Method of determining operating parameters of beamformers for over-the-air computing

Abstract

A method (100) of determining operating parameters (v*k, u*) of a precoder stage and a combiner stage of a transmitter (500) and a receiver (600), respectively, configured for over-the-air computing, comprises determining (130) an initial combiner (u(0)) based on an aggregated channel matrix (H) that comprises all channel vectors (hk) of all transmitters (500) The method further comprises iteratively refining (101) the combiner (u(li)) by minimising the square Euclidian vector norm of the combiner ( || u || 2), until a termination criterion is met, yielding an optimised combiner (u*). Optimised precoder parameters (v*k) for all transmitters (500) are computed (190) based on the previously optimised combiner (u*) while ensuring that the square (|vk| 2 ) of the precoding scalars of each transmitter (500) does not exceed a predetermined power value (P).

Description

METHOD OF DETERMINING OPERATING PARAMETERS OF BEAMFORMERS FOR OVER-THE-AIR COMPUTINGFIELD OF THE INVENTIONThe present invention relates to wireless communication, in particular to over-the-air computing in systems comprising multiple edge devices (ED) and at least one base station (BS).NOTATIONThe following notation is used throughout this specification. Real vectors and matrices are represented in bold small and bold capital letters, respectively, like in v and V, while their complex counterparts are represented bold italic small and bold italic capital letters, respectively, like in v and V. N × N identity matrix and N × 1 all-one column vector are denoted by In and 1n, respectively. Vector norm and the absolute value of a scalar are respectively denoted by || || and ||. Transposition and conjugate transposition are represented as T and H, respectively, while R{},{} and min() denote the real part, imaginary part, and minimum operators. Finally, N(μ,Σ) and XN(μ, Σ) respectively denote the real- and complex-valued multidimensional Gaussian distribution with mean µ and covariance matrix Σ.BACKGROUNDWith the expanding adoption of Internet-of-things (IoT) applications in smart homes and industrial settings, telemedicine, and autonomous driving, the coordination of the communication of distributed IoT devices and the aggregation of their data become increasingly important, notably for satisfying stringent requirements such as low latency, low power consumption, efficient spectral usage, reliability, and scalability, which are mandated by certain applications and certain use cases.Aiming to allow communication and computation tasks to be performed simultaneously, over-the-air-computation (AirComp) has emerged as a promising technique for data aggregation in IoT systems as an alternative to the conventional sequential communication and processing, which also lays the ground for a paradigm shift towards distributed computing over wireless systems. Figure 1 shows a simplified diagram of an AirComp system. In the system K EDs are wirelessly connected to an AP. The individual wireless signals interfere on their paths from the EDs to the AP, the interference being used for computation, as will be discussed hereafter.The basic principle of AirComp is to exploit the waveform superposition property of a wireless channel to realize over-the-air aggregation of data simultaneously transmitted by devices. The simultaneous transmission by multiple synchronised devices and the analogue-wave superposition property of such multiple-access channel results in an adding of the simultaneously transmitted signals "over-the-air". The added signals arrive at the receiver as a weighted sum, also referred to as the "aggregated signal", with weights being the channel coefficients. AirComp relies on linear-analogue modulation and channel pre-compensation at each transmitter. The former modulates the data values into the magnitudes of the carrier signals; the latter compensates for heterogeneous channel fading of different links. As a result, each component part of received signal is the transmitted data scaled by a pre-determined factor. Setting the factor uniform for all signals, called magnitude alignment, reduces the aggregated signal to the desired average of transmitted distributed data, realizing an averaging function through AirComp.With appropriate data pre- and post-processing the capability of AirComp can go beyond averaging to computing a class of so-called nomographic functions, which can generally be expressed as a post-processed summation of multiple pre-processed data-values. Typical functions in this class include arithmetic mean, weighted sum, geometric mean, polynomial, and Euclidean norm. For example, to compute the geometric mean, the pre-processing is a logarithm function and post-processing an exponential function. It has been proven that any function can be decomposed as a summation form of nomographic functions, indicating that any function can be computed via AirComp in general.Figure 2 shows an exemplary simplified block diagram of an AirComp transmitter 500 and an AirComp receiver 600. Transmitter 500 receives data to be transmitted at pre-processing block 550, which processes the data for over-the-air computing. The processed data is then provided to precoder 560, which precodes the data for transmission over the wireless channel, indicated by the dashed-line arrow. Receiver 600 provides the received signal to a combining block 660 that is configured for over-the-air computing. Combining block 660 receives the required combiner design from design block 650. Combining block 660, which performs the actual over-the-air computing and implements the target function, provides the output of the target function to processing block 670 for further processing. Note that the conventional functional blocks of a wireless transmitter and receiver are omitted in the figure.While simultaneous transmission in AirComp allows each device to access all radio resources instead of only a fraction of them as in the conventional orthogonal multiple access schemes, enabling high spectrum efficiency, the most promising feature of AirComp for wireless data acquisition and other purposes is that it enables the direct computation of nomographic functions by multiplexing multiple streams of data over the wireless multi-access channel. Thanks to this parallel processing, AirComp systems can achieve low-complexity and energy-efficient data aggregation.However, AirComp is still in its infancy, with the bulk of work in the topic focusing on the fundamental issue of mitigating distortions in the computation that are caused by the random nature of the wireless channel. To cite a few examples, the computation-optimal transmitter (TX)-receiver (RX) scaling policy for a multi-point single input single output (SISO) AirComp setting, i.e., with a single-antenna access point (AP) and multiple single-antenna EDs, was investigated by W. Liu, X. Zang, Y. Li, and B. Vucetic, in "Over-the-Air Computation Systems: Optimisation, Analysis and Scaling Laws," IEEE Trans. Wireless Comm., vol. 19, no. 8, pp. 5488-5502, 2020, and later generalised to multiple frequency AirComp schemes by T. Qin, W. Liu, B. Vucetic, and Y. Li, in "Over-the-Air Computation via Broadband Channels," IEEE Wireless Commun. Lett., vol. 10, no. 10, pp. 2150-2154, 2021.Work similar to those mentioned above, but considering fading channels and the single input multiple output (SIMO) case when the AP has multiple antennas, can also been found in the related literature, which shows that the performance of SIMO AirComp is highly dependent on the precoders and combiners, whose optimal design is non-trivial.Focusing on such cases, a paired uniform-forcing precoder-combiner design scheme to minimize the AirComp mean square error (MSE) was proposed by L. Chen, X. Qin, and G. Wei, in "A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation," IEEE Wireless Commun. Lett., vol. 7, no. 6, pp. 942-945, 2018, where the precoder is obtained via a Lagrangian method given the combiner, while the combiner is first initialised via an semi-definite relaxation (SDR)-based MSE-minimizing algorithm and subsequently refined iteratively via successive convex approximation (SCA). Building on the latter, it was recently shown by W. Fang, Y. Zou, H. Zhu, Y. Shi, and Y. Zhou, in "Optimal Receive Beamforming for Over-the-Air Computation," 2021 IEEE 22nd International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2021, pp. 61-65, that the optimal combiner to a uniform-forcing precoder can be obtained via the branch and bound (BnB) method, albeit at the penalty of a significant increase in computational complexity, which grows further with the number of EDs in the system. As an extensional work of the above, two types of initialisation, one having much lower complexity albeit with performance loss and the other one having better performance albeit with higher complexity are proposed by K. Ando and G. T. F. de Abreu, in "Low-Complexity and High-Performance Combiners for Over the Air Computing," in 2023 IEEE 9th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2023, pp. 126-130. Nevertheless, even with the conventional lower complexity initialisation, the overall complexity order is still high due to the subsequent refinement via SCA, which is likely to hinder the widespread adoption of AirComp in low-cost IoT applications.It is, therefore, desirable to provide a combiner design stage for AirComp systems, e.g., multi-point SIMO systems, which enables both low complexity as well as high performance and accuracy.SUMMARY OF THE INVENTIONThis object is achieved by the methods of claims 1 and 5, the wireless transmitter of claim 2, the wireless receiver of claim 3, and the wireless communication system of claim 6. A computer program product and a computer-readable medium are provided in claims 7 and 8, respectively. Embodiments and developments are described in respective dependent claims.The combiner design method in accordance with the present invention comprises an initialisation stage and a subsequent refinement stage. The exemplary initialisation stage used herein for comparison purposes is targeted to provide low complexity yet better performance than conventional approaches. This