Wirelessly Powered Data Aggregation for IoT via OvertheAir Functional Computation: Beamforming and Power Control
Abstract
As a revolution in networking, Internet of Things (IoT) aims at automating the operations of our societies by connecting and leveraging an enormous number of distributed devices (e.g., sensors and actuators). One design challenge is efficient wireless data aggregation (WDA) over tremendous IoT devices. This can enable a series of IoT applications ranging from latencysensitive highmobility sensing to dataintensive distributed machine learning. Overtheair (functional) computation (AirComp) has emerged to be a promising solution that merges computing and communication by exploiting analogwave addition in the air. Another IoT design challenge is battery recharging for dense sensors which can be tackled by wireless power transfer (WPT). The coexisting of AirComp and WPT in IoT system calls for their integration to enhance the performance and efficiency of WDA. This motivates the current work on developing the wirelessly powered AirComp (WPAirComp) framework by jointly optimizing wireless power control, energy and (data) aggregation beamforming to minimize the AirComp error. To derive a practical solution, we recast the nonconvex joint optimization problem into the equivalent outer and inner subproblems for (inner) wireless power control and energy beamforming, and (outer) the efficient aggregation beamforming, respectively. The former is solved in closed form while the latter is efficiently solved using the semidefinite relaxation technique. The results reveal that the optimal energy beams point to the dominant eigendirections of the WPT channels, and the optimal power allocation tends to equalize the closeloop (downlink WPT and uplink AirComp) effective channels of different sensors. Simulation demonstrates that controlling WPT provides additional design dimensions for substantially reducing the AirComp error.
I Introduction
In the near future, tens of billions of Internetofthings (IoT) devices (e.g., sensors and actuators) are expected to be deployed to automate the operations of our societies and make the ambient environment smart. Among others, there exist two design challenges for IoT. The first is fast wireless data aggregation (WDA), namely fast collection and processing of data distributed at tremendous IoT devices by wireless transmission. WDA is an enabling operation for a series of IoT applications such as fusion of sensing values in environmental monitoring [1], aggregation of mobile updates in federated machine learning [2], and distributed consensus in fleet driving [3]. Fast WDA is needed to regulate latency in cases with ultradense devices and/or high mobility (e.g., for sensors carried by drones or vehicles). A promising solution is overtheair (functional) computation (AirComp), which realizes fast WDA by simultaneous transmissions and exploiting analogwave addition in a multiaccess channel [22]. The other design challenge for IoT is powering dense energyconstrained sensors for WDA and other operations. One attractive solution is wireless power transfer (WPT) using microwaves, whose feasibility has been proven in practical sensor networks [4].
To facilitate efficient implementation for IoT, it is natural to tackle the two said challenges simultaneously by pursuing the fusion of two corresponding technologies: AirComp and WPT. The resultant design challenge lies in the joint optimization of their key operations at servers (or fusion centers), namely energy beamforming and power control for WPTï¼ and (data) aggregation beamforming for AirComp. This motivates the current work on developing a framework called wirelessly powered overtheair computation (WPAirComp).
Ia WDA via OvertheAir Computation
Data aggregation in IoT can be posed as the mathematical problem of computing at a server a function of distributed data samples, denoted as , which is characterized by the following form:
(1) 
where and represent postprocessing at the server and preprocessing at a device, respectively. The class of functions having the above form is known as nomographic functions such as averaging and geometric mean. Typical functions in this class are summarized in Table I. The conventional approach for WDA decouples data collection, namely distributed transmission of , and functional computation, namely computing in (1). The approach is incapable of supporting fast WDA as the application of any traditional orthogonal multiaccess scheme for data collection results in the linear scaling of latency with the number of devices. In contrast, AirComp merges data collection (or else radio resources) and computation. Specifically, the summation in (1) is implemented by simultaneous analog transmission to exploit the waveaddition of the multiaccess channel. Consequently, the functional computation is performed ”overtheair” and the result is directly received by the server, thus giving the name of the technology. Simultaneous transmission in AirComp achieves low latency independent of the number of devices, and thereby enables fast WDA.
