METHOD FOR DETERMINING OPERATING PARAMETERS OF BEAM FORGERS FOR OVER-THE-AIR COMPUTING

Abstract

A method (100) for determining operating parameters (vk*, u*) a pre-encoder phase and a combiner phase of a transmitter (500) or a receiver (600) configured for over-the-air computing, includes determining (130) a first combiner (u (0) ) based on an aggregated channel matrix (H) containing all channel vectors (h k ) of all transmitters (500)). The procedure further includes the iterative refinement (101) of the combiner (u (i i) by minimizing the square of the Euclidean vector norm of the combiner (||u|| 2 ), until a termination criterion is met, resulting in an optimized combiner (u*). Based on the previously optimized combiner (u*), optimized precoder parameters are calculated. (vk * ) calculated for all transmitters (500) (190), while ensuring that the square (|v k | 2) the precoding scalar of each transmitter (500) does not exceed a predetermined power value (P).

Description

AREA OF INVENTION

[0001] The present invention relates to wireless communication, in particular over-the-air computing, in systems comprising multiple edge devices (ED - edge devices) and at least one base station (BS - base station). notation

[0002] The following notation is used throughout the description. Real vectors and matrices are represented in bold lowercase and bold uppercase letters, respectively, as in v and V, while their complex counterparts are represented in bold italic lowercase and bold italic uppercase letters, respectively, as in v and V. An N × N identity matrix and an N × 1 vector with one column consisting only of ones are denoted by I. N or 1 N The vector norm and the absolute value of a scalar are denoted by ||·|| and |·|, respectively. Transposition and its conjoined transposition are denoted by ▪ T or ▪ Hdepicted, whereby ℜ{⋅},ℑ{⋅} and min(·) denote the real part, the imaginary part, and minimum operators. Finally, N(µ,Σ) and XN(µ,Σ) denote the Gaussian normal distribution and the multidimensional Gaussian distribution with complex numbers, respectively, with mean µ and covariance matrix Σ. BACKGROUND

[0003] With the increasing acceptance of Internet of Things (IoT) applications in smart homes and industrial environments, telemedicine and autonomous driving, the coordination of communication between decentralized IoT devices and the aggregation of their data is becoming increasingly important, especially to meet high requirements such as low latency, low power consumption, efficient spectrum use, reliability and scalability, which are essential for certain applications and use cases.

[0004] With the aim of performing communication and computing tasks simultaneously, over-the-air computing (AirComp) emerged as a promising technique for data aggregation in IoT systems as an alternative to conventional sequential communication and processing, which also forms the basis for a paradigm shift towards decentralized computing via wireless systems. Fig. Figure 1 shows a simplified representation of an AirComp system. In the system, K EDs are wirelessly connected to an AP. The individual wireless signals interfere with each other on their paths from the EDs to the AP, and this interference is used for computation, as discussed below. The basic principle of AirComp is to exploit the

[0005] AirComp is a waveform superposition property of a wireless channel used to implement over-the-air aggregation of data simultaneously transmitted by devices. Simultaneous transmission by multiple synchronized devices, combined with the analog waveform superposition of such a multi-access channel, results in the simultaneous addition of the simultaneously transmitted signals over-the-air. The added signals arrive at the receiver as a weighted sum, also known as the "aggregated signal," where the weights are the channel coefficients. AirComp relies on linear analog modulation and channel pre-compensation at each transmitter. The former modulates the data values ​​into the strengths of the carrier signals; the latter compensates for heterogeneous channel attenuation across different links. As a result, each component of a received signal consists of the transmitted data scaled by a predefined factor.A uniform setting of the factor for all signals, referred to as size adjustment, reduces the aggregated signal to the desired average of transmitted decentralized data, thereby implementing an averaging function through AirComp.

[0006] With appropriate data pre- and post-processing, AirComp's capabilities can extend beyond averaging to the computation of a class of so-called nomographic functions, which can generally be expressed as a post-processed summation of several pre-processed data values. Typical functions in this class include the arithmetic mean, the weighted sum, the geometric mean, the polynomial, and the Euclidean norm. For example, to compute the geometric mean, the pre-processing is a logarithmic function and the post-processing is an exponential function. It has been proven that any function can be decomposed into a summation form of nomographic functions, indicating that any function can generally be computed using AirComp.