is enabled by a reformulation of the MSE minimisation problem into a canonical Rayleigh quotient (RQ) problem. In particular, unlike the load-aware approximation discussed in "Low-Complexity and High-Performance Combiners for Over the Air Computing" (full citation further above), the computationally costly min-operator of the conventional combiner design method is avoided by using the less costly average of the multi-access channel. In accordance with the present invention the refinement stage relaxes the computationally costly MSE minimisation through the convex concave procedure (CCP), and applies a proximal gradient (PG) method for obtaining a low-complexity solution for the minimisation.Prior to describing the invention the underlying problem encountered when designing precoders for AirComp will be described. Consider an AirComp scheme over a multi-point SIMO communication system composed of K single-antenna EDs and one AP equipped with N antennas. Let the AirComp target function f be given byKf(s) = Σ Skk=1where s [S1,…, SK]T is a vector containing all symbols Sk ~ N(0,1) transmitted by the EDs.Under perfect synchronisation among all EDs, the received signal y ∈ CN×1 subjected to fading and noise is given byKy = ΣhkVkSk + nk=1where hk ∈ CN×1 denotes the channel vector between the k th ED and the AP, Vk EC is the k-th ED's precoding scalar, and n ∈ CN×1 ~ XN(0,σ²IN) denotes the additive white Gaussian noise (AWGN) at the AP.In possession of the received signal y, an estimated target function is constructed at the AP via the combinerKf(u,v;H|s) = uty = uHhkVkSk + nk=1where v▲ [V1,…,VK]T is a vector containing all precoding scalars Vk employed by the EDs, u ∈ CN×1 denotes the combiner vector applied by the AP, and H▲ [h₁,…,hҝ] ∈ CNXK is the channel matrix from all EDs to the AP.The notation in equation (3) is meant to emphasize that, from the perspective of the AP, f is a function of the combining and precoding vectors u and v, which are the variables that can be optimised, while the known channel matrix H is a fixed parameter, and the unknown vector of transmitted symbols s is a conditioning quantity.In light of the above, the MSE between the target and reconstructed functions averaged over transmissions of multiple distinct symbol vectors s and under constant channel H, precoders v and combiners u, can be concisely expressed asKε(u,v;H) ^ E[|f - f|2] = ∑ |1 - uhkvk|2 + σ2uHu,k=1where the arguments of the functions f and f are omitted for simplicity of notation.As discussed in "A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation" (full citation further above), for a given, i.e., fixed, combining vector u, the optimal precoding scalars V that the EDs must apply in order to minimize the average MSE while satisfying the power constraint |vk|2 ≤ P is given byv = ΝηHKza)니n = P. min (hub)where η denotes a scaling coefficient and P denotes the maximum transmit power constraint, both common to all EDs.From equations (3) and (5)* (u;v*,H|s) f(u,v*;H|s)ΝηuHn= f(u,v*;H|s) +can be obtained, where again it is emphasised that the notation is meant to remind that the estimated target function f, partially optimised by employing the precoders v*, is still a function of the combiner vector u, which can also be further optimised.To that end, one might be inclined to simply seek to minimize the average MSE between f and f, namely*212น6(u;n) ▲ E[|f - f*|²] = σ² || u || 2ηIt is noted, however, that minimizing the average MSE δ(u;η) with respect to u is non-trivial since the parameter η is, as evidenced by equation (5b), a quantity that is itself dependent on u and coupled to v*, besides being dependent also on the channel matrix H and power constraint P. In other words, an MSE-optimal combiner design must simultaneously minimize equation (7) and satisfy equation (5b), given H and P.In view of the aforementioned challenge, in the next subsection the best feasible combiner design approach currently known will be discussed.A straightforward optimisation problem formulation that puts together the objective given by equation (7) and the condition imposed by equation (5b) isminimizeuσ2 || u || 2ECNXI p. min (hu)hkwhich, although difficult to solve directly due to the presence of the min operation in the denominator, can be reformulated into the standard constrained minimisation problemminimizeUE CNx1subject to|| 4 |||| u || 2a)2Hku≥ 1,∀k,b)2where the constant σ² / P was ignored and attention is called to the fact that the min operator in equation (8) is relaxed into the K constraints described by inequality (9b).Although the problem represented in equation (9) is still NP-hard due to the non-convex constraints (9b), it can be solved via successive convex approximation (SCA)-based iterative refinement of the combiner.The SCA procedure will be only briefly described below and summarised as follows. Details on the derivation of the method can be found in "A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation" (full citation further above).First, for the sake of convenience, define the projectionC(u,h) Ruh}I{uHh}Then, the i-th iteration of the SCA procedure proposed reduces to computing(1-1) C(u(i-1), hk),vka)Ckand subsequently solving the problemminimizeu(i) E CNx1|| u(i) || 2(11b)subject toT2(-1))C(u(i), hk) - || (-1) || ≥ 1,∀k(11c)CkAs can be seen from equation (11), an initial combiner u(0) is required. A relatively accurate conventional approach designs the initial combiner u(0) based on a semi-definite relaxation (SDR) of the problem represented in equation (9). A low complexity, albeit heuristic RQ-based conventional alternative exists. Both approaches are briefly described hereafter.The SDR variation of the problem represented in equation (9) is given byminimizeUE CNXNTr(U)(12a)subject to Tr(UHk) ≥ 1,∀k(12b)U≥0(12c)where U uuH and Hk hk.The solution U of the problem represented in equation (12) can be obtained efficiently via interior point methods, and the corresponding combiner vector u(0) can be extracted from the latter as its largest eigenvector, which can be computed efficiently via principal component analysis (PCA). Thereforeu(0) = maxeigv(U)is set.The computational complexity of the aforementioned conventional method to design AirComp combiners is dominated by the SDR-based initialisation scheme. Indeed, it can be inferred that an approach in which one first obtains a positive semi-definite solution U E CNXN CN×N, only to immediately thereafter truncate it to the SCA initialisation point u(0) ∈ CN×1, is not very efficient.Therefore, low-complexity alternatives to the SDR procedure are needed. To this end, first notice that as long as || u || 2 ≠ 0 the optimisation problem represented in equation (8) can, without loss of generality or any other penalty, be rewritten asH2P. min (hu²)maximizehkU E CNx1σ2 || u || 2where the nominator and denominator of the objective function of the problem represented in equation (8) are merely "flipped", motivated by the resemblance between the latter and the classic Rayleigh quotient (RQ), as shall soon become clearer.Of course, just as in the problem represented in equation (8), the presence of the min operator in the objective of the problem represented in equation (14) is problematic. However, unlike the approach followed in "A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation" (full citation further above), where such operator was relaxed into the set of constraints as shown in equation (9b), the load-aware approximation was considered asmin (hu) ≈ hula)hkwheremin (|hk|)if N ≤ K,hk1b)Κ.Σ. κif N > K.k=1Substituting the approximation represented in equation (15a) into equation (14) and ignoring the constant P / σ2, the standard rank-one RQ problem can be obtained asHulmaximizeu E CNx1 || u || 2whose solution is straightforwardly given by the only non-zero eigenvector of HH, which in turn is merely the normalised vector h, that isu(0) = maxeigv(ҺҺ) =|| h || 2The performance of the conventional iterative refinement-based design heavily depends on the performance of the combiner. As a consequence, a heavy penalty is paid in exchange for the linear-order complexity of the conventional initialisation methods, in terms of degradation of the performance of the converged solution compared to the high-complexity conventional alternative via SDR.A better initial point with lower complexity than the conventional approaches can be obtained by using yet another alternative solution of the optimisation problem represented in equation (14). Starting with the load-aware approximation represented in equation (15b), and taking into account that all ED channel vectors should contribute to the design of the initial combiner, a design based on a variation of the problem represented in equation (16) is considered:maximize-|| HHu || 2-a)U ECNx1 || u || 2which uses the aggregated channel matrixH▲ [h₁,...,hk] ∈ CN×Kb)rather than the individual channel matrices.Since the optimisation problem represented in equation (18a) is a simple RQ problem, the initial combiner can be designed by the maximum eigenvector of HHH as,u(0) = maxeigv(HHH)which in turn can be computed efficiently via the well-known power method, e.g., as discussed by G. H. Golub and C. F. V. Loan, "Matrix Computations", Johns Hopkins University Press, 1996.In hand of the optimised initial combiner the refined combiner can be obtained by solving the minimisation problem represented in equation (9) through iterative refinement. Finally, the precoder parameters can be computed based on the refined combiner. However, the computational complexity of the conventional methods for solving the minimisation problem, e.g., the SCA method discussed above, may still be too high to be widely used, and some reduction of the complexity is needed, while still providing at least the same MSE performance than conventional methods.In accordance with the present invention, the reduction of the complexity of solving the optimisation problem represented in equation (9) is achieved by reformulating the equation as a difference of convex (DC) functions problem and solving the DC problem via the CCP. The CCP is a heuristic method for finding local optimum solutions to DC programming problems. Using the CCP method may seem counterintuitive, since conventional solutions