Name  Expression 

Arithmetic Mean  
Weighted Sum  
Geometric Mean  
Polynomial  
Euclidean Norm 
The idea of AirComp can be traced back to the pioneering work studying functional computation in sensor networks [23]. In [23], structured codes (e.g., lattice codes) are designed for reliable functional computation at a server based on distributed sensing values analog modulated and transmitted over a ãmultiaccess channel. The importance of the work lies in the counterintuitive finding that interference caused by simultaneous transmission can be exploited to facilitate computation. Subsequently, it was proved that the simple analog transmission without coding is optimal in terms of minimizing functional distortion in the case of independent Gaussian data sources [24]. Nevertheless, coding is still useful if the sources follow more complex distributions, such as bivariate Gaussian [25], correlated Gaussian [26], and sum of independent Gaussian [23]. The promising performance (with optimality in certain cases) of simple analog AirComp has led to an active area focusing on its robustness and implementation [27, 28, 29, 30, 22, 31]. In particular, techniques for distributed power control and robust AirComp against channel estimation errors are proposed in [27] and [28], respectively. Another vein of research focuses on transforming AirComp from theory into practice by prototyping [22] and addressing practical issues such as transmission synchronization over sensors [31, 30].
It is also worth mentioning that inspired by the success of AirComp in computationcentric sensor networks, the relevant principles have been applied to design new schemes for ratecentric communication networks. The computeandforward relaying schemes as proposed in [32] decodes and forwards linear functions of the transmitted messages. The integerforcing linear receiver designed in [33] spatial multiplexes lattice codewords. Furthermore, the wellknown physical layer network coding leveraging analog wave addition generalizes the celebrated network coding schemes invented for wired networks to wireless networks [34].
IoT will feature multimodal sensing and the prevalence of antenna arrays (especially highly compact millimeterwave arrays) at both servers and devices [35]. A multimodal sensor can simultaneous monitor multiple parameters of the environment (e.g., pressure, light, humidity, and pollution), or of a user state (e.g., motion, location, and speech). The need of WDA over multimodal sensors and other dataintensive IoT applications (e.g., federated machine learning) calls for the acceleration of computation rates in AirComp. While prior works mostly target singleantenna sensor network and support scalarfunction AirComp, recent research has started to explore the possibility of using antenna arrays to enable the vectorfunction AirComp. A technique called multipleinputmultipleoutput (MIMO) AirComp leverages spatial degreesoffreedom to spatially multiplex multifunction computation (i.e., vectorfunction computation) and reduce computation errors by noise suppression [29]. The key design challenge lies in the optimization of the receive beamformer, called aggregation beamformer, with the objective of minimizing the error of computed functions, w.r.t., the desired result in (1). On the other hand, traditional multiuser MIMO beamforming aims at a different objective that is to minimize the sum distortion of individual data symbols. The difference in objective results in a fundamental shift in the beamforming design principle. To be specific, multiuser beamforming attempts to decouple simultaneously transmitted symbols (or equivalently sumrate maximization), yielding the classic zeroforcing or minimizing meansquarederror (MMSE) design [36]. In contrast, aggregation beamforming tries to balance the effects of individual MIMO channels so as to accurately compute the function in (1). The principle is reflected in the beamformer design in [29] that is obtained from the weighted sum of multiuser MIMO channels projected onto a Grassmann manifold.
IB Energy Beamforming for Wirelessly Powered Communication
WPT was originally developed for pointtopoint power delivery using radio waves [5]. Recently, the technology has been further developed to power communication networks [6]. One research focus is energy beamforming, referring to the use of an antenna array at a transmitter to beam energy in the direction of a targeted receiver [7, 8]. The integration of WPT with traditional wireless communication has created a new area, called simultaneous wireless information and power transfer (SWIPT), which remains active. Recent studies focus on applying SWIPT to a series of communication systems, including MIMO communication [9, 10], orthogonal frequencydivision multiple access (OFDMA) [11, 12], twoway transmission [13], relaying [14, 15, 16], and cognitive networking [17]. Practical SWIPT designs accounting for imperfect channel state information were developed in [18]. More recently, WPT was also considered for unmanned aerial vehicle (UAV) assisted communication [19], where an UAV serves as a mobile relay and its transmit power and trajectory are jointly optimized to maximize the throughput.