[0007] Fig.Figure 2 shows an exemplary simplified block diagram of an AirComp transmitter 500 and an AirComp receiver 600. The transmitter 500 receives data to be transmitted at the preprocessing block 550, which processes the data for over-the-air computing. The processed data is then provided to the precoder 560, which precodes the data for transmission over the wireless channel, indicated by the dashed arrow. The receiver 600 provides the received signal to a combiner block 660, which is configured for over-the-air computing. The combiner block 660 receives the required combiner design from a design block 650. The combiner block 660, which performs the actual over-the-air computing and implements the objective function, provides the output of the objective function to the processing block 670 for further processing.It should be noted that the conventional functional blocks of a wireless transmitter and receiver have been omitted from the drawing.

[0008] While AirComp's simultaneous transmission allows each device to access all radio resources, instead of just a fraction as with conventional orthogonal multiple-access schemes, thus enabling high spectrum efficiency, AirComp's most promising feature for wireless data acquisition and other applications is its ability to directly compute nomographic functions by multiplexing multiple data streams over the wireless multiple-access channel. This parallel processing enables energy-efficient, low-complexity data aggregation with AirComp systems.

[0009] However, AirComp is still in its infancy, with the main part of the work on the subject focused on the fundamental problem of mitigating distortions in the calculation caused by the random nature of the wireless channel. To cite just a few examples, the computationally optimal transmitter (TX)-receiver (RX) scaling policy for a multi-point single-input, single-output (SISO) AirComp environment, 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 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, p.2150-2154, 2021, generalized to multi-frequency AirComp schemes.

[0010] A study similar to the one mentioned above, but which takes into account channels affected by attenuation and the case of single input multiple output (SIMO) where the AP has multiple antennas, can also be found in the literature; it shows that the performance of SIMO-AirComp is highly dependent on the precoders and combiners, whose optimal design is therefore not insignificant.

[0011] With such cases in mind, a paired uniform-forcing precoder-combiner design scheme for minimizing the mean square error (MSE) in AirComp 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. In this scheme, the precoder is obtained via a Lagrange method for a given combiner, while the combiner is first initialized using a semi-definite relaxation (SDR)-based MSE-minimizing algorithm and then iteratively refined using a successive convex approximation (SCA). Building on the latter, W. Fang, Y. Zou, H. Zhu, Y. Shi and Y. recently published a study.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, showed that the optimal combiner for a uniformity-enforcing precoder can be obtained via the BnB (Branch-and-Bound) method, but at the cost of a considerable increase in computational complexity, which increases further with the number of EDs in the system. As an extension to the above, two types of initialization, one with much lower complexity but with performance loss, and the other with better performance but with higher complexity, are proposed by K. Ando and GTF de Abreu in “Low-Complexity and High-Performance Combiners for Over the Air Computing” in IEEE 9th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2023, pp. 126-130.Nevertheless, even with conventional initialization with low complexity, the overall level of complexity is still high due to the subsequent refinement via SCA, which probably hinders the widespread use of AirComp in inexpensive IoT applications.

[0012] It is therefore desirable to provide a combiner design phase for AirComp systems, e.g., multi-point SIMO systems, that enables both low complexity and high performance and accuracy. BRIEF SUMMARY OF THE INVENTION

[0013] This problem is solved 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 further developments are described in the respective dependent claims.

[0014] The combiner design method according to the present invention comprises an initialization phase and a subsequent refinement phase. The exemplary initialization phase used here for comparison purposes aims to provide low complexity but better performance than conventional approaches. This is achieved by reformulating the MSE minimization problem into a canonical Rayleigh quotient (RQ) problem. In particular, unlike the load-sensitive approximation discussed in "Low-Complexity and High-Performance Combiners for Over the Air Computing" (full citation above), the computationally intensive min operator of the conventional combiner design method is avoided by using the less expensive multi-access channel averaging.According to the present invention, the refinement phase relaxes the computationally intensive MSE minimization by the convex-concave procedure (CCP) and applies a proximal gradient (PG) method to obtain a low-complexity solution for minimization.

[0015] Before describing the invention, the underlying problem that arises in the design of pre-encoders for AirComp is described. An AirComp scheme is assumed to be based on a multi-point SIMO communication system consisting of K single-antenna EDs and an AP equipped with N antennas. The AirComp objective function f is given by f(s)=∑k=1Ksk where s≜[s1,⋯,sK]T a vector that contains all symbols s k ~ N(0,1), which are transmitted by the EDs.

[0016] With perfect synchronization among all EDs, the received signal y ∈ ℂ N×1 , which is subject to fading and noise, indicated by y=∑k=1Khkvksk+n where h k ∈ ℂ N×1 denotes the channel vector between the k-th ED and the AP, v k ∈ ℂ is the precoding scalar of the k-th ED and n ∈ ℂ N×1 ~ XN(0,σ 2 I N ) the additive white Gaussian noise (AWGN - additive white Gaussian noise) at the AP is referred to.