of the CCP involve applying interior point methods, which have a high computational complexity. The present invention, however, proposes a solution that avoids the computationally costly interior point methods.Consider a general real domain DC programming problem given byminimizeX ERN×1subject toho(x) - go(x)(20a)hm(x) - gm(x) ≤ 0,m = 1,...,M(20b)under the assumptions of real domain convex functions hm : RN×1→R1×1 and gm: RN×1 R1×1 for m = 0,...,Μ.According to the CCP, the solution of DC problems can be approximately obtained by iteratively solving an alternative convexised optimisation problem. In particular, the convexised optimisation problem for obtaining the solution of the ic +1-th iteration with the previous solution x(ic) is given byminimizeX ERNx1ho(x) - go(x,x(ic)) (21)subject to hm(x) - 9m (x,x(ic)) ≤ 0,m = 1,...,M(21b)where gm(x,x(ic)) is an approximation of the function gm(x) from equation (20), defined as9m(x,x(ic)) gm(x(ic)) + √gm(x(ic)) (x -x(ic))c)From the optimisation problem represented in equations (9) and (20) it can be seen that the MSE minimisation problem can be regarded as a DC problem. In particular, setting ho(x)→ || u || 2 with go(x)→ 0, and gm(x)→ || 樅u || 2 -1 with hm(x)→0, suffices to relate equation (9) to equation (20). Then, taking into consideration that the desired CCP is in the complex domain, the corresponding reformulated MSE minimisation problem for obtaining the solution in ic +1 CCP iterations becomesminimizeu E CNx1|| u || ² (22a)subject to 1 - ğk(u,u(ic)) ≤ 0,∀k(22b)where the complex domain 9k:CN×1→R1×1 isHÖk(u,ulic)) =|| huic) || 2 + 2R {(hkhulic)). (u - u(ic))}=(i)) Hu} -|| (ic) || 22R {(hkhu(ic))(23)For the sake of later convenience, the optimisation (22) can be rewritten asminimizeu E CNx1|| u || 2(24a)withsubject to aru + utak ≤ bk,∀k(24b)ak-hkhu(ic), k(24c)bk- (1+|| huic) || 2), ∀k(24d)The optimisation represented in equation (24) is obviously convex, with a quadratic objective function and half-space constraint. In accordance with the present invention the optimal solution can be obtained, at a low computational complexity, via a proximal gradient (PG) descent method. By applying PG for the ic-th iteration of the optimisation problem, the solution can be iteratively updated by the gradient of the objective function and the proximal operator for the constraint. Thus the i¡ +1-th update of the PG steps can be expressed asu(i+1) = prox(u(ii) - q(ic)u(i))a)with the proximal operator for half-space constraint over all different k given byprox(u) umax(0,aku + utak - bк)|| ak || 2ak,kb)where i¡ and α(ic) denote the iteration index of the PG iteration, and the step size of the gradient in the ic-th iteration, respectively.All in all, a method for designing the uniform-forcing (UF) precoder and combiner using the iterative refinement in accordance with the invention can be summarised as shown below and as exemplarily represented in the flow diagram of figure 3. The method has a nested loop structure whose outer loop is for the CCP procedure and whose inner loop is using the PG method for solving the approximated minimisation with low computational complexity. In the method, since the termination criterion is exemplarily given by the maximum number of iterations, the outer and inner loops can be terminated after Imax and Imax iterations, respectively. Note that other termination criteria may be used, e.g., convergence of the eigenvector values or the MSE below a threshold, the improvement of the objective value from one iteration step to the next lying below a threshold value, or the like.CCPINNINNInputs to the method 100 are the channel matrix H∈ CNXK from all EDs to the AP, the number of iterations for the outer loop and inner loop, Imax and max, and the step size α(ic), Vic of the gradient for each iteration. Outputs of the method are the optimal precoding scalars v ∈ C Vk and the optimal combining vector u* ∈ CN×1. After receiving the inputs in step 110, the initialisation point u(0) is computed in step 120 via equation (19). The iterative refinement for obtaining the optimised output values, which comprises the steps inside the dashed-line box 101, is then carried out in the nested iteration loops. Starting with step 130 in the outer iteration loop, ak and bk are computed via equation (24), using u(0) as the starting point for the optimisation. In step 140 the inner iteration loop starts, where u(i) is updated via equation (25). Step 150 checks whether a termination criterion is met for the inneriteration loop. As mentioned further above, the termination criterion may comprise reaching a predetermined number Imax of iterations, or convergence towards a stable value, or the like. Termination on convergence may be triggered, e.g., when the difference or a gradient in the output values between a predetermined number of subsequent iterations lies below a predetermined value. Other convergence criteria are also conceivable. If the termination criterion is not met, "no"-branch of step 150, the method returns to step 140. Otherwise, "yes"-branch of step 150, in step 160 the output u(i) of the inner loop replaces the previous initialisation point, e.g., u(0), for the next execution of the inner loop, and the method checks, in step 170, if a termination criterion for the outer loop is met. In the negative case, "no"-branch of step 170, the method returns to step 130, executing the inner loop a further time. Otherwise, "yes"-branch of step 170, the last output of the inner loop is output as the optimal combiner u* in step 180, and the optimal precoders v are computed thereon, for all k EDs, via equation (5) and using u* in step 190, and output as vr all k EDs in step 195.INNWhile the exemplary method presented above uses an initialisation that differs from the conventional high-performance, high complexity SDR method and the low-complexity, low-performance RQ method initially described, it is possible to use any suitable initialisation. This is due to the separation of the refinement process from the initialisation, which provides a larger freedom in designing the full process.In the following section a variant of the novel approach concerning accelerating the convergence of the iteration will be discussed, which will be used further below for comparison purposes.This variant pertains to a Bayesian optimisation (BO)-based hyperparameter tuning for the proximal gradient (PG) method for finding the optimal combiner u*.As shown above for the exemplary CCP- and PG-based method 100, it is possible to design the AirComp precoder and combiner exclusively relying on low-complexity mathematical operations. In PG, since the convergence property and performance of the converged solution heavily relies on the value of hyperparameters, i.e., step sizeα(1),...,α(m), all the hyperparameters must be carefully determined. Classically, the tuning of hyperparameter can be carried out by brute force grid search over the possible combination of all the parameters. However, the complexity cost of grid search is comparatively high, not only because the algorithm must be run for the evaluation of each hyperparameter combination, but also because the number of possible hyperparameter combinations increases with the number of outer-loop iterations I Imax. In order to achieve a reasonable hyperparameter tuning with limited searching effort, a BO-based offline hyperparameter tuning is proposed.CCPIn AirComp, as can be seen from equation (7) and the method 100 for designing the UF precoder and combiner presented above, hyperparameter tuning can be regarded as the following MSE minimisation problemCCPminimizeqsubject toq=δ(u;η)(26a)u← method 100 (H,q)(26b)CCPmax[α(1),...,α( )](26c)where q ∈ R1xImax denotes the vector of collected hyperparameters.Unfortunately, this problem cannot be solved analytically and, as mentioned before, grid-based brute-force search is too computationally expensive to be practically used. To tune the hyperparameters with reasonable complexity, a BO-based approach is proposed. BO is a black-box optimisation method that efficiently searches for the optimal hyperparameters based on the evaluation results and Gaussian Process regression (GPR). In particular, this variant discusses employing BO for modelling the objective function (u;n) as a Gaussian process (GP), such that after an initial dataset is constructed, this method iterates(i) modelling of the objective function by GPR,(ii) hyperparameter selection based on the acquisition function, and(iii) performance evaluation.In the this process, prior to actually modelling the objective function the data set to be used in the BO method must be initialised. To this end, the initial dataset for the parameter search is constructed by selecting linit of the hyperparameter vectors by random sampling from the feasible region, yielding the setBOBOlinitQinitU {q(i)}i=1,whereq(i) ∈ [0,1]1×1CCPmaxNext, the value of the objective function, i.e., observation, is computed using method 100 described further above and the inputs in Qinit constructed above.To find a robust hyperparameter set against the statistically random channel matrix H this step considers averaging Imhan observations on random channel realisations.For an input q(i) with the j-th channel realisation, its MSE performance can be evaluated by0;(q(i)) = (u← method 100 (H(i), q(i));η)After the MSE values are evaluated for Ichan of different channel realisations, the average MSE can be obtained bymaxChanmax1θ(q(i)) =maxIchan j=1Σθ;((i))For future convenience, the concatenated averaged MSE vector of all the hyperparameters in a set Q with cardinality |Q| = IQ can be constructed byθ(Q) = [θ((1)),...,θ((10))]Ta)loQ = U {q(i)}b)i=1where q(i) ∈ [0,1]1x denotes the i-th hyperparameters.Then, the initial dataset can be constructed asBOinit(0)DBO = U {[q(i),8(q(i))]}i=1Now the actual function modelling based on GPR can be carried out. This step models the average MSE for unknown inputs based on the current dataset and GPR, a non-parametric regression method assuming that the modelling target follows a GP. It is assumed that the parameter tuning is performed via BO with a total number of Imax iterations. At the in-th iteration, the task in this step is to estimate the probability distribution of the averaged MSE vector θhyperparameter set(ів)ItestQtest = U {q)}i=1(Q) test) by a given testingwhere (B) ∈ [0,1]1× are sampled testing hyperparameters as in equation (28).maxAt the iß-th trial, the joint distribution of the averaged MSE in the current dataset D(ів-1)во and the expected observation corresponding to the given testing(ів)hyperparameter set Qtest are modeled by the following multivariate normal distributionwith(ів-1)10(Q(18-1))θ testΣ(ів)~ Ν(μ,Υ),a)μ =(Q(8-1))(ів)test(Q(8-1),(18))Qtest(Q(8-1),(i))))(ів-1),test(ів) (ів)Qtest, QtestY =|(Q(ів-1)) T|ψ(Q(ів-1), (ів)Qtestwhere the covariance matrix Y, the matrix ∑ can be given as∑ = Ψ(Q(ів-1),Q(ів-1)) + (i-1)with σε denoting the variance for the observation noise, while the hyperparameter set for the past evaluation is given byBOlinit+ів-1Q(ів-1) =U {q(i)}i=1Without loss of generality, the kernel matrix between hyperparameter sets of arbitrary cardinalities Q and Q composed byQ = U ∪{q ∈ [0,1]1×Q1xICCP max}, and Q = U {àj ∈ [0,1]1×}j=1i=1can be defined as Ψ(Q,Ò) ∈ RQ×ÒAssuming automatic relevance determination (ARD) Matérn 5 / 2 kernel, (i,j)-th element of the kernel matrix denoted (Q,Q)(i,j) is computed by2Ψ(Q,Ò)(i,j) = vỏ 1 + 15. Гіj +5.2i,j3 · expwith the biased distance among hyperparameters given by-2rij = || diag....