Another important application of WPT is in sensor networks, with energy constrained sensors. Recharging the batteries of hundreds to thousands of sensors is a challenging problem that can be solved efficiently by WPT [4]. For multiuser WPT systems, energy beamforming at different power beacons needs to be coordinated to enable efficient energy multicasting. To this end, a collaborative energybeamforming scheme is proposed in [20] for efficiently powering a sensor network. For largescale wirelessly powered sensor networks, a novel framework of backscatter sensing was recently proposed in [21], where lowcost passive sensors upload their sensing data to a drone mounted reader by concurrently reflecting the beamed power signal from a power beacon in a designed probabilistic manner. Then statistical inference algorithms can be devised for sensing value recovery without the knowledge of channel state information.
Though energy beamforming in wirelessly powered communication and sensor networks have been widely investigated, most of prior work focuses on rate maximization via optimizing the WPT efficiency. The design of wirelessly powered innetwork computation, such as WPAirComp in this work, remains as uncharted area.
IC Contributions and Organization
We consider an IoT system supporting downlink WPT and uplink AirComp. To be specific, a multiantenna server transmits energy to power multiple sensors (or other types of devices) so that they can transmit sensing data for WDA at the server based on AirComp. The server controls the amount of energy harvested by sensors via energy beamforming and power allocation to different beams. Since the transmit power of sensors depends on the harvested energy, the two WPT operations affect the analogsignal superposition in the array observations at the server. This introduces coupling between WPT and AirComp, and hence necessitates the joint design of wireless power control, energy and (data) aggregation beamforming, which is the core of WPAirComp development. Via the joint design, WPT contributes additional dimensions for reducing the computation error in AirComp. The contributions of this work are summarized as follows.

Decomposition Based Design Approach: The said joint design is formulated as an optimization problem for minimizing the computation error in AirComp under a power constraint for WPT. Without compromising optimality, a decomposition approach is derived that decomposes the nonconvex problem into two tractable nested subproblems. The inner subproblem concerns WPT optimization, involving the joint design of power control and energy beamforming, given the aggregation beamformer. The outer subproblem is the optimization of aggregation beamformer. The approach is applied to develop an efficient convexoptimization algorithm for solving the nonconvex joint design problem of the simplified multipleinput singleoutput (MISO) case with singleantenna sensors, corresponding to scalarfunction AirComp. The results are then extended to the general MIMO case with multiantenna sensors, corresponding to vectorfunction AirComp.

Wireless Power Control and Energy Beamforming: The said inner subproblem is solved in closed form. The solution has simple structures that facilitate the implementation of WPAirComp. First, the optimal policy for wireless power control attempts to equalize the multiple cascaded WPTAirComp (or downlinkuplink) channels to facilitate the outer subproblem of designing aggregation beamformer. Second, each optimal energy beam points to the direction of the corresponding WPT channel vector in the MISO case or the dominant eigenvector of the channel matrix in the MIMO case.

Data Aggregation Beamforming: The said outer subproblem is shown to be solvable by the powerful technique of semidefinite relaxation (SDR). It is demonstrated by simulation that the joint design with WPT leads to significant gain in computation accuracy compared with decoupled designs.
Organization: The remainder of the paper is organized as follows. Section II introduces the WPAirComp system model. Section III presents the problem formulation for the joint design of power control, energy and aggregation beamforming. The solution for the MISO case is presented in Section IV. The extension to the MIMO case is given in Section V. Section VI further analyzes the difference between WPAirComp framework and traditional designs. Simulation results are provided in Section VII, followed by concluding remarks in Section VIII.
Notation: Boldface lowercase symbols denote vectors (e.g., ) and boldface uppercase symbols denote matrices (e.g., ). , and denote the real domain, positive real domain and complex domain, respectively. denotes the space of vector/matrix. and represent the conjugate transpose and inverse operations on a matrix , respectively. and represent the trace and minimum eigenvalue of the square matrix , respectively. denotes the expectation of a vector/matrix, denotes the identity matrix, denotes the Euclidean norm. Last, denotes the complex Gaussian distribution with mean and variance .
