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

Abstract

A method (100) of determining operating parameters formula (I) 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 formula (II) by minimising the square Euclidian vector norm of the combiner (II u II2), until a termination criterion is met, yielding an optimised combiner (u*). Optimised precoder parameters formula (III) for all transmitters (500) are computed (190) based on the previously optimised combiner (u*) while ensuring that the square formula (IV) of the precoding scalars of each transmitter (500) does not exceed a predetermined power value (P).

Description

[0001]202403565 -1- METHOD OF DETERMINING OPERATING PARAMETERS OF BEAMFORMERS FOR OVER-THE-AIR COMPUTING FIELD OF THE INVENTION The 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). NOTATION The following notation is used throughout this specification. Real vectors and matrices are represented in bold small and bold capital letters, respectively, like in ^ and ^, while their complex counterparts are represented bold italic small and bolditalic capital letters, respectively, like in ^ and ^. ^ × ^ identity matrix and ^ × 1 all-one column vector are denoted by ^^and ^^, respectively. Vector norm and theabsolute value of a scalar are respectively denoted by ∥⋅∥ and | ⋅ |. Transposition andconjugate transposition are represented as ·T and ·H, respectively, while ℜ{⋅}, ℑ{⋅}and min(⋅) denote the real part, imaginary part, and minimum operators. Finally,^(^, ^) and ^^(^, ^) respectively denote the real- and complex-valuedmultidimensional Gaussian distribution with mean ^ and covariance matrix ^. BACKGROUND With 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 Internal 202403565 -2- 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 Internal 202403565 -3- 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. Internal 202403565 -4- 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 INVENTION This object is achieved by the methods of claims 1 and 6, the wireless transmitter of claim 3, the wireless receiver of claim 4, and the wireless communication system of claim 7. A computer program product and a computer-readable medium are provided in claims 8 and 9, respectively. Embodiments and developments are described in respective dependent claims. Internal 202403565 -5- 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. The refinement stage upon which the present invention is based relaxes the computationally costly MSE minimisation through the convex concave procedure (CCP), which applies a proximal gradient (PG) method for obtaining a low-complexity solution for the minimisation. The present invention proposes ways of efficiently designing the hyperparameters of the refinement process for improving its performance and further reducing the computational complexity over conventional refinement processes. 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 ^ single-antenna EDs and one AP equipped with ^ antennas. Let the AirComp target function ^ be given by where $ &$#, ⋯ , $ (T is a vector containing all symbols $! ∼ ^(0,1) transmitted bythe EDs.Under perfect synchronisation among all EDs, the received signal + ∈ ℂ^×# subjectedto fading and noise is given by Internal 202403565 -6-where .! ∈ ℂ^×# denotes the channel vector between the 3 th ED and the AP, / ! ∈ ℂis the 3-th ED's precoding scalar, and 1 ∈ denotes the additivewhite Gaussian noise (AWGN) at the AP. In possession of the received signal +, an estimated target function is constructed at the AP via the combiner where ^ ≜ & / #, ⋯ , / (T is a vector containing all precoding scalars / ! employed bythe EDs, 7 ∈ ℂ^×# denotes the combiner vector applied by the AP, and9 ≜ &. , ⋯ , . ( ∈ ℂ^×# 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, ^̂ is a function of the combining and precoding vectors 7 and ^, which are the variables that can be optimised, while the known channel matrix 9 is a fixed parameter, and the unknown vector of transmitted symbols $ 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 $ and under constant channel 9, precoders ^ and combiners 7, can be concisely expressed as where the arguments of the functions ^ and ^̂ 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 7, the optimal precoding scalars / !