[0017] Possessing the received signal y, an estimated objective function is constructed at the AP via the combiner. f^(u,v;H|s)=uHy=uH(∑k=1Khkvksk+n), where v≜[v1,⋯,vK]T a vector that includes all precoding scalars used by the EDs v k , contains, u ∈ ℂ N×1 denoted the combiner vector applied by the AP and H≜[h1,…,hK]∈ℂN×K The channel matrix of all EDs to the AP is.

[0018] The notation in equation (3) is intended to emphasize that, from the perspective of the AP, f̂ is a function of the combination and precoding vectors u and v, which are the variables that can be optimized, while the known channel matrix H is a fixed parameter and the unknown vector of transmitted symbols s is a condition variable.

[0019] Based on the above, the MSE between the target function and the reconstructed function, averaged over transmissions of several different symbol vectors s and under the constant channel H, the pre-coders v and the combiners u, can be conclusively expressed as follows. ε(u,v;H)≜E[|f−f^|2]=∑k=1K|1−uHhkvk|2+σ2uHu, for the sake of simplicity of notation, the arguments of the functions f and f̂ have been omitted.

[0020] As discussed in “A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation” (full quote above), for a given, i.e., a fixed, combination vector u, the optimal precoding scalars are vk*, which the EDs must apply to minimize the average MSE, while the performance constraint |v k | 2 ≤ P is satisfied by the following vk*=η⋅hkHu|hkHu|2, η=P⋅minhk(|hkHu|2), where η denotes a scaling coefficient and P denotes the maximum transfer power constraint, both of which are common to all EDs.

[0021] From equations (3) and (5) it can be f^*(u;v*,H|s)≜f^(u,v*;H|s)η=f(u,v*;H|s)+uHnη to be obtained, again emphasizing that the notation is meant to remind us that the estimated objective function f̂*, which has been partially optimized by substituting the precoders v*, is still a function of the combiner vector u, which can also be further optimized.

[0022] One might think that this would simply require minimizing the average MSE between f and f*, namely δ(u;η)≜E[|f−f^*|2]=σ2‖u‖2η.

[0023] However, it is noted that minimizing the average MSE δ(u;η) with respect to u is not insignificant, since the parameter η, as shown by equation (5b), is a quantity that is itself dependent on u, coupled to v*, and also depends on the channel matrix H and the power constraint P. In other words, an MSE-optimized combiner design must simultaneously minimize equation (7) and satisfy equation (5b) if H and P are given.

[0024] In light of the aforementioned challenge, the next section will discuss the best practical approach currently known for a combiner design.

[0025] A simple formulation of the optimization problem, which combines the goal set by equation (7) and the condition imposed by equation (5b), is: minimizeu∈ℂN×1σ2‖u‖2P⋅minhk(|hkHu|2), which, although difficult to solve directly due to the presence of the min operation in the denominator, can be reformulated into the standard constrained minimisation problem. minimizeu∈ℂN×1‖u‖2 subject to ‖hkHu‖2≥1,∀k, where the constant σ 2 / P was ignored and it is pointed out that the min operator in equation (8) is relaxed into the K constraints described by inequality (9b).

[0026] Although the problem represented in equation (9) is still NP-hard due to the non-convex constraints (9b), it can be solved via an iterative refinement of the combiner based on successive convex approximation (SCA).

[0027] The SCA method is only briefly described below and summarized 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 quote above).

[0028] First, for the sake of simplicity, the projection is defined. C(u,h)≜[ℜ{uHh}ℑ{uHh}]

[0029] The i-th iteration of the proposed SCA method is then reduced to calculating the following. ck(i−1)≜C(u(i−1),hk),∀k, and then the problem will be solved. minimizeu(i)∈ℂN×1‖u(i)‖2 subject to 2(ck(i−1))TC(u(i),hk)−‖ck(i−1)‖2≥1,∀k

[0030] As can be seen from equation (11), a first combiner u (0) required. A relatively accurate conventional approach designs the first combiner u (0) based on a semidefinite relaxation (SDR) of the problem represented in equation (9). There is a conventional RQ-based, albeit heuristic, alternative with low complexity. Both approaches are briefly described below.

[0031] The SDR version of the problem represented in equation (9) is given by minimizeU∈ℂN×N Tr(U) subject to Tr(UHk)≥1,∀k U≽0 where U≜uuH and Hk≜hkhkH.

[0032] 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 efficiently calculated via principal component analysis (PCA). Therefore, u(0)=maxeigv(U) determined.