( - ġj) || 2b)Гі,ј-21where Yo Y₁CCP are trainable parameters.Imaxmax(ів-1)With the knowledge of the current dataset DBoand the maximum likelihoodestimation (MLE), the mean and covariance matrix (Q(s) of the conditional(ів)testprobability distribution for the desired average MSE can be respectively derived asTμα) = (Q) + Ψ(Q(18-1), Q (18)) 2-10)a)(ів)testΨ(ів)Qtest(ів)test=Qtest(ів)(ів)Qtest ΣQtest Θ (ів) 2-10 (B)b)zwhereθ(ів)өв) = (0(Q(ів-1)) - μ(Q(ів-1)))a)Θ (ів) = (Q(ів-1),Q(IB) (is))b)zSince the prior mean of the averaged MSE for testing the hyperparameters μ(Qs)testcannot be observed, it is approximated by the arithmetic mean of the observation values, i.e.,(ів)test≈Zi=1Q(8-1) (q(i))|Q(8-1)|1(ів)QtestNote that the parameters Yo,･･･, Y₁CCP and σε, which are used in the computation of equation (39), can be trained by maximizing the log-likelihood functionmaxlog p(0(Q(i-1))|σε, ο,...,'(ів-1)DBO1((B))2π- Σ-1.08).log 2m 2(ів)- log det() - 10g 2-(01일). 2-1.01%) (42)-log2Z·zThis function can be maximised, e.g., using gradient-based methods such as the Quasi-Newton method or the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm.The following section describes searching the next hyperparameters q(iB) based on the GPR results and an acquisition function. The acquisition function measures the impact of a given input on the performance. When measured by the expected improvement (El), e.g., as described by M. Balandat, B. Karrer, D. R. Jiang, S. Daulton, B. Letham, A. G. Wilson, and E. Bakshy, in "BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization,” Advances in Neural Information Processing Systems 33, 2020. [Online]. Available: http: / / arxiv.org / abs / 1910.06403, the acquisition function at an input q can be expressed asv(q) = E[max(-0(q) + (ів-1)(ів-1)0(ів-1) 01) denotes the minimum MSE in the current dataset Dowhere min(ів)After v(q) is evaluated for all q EQfest, the next hyperparameter can be determined asq(ів)=argmax v(q).qEQtestAfter the average MSE with the updated input, i.e., (q(is)), is evaluated in the(ів)performance evaluation step, the dataset is updated into Dво via(ів)BODBO(ів) 8-1) ∪ {[q(is), (q(ів))]}Finally, after the function modelling, the hyperparameter selection and theBOmaxperformance evaluation steps are iterated for Inox times the tuned hyperparameters can be obtained byqopt = argmine(q).EQ(x)qNote that the index max may be correspondingly replaced in case a different termination criterion than a maximum iteration count is used. Alternative or additional termination criteria may comprise, inter alia, convergence of a suitable parameter or such parameter reaching a predetermined threshold, such as the expected improvement falling below a preset value.The overall method of tuning the hyperparameters via BO can be summarised as shown below and as exemplarily represented in figure 4.BOINNCCP ВО ChanInputs to the method 200 are the maximum iteration counts Immaxmax max' max , theBO initial value linit and the BO test value I Itest. It is noted that further initialisation values may be provided, e.g., for alternative or further termination criteria. The output of the method 200 is the optimised qopt ∈ R max X1CCPAn initialisation phase of the method 200 comprises sampling, in step 210, a set of initial parameters Qinit as per equation (27), evaluating the initial average MSE performance (q ∈ Qinit) as per equation (30) in step 220, and initialising the dataset(0)DBO as per equation (32) in step 230. The actual iterative BO hyperparameter tuning starts in step 240 with modelling the function. The function modelling comprises tuning the trainable parameters Yo Y₁CCP and the variance for the observationmax(ів)noise σε by maximising equation (42). Next, in step 250, the set of test inputs Qtestis selected as per equation (28). Now the mean(ів)(es) and covariance matrixtest(Q(s) of the conditional probability distribution for the desired average MSE can be computed as per equation (39a) and (39b), respectively, in step 260. Then, the hyperparameters for the current iteration are selected in step 270, in which first an acquisition function is computed as per equation (43) and subsequently q(is) is determined in accordance with equation (44). In step 280 the average MSE performance e for the current iteration's input vector q(is) of collected hyperparameters is determined in accordance with equation (30), and the current(ів)iteration's dataset Deo is updated as per equation (45). The method then checks, in step 290, if a termination criterion is met, e.g., a predetermined number of iterations has been carried out. In the negative case, "no"-branch of step 290, the method returns to step 240, executing the next iteration. In the positive case, "yes"-branch of step 290, the method determines, in step 295, the optimally tuned hyperparameters as per equation (46) outputs them.testThe BO-based hyperparameter tuning of the hyperparameters discussed above provides an efficient way of observing well-designed hyperparameters with limited evaluation effort, yielding a reduction or minimisation of the expected MSE, which leads to faster convergence. However, the performance of BO-based searching can deteriorate with an increasing dimension of the hyperparameter vector. Furthermore, the computational complexity of the PG-based scheme may be problematic, as it requires a high number of inner-loop iterations to yield a good solution, due to its poor convergence behaviour, despite well-designed hyperparameters.A development of the variant addresses this issue by applying convergence acceleration for obtaining an accurate solution with fewer hyperparameters and inner-loop iterations compared to conventional methods. An exemplary suitable method whose application is proposed herein is adaptive moment estimation, alsoknown as Adam, which is an algorithm for gradient-based optimisation of stochastic objective functions. Details on Adam are provided, inter alia, by D. P. Kingma and J. Ba in "Adam: A Method for Stochastic Optimisation," CoRR, vol. abs / 1412.6980, 2014, and by P. Melchior, R. Joseph, and F. Moolekamp, in "Proximal Adam: Robust Adaptive Update Scheme for Constrained Optimisation," 2020.Here, in particular, the first and second moment of the objective function's gradient are used according to the Adam scheme, reducing the number of hyperparameters and for improving the convergence behaviour. Adam as applied herein also increases the numerical stability. To this end, for the i¡-th iteration, the first moment vector m₁ ∈ CN×1 is computed as a moving average of the previous first moment and the gradient of the objective function u(i) with exponential decay rates ẞ1=[0,1) asm(i) = B1m-1) + (1-1)u(i-1)m1=-a)(1-β)Similarly, in the i¡-th iteration the second moment m2 ∈ R is computed as a moving average of the previous second moment and the squared gradient || u(i-1) || 2 with exponential decay rates ẞ2 = [0,1)m2(i-1)(i) B2m2) + (1-β₂)· ||=|| u(i;-1) || 2b)(1-β)Incorporating the two types of moments derived above the i¡-th update in Adam is then given asu(ii) = prox u(i-1) - α(i)m1VmIm₂ + € / where € = 10-6 denotes a small-valued scaler introduced for avoiding numerical issues.The Adam accelerated PG descent for the MSE minimisation can be summarised as shown below and as exemplarily represented in figure 5. In this method, the three hyperparameters, i.e., α,β₁, and ẞ2, are used over different outer loops so that the number of hyperparameters is kept at three even when the number of outer loops is changed.Inputs to the method 300 are the channel matrix H∈ CNXK, maximum iteration counts Imax and Imax, the three hyperparameters α, β₁, and β2, and the scaler e. It is noted that further initialisation values may be provided, e.g., for alternative or further termination criteria. The output of the method 300 are the optimised combining vector u* E CN×1 and the optimised precoding scalars VK E C, Vk.