⋆that the EDs must apply in order to minimize theaverage MSE while satisfying the power constraint ≤ G is given byInternal 202403565 -7- (5K)(5O) where I denotes a scaling coefficient and G denotes the maximum transmit power constraint, both common to all EDs. From equations (3) and (5) can be obtained, where again it is emphasised that the notation is meant to remindthat the estimated target function ^̂⋆, partially optimised by employing the precoders^⋆, is still a function of the combiner vector 7, which can also be further optimised.To that end, one might be inclined to simply seek to minimize the average MSE between ^ and ^̂⋆, namely (7) It is noted, however, that minimizing the average MSE Q(7; I) with respect to 7 isnon-trivial since the parameter I is, as evidenced by equation (5b), a quantity that is itself dependent on 7 and coupled to ^⋆, besides being dependent also on the channel matrix 9 and power constraint G. In other words, an MSE-optimal combiner design must simultaneously minimize equation (7) and satisfy equation (5b), given 9 and G. In view of the aforementioned challenge, in the next subsection the best feasible combiner design approach currently known will be discussed. Internal 202403565 -8- A straightforward optimisation problem formulation that puts together the objective given by equation (7) and the condition imposed by equation (5b) is m7in∈ℂimX×iYze (8) which, although difficult to solve directly due to the presence of the min operation in the denominator, can be reformulated into the standard constrained minimisationproblemm 57in∈ℂimX×iYze ∥ 7 ∥(9K)subject to (9O) where the constant 45 / G was ignored and attention is called to the fact that the min operator in equation (8) is relaxed into the ^ 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 projection Then, the e-th iteration of the SCA procedure proposed reduces to computing Internal 202403565 -9- and subsequently solving the problem m7(ikn)∈imℂXi×zYe subject to As can be seen from equation (11), an initial combiner 7(m)is required. A relatively accurate conventional approach designs the initial combiner 7(m)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 by mnin∈ℂimX×iXze Tr(n) (12K)subject to Tr(n9!) ≥ 1, ∀3 (12O)n ^ 0 (12l)where n ≜ 77H and 9 H! ≜ .!.!.The solution n of the problem represented in equation (12) can be obtained efficiently via interior point methods, and the corresponding combiner vector 7(m)can be extracted from the latter as its largest eigenvector, which can be computed efficiently via principal component analysis (PCA). Therefore maxeigv (n) (13)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 Internal 202403565 -10-solution n ∈ ℂ^×^, only to immediately thereafter truncate it to the SCA initialisationpoint 7(m) ∈ ℂ^×#, is not very efficient.Therefore, low-complexity alternatives to the SDR procedure are needed. To thisend, first notice that as long as ∥ 7 ∥5≠ 0 the optimisation problem represented inequation (8) can, without loss of generality or any other penalty, be rewritten as m7a∈xℂimX×iYze where 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 as where Substituting the approximation represented in equation (15a) into equation (14) and ignoring the constant G / 45, the standard rank-one RQ problem can be obtained as C.xH7C5m7a∈xℂimX×iYze∥ 7 ∥5 (16)Internal 202403565 -11- whose solution is straightforwardly given by the only non-zero eigenvector of .x.xH,which in turn is merely the normalised vector ℎ‾ , that is7(m) = maxeigv The 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: m7a∈xℂimX×iYze which uses the aggregated channel matrix 9≜ &.#, … , . ( ∈ ℂ^× (18O)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 99Has,7(m) = maxeigvi99Hj , (19)Internal 202403565 -12- 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 can be achieved, while still providing at least the same MSE performance than conventional methods. Such reduction of the complexity of solving the optimisation problem represented in equation (9) can exemplarily be 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. The computationally costly interior point methods conventionally used can be replaced by a different approach that is briefly elucidated below.Consider a general real domain DC programming problem given by subject to ℎ^(^) − ^^(^) ≤ 0, } = 1, … , ^ (20O)under the assumptions of real domain convex functions ℎ ^×# #×#^ : ℝ → ℝ and According to the CCP, the solution of DC problems can be approximately obtained by iteratively solving an alternative convexised optimisation problem. In particular, theconvexised optimisation problem for obtaining the solution of the e^ + 1-th iterationwith the previous solution ^(g^)is given by Internal 202403565 -13- m^i∈nℝimX×iYze subject to where ^^̂i^, ^(g^)j is an approximation of the function ^^(^) from equation (20),defined as 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 suffices to relate equation (9) to equation (20). Then, taking into consideration that the desired CCP is in the complex domain, the corresponding reformulated MSEminimisation problem for obtaining the solution in e^ + 1 CCP iterations becomesm 57in∈ℂimX×iYze ∥ 7 ∥ (22K)subject to where the com domain For the sake of later convenience, the optimisation (22) can be rewritten as minimize 57∈ℂX×Y ∥ 7 ∥ (24K)subject to ^H H!7 + 7 ^! ≤ O!, ∀3, (24O)with Since the optimisation represented in equation (24) is obviously convex, with a quadratic objective function and half-space constraint, the optimal solution can be Internal 202403565 -14-obtained, e.g., via a proximal gradient (PG) descent method. By applying PG for thee^-th iteration of the optimisation problem, the solution can be iteratively updated bythe gradient of the objective function and the proximal operator for the constraint.Thus the e^ + 1-th update of the PG steps can be expressed as with the proximal operator for half-space constraint over all different 3 given by maxi0, ^ 7 + 7 ^ − Orox 7 ≜ 7 − ! !jp ( ) !5 ⋅ ^! , ∀3, (25O)∥^!∥where e^and ^(g^)denote the iteration index of the PG iteration, and the step size of the gradient in the e^-th iteration, respectively. All in all, a method for designing the uniform-forcing (UF) precoder and combiner using the iterative refinement introduced above 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 and ¡¤§¨¥¨¦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.Inputs to the method 100 are the channel matrix 9 ∈ ℂ^× from all EDs to the AP, thenumber of iterations for the outer loop and inner loop, and ¡¤§¨¥¨¦, and the stepsize ^(g^), ∀e^ of the gradient for each iteration. Outputs of the method are the optimalprecoding scalars / ⋆ ⋆ ^×#! ∈ ℂ ∀3 and the optimal combining vector 7 ∈ ℂ . Afterreceiving the inputs in step 110, the initialisation point 7(m)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 Internal 202403565 -15- nested iteration loops. Starting with step 130 in the outer iteration loop, ^!and O!are computed via equation (24), using 7(m)as the starting point for the optimisation. In step 140 the inner iteration loop starts, is updated via equation (25). Step 150 checks whether a termination criterion is met for the inner iteration loop. As mentioned further above, the termination criterion may comprise reaching a predetermined number 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 of the inner loop replaces the previous initialisation point, e.g., 7(m), 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 7⋆in step 180, and the optimal precoders / !⋆are computed thereon, for all k EDs, via equation (5) and using 7⋆in step 190, and output as ^⋆! all k EDs in step 195. While 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. As initially mentioned, any reduction of the computational complexity is beneficial, inter alia, for promoting the deployment of over-the-air computation systems. In accordance with the present invention a novel approach concerning accelerating the convergence of the CCP- and PG-based iterative refinement will be presented, which achieves a further reduction of the computational complexity. Internal 202403565 -16- In accordance with the present invention Bayesian optimisation (BO)-based hyperparameter tuning for the proximal gradient (PG) method for finding the optimal combiner 7⋆is proposed. As shown above for the 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 theconverged solution heavily relies on the value of hyperparameters, i.e., step size all the hyperparameters must be carefully determined. Classically, thetuning 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 ¡¤¢¢¥£¦. In order to achieve a reasonable hyperparameter tuning with limited searching effort, a BO-based offline hyperparameter tuning is proposed. In 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 problem mini¯mize subject to 7 ← method 100 (9, ¯) (26O) where ¯ ∈ ℝ#ש ­­®max 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, the present invention proposes employing BO for Internal 202403565 -17-modelling the objective function as a Gaussian process (GP), such that afteran 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 of the hyperparameter vectors by random sampling from the feasible region, yielding the set where¯(g) ∈ &0,1(#שª­­«®. (28)Next, the value of the objective function, i.e., observation, is computed using method 100 described further above and the inputs in ³initconstructed above.To find a robust hyperparameter set against the statistically random channel matrix9, this step considers averaging ¡ Chanmax observations on random channel realisations.For an input ¯(g)with the ·-th channel realisation, its MSE performance can be evaluated by i¯(g)j = Qi7 ← (¹) (g)¹̧ method 100 i9 , ¯ j; Ij. (29)After the MSE values are evaluated for ¡¤¢º¥¥¦»of different channel realisations, the average MSE can be obtained by Internal 202403565 -18- For future convenience, the