating point can be expressed as
where . Hence, can be approximated in the vicinity of a fixed operating point as
Thus, by iteratively solving problem while updating with the solution at each iteration, we can find the best local optimal solution . The proposed low-complexity iterative method for problem is summarized in Algorithm 1.
Iv-B MRT based Precoder Design
In this subsection, we exploit further complexity reduction by adopting an approach similar to the proposed in [10, 11, 12, 7]. The key lies in setting and adopting a precoder design akin to MRT in MIMO communications as follows
where represents the power budget of for . Then, with perfect CSI knowledge, the th signal transmitted from all radio stripes antennas arrives at with constructive superposition. Meanwhile, note that the impact of the signals meant to other devices on the energy harvested at is not considered for the phases’ design, thus leading to an unavoidable performance degradation with respect to the SDP-based and/or SCA-based optimal solution previously discussed. With above precoder, the estimated RF incident power available at each per channel use of a WET phase given in (7) transforms to
where becomes now the optimization variable, and denotes the normalized precoder.
We can observe that by using expression (19) in problem , the constraint (10b) is a difference of convex functions, and hence is still non-convex problem. We again resort to SCA framework  to find a solution. The first-order Taylor approximation of around a fixed operating point can be expressed as
Hence, by using (20), the MRT-optimum precoder can be found by solving
where the fixed operating point is iteratively updated as summarized in Algorithm 2.
Iv-C Complexity Analysis
The approximated convex problems and can be efficiently solved using generic convex optimization tools, i.e., as a sequence of second-order cone program (SOCP) . Interior points methods are usually adopted to efficiently solve SOCP formulations, and would require a number of iterations in the order of and in case of and , respectively. Note that in case of , such required number of iterations matches that of the SDP formulation in . Meanwhile, does require less iterations, but such gain may disappear once the SCA iterative procedure of Algorithm 2, by which needs to be repeatedly solved, is taken into account. However, the clear and definite advantage behind using the optimization methods proposed in this section lies on the significantly reduced worst-case computational requirement of each SCA iteration. Note that solving requires arithmetic operations, while is solvable using arithmetic operations.
V Single UE Case
Herein, we study a scenario of particular interest: the single UE case. Note that the system may serve the UEs in a TDMA fashion, in which case the derivations and performance analysis here hold accurate. Alternatively, the performance results for this scenario correspond to upper bounds of what it is expected if more UEs are simultaneously served in the same time interval.
V-a Optimum Precoder
In case of a single UE, the MRT-like precoder discussed in Section IV-B is the optimum. Then, let us relax the per-PB power constraint. The optimum concatenated precoder is
thus . However, such solution may demand a transmit power above the per-PB power constraint at some PB since would need to be
and has not been bounded.
Let us assume that the set of PBs operating with transmit power below is known beforehand. Then, we have that
where . Using this result, one can transform to obtain
Now, let us write the Lagrangian of as
where is the Lagrangian multiplier. Observe that (27) is a convex quadratic function of every , thus minimized at
Then, the Lagrange dual of is given by
This means that if the set of PBs are known to operate with transmit power , the remaining PBs optimum transmit power is given by , where is given in (31).
Finally, the optimum precoder can be constructed as shown in Algorithm 3. In a nutshell, the transmit power of each PB is computed by exploiting (31) (line 8). In case a power allocation exceeds the maximum allowed power , it is immediately reduced to match (lines 4-6). The process is repeated until all PBs are allocated a transmit power not greater than . Note that this is an extremely simple optimization algorithm, where at most iterative computations of (31) are required, but still provides the optimum WET precoder.
V-B Problem Feasibility
The feasibility of a single UE optimization problem is rather simple to verify. It is enough setting , and determine whether holds or not. In this case, can be evaluated departing from (19) as
Then, if , we can conclude the problem is feasible and the solution can be obtained from Algorithm 3. Finally, note that in the scenario with UEs it is sufficient (but not necessary) that for some to declare that is not feasible.
V-C Minimum Average Radio Stripes Transmit Power
One interesting question is how much average power is required to serve the UE. The minimum average transmit power required by the radio stripes system can be computed by assuming no per-PB power constraint as follows
where comes from applying Jensen’s inequality.
Vi EMF-Aware Optimization
In this section, we present an analytical framework to determine, analyze and limit EMF exposure levels. Different from previous sections where results are generic, herein we rely on the assumption of quasi-static Rician fading. However, the accuracy of our approaches and claims here should still approximately hold when considering other (more practical) fading distributions with the same first and second order statistics, i.e, same and . The latter is because the channel hardening and favorable propagation phenomena enabled by mMIMO, which makes the communication performance almost independent of the small-scale fading realizations .
We assume each PB of the radio stripes (Fig. 1) is equipped with a relative small number of antennas such that is bounded to a desired level555Assuming a small number of antennas per PB also implies that each UE is in the far field with respect to all PBs., and we impose a separation restriction between consecutive PBs. To ease the deployment and system design, we assume that no PB expands over any consecutive two sides of the perimeter, thus, the upper bound in (1) further tightens to
As an example, Fig. 3 illustrates the maximum number of PBs that could be equipped in the radio stripes system as a function of the central operation frequency .
Note that the focus of our work is on indoor scenarios where LOS channels are predominant. Thus, we first characterize the LOS channels and then introduce the EMF exposure metrics.
Vi-a LOS Geometry
The geometry of the discussed scenario is illustrated in Fig. 3. The LOS (geometric) channel component between and is given by
where denotes its large-scale fading component. Herein we adopt the following 3GPP indoor LOS path-loss model 
where and are the 3D coordinate position of the ’s antenna array and , is given in GHz, and distance is measured in meters. Note that the non-LOS (NLOS) channel component , such that , would be distributed as , where
Moreover, the Rician LOS factor as a measure of how strong the LOS channel is with respect to the NLOS components is thus defined as , and dB holds.
where is the azimuth angle relative to the boresight of the ’s antenna array, thus
where or if the antenna array of the corresponding PB is aligned with the or axis, respectively, while is the distance component between and . Finally, accounts for an initial phase shift, which in our case depends on the PB and UE position as
Vi-B EMF Level in the Proximity of the UEs
Herein, we analyze the EMF exposure level for the user holding the device. In the proximity of the UEs, high EMF exposure levels may come from the coherent combination of the signal, specially if channels have not been accurately estimated. In this case, an appropriate performance figure could be how fast does the RF power density falls as a spatial point moves away from the device antenna.
Using the derivations in previous section, the first-order statistics of the channel at a distance from , and with angular position defined by (measured in the plane) and (measured in the axis), can be defined as in a similar way to (35) but with a virtual measurement point located at
This point lies in the radius sphere centered at as illustrated in Fig. 3. Then, the expected RF power density at a distance from , herein called proximity region, is given by