*After receiving the inputs in step 310, the initialisation point u(0) is computed in step 320 via equation (19). The refinement for obtaining the optimised output values is then carried out in nested iteration loops, starting with step 330 in the outer iteration loop, in which ak and bk are computed via equation (24), using u(0) as the starting point for the optimisation. The moments m₁ and m₂ are initialised as m₁ = ON×1 and m₂ = 0, respectively, in step 340. The inner iteration loop, Adam-iteration, starts with step 350, where the first moment m₁" is updated via equation (47a), the second moment m2 is updated via equation (47b), and u(ii) is updated via equation (48). Step 360 checks whether a termination criterion is met for the inner iteration loop. The termination criterion may comprise reaching a predetermined number max of iterations, or convergence towards a stable value, or the like. Termination on convergence may be triggered, e.g., when the difference or a gradient in the output values between a predetermined number of subsequent iterations lies below a predetermined value. Other convergence criteria are also conceivable. If the termination criterion is not met, "no"-branch of step 360, the method returns to step 350 for the next iteration. Otherwise, “yes”-branch of step 360, in step 370 the last updated u(ii) of the inner loop, in the exemplary case u(max), replaces the previous initialisation point, e.g., u(0) for the first iteration, and the method checks, in step 380, if a termination criterion for the outer loop is met. In the negative case, "no"-branch of step 380, the method returns to step 330 for the next iteration. Otherwise, “yes”-(0)2(i)(i)CCPINN(0)2(i)branch of step 380, the last output of the inner loop is output as the optimal combiner u* in step 390, and the optimal precoders v are computed thereon, for all k EDs, via equation (5) and using u* in step 400 and output as v in step 410.It is noted that three hyperparameters tuned in the method 300 can also be tuned using method 200, replacing method 100 with method 300, setting the hyperparameter dimension I Imax to three, and changing Yo... Y₁cce to Yo,･･･, Y3, respectively.CCPmaxIn the following section a precoder and combiner design using the proposed low-complexity refinement is compared with the conventional alternatives presented in "A uniform-forcing transceiver design for over-the-air function computation," (full citation further above), and by K. Ando and G. T. F. de Abreu, in "Low-complexity and high-performance combiners for over the air computing," (full citation further above). The comparisons will take into account not only the MSE performances as determined via equation (7) but also the order of computational complexity. For the sake of later convenience, the initialisation via equation (17) is referred to as RQ, and the initialisation via equation (19) is referred to as MRQ.Since all techniques to be compared utilize the same UF precoder design, it is sufficient to solely consider the costs for designing the combiners, i.e., the complexity order.First, the complexity for designing the combiner by the best available initialisation and the novel low-complexity refinement, and further exploiting the PG based variant as summarised in method 100 is evaluated. For the initialisation as per equation (19), the complexity order is O(KN2), due to the computation of the covariance matrix H HH and its maximum eigenvector. Focusing on the per-iteration complexity of the inner loop over i¡, the required computations only pertain to the gradient and proximal operation as in equation (25). Since there is no computation to obtain the gradient via equation (25a), the per-iteration complexity of the gradient descent is O(N) for (1 - α)· u(ii). For the proximal operation in equation (26b), O(NK) computations are required.In the outer loop, indexed by ic, the additional per-iteration computations, i.e., without the computations of the inner loop, pertain to computing ak and bk as per equation (24). The complexity order to construct ak for all k as per equation (24c) is O(KN). The complexity order for computing bk as per equation (24d) is of the order O(NK).For the accelerated refinement using Adam presented with reference to method 300 the required computational complexity is increased by the computation of the first and second moment as per equation (47). In the computation of both first and second moments the additional per-inner-loop complexity is of the order O(N). Since the additional complexity from Adam accelerated PG is negligible in terms of complexity order, compared to the complexity of the other algorithmic components already mentioned.From the above the overall complexity order for the combiner design proposed in methods 100 and 300 isO(KN2 + IC (KN+IKN))maxThe complexity of the two conventional combiner designs mentioned herein will be determined hereafter for comparison purposes, starting with the conventional combiner design with initialisation by SDR and SCA based refinement using the interior point method as discussed in "A uniform-forcing transceiver design for over-the-air function computation” (full citation further above). Assuming that the PCA procedure of equation (13) used to extract the largest eigenvector of the SDR solution U is implemented based on the efficient power iteration algorithm as presented in "Matrix Computations" (full citation further above), the computational complexity is of the order O(N2). In the SDR optimisation problem for obtaining the solution matrix U, as represented by equation (12), the computational complexity is of the order O((K + N2)3.5). Thus, the complexity order for the initial combiner of this conventional combiner design is O((K + N2)3.5).The complexity of the refinement via iterative SCA is dominated by the cost of the optimisation problem as represented in equation (11), whose worst-case complexity is determined by the number of iterations lopt, the number of scalar constraints Kcst, and the number of variables Nvar. As shown by S. Boyd and L. Vandenberghe, in "Convex Optimisation", Cambridge University Press, March 2004, the number of iterations lopt, which is required to solve a convex optimisation problem is,Kcstlogloptlog(54)(Kcst(ξ4 - 1 - log(ξ4))ξ2+ ξ3 (50a)where §1,..., §4 denote constants related to numerical tolerance and convergence rate of the solver.As has been shown by C. Roos, T. Terlaky, and J. P. Vial, in "Interior point methods for linear optimisation", 2nd ed. New York: Springer, 2006, in each iteration, the complexity order associated with solving a convex optimisation with Nyar variables can be estimated to be at least O(Nar). From equation (11), the number of constraints and the number of optimisation variable are respectively given asandKcst = Kb)NvarNc)such that the estimated complexity order for solving the optimisation as represented by equation (11) is O(KN³log(K)).Since each loop of an SCA computation, as represented in equation (11a), has a complexity order of O(KN), the overall complexity order of the first conventional combiner design method considered here is estimated asO((K+ N2)3.5 + ICA (KN + KN³log(K)))maxSCAmaxwhere denotes the number of SCA-based refinements.In the second conventional combiner design, referred to herein as RQ / SCA the cost of obtaining the initial combiner u(0) is that of computing the load-aware approximation as represented in equation (15b) and normalizing the latter as per equation (17), which has a complexity order of O(KN). Since the SCA-based refinement of in the second conventional combiner design is the same as in the first conventional combiner design, the total complexity order can be estimated asO((KN) + ISC(KN + KN³log(K)))Table I shows an overview of the complexity orders of the conventional methods and the variants of the proposed methods presented herein.Table I: Computational Complexities of AirComp CombinersThe following section provides a numerical assessment of the convergence behaviour of the proposed methods in terms of MSE. For the computer simulation the following system parameters are used: N = 10, K = 5,10,15, ICCP 10, IIND= 10, 60. The signal-to-noise-ratio (SNR) is defined as PK / σ2 = 20[dB]. In the computation of the two methods 100, 300 each hyperparameter is tuned by BO as per method 200. An overview of the setup is shown in Table II.DefinitionFunction / ParameterAcquisition FunctionExpected Improvement eq. (43)Kernel FunctionARD Matérn 5 / 2 Kernel eq. (38)Num. of Initial ObservationBOlin= 2~7initNum. of Total ObservationBOBO= 20initmax+Num. of Random ChannelChan = 1000maxNum. of Testing HyperparameterBOtest= 10000Table II: Setup of the BO-based hyperparameter tuningFigures 6 and 7 show the convergence behaviour of the averaged MSE achieved by the proposes methods 100 and 300, respectively, as a function of the total iterations, with the hyperparameters determined by BO using method 200. In order to point out the performance improvement achieved by the BO-based hyperparameter tuning, the performance of method 100 with grid-search based hyperparameter tuning is also depicted. For the sake of fair comparison of grid-search and BO-based tuning, the number of grid points for searching is set to be rid = 20, and the same outer-loop parameters are used in all compared approaches, i.e., α(ic) = α,Vic.maxIn the grid-search, the averaged MSE performance as determined per equation (30) by parameter α is evaluated for 20 grid points uniformly taken in the range [10-3,1], with Imax Imaan random channel realisations, and using the hyperparameters achieving the lowest MSE as the solution. For the hyperparameter tuning through methods 100 and 300, respectively, the feasible regions of each hyperparameter are respectively set toα(ic) ∈ [10-3,1], Vicandα Ε [10-3,1],β₁ ∈ [10-3,1],β2 ∈ [10-3,1]Figure 8 shows the effect of the system load, namely the number of EDs K compared to the number of antennas N, onto the MSE performance. In particular it is visible that the MSE value grows with K for fixed N, while the convergence of the proposed methods is not affected by the loading conditions. It is noted that the MSE values are normalised by K, yielding normalised mean square error (NMSE), the normalisedMSE results with K=5 (underloaded) and K=10 (fully loaded) are the same, while the results with K=15 (overloaded) show a higher NMSE.The performance gains achieved by the variants using BO-based hyperparameter tuning and Adam-based acceleration are clearly visible, confirming that, thanks to BO, hyperparameter design can be effectively performed even when the number of total observations is limited. In addition, owing to the Adam acceleration, the number of hyperparameters can be reduced, which can further contribute to the reduction of the computational complexity for BO-based tuning and the accurate searching.Having observed that the MSE performance can be improved by BO and Adam, in the following section a performance-complexity trade-off for the number of inner-loop iterations IINN Imax is evaluated hereafter.INNSince a reasonable design of the maximum number of iterations Imax is mandatory to obtain a sufficiently accurate solution with low complexity, the performance loss ratio is illustrated in figure 8 in terms of a relative fraction of averaged MSE. The convexised optimisation