concatenated averaged MSE vector of all thehyperparameters in a set ³ with cardinality |³| = ¡¼ can be constructed by ©¾³ = ´     b¯(̃g)c, (31O)g"#where ¯(̃g) ∈ &0,1(#ש ­­®max denotes the e-th hyperparameters.Then, the initial dataset can be constructed as Now 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 iterations. At the e±-th iteration, the task in this step is to estimate theprobability distribution of the averaged MSE vector ½hyperparameter set where x̄(gµ)g ∈ &0,1(#ש CCPmax are sampled testing hyperparameters as in equation (28).At the e±-th trial, the joint distribution of the averaged MSE in the current dataset¿(gµh#)±² and expected observation corresponding to the given testinghyperparameter set ³(gteµ)stare modeled by the following multivariate normal distribution with Internal 202403565 -19- where the covariance matrix Â, the matrix ^ can be given as with 4Æ5denoting the variance for the observation noise, while the hyperparameter set for the past evaluation is given by Without loss of generality, the kernel matrix between hyperparameter sets of arbitrary cardinalities Ê and Ễ composed by can be defined Assuming automatic relevance determination (ARD) Matérn 5 / 2 kernel, the (e, ·)-thelement of the kernel matrix denoted Ä(³, computed by with the biased distance among hyperparameters given by Ï= ∥d h5 h5g,¹ ∥ iag MÌ# , … , Ì©ª­­«®N Internal 202403565 -20-where Ìm, … , Ì© CCPmax are trainable parameters.With the knowledge of the current dataset and the maximum likelihoodestimation (MLE), the mean ^x and covariance matrix Ä‾ conditionalprobability distribution for the desired average MSE can be respectively derived as where Since the prior mean of the averaged MSE for testing the hyperparameters ^ M³ (gµ)test Ncannot be observed, it is approximated by the arithmetic mean of the observation values, i.e., Note that the parameters Ìm, … , Ì©­­® and 4×, which are used in the computation ofequation (39), can be trained by maximizing the log-likelihood function log1= − log det(^) This 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. Internal 202403565 -21- The following section describes searching the next hyperparameters 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 (EI), 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 ¯ can be expressed as Û(¯) = ? Rmax denotes the minimum MSE in the current dataset After Û(¯) is evaluated for all ¯ ∈ ³ (gµ)test , the next hyperparameter can be determinedas After the average MSE with the updated input, i.e., ¸i¯(gµ)j, is evaluated in the performance evaluation step, the dataset is updated into Finally, after the function modelling, the hyperparameter selection and the performance evaluation steps are iterated for times the tuned hyperparameterscan be obtained by¯ÜÝÞ = argmin ¸(¯). (46)¯∈³Mßµª¶«NNote 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 Internal 202403565 -22- 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.Inputs to the method 200 are the maximum iteration counts ¢¢£ ±² ¢º¥» ¡max, ¡max , ¡max , theBO initial value ¡i±ni²t and the BO test value ¡t±e²st. It is noted that further initialisation values may be provided, e.g., for alternative or further termination criteria. The outputof the method 200 is the optimised ¯ ©ÜÝÞ ∈ ℝ m­­ax®×#.An initialisation phase of the method 200 comprises sampling, in step 210, a set of initial parameters ³initas per equation (27), evaluating the initial average MSEperformance ¸(¯ ∈ ³init) as per equation (30) in step 220, and initialising the dataset¿(m)±² as per equation (32) in step 230. The actual iterative BO hyperparameter tuningstarts in step 240 with modelling the function. The function modelling comprisestuning the trainable parameters Ìm, … , Ì© CCPmax and the variance for the observationnoise 4 by maximising equation (42). Next, in step 250, the set of test inputs selected as per equation (28). Now the mean and covarianceÄ of the conditional probability distribution for the desired average MSEbe 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 is determined in accordance with equation (44). In step 280 the average MSE performance ¸ for the current iteration’s input vector collected hyperparameters is determined in accordance with equation (30), and the current iteration’s dataset is updated as per equation (45). The method then checks, 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. Internal 202403565 -23- The 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 can require 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 proposed method 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, also known 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 e^-th iteration, the first momentvector ∈ ℂ^×# is computed as a moving average of the previous first moment andthe gradient of the objective function with exponential decay rates â#"&0,1)as Similarly, in the e^-th iteration the second moment }5 ∈ ℝ is computed as a movingaverage of the previous second moment and the squared gradient∥∥7(gkh#)∥∥5withexponential decay rates â5 = &0,1)Internal 202403565 -24- Incorporating the two types of moments derived above the e^-th update in Adam is then given as where æ = 10hé denotes a small-valued scaler introduced for avoiding numericalissues. 