Ii System Model
We consider the mobile IoT system illustrated in Fig. 1, where WPAirComp is deployed for fast WDA. As shown in Fig 1 (c), the mobile server is multifunctional serving as a power beacon, an access point and a data fusion center. The server is provisioned with an array of antennas. We consider both the cases of singleantenna and multiantenna sensors, corresponding to the equivalent cases of MISO and MIMO channels. Let denote the number of antennas at each of total sensors. Time is divided into symbol durations, each of which lasts seconds and is called a (time) slot. WPAirComp is implemented based on the harvestedthentransmit protocol that alternates WPT and AirComp phases with corresponding fixed durations. The energy a sensor harvests in a WPT phase is applied to power transmission in the following AirComp phase. The operations of different sensors are synchronized using a reference clock broadcast by the server (see e.g., [30]). For simplicity, channels are assumed to vary following the blockfading model. In other words, each channel remains fixed within a (WPT/AirComp) phase and varies over different phases.
Iia WPT Phase
Consider an arbitrary WPT phase. The server serves as a power beacon that allocates power for different sensors and delivers the power by energy beamforming. Let represent the MIMO channel for the link from the server to sensor , denote the power allocated to sensor , and with denote the transmit beamforming vector. The server transmission power is limited by a fixed value , giving the following total power constraint:
(2) 
At sensor , a receive beam is steered to harvested the transferred energy with the beamforming vector denoted as and .
Assuming energy beams are sufficiently sharp such that harvesting the side lobes of unintended beams gives a sensor negligible energy. Moreover, the energyconversion process at a sensor, say sensor , is characterized by a fixed efficiency factor, denoted as . Then the amount of energy harvested by sensor in a WPT phase of slots is given by
(3) 
IiB AirComp Phase
Consider an arbitrary AirComp phase. Each sensor records the values of heterogeneous parameters of an external timevarying process such as the ambient environment, or a human being. The measurement generates a vector symbol in each slot, represented by for sensor . Powered by WPT, each sensor transmits vector symbols using its array to the server. On the other hand, the server doubly serves as an access point and a fusion center in the current phase. For WDA, the server aims at computing a vector function of the distributed vector symbols . Let denote the desired computation output, called the targetfunction vector. Then is a nomographic function, denoted as , of simultaneous observations of parameter , namely . Following the definition in (1), we have
(4) 
where and represent the postprocessing and preprocessing functions (see Table I for examples). Let the vector symbol preprocessed and transmitted by sensor using linear analog modulation be denoted as . To facilitate power control but without loss of generality, is assumed to have unit variance, i.e., . The WDA process and performance metric are described as follows.
Given synchronized symbol boundaries, all sensors transmit their vector symbols simultaneously. The distortion of array observations at the server with respect to the targetfunction vectors due to channel noise is suppressed using aggregation beamforming. To this end, the WDA process attempts to achieve coherent combining of vector symbols at the server in each slot. Let denote the aggregation beamforming matrix and the data precoding matrix at sensor . Then the vector symbol received by the server after beamforming is given by:
(5) 
where represents the MIMO channel for the link from sensor to the server, and is the additive white Gaussian noise (AWGN) vector with independent and identically distributed (i.i.d.) elements. The distortion of with respect to the desired vector , called the computation error, quantifies the AirComp performance and is measured by the meansquarederror (MSE) defined as
(6) 
Substituting (5) into (6), the MSE can be explicitly written as a function of the aggregation beamforming and precoding as follows:
(7) 
The aggregation beamforming and precoding are jointly optimized in the sequel under the criteria of MMSE. Since the transmission energy of each sensor cannot exceed the harvested energy, the transmission power constraint for sensor can be obtained using (3) as
(8) 
where denotes the number of vector symbols transmitted within each AirComp phase and the factor represents the fraction of harvested energy allocated for transmission. For ease of notation, we denote as the effective power conversion efficiency accounting for all the power discounting factors in the subsequent analysis.
Iii Joint Design of WPT and AirComp: Problem Formulation
The joint design of the WPT and AirComp phases in the WPAirComp system can be formulated as a joint optimization problem over wireless power control, energy and aggregation beamforming. Specifically, given the computation error in (7) and the two power constraints in (2) and (8), the problem can be formulated as:
Problem P1 is difficult to solve due to its nonconvexity. The lack of convexity arises from the coupling between the transmit and receive beamformers for both the WPT and AirComp phases. To simplify the problem and shed light on the optimal solution structure, we first consider the simplified case of singleantenna sensors, resulting in MISO channels, and thus called the MISO case. Note that in the MISO case, only scalarfunction AirComp is feasible. In the optimization problem for this case, the receive energy beamformers at sensors varnish and the data precoders reduce to scalars denoted as , making the solution tractable. Let the th vector channels for AirComp and WPT be denoted as and respectively, and the aggregation beamformer vector as . Then the original problem in P1 is simplified for the MISO case as:
Problem P2 is solved in the next section. The insights are leveraged to develop a practical solution in Section V. Last, we remark that an alternative and also natural formulation based on maximization of the receive signaltonoise ratio (SNR) leads to strategies fundamentally different from computationerror minimization. More details are given in Section VI.