problem as represented by equation (24) can be solved by recent convex optimisation techniques such as the interior-point method discussed by M. Grant and S. Boyd, in "CVX: Matlab Software for Disciplined Convex Programming,” http: / / cvxr.com / cvx, Version 2.2 January 2020, and by the same authors in "Graph Implementations for Nonsmooth Convex Programs," Recent Advances in Learning and Control, ser. Lecture Notes in Control and Information Sciences, V. Blondel, S. Boyd, and H. Kimura, Eds. Springer-Verlag Limited, 2008, pp. 95-110. The accuracy of the solution by method 300 can be evaluated by the performance loss ratio δ(η)δ(η) =E[6(u-method 300; η)]E[6(u← solution of eq. (24);η)]Figure 8 shows that the MSE performance loss is significant for a lower number of inner-loop iterations and can be improved by increasing the number of inner-loop iterations. Specifically, the gap in MSE performance loss for different system load conditions is found to be smaller as the number of inner-loop iterations increases. Itis assumed that about 1% MSE performance loss is acceptable for the lower complexity design so that a number of inner-loop iterations I = 60 is adopted for further comparisons.maxIn the following section the performance of the methods proposed herein is compared with conventional approaches. The first comparison uses a median of the achievable MSE performance over random channel realisation with different numbers of EDs and corresponding quantitative complexity order as illustrated in figures 9 and 10.From figure 9 it is seen that the refinement stage is fundamental for achieving a good performance, and that the all three methods with refinement are equivalent in terms of the achieved median MSE.In turn, from figure 10 it is seen that the proposed method based on equation (19) for initialization, followed by method 200 for hyperparameter tuning and method 300 for convergence acceleration, is the best-performing method among the high-performance alternatives, at a lower complexity then the conventional techniques used for comparison.In the quantitative assessment shown in figures 9 and 10, for the case K=10 with refinement, the median of the MSE performance for the proposed, high-complexity conventional design and the low-complexity conventional design are respectively 5.05×10-2, 4.86×10-2, and 5.13 × 10-2, while the complexity orders are 6.2 × 104, 1.5 × 107, and 2.6 × 105, respectively. In other words, in comparison with the high-complexity conventional design the proposed combiner offers a computational complexity reduced by about 230 times, while merely sacrificing MSE performance by a factor of 1.9 × 10-3. On the other hand, in comparison with the low-complexity conventional design, the proposed combiner offers a computational complexity that is about 3.7 times smaller while providing an additional MSE performance gain of 7.8 × 10-4.In summary, the insight obtained from figures 9 and 10 altogether is that spending computation power at initialization, as is the case of the conventional SDR-basedapproach, is not as good a strategy as spending computational resources for a better refinement stage. This conclusion is further corroborated by the results shown in figures 11 to 13, where the cumulative distribution functions (CDFs) of the MSE achieved by the conventional methods and the proposed methods under different system load conditions are compared. Taking the 90-percentile of MSE performance in the fully load case as a reference, it can be seen from figure 5, for instance, that the proposed combiner design, the high-complexity conventional combiner design, and the low-complexity conventional combiner design, all with refinement (black markers) yield MSEs of 6.37×10−2, 5.93 × 10-2, and 6.38 × 10-2, respectively, such that the MSE performance gap of the proposed and the conventional designs are only 4.4×10-4, and 1.7×10-3, respectively. In other words, the proposed method has fundamentally the same performance of the conventional schemes with SCA-based refinement, although the latter techniques have higher complexity.In light of the foregoing description, in accordance with a first aspect of the present invention a method of determining operating parameters v, u* of respective precoder stages of a plurality of transmitters and a combiner stage of a receiver, respectively, configured for over-the-air computing is presented. The operating parameters v, u* are targeted to minimise the average MSE between the target and reconstructed over-the-air computing functions. The method comprises determining an initial combiner u(0), and iteratively refining the combiner u(i) by minimising the square Euclidian vector norm of the combiner || u || 2, until a termination criterion is met, yielding an optimised combiner u*. The initial combiner may be based on an aggregated channel matrix H that comprises all channel vectors hk of all transmitters, although other ways of determining the initial combiner may be used. The minimisation in the refinement stage is performed under the constraint that, at each iteration, the Euclidian vector norm of the product of the transposed channel matrix and the respective iteration's combiner must be equal to or larger than 1, for each of the plurality of transmitters. As per the present invention, iteratively refining comprises approximating the constraint, based on the product of the derivative of the original constraint and the previous iteration's solution thereof. The resulting inequality representing the constraint must not exceed the negative value of the Euclidian vector norm of the product of the transposed channel matrix and the respective iteration's combiner, minus 1. Further in accordance with theinvention, the minimisation objective represented by the approximation is efficiently solved by applying a PG function. Based on the previously optimised combiner u while ensuring that the square |vk|2 of the precoding scalars of each transmitter does not exceed a predetermined power value P, the method computes optimised precoder parameters V for all transmitters. The so-computed operating parameters v, u* are provided to the transmitters and the receiver, respectively, for precoding pre-processed data to be transmitted in the transmitters or for combining received signals representing transmitted data from multiple transmitters in the receiver, respectively.*Determining the initial combiner u(0) may comprise determining the maximum eigenvector of the product of the aggregated channel matrix H and its conjugate transposition HH.Hyperparameter tuning may be applied for further reducing the computational complexity of the iterative refining step. The hyperparameter tuning can be carried out, e.g., in accordance with the method 200 described herein.Hyperparameter reduction may be applied for further reducing the computational complexity of the iterative refining step. The hyperparameter reduction can be carried out, e.g., in accordance with the Adam-based method 300 described herein.In accordance with a second aspect of the present invention a transmitter configured for wirelessly transmitting data in accordance with an over-the-air computing process is presented. The transmitter comprises an antenna, circuitry for processing radio frequency signals, one or more microprocessors, and volatile and non-volatile memory. The aforementioned components and elements are connected via one or more data and / or signal lines or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure components of the transmitter to implement or carry out a method comprising:- receiving optimised precoder parameters v determined via the method in accordance with the first aspect of the invention, and data to be transmitted,pre-processing the data to be transmitted for over-the-air computing in accordance with a target function,- precoding the pre-processed data using the received precoder parameters vr, and- transmitting the precoded pre-processed data via an antenna of the transmitter.In accordance with a third aspect of the present invention a receiver configured for wirelessly receiving signals representing data processed in accordance with an over-the-air computing process is presented. The receiver comprises at least one antenna, circuitry for processing radio frequency signals, one or more microprocessors, and volatile and non-volatile memory. The aforementioned components and elements are connected via one or more data and / or signal lines or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure components of the receiver to implement or carry out a method comprising:*- receiving an optimised combiner u^ determined via the method in accordance with the first aspect of the invention, and signals representing data transmitted from two or more transmitters,- combining the received signals representing data using the received optimised combiner u, and*- outputting the combined signals, which correspond to an output of the target function determined by the pre-processing in the transmitters.