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 threehyperparameters, i.e., ^, â#, and â5, are used over different outer loops so that thenumber of hyperparameters is kept at three even when the number of outer loops is changed.Inputs to the method 300 are the channel matrix 9 ∈ ℂ^×, maximum iteration counts¡§¨¨max and ¡¢¢£max, the three hyperparameters ^, â#, and â5, and the scaler æ. It is notedthat further initialisation values may be provided, e.g., for alternative or furthertermination criteria. The output of the method 300 are the optimised combining vector7∗ ∈ ℂ^×# and the optimised precoding scalars / ∗! ∈ ℂ, ∀3.After receiving the inputs in step 310, the initialisation point 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 ^!and O!are computed via equation (24), using 7(m)as the starting point for the optimisation. The moments m and m2are initialised = ë=0, respectively, in step 340. The inner iteration loop, Adam-iteration,with step 350, where the first moment is updated via equation (47a), the moment }(g5^)is updated via equation (47b), and 7(g^)is updated via equation (48). Internal 202403565 -25- Step 360 checks whether a termination criterion is met for the inner iteration loop. The termination criterion may comprise reaching a predetermined number 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 of the inner loop, in the exemplary case replaces the previous initialisation point, e.g., 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”-branch of step 380, the last output of the inner loop is output as the optimal combiner7⋆ in step 390, and the optimal precoders / ⋆! are computed thereon, for all k EDs, viaequation (5) and using 7⋆in step 400 and output as ^⋆! 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 thehyperparameter dimension ¡¢¢£¤¥¦ to three, and changing Ìm, … , Ì©ª­­«®to Ìm, … , Ìî,respectively. In the following section the proposed 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. Internal 202403565 -26- First, the complexity for designing the combiner by the best available initialisation and the PG based refinement, and further exploiting the novel low-complexity refinement,is evaluated. For the initialisation as per equation (19), the complexity order isï(^^5), due to the computation of the covariance matrix 99  and its maximumeigenvector. Focusing on the per-iteration complexity of the inner loop over 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 ï(^) for(1 − ^) ⋅ 7(gk). For the proximal operation in equation (26b), ï(^^) computations arerequired. In the outer loop, indexed by e^, the additional per-iteration computations, i.e., without the computations of the inner loop, pertain to computing ^!and O!as per equation (24). The complexity order to construct ^!for all 3 as per equation (24c) is ï(^^). The complexity order for computing O!as per equation (24d) is of the order ï(^^). 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 ï(^). 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 is The 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 Internal 202403565 -27- 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 n 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 ï(^5). In the SDR optimisation problem for obtaining the solution matrix n, as represented by equation (12), the computational complexity is ofthe order + ^5)î.ð). Thus, the complexity order for the initial combiner of thisconventional combiner design is ï((^ + ^5)î.ð).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 ¡ÜÝÞ, the number of scalar constraints ^^ñÞ, and the number of variables ^ò¥ó. As shown by S. Boyd and L. Vandenberghe, in “Convex Optimisation”, Cambridge University Press, March 2004, the number of iterations ¡ÜÝÞ, which is required to solve a convex optimisation problem is, denote constants related to numerical tolerance and convergence rateof 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 ^varvariables can be estimated to be at least ï(^vîar). From equation (11), the number of constraints and the number of optimisation variable are respectively given as ^^ñÞ = ^, (50O)and^var = ^, (50l)Internal 202403565 -28- such that the estimated complexity order for solving the optimisation as represented by equation Since each loop of an SCA computation, as represented in equation (11a), has a complexity order of ï(^^), the overall complexity order of the first conventional combiner design method considered here is estimated as where 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 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 ï(^^). 