Iv Joint Design of WPT and AirComp: MISO Case
In this section, we consider the MISO case corresponding to singleantenna sensors and solve the design problem in P2. To this end, a decomposition based solution approach is developed. The approach and the solution of Problem P2 are discussed in the following subsections.
Iva Decomposition Approach
Problem P2 remains nonconvex and difficult to solve directly. We overcome the difficulty by decomposing it into two solvable subproblems. To begin with, channelinversion precoding conditioned aggregation beamforming is shown to be optimal as follows.
Lemma 1 (Optimal Sensor Precoder).
For the MISO case, given a data aggregation beamformer , the computation error is minimized by the following channelinversion precoders at sensors:
(9) 
where is defined by normalizing the aggregation beamformer, and is chosen to satisfy the sensor transmission power constraints in Problem P2, i.e., to guarantee that .
Proof: See Appendix A.
By substituting the result in Lemma 1, Problem P2 can be transformed into the equivalent problem of minimizing the variable , called the denoising factor:
The name of is given based on the fact that reducing it suppresses the effect of channel noise by increasing symbol energy (or equivalently as observed from (9)). Since minimizing is equivalent to maximizing , one can incorporate the power constraint into the objective function of Problem P3 and have the following equivalent maxmin problem:
Thought Problem P4 has a simpler form than Problem P3, it remains nonconvex. To tackle the challenge, we recast the problem into an equivalent outerinner subproblems formulation, having the following form:
The inner subproblem solves the optimal power allocation and energy beamforming, while the outer subproblem aims at further optimizing the aggregation beamformer. The two subproblems are tractable and solved separately in the following subsections.
IvB Optimal Wireless Power Control and Energy Beamforming
A close observation of the inner subproblem in P5 indicates that the optimal energy beamforming can be firstly solved in closed form, based on which the optimal power allocation can be then derived as a function of the normalized aggregation beamformer . The first key result of this subsection is presented as follows.
Proposition 1 (Optimal Energy Transmit Beamformer).
Consider the MISO case. For each sensor, the optimal power beam should point to the direction of the corresponding WPT channel, i.e.,
(10) 
Proof: In the inner subproblem of P5, with the equality holds if and only if has the same direction as .
By substituting (10), the inner subproblem is transformed into the following problem:
To solve Problem P6, a necessary condition for the optimal solution is derived as follows.
Lemma 2 (Optimal Wireless Power Control).
For the MISO case, given the aggregation beamformer , the optimal power control should have the following form:
(11) 
where is some constant.
Proof: See Appendix B.
Based on Lemma 2, Problem P6 reduces to the following:
Problem P7 is trivial and the solution is
(12) 
Then combining (11) and (12) gives the following second key result of this subsection.
Proposition 2 (Optimal Power Allocation).
For the MISO case, given the aggregation beamformer , the optimal power allocation is given by
(13) 
Remark 1 (Optimal WPT Strategies).
It can be observed form Proposition 1 that the optimal power beam points in the direction of its corresponding WPT channel. With the beams thus steered, the optimal power allocation attempts to equalize the effective closeloop channels of different sensors that cascade the WPT channels specified by the gains and AirComp channels specified by the gains .
Remark 2 (Sensor Scheduling).
The current design focuses on a set of sensors selected for WDA. The issue of scheduling is not addressed but important. To be specific, deep fading in a cascading WPTAirComp channel will lead to highly noisy data collected from the corresponding sensor and thereby amplify the computation error. This is reflected in high transmit power for inverting a poor channel (see (13)) and thereby reduces the average SNR of aggregated data. Thus, it is important to select sensors with both reliable WPT and AirComp channels.