[0033] The computational complexity of the aforementioned conventional method for designing AirComp combiners is dominated by the SDR-based initialization scheme. This is because it can be inferred that an approach in which a positive semidefinite solution U ∈ ℂ is first obtained... N×N is received, only to immediately afterwards send it to the SCA initialization point u (0) ∈ ℂ N×1 Truncation is not very efficient.

[0034] Therefore, low-complexity alternatives to the SDR method are needed. It should first be noted that, as long as ||u|| 2Since ≠ 0, the optimization problem represented in equation (8) can be rewritten without loss of generality or any other disadvantage as follows. maximizeu∈ℂN×1P⋅minhk(|hkHu|2)σ2‖u‖2, where the numerator and denominator of the objective function of the problem represented in equation (8) are merely ‘reversed’, which is motivated by the similarity between the latter and the classical Rayleigh quotient (RQ), as will soon become clearer.

[0035] Of course, as in the problem presented in equation (8), the presence of the min operator in the goal of the problem presented in equation (14) is problematic. However, unlike the approach taken in “A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation” (full quote above), where such an operator was relaxed into the set of constraints, as shown in equation (9b), the load-sensitive approximation was taken into account as follows. minhk(|hkHu|)≈|h¯Hu|, where h¯≜{minhk(|hk|)if N≤K,1K∑k=1Khkif N>K.

[0036] By substituting the approximation shown in equation (15a) into equation (14) and ignoring the constant P / σ 2 The standard rank one RQ problem can be obtained as follows. maximizeu∈ℂN×1|h¯Hu|2‖u‖2 whose solution is simply given by the only non-zero eigenvector of hh H is given, which in turn is merely the normalized vector h, that is u(0)=maxeigv(h¯h¯H)=h¯‖h¯‖2

[0037] The performance of conventional iterative refinement-based design depends heavily on the combiner's performance. Consequently, compared to the highly complex conventional alternative using SDR, a high price is paid in exchange for the linear-order complexity of conventional initialization procedures in terms of the degradation of the converged solution's performance.

[0038] An improved starting point with less complexity than the conventional approaches can be obtained by using yet another alternative solution to the optimization problem presented in equation (14). Starting with the load-sensitive approximation shown in equation (15b), and considering that all ED channel vectors should contribute to the design of the first combiner, a design based on a variant of the problem shown in equation (16) is considered: maximizeu∈ℂN×1‖HHu‖2‖u‖2, that the aggregated channel matrix H≜[h1,…,hK]∈ℂN×K used in place of individual channel matrices.

[0039] Since the optimization problem represented in equation (18a) is a simple RQ problem, the first combiner can be determined by the maximum eigenvector of HH. H are designed as u(0)=maxeigv(HHH), which in turn can be efficiently calculated using the well-known power method, as discussed by GH Golub and CFV Loan, “Matrix Computations”, Johns Hopkins University Press, 1996.

[0040] Using the optimized first combiner, the refined combiner can be obtained by iteratively refining the minimization problem represented in equation (9). Finally, the precoder parameters can be computed based on the refined combiner. However, the computational complexity of conventional methods for solving the minimization problem, such as the SCA method discussed above, may still be too high for widespread use, and some reduction in complexity is necessary while providing at least the same MSE performance as conventional methods.

[0041] According to the present invention, the complexity of solving the optimization problem represented in equation (9) is reduced by reformulating the equation as a difference-of-convex (DC) problem and solving the DC problem using the CCP. The CCP is a heuristic method for finding local optimal solutions to DC programming problems. Using the CCP method may seem illogical, since conventional solutions to the CCP involve applying interior-point methods, which are computationally complex. However, the present invention proposes a solution that avoids the computationally intensive interior-point methods.

[0042] A general DC programming problem in the realm of real numbers is assumed, given by minimizex∈ℝN×1 h0(x)−g0(x) subject to hm(x)−gm(x)≤0, m=1,…,M assuming convex functions in the domain of real numbers h m : ℝ N×1 → ℝ 1×1 and g m : ℝ N×1 → ℝ 1×1 for m = 0,...,M.

[0043] According to the CCP, the solution of DC problems can be obtained approximately by iteratively solving an alternative convexified optimization problem. In particular, the convexified optimization problem is used to obtain the solution of i c + 1st iteration with the previous solution x (ic) given by minimizex∈ℝN×1 h0(x)−g^0(x,x(ic)) subject to hm(x)−g^m(x,x(ic))≤0, m=1,…,M, where ĝ m (x,x (ic) ) an approximation of the function g m (x) from equation (20) is and is defined as g^m(x,x(ic))≜gm(x(ic))+∇gm(x(ic))T⋅(x−x(ic)).