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 as Table I shows an overview of the complexity orders of the conventional methods and the variants of the proposed methods presented herein. Method Complexity Order conventional SDR / SCA ï M(^ + ^5)î.ð + ¡SCA¤¥¦ ⋅ (^^ + ^^îlog (^))Nconventional RQ / SCA ï M(^^) + ¡ ø¢ùmax ⋅ (^^ + ^^îlog (^))Nproposed method 100 PG ï M^^5 + ¡ ¢¢£max ⋅ (^^ + ¡§¨¨¤¥¦ ⋅ ^^)Nproposed method 300 Adam ï M^^5 + ¡ ¢¢£max ⋅ (^^ + ¡ §¨¨max ⋅ ^^)NTable I: Computational Complexities of AirComp Combiners Internal 202403565 -29- The following section provides a numerical assessment of the convergence behaviour of the proposed methods in terms of MSE. For the computer simulation thefollowing system parameters are used: ^ = 10, ^ = 5,10,15, = 10, = 10,60. The signal-to-noise-ratio (SNR) is defined as G^ / 45 ≜ 20& dB(. In thecomputation 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. Definition Function / Parameter Acquisition Function Expected Improvement eq. (43) Kernel Function ARD Matérn 5 / 2 Kernel eq. (38) Num. of Initial Observation ¡±²^»^Þ = 2 ∼ 7Num. of Total Observation ¡±² ±²^»^Þ + ¡¤¥¦ = 20Num. of Random Channel ¡¢º¥»¤¥¦ = 1000Num. of Testing Hyperparameter ¡ ±²test = 10000Table II: Setup of the BO-based hyperparameter tuning Figures 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, thenumber of grid points for searching is set to be = 20, and the same outer-parameters are used in all compared approaches, i.e., ^(g^) = ^, ∀eý.In the grid-search, the averaged MSE performance as determined per equation (30)by parameter ^ is evaluated for 20 grid points uniformly taken in the range &10hî, 1(,with ¡m¢ºa¥x»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 to ^(g^) ∈ &10hî, 1(, ∀e^ (53)Internal 202403565 -30-and^ ∈ &10hî, 1(, â# ∈ &10hî, 1(, â5 ∈ &10hî, 1( (54)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 normalised MSE 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 BO-based hyperparameter tuning and Adam-based acceleration in accordance with the invention 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 further contributes 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 is evaluated hereafter. Since a reasonable design of the maximum number of iterations 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, Internal 202403565 -31- pp.95–110. The accuracy of the solution by method 300 can be evaluated by the performance loss ratio Q‾(I) ?&Q(7 ← method 300; I)( =?&Q(7 ← solution of eq. (24); I)( . (55)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. It is assumed that about 1% MSE performance loss is acceptable for the lowercomplexity design so that a number of inner-loop iterations ¡§¨¨¤¥¦ = 60 is adopted forfurther comparisons. In 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 × Internal 202403565 -32- 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-based approach, 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 presentinvention a method of determining operating parameters / ⋆ ⋆!, 7 of respective precoderstages of a plurality of transmitters and a combiner stage of a receiver, respectively,configured for over-the-air computing is presented. The operating parameters / ⋆ ⋆! , 7are targeted to minimise the average MSE between the target and reconstructed over-the-air computing functions. The method comprises determining an initial combiner 7(m), and iteratively refining the combiner 7(g^)by minimising the squareEuclidian vector norm of the combiner ∥ 7 ∥5, until a termination criterion is met,yielding an optimised combiner 7∗. The initial combiner may be based on an aggregated channel matrix H that comprises all channel vectors .!of all transmitters, Internal 202403565 -33- 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 a starting point of the present invention, iteratively refining comprises approximating, for all of the plurality of transmitters, 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 required 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. A PG function is applied for effectively solving the minimisation objective represented by the approximation. Based on the previously optimised combiner 7∗while ensuring that the square of the precoding scalars of each transmitter does not exceed a predetermined power value G, the method computes optimised precoder parameters / !