*Receiving optimised precoder parameters V or an optimised combiner u in the transmitter or the receiver, respectively, may comprise receiving said information from a remote computing unit or from a local computing unit or a local computing process executing the method in accordance with the first aspect of the invention.In accordance with a fourth aspect of the invention a method of over-the-air computing in a system comprising two or more transmitters wirelessly connected to a receiver is presented. The method comprises, at each transmitter, receiving optimised precoder parameters v determined via the method in accordance with the first aspect of the invention, and data to be transmitted. The data to be transmitted is then pre-processed for over-the-air computing in accordance with a target function. The pre-processed data is precoded using the received precoder parameters V,prior to transmitting the precoded and pre-processed data via an antenna of the transmitter. At the receiver the method comprises receiving an optimised combiner u* determined via the method in accordance with the first aspect of the invention, and signals representing data transmitted from two or more transmitters. The received signals representing data are combined using the received optimised combiner u*, prior to outputting the combined signals, which correspond to an output of the target function determined by the pre-processing in the transmitters.Two or more transmitters in accordance with the second aspect of the invention and at least one receiver in accordance with the third aspect of the invention may form a system that is configured for executing the method in accordance with the fourth aspect for implementing over-the-air computing.In one or more embodiments of the transmitter and / or the receiver the circuitry for processing radio frequency signals comprises a low noise amplifier and / or a mixer configured for providing a representation of a received signal at an intermediate frequency. The mixer preferably uses a same oscillator signal as a transmitter co-located with the receiver in the wireless apparatus.As will be appreciated by one skilled in the art, aspects of the embodiments may be embodied as a system, apparatus, method, or program product. Accordingly, embodiments may take the form of an entirely hardware embodiment, an entirely software-implemented embodiment, including firmware, resident software, micro-code, etc., or an embodiment combining software and hardware aspects.For example, the disclosed embodiments may be implemented as a hardware circuit comprising custom very-large-scale integration (VLSI) circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. The disclosed embodiments may also be implemented in programmable hardware devices such as field programmable gate arrays, programmable array logic, programmable logic devices, or the like. As another example, the disclosed embodiments may include one or more physical or logical blocks of executable code which may, for instance, be organised as an object, procedure, or function.The method presented hereinbefore may be represented by computer program instructions. Accordingly, in accordance with a further aspect of the invention, a computer program product comprises computer program instructions which, when executed by a microprocessor of a wireless apparatus in accordance with the second aspect of the invention, cause the microprocessor to execute methods in accordance with the first aspect of the present invention, and to accordingly control hardware and / or software blocks or modules of the wireless apparatus.Computer program instructions, or code, for carrying out operations for embodiments may be any number of lines and may be written in any combination of one or more programming languages including an object-oriented programming language such as Python, Ruby, Java, Smalltalk, C++, or the like, and conventional procedural programming languages, such as the "C" programming language, or the like, and / or machine languages such as assembly languages. The code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN), wireless LAN (WLAN), or a wide area network (WAN), or the connection may be made to an external computer, for example, through the Internet using an Internet Service Provider (ISP).The computer program instructions may be retrievably stored or transmitted on a computer-readable medium or data carrier. The medium or the data carrier may by tangibly or physically embodied, e.g., in the form of a hard disk, solid state disk, flash memory device or the like. However, the medium or the data carrier may also comprise a modulated electro-magnetic, electrical, or optical signal that is received by the computer by means of a corresponding receiver, and that is transferred to and stored in a memory of the computer.The described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In this description, numerous specific details are provided, such as examples of programming, software modules, user selections, network transactions, database queries, database structures, hardware modules,hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments. One skilled in the relevant art will recognize, however, that embodiments may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment. Reference throughout this specification to "one embodiment," "an embodiment," or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, appearances of the phrases "in one embodiment,” "in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment, but mean "one or more but not all embodiments" unless expressly specified otherwise. The terms "including,” "comprising," "having,” and variations thereof mean "including but not limited to," unless expressly specified otherwise. An enumerated listing of items does not imply that any or all of the items are mutually exclusive, unless expressly specified otherwise. The terms "a," "an,” and “the” also refer to "one or more" unless expressly specified otherwise.Where aspects of the embodiments are described in this specification with reference to schematic flowchart diagrams and / or schematic block diagrams of methods, apparatuses, systems, and program products according to embodiments it will be understood that each block of the schematic flowchart diagrams and / or schematic block diagrams, and combinations of blocks in the schematic flowchart diagrams and / or schematic block diagrams, can be implemented by code. This code may be provided to a processor of a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart diagrams and / or block diagrams.It should be noted that, in some implementations or embodiments, the functions noted in the exemplary embodiments shown in the figures may occur out of the order shown in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in thereverse order, depending upon the functionality involved. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more blocks, or portions thereof, shown in the figures.The present invention proposes a low-complexity combiner design for MSE minimization for AirComp systems. The novel refinement process can be used along with sophisticated RQ-based initialization and may be further enhanced with BO-based hyperparameter tuning, and Adam-based acceleration. The novel refinement is obtained from a CCP-based reformulation of the average MSE minimization problem, which can be solved at low complexity using a PG method. Numerical results demonstrate that embodiments and variants of the proposed design can achieve an MSE performance equivalent to the best conventional alternatives currently known, with about 200-times smaller complexity than the highest-performing conventional scheme, and about 4-times smaller complexity than its conventional low-complexity counterpart.BRIEF DESCRIPTION OF THE DRAWINGIn the following section the invention will be described with reference to the drawings, in whichFig. 1 shows a simplified diagram of an AirComp system,Fig. 2 shows an exemplary simplified block diagram of an AirComp transmitter and an AirComp receiver,Fig. 3 shows a flow diagram of an exemplary method in accordance with the invention,Fig. 4 shows a flow diagram of a first embodiment of the exemplary method in accordance with the invention for accelerating the convergence of the iteration,Fig. 5 shows a flow diagram of a second embodiment of the exemplary method in accordance with the invention for accelerating the convergence of the iterationFig. 6 depicts the MSE convergence behaviour of the proposed method and variants thereof for a low number of inner-loop iterations,Fig. 7 depicts the MSE convergence behaviour of the proposed method and variants thereof for a high number of inner-loop iterations,Fig. 8 illustrates the effect of the system load on the MSE performance for the proposed method and variants thereof over the number of inner-loop iterations,Fig. 9 shows a comparison of the MSE performance of the proposed method with conventional approaches over the system load,Fig. 10 shows a comparison of the computational complexity of the proposed method with conventional approaches over the system load,Fig. 11 shows a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for an underloaded system,Fig. 12 shows a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for a fully loaded system,Fig. 13 shows a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for an overloaded system,Fig. 14 shows an exemplary block diagram of a transmitter in accordance with the invention,Fig. 15 shows an exemplary block diagram of a receiver in accordance with the invention, andFig. 16 shows an exemplary and schematic flow diagram of a method of over-the-air-computing in accordance with a fourth aspect of the invention.In the figures identical or similar elements may be referenced using the same reference designators.DETAILED DESCRIPTION OF EMBODIMENTSFigures 1 to 13 have been described further above and will not be discussed again.Figure 14 shows an exemplary block diagram of a wireless transmitter 500 in accordance with the invention. The wireless receiver 500 comprises at least one antenna 502, circuitry 504 for processing radio frequency signals, one or more microprocessors 505, volatile 508 and non-volatile memory 510. The aforementioned components or elements are connected via one or more data and / or signal lines or buses 512. The non-volatile memory 510 stores computer program instructionswhich, when executed by the one or more microprocessors 505, configure components of the receiver 500 to implement or carry out a method comprising:*- receiving an optimised combiner u^ determined in via the method 100 in accordance with the first aspect of the invention, and data to be transmitted,- pre-processing the data to be transmitted for over-the-air computing in accordance with a target function,- precoding the pre-processed data using the received precoder parameters vr, and- transmitting the precoded pre-processed data via an antenna of the transmitter.Figure 15 shows an exemplary block diagram of a wireless receiver 600 in accordance with the invention. The wireless receiver 600 comprises at least one antenna 602, circuitry 604 for processing radio frequency signals, one or more microprocessors 606, volatile 608 and non-volatile memory 610. The aforementioned components or elements are connected via one or more data and / or signal lines or buses 612. The non-volatile memory 610 stores computer program instructions which, when executed by the one or