IvC Optimal Aggregation Beamforming
Given the solution of the inner subproblem in the preceding subsection, the outer subproblem in P5 for aggregation beamformer optimization can be simplified by substituting the optimal energy beamformer in (10) and optimal power allocation in (13):
Note that , Problem P8 is equivalent to the following problem:
Though having a simple structure, Problem P9 is still challenging due to the nonconvex norm constraint on . To tackle the constraint, the celebrated SDR technique is applied where the nonconvex constraint in Problem P9 is relaxed by its convex hull.
Lemma 3 (Convex Hull Relaxation [37]).
Let and , wherein is of the size by and has the dimension of by . The second condition means that both and are positive semidefinite. Then, is the convex hull of , and is the set of extreme points of .
Thereby relaxing Problem P9 gives:
where . Then, the convexity of Problem P10 is established in the following lemma.
Lemma 4 (Convexity of Problem P10).
Problem P10 is a convex problem.
Proof: See Appendix C.
Upon solving the Problem P10 via a convex problem solver (e.g., the cvx toolbox in MATLAB) and attaining the globally optimal solution , the next task is to retrieve from it a feasible solution to Problem P9 denoted by . Since the rank of might be larger than one, the Gaussian randomization algorithm proposed in [38] can be applied to extract from . The main procedure of applying the algorithm to the current design is summarized in Algorithm 1.
Remark 3 (Optimality of SDR Solution).
It is worth pointing out that the SDR technique can lead to an optimal solution. If a rank one is solved, the global optimal solution can be immediately achieved by extracting the dominant eigenvector of as the normalized data aggregation beamformer, i.e., . As shown in the simulation later, has a high probability to be rank one.
V Joint Design for WPT and AirComp: MIMO Case
In the preceding section, we consider the MISO case with singleantenna sensors. In this section, the results are extended to the general MIMO case with multiantenna sensors. In particular, the original vectorfunction WPAirComp problem in Problem P1 is solved. The solution builds on the outerinner framework developed in the previous section.
To further develop the framework for the MIMO case, the nonconvex Problem P1 is first simplified by showing the optimality of the zeroforcing (channel inversion) precoder conditioned on the aggregation beamformer as follows.
Lemma 5 (Optimal MIMO Precoder).
Given an aggregation beamformer , the computation error is minimized by the following zeroforcing precoders:
(14) 
The proof of Lemma 5 is similar to that of Lemma 1 shown in Appendix A, thus omitted for brevity. Let denote the normalized aggregation beamformer with and thus with being the norm of . Then Problem P1 can be reduced to the following form by substituting (14):
Note that the first set of power constraints in Problem P11 can be rewritten as:
(15) 
Note that the minimum in Problem P2 is achieved when the above constraint is active (i.e., the equality holds). Since minimizing is equivalent to maximizing , one can move the power constraint to the objective function and have the following equivalent maxmin problem:
Similar to the MISO counterpart, Problem P12 can be recast as the following outerinner subproblems:
where the inner subproblem contains MIMO power control and energy beamforming optimization, and the MIMO aggregation beamformer design gives the outer subproblem.
Va Optimal Wireless Power Control and Energy Beamforming
We firstly tackle the inner subproblem in Problem P13. The optimal energy transmit and receive beamformers are solved as a function of the normalized aggregation beamformer as shown below.
Proposition 3 (Optimal MIMO Energy Beamformers).
For each sensor, the optimal energy transmit/receive beamformers point to the left/right dominant singular vector of the WPT channel matrix, namely,
(16) 
where and denote the left and right dominant singular vectors of matrix , respectively.
Proof: Starting from the wellknown Rayleighquotient inequality
(17) 
where the equality holds if and only if and have the same direction with the left and right dominant singular vector and . Therefore, one can readily note that the objective in Problem P13 can be maximized with respect to and by setting them as shown in (16), which completes the proof.
By substituting (16) in Lemma 3, the inner subproblem is transformed into the following problem:
where denotes the dominant singular value of the WPT channel matrix . To solve Problem P14, a necessary condition for the optimal solution is provided as follows.
Lemma 6 (Optimal Wireless Power Control).
For the MIMO case, the optimal power control should have the following form:
(18) 
where is some constant.
Proof: See Appendix D.
Based on Lemma 6, Problem P14 reduces to the following:
Problem P15 can be easily solved with the optimal solution:
(19) 
Thus the optimal power control policy is given as follows.
Proposition 4 (Optimal MIMO Power Allocation).
For the MIMO case, the optimal power allocation is given by
(20) 
Remark 4 (Optimal WPT for MIMO Case).