⋆for all transmitters.The so-computed operating parameters / ⋆!, 7⋆ are provided to the transmitters andthe 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. In accordance with the invention a hyperparameter tuning and / or a hyperparameter reduction are applied for further reducing the computational complexity. Hyperparameter tuning for the iterative refining step comprises employing a BO process that iteratively models the objective over-the-air computing function by a Gaussian process regression (GPR), selects hyperparameters based on an acquisition function that measures the impact of a given input on the performance, and evaluates the performance at each iteration, until a termination criterion is met. The BO process may be designed, e.g., in accordance with the method 200 described further above. Termination criteria may comprise a predetermined number of iterations, an expected performance gain falling below a threshold, or the like. Determining the initial combiner 7(m)may comprise determining the maximum eigenvector of the product of the aggregated channel matrix 9 and its conjugate transposition 9H. Internal 202403565 -34- In one or more embodiments of the method the step of hyperparameter reduction for the iterative refining step comprises applying 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 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 / !⋆, 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 7∗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 7∗, and Internal 202403565 -35- - 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 or an optimised combiner 7∗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 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 / !⋆, 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 7∗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 7∗, 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. Internal 202403565 -36- 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 Internal 202403565 -37- 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. Internal 202403565 -38- 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 the reverse 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 refinement process can be used along with sophisticated RQ-based initialization and is based on BO-based hyperparameter tuning, which can be further enhanced by Adam-based acceleration. The novel refinement is enabled by a CCP-based reformulation of the average MSE minimization problem, which can be solved at low complexity through a PG process. 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 DRAWING Internal 202403565 -39- In the following section the invention will be described with reference to the drawings, in which Fig.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 iteration Fig.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, Internal 202403565 -40- Fig.15 shows an exemplary block diagram of a receiver in accordance with the invention, and Fig.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 EMBODIMENTS Figures 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 instructions which, 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 7∗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 / !⋆, 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: Internal 202403565 -41- - receiving an optimised combiner 7∗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 7∗, 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 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 / !⋆, 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 7∗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 7∗, 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. Internal 202403565 -42- LIST OF REFERENCE NUMERALS (PART OF THE DESCRIPTION) 100 method 320 calculate initialisation point 101 iterative refinement 330 compute ^!and O!110 receive inputs 340 initialise moments m1 and m2 120 calculate initialisation point 350 update first and second 130 compute ^!and O!moments m1 and m2 140 update 360 inner-loop termination criterion 150 inner-loop termination criterion met? met? 370 replace previous initialisation 160 replace previous initialisation point point 380 outer-loop termination criterion 170 outer-loop termination criterion met? met? 390 output optimal combiner 7⋆180 output optimised combiner 400 compute optimal precoders / !⋆190 compute optimal precoder 410 output optimal precoders / !⋆195 output optimal precoder 500 transmitter 200 method 550 pre-processing 210 sampling initial parameters 560 precoding 220 evaluating initial MSE 600 receiver performance 650 combiner design 230 initialising dataset 660 combiner 240 modelling function 670 processing 250 select input test set 700 method 260 compute average MSE and 710 receive optimised parameters mean variance and data 270 select hyperparameters 720 pre-processing 280 determine MSE performance & 730 precoding update current iteration dataset 740 transmitting 290 termination criterion met? 750 receive optimised parameters 295 determine and output optimised and signals hyperparameters 760 combining 300 method 770 outputting 310 receive inputs Internal