more microprocessors 606, configure components of the receiver 600 to implement or carry out a method comprising:- receiving an optimised combiner u* determined in via the method 100 in accordance with the first aspect of the invention, and signals representing data transmitted from two or more transmitters 500,- combining the received signals representing data using the received optimised combiner u*, and- outputting the combined signals, which correspond to an output of the target function determined by the pre-processing in the transmitters.Figure 16 shows an exemplary and schematic flow diagram of a method 700 of over-the-air-computing in accordance with the fourth aspect of the invention. The method comprises, at two or more transmitters 500, receiving, in step 710, optimised precoder parameters v determined in via the method 100 in accordance with the first aspect of the invention, and data to be transmitted (not shown in the figure). The two or more transmitters 500 pre-process, in step 720, the data to be transmitted for over-the-air computing in accordance with a target function, and precode, in step 730, the pre-processed data using the received precoder parameters V, prior to transmitting the precoded and pre-processed data via an antenna 502 of the*transmitter 500 in step 740. The transmission is indicated by the dashed zig-zagged arrows. The method comprises, at the receiver 600, receiving, in step 750, an optimised combiner u^ determined via the method 100 in accordance with the first aspect of the invention, and signals representing data transmitted from two or more transmitters 500, represented by the dashed zig-zagged arrows. In step 760 the method comprises combining the received signals representing data using the received optimised combiner u*, prior to outputting, in step 780, the combined signals, which correspond to an output of the target function determined by the pre-processing in the transmitters 500.*LIST OF REFERENCE NUMERALS (PART OF THE DESCRIPTION)100 method101 iterative refinement110 receive inputs120 calculate initialisation point130 compute ak and bk140 update u(i)150 inner-loop termination criterion met?160 replace previous initialisation point170 outer-loop termination criterion met?180 output optimised combiner190 compute optimal precoder195 output optimal precoder200 method210 sampling initial parameters220 evaluating initial MSE performance230 initialising dataset240 modelling function250 select input test set260 compute average MSE and mean variance270 select hyperparameters280 determine MSE performance & update current iteration dataset290 termination criterion met?295 determine and output optimised hyperparameters300 method310 receive inputs320 calculate initialisation point330 compute ak and bk340 initialise moments m₁ and m₂350 update first and second moments m, and m₂360 inner-loop termination criterion met?370 replace previous initialisation point380 outer-loop termination criterion met?390 output optimal combiner u*400 compute optimal precoders v410 output optimal precoders v500 transmitter550 pre-processing560 precoding600 receiver650 combiner design660 combiner670 processing700 method710 receive optimised parameters and data720 pre-processing730 precoding740 transmitting750 receive optimised parameters and signals760 combining770 outputting

Claims

1. Method (100) of determining operating parameters (v, u*) of respective precoder stages of a plurality of transmitters (500) and a combiner stage of a receiver (600), respectively, configured for over-the-air computing, the operating parameters (v, u*) being targeted to minimise the average mean square error (MSE) between the target and reconstructed over-the-air computing functions, the method comprising:- determining (130) an initial combiner (u(0)),- iteratively refining (101) the combiner (u(ii)) by minimising the square Euclidian vector norm of the combiner ( || u || 2) while maintaining, at each iteration, the Euclidian vector norm of the product of the transposed channel matrix and the respective iteration's combiner equal to or larger than 1, for each of the plurality of transmitters (500), until a termination criterion is met, yielding an optimised combiner (u*),- computing (190) optimised precoder parameters (v) for all transmitters (500), based on the previously optimised combiner (u*) while ensuring that the square (|vk|2) of the precoding scalars of each transmitter (500) does not exceed a predetermined power value (P),wherein iteratively refining (101) comprises:- approximating, for all of the plurality of transmitters (500), the respective constraint, i.e., the requirement that the Euclidian vector norm of the product of the transposed channel matrix and the respective iteration's combiner must be equal to or larger than 1, based on the product of the derivative of the original constraint and the previous iteration's solution thereof, and- applying a proximate gradient (PG) function for solving the minimisation objective represented by the approximation.

2. Wireless transmitter (500) comprising an antenna (502), circuitry (504) for processing radio frequency signals, one or more microprocessors (506), volatile (508) and non-volatile memory (510), connected via one or more data and / or signal lines or buses (512), wherein the non-volatile memory (510) stores computer program instructions which, when executed by the one or more microprocessors (506), configure components of the transmitter (500) to implement or carry out a method comprising:- receiving (710) optimised precoder parameters (v) determined in accordance with the method (100) of claim 1, and data to be transmitted,- pre-processing (720) the data to be transmitted for over-the-air computing in accordance with a target function,- precoding (730) the pre-processed data using the received precoder parameters (V), and- transmitting (740) the precoded pre-processed data via an antenna (502) of the transmitter (500).

3. Wireless receiver (600) comprising at least one antenna (602), circuitry (604) for processing radio frequency signals, one or more microprocessors (606), volatile (608) and non-volatile memory (610), connected via one or more data and / or signal lines or buses (612), wherein the non-volatile memory (610) stores computer program instructions which, when executed by the one or more microprocessors (606), configure components of the receiver (600) to implement or carry out a method comprising:- receiving (750) an optimised combiner (u*) determined in accordance with the method (100) of claim 1, and signals representing data transmitted from two or more transmitters (500),- combining (760) the received signals representing data using the received optimised combiner (u*), and- outputting (780) the combined signals, which correspond to an output of the target function determined by the pre-processing (720) in the transmitters (500).

4. The wireless transmitter (500) of claim 2, or the wireless receiver of claim 3, wherein receiving (710; 750) optimised precoder parameters (V) and / or an optimised combiner (u*) comprises receiving said information from a remote computing unit or from a local computing unit or a local computing process executing the method in accordance with claim 1.

5. Method (700) of over-the-air computing in a system comprising two or more transmitters (500) wirelessly connected to a receiver (600) comprising, at each transmitter (500):- receiving (710) optimised precoder parameters (v) determined in accordance with the method (100) of claim 1, and data to be transmitted,- pre-processing (720) the data to be transmitted for over-the-air computing in accordance with a target function,- precoding (730) the pre-processed data using the received precoder parameters (V), and- transmitting (740) the precoded and pre-processed data via an antenna (502) of the transmitter (500),and comprising, at the receiver (600):- receiving (750) an optimised combiner (u*) determined in accordance with the method (100) of claim 1, and signals representing data transmitted from two or more transmitters (500),- combining (760) the received signals representing data using the received optimised combiner (u*), and- outputting (770) the combined signals, which correspond to an output of the target function determined by the pre-processing (720) in the transmitters (500).

6. Wireless communication system comprising two or more transmitters (500) in accordance with claim 2 and at least one receiver (600) in accordance with claim 3, wherein the system is configured for implementing the method of claim 5.

7. Computer program product comprising computer program instructions which,- when executed by a computer, configure the computer to execute the method in accordance with claim 1, or- when executed by a microprocessor of a wireless transmitter (500) according to claim 2, cause the wireless transmitter (500) and / or control hardware blocks, modules or components of the wireless transmitter (500), respectively, to implement or carry out a method comprising:- receiving (710) optimised precoder parameters (v) determined in accordance with the method (100) of claim 1, and data to be transmitted,- pre-processing (720) the data to be transmitted for over-the-air computing in accordance with a target function,- precoding (730) the pre-processed data using the received precoder parameters (V), and- transmitting (740) the precoded pre-processed data via an antenna (502) of the transmitter (500), or- when executed by a microprocessor of a wireless receiver (600) according to claim 3, cause the wireless receiver (600) and / or control hardware blocks, modules or components of the wireless receiver (600), respectively, to implement or carry out a method comprising:- receiving (750) an optimised combiner (u*) determined in accordance with the method (100) of claim 1, and signals representing data transmitted from two or more transmitters (500),- combining (760) the received signals representing data using the received optimised combiner (u*), and- outputting (770) the combined signals, which correspond to an output of the target function determined by the pre-processing (720) in the transmitters (500).

8. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 7.

Images