Similar to its MISO counterpart, (20) indicates that the optimal transmit beam points to the left dominant eigendirection of the WPT channel matrix, and the optimal receive beam to the right dominant eigendirection. Moreover, the allocated power to a sensor is inversely proportional to its effective closeloop channel gain that combines the dominant singular value of WPT channel matrix and the beamed AirComp channel after aggregation beamforming, i.e., (see Proposition 4).
VB Optimal Aggregation Beamformer
Given the solution of the inner subproblem presented above, the outer subproblem in P13 for aggregation beamformer optimization can be obtained by substituting the optimal WPT strategies in (16) and (20) into Problem P13. It follows that:
Problem P16 is difficult to solve due to the nonconvex objective involving . To overcome the difficulty, we adopt the following inequality relaxation of the objective function:
(21) 
where the equality holds given a wellconditioned channel, i.e., the singular values of are identical. Using (21), a relaxed version of Problem P16 is posed as follows:
According to [38], by introducing a new variable , an equivalent formulation of Problem P17 is obtained as follows:
It can be observed that the only nonconvex constraint in Problem P18 is , thus we may remove it to obtain the following relaxed version:
The convexity of Problem P19 is shown in the following lemma.
Lemma 7 (Convexity of Problem P19).
Problem P19 is a convex problem.
Proof: See Appendix E.
Upon attaining the globally optimal solution of Problem P19, denoted by , the remaining task is to convert it into a feasible solution of Problem P16, denoted by , of rank . To this end, the Gaussian randomization algorithm for the MISO case in Algorithm 1 is generalized to the MIMO case for searching the closetooptimal approximate solution for Problem P16, as summarized in Algorithm 2.
Vi Further Discussion
In this section, we provide further discussion to gain more insights into the properties of the WPAirComp design in the preceding sections.
Via Computation Error Minimization versus SNR Maximization
An intuitive and alternative design criterion for WPAirComp could be one that maximizes the (total) receive SNR at the server. This criterion, however, leads to a completely different strategy from the counterpart that minimizes the computation error, as shown in the sequel. To clarify this point, we consider the simple MISO case for example. Let denote the receive SNR and it can be defined based on the channel model in (5) as follows:
where the second equality is attained by substituting , and the power constraint . Thus, the SNR maximization problem can be casted as:
Although the above problem and P4 differ slightly only in the objective function, the resultant resource allocation strategies are fundamentally different. The strategy from SNR maximization tends to allocate more power to a sensor with better channel condition so as to maximize the sum effective channel gain. In contrast, the strategy from computation error minimization in Problem P4 attempts to equalize effective channel gain across different sensors (see Proposition 2). This suggests the fundamental difference between WDA and conventional data communication.
ViB How does WPT Help AirComp?
As mentioned, wireless power control provides an additional design dimension for reducing the AirComp error. In the sequel, we provide insights to understand the performance gain from relevant design presented in the preceding sections. To help exposition, a benchmark scheme is considered that equally allocates wireless power, i.e., . Consider the aggregation beamformer design problem in P4. Note that under equal power allocation, P4 is converted to a NPhard problem as shown below:
Comparing the objective functions of Problems and P9 (with wireless power control) gives:
(22) 
In other words, wireless power control reduces the computation error.
Vii Simulation
In this section, the performance of our proposed WPAirComp framework is evaluated by simulation. The performance metric is the normalized computation error, defined by with MSE defined in (7). The simulation parameters are set as follows unless specified otherwise. The number of sensors is . In the MISO case, the number of antennas at the server is set as . The number of computed functions is set to be as only antenna is available at sensor side. In the MIMO case, the antenna setting at the server is given by , while at sensor side we assume antennas for energy receive beamforming or data precoding, which is equal to the number of computed function . All the WPT and Aircomp channels are assumed to be i.i.d. Rician fading, modeled as i.i.d. complex Gaussian random variables with nonzero mean and variance . In addition, the maximum transmission power is set as . The effective power conversion efficiency follows a uniform distribution with , and the noise variance is assumed to be .