Claims

202403565 -43- CLAIMS1. Method (100) of determining operating parameters ( / ⋆ ⋆!, 7 ) of respectiveprecoder 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 ( / ⋆ ⋆!, 7 ) being targeted to minimise the average meansquare error (MSE) between the target and reconstructed over-the-air computing functions, the method comprising: - determining (130) an initial combiner (7(m)), - iteratively refining (101) the combiner (7(g^)) by minimising the square Euclidian vector norm of the combiner (∥ 7 ∥5) while maintaining, at eachiteration, 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 (7∗), iteratively refining (101) comprising approximating, for all of the plurality of transmitters (500), the respective constraint 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, - computing (190) optimised precoder parameters ( / !⋆) for all transmitters (500), based on the previously optimised combiner (7∗) while ensuring that the square (| / !|5) of the precoding scalars of each transmitter (500) does not exceed a predetermined power value (G), wherein the method further comprises: - employing a Bayesian optimisation (BO) process comprising iteratively modelling the objective over-the-air computing function by a Gaussian process regression (GPR), selecting hyperparameters based on an acquisition function that measures the impact of a given input on the performance, and evaluating the performance at each iteration, until a termination criterion is met, and / or - employing a process for reducing the number of hyperparameters. Internal202403565 -44- 2. The method of claim 1, wherein the process (300) for reducing the number of hyperparameters comprises applying the Adam scheme.

3. 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 ( / !⋆) determined in accordance with the method (100) of claim 1 or 2, 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 ( / !⋆), and - transmitting (740) the precoded pre-processed data via an antenna (502) of the transmitter (500).

4. 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 (7∗) determined in accordance with the method (100) of claim 1 or 2, and signals representing data transmitted from two or more transmitters (500), - combining (760) the received signals representing data using the received optimised combiner (7∗), 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). Internal202403565 -45- 5. The wireless transmitter (500) of claim 3, or the wireless receiver of claim 4, wherein receiving (710; 750) optimised precoder parameters ( / !⋆) and / or an optimised combiner (7∗) 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 or 2.

6. 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 ( / !⋆) determined in accordance with the method (100) of claim 1 or 2, 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 ( / !⋆), 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 (7∗) determined in accordance with the method (100) of claim 1 or 2, and signals representing data transmitted from two or more transmitters (500), - combining (760) the received signals representing data using the received optimised combiner (7∗), 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).

7. Wireless communication system comprising two or more transmitters (500) in accordance with claim 3 and at least one receiver (600) in accordance with claim 4, wherein the system is configured for implementing the method of claim 6. Internal202403565 -46- 8. 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 2, or - when executed by a microprocessor of a wireless transmitter (500) according to claim 6, 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 ( / !⋆) determined in accordance with the method (100) of claim 1 or 2, 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 ( / !⋆), 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 4, 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 (7∗) determined in accordance with the method (100) of claim 1 or 2, and signals representing data transmitted from two or more transmitters (500), - combining (760) the received signals representing data using the received optimised combiner (7∗), 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).

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

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