Viia Baseline Schemes
We consider three baseline schemes designed based on two classic approaches: antenna selection (AS) and eigenmode beamforming (EB). All three schemes assume the channelinversion data precoding in (9) or (14) and also the optimal energy beamforming in (10) or (16) depending on whether MISO or MIMO case is considered. The difference between the three schemes lie in the aggregation beamformer and the wireless power allocation policy. Define the sumchannel matrix . For the first two baseline schemes, the optimal power allocation is used by solving the inner subproblems of Problem P5 and P13 under the condition that the aggregation beamformer is set to be AS or EB to enhance the receive SNR. The AS scheme selects the receive antenna observing the largest channel gains in the sum channel , while the EB scheme selects the largest eigenmode of for AirComp and thus consists of the dominant left eigenvectors of . The third baseline scheme assumes equal power allocation and solves the resultant outer subproblems of Problem P5 and P13 to attain the aggregation beamformer. For fair comparison, all aggregation beamformers in the baseline schemes are scaled to have the same norm.
ViiB Optimality of the SDR Solution
Consider the MISO case in Section IV, the probability of SDR solution for Problem P10 to be globally optimal, i.e., , is evaluated under various antenna settings at the server as summarized in Table II. It can be observed that the probability of SDR solution to be optimal is more than for different settings. Moreover, the probability is observed to increase with the growth of the antenna numbers at the server, which implies that equipping the server with more antennas will boost the AirComp accuracy to some extent.
ViiC Computation Performance of WPAirComp
First, the performance of the normalized computation error under varying maximum transmission power are evaluated in Fig. 4 for both the MISO and MIMO cases. It can be observed that the normalized computation error decreases with the increasing maximum transmission power, which coincides with the intuition that larger transmitting power will result in higher SNR and thus less estimation error. Moreover, our proposed solution outperforms all other baseline schemes throughout the whole considered range of transmission power. Particularly, there are significant gaps between the baseline schemes of AS, EB and our proposed WPAirComp scheme, which demonstrate the merit of solving the outer subproblem for aggregation beamformer optimization. Interestingly, the gap between the baseline scheme of EPA and the WPAirComp scheme is relatively narrow, suggesting that optimizing the power allocation is not so critical in terms of AirComp accuracy.
Fig. 5 further compares the proposed solution with all three baseline schemes under varying number of antennas at the server. It is observed that, for all four schemes, the normalized computation error monotonically decreases with the growth of the antenna numbers at the server due to the increasing diversity gain. Similar observations to those in Fig. 4 also apply here, showing that our proposed WPAirComp scheme performs consistently well in various settings of the system parameters.
ViiD Effects of Different System Parameters on WPAirComp Performance
We further quantify the effects of different system parameters on the proposed WPAirComp framework in Fig. 6, which shows the curves of the normalized computation error versus the number of sensors under different antenna settings at the server. One can observe that the computation error increases with the number of sensors but decreases with the number of antennas at the server. This aligns with our intuition that more connected sensors makes it harder to design one common data aggregation beamformer to equalize all different sensors’ channels, while having more antennas at the server can significant boost the computation accuracy by exploiting the spatial diversity gain.
Viii Concluding Remarks
In this paper, we have proposed the WPAirComp framework for the joint design of wireless power allocation, energy and aggregation beamforming for effective WDA in IoT systems. The framework design is tractable via an intelligent decomposition of the original nonconvex problem into an outerinner form separating the complex design into subproblems. The decomposition approach not only yields useful insights into the optimal solution structure, but also approaches globally optimum with a high probability. The additional design dimension created by wireless power control is shown to be able to boost the AirComp accuracy. The work points to the promising new research area of WPAirComp where many interesting research issues warrant further investigation, such as sensor scheduling, sensor clustering, and multiple servers cooperation.
Acknowledgment
This work was supported in part by Hong Kong Research Grants Council under the Grants 17209917 and 17259416, and by Shenzhen Science and Technology Program under Grant No. JCYJ20170817110410346.
a Proof of Lemma 1
Given the computationerror minimization objective provided in (7), it is easy to note that both the first and the second terms within, i.e., and are positive. As a result, for any given data precoder , we have the following inequality:
(23) 
It is easy to verify that setting to have the zeroforcing structure in (9) enforces
and thus achieves the equality in (23), combining with completes the proof.
B Proof of Lemma 2
Since , there exists such that,
If are not equal at the optimal point. Assume that is the minimum and is the maximum. Let and , where is small enough to ensure that . Without violating the power constraint,
By such contradiction, equal