[0044] From the optimization problem presented in equations (9) and (20), it is evident that the MSE minimization problem can be viewed as a DC problem. In particular, setting h0(x) → ||u|| suffices. 2 with g0(x) → 0 as well as gm(x)→‖hkHu‖2−1 with h m (x) → 0 to relate equation (9) to equation (20). Then, taking into account that the desired CCP lies in the domain of complex numbers, the corresponding reformulated MSE minimization problem is used to obtain the solution in i. c + 1 CCP iteration to minimizeu∈ℂN×1 ‖u‖2 subject to 1−g˜k(u,u(ic))≤0,∀k, where the domain of complex numbers g̃ k : ℂ N×1 → ℝ 1×1 as follows g˜k(u,u(ic))=‖hkHu(ic)‖2+2ℜ{(hkhkHu(ic))H⋅(u−u(ic))}=2ℜ{(hkhkHu(ic))Hu}−‖hkHu(ic)‖2

[0045] For the sake of later simplicity, the optimization (22) can be rewritten as follows. minimizeu∈ℝN×1 ‖u‖2 subject to akHu+uHak≤bk,∀k, where ak≜−hkhkHu(ic),∀k bk≜−(1+‖hkHu(ic)‖2),∀k.

[0046] The optimization shown in equation (24) is obviously convex with a quadratic objective function and a half-space constraint. According to the present invention, the optimal solution can be obtained with low computational complexity via a proximal gradient (PG) descent method. By applying the PG to the i c In the --th iteration of the optimization problem, the solution can be iteratively updated using the gradient of the objective function and the proximal operator for the constraint. Thus, the i i + The first update of the PG steps can be expressed as follows. u(ii+1)=prox(u(ii)−α(ic)u(ii)), where the proximal operator for the half-space constraint over all different k is given by prox(u)≜u−max(0,akHu+uHak−bk)‖ak‖2⋅ak,∀k, is given, where i i and α (ic) the iteration index of the PG iteration or the step size of the gradient in the i c denotes the -th iteration.

[0047] In summary, a method for designing the uniform-forcing (UF) precoder and combiner using the iterative refinement according to the invention can be described as shown below and as exemplified in the flowchart of Fig.Figure 3 summarizes the procedure. The method features a nested loop structure, the outer loop of which is used for the CCP method, and the inner loop of which employs the PG method to solve the approximate minimization with low computational complexity. Since the termination criterion is exemplified by the maximum number of iterations, the outer and inner loops can be configured according to... ImaxCCP or ImaxINN Iterations are terminated. It is noted that other termination criteria can be used, e.g., convergence of the eigenvector values ​​or the MSE below a threshold, where the improvement of the target value from one iteration step to the next is below a threshold, or the like.

[0048] The inputs to procedure 100 are the channel matrix H ∈ ℂ N×Kof all EDs to the AP, the number of iterations for the outer loop and the inner loop ImaxCCP and ImaxINN and the step size α (ic) , ∀i c of the gradient for each iteration. The outputs of the procedure are the optimal ones. Precoding scalarevk*∈ℂ∀k and the optimal combination vector u* ∈ ℂ N×1 After receiving the inputs at step 110, the initialization point u is reached at step 120. (0) calculated using equation (19). The iterative refinement to obtain the optimized output values, which includes the steps within box 101 with the dashed lines, is then performed in the nested iteration loops. Starting with step 130 in the outer iteration loop, a k and b k via equation (24) using u (0)calculated as the starting point for optimization. The inner iteration loop begins at step 140, where u (ii) The value is updated via equation (25). At step 150, it is checked whether a termination criterion is met for the inner iteration loop. As mentioned above, the termination criterion can be the reaching of a predefined value. Number of ImaxINN The termination criteria include iterations or convergence to a stable value, or similar conditions. Termination upon convergence can be triggered, for example, when the difference or gradient in the output values ​​between a predetermined number of consecutive iterations falls below a specified value. Other convergence criteria are also conceivable. If the termination criterion is not met (the "No" branch of step 150), the procedure returns to step 140. Otherwise (the "Yes" branch of step 150), the output at step 160 is replaced with u. (ii)the inner loop the previous initialization point, e.g. u (0) For the next execution of the inner loop, the procedure checks at step 170 whether a termination criterion for the outer loop is met. If negative (“No” branch at step 170), the procedure returns to step 130 and executes the inner loop again. Otherwise (“Yes” branch at step 170), the last output of the inner loop is output as the optimal combiner u* at step 180, and based on this, the optimal precoders are determined at step 190. vk* for all k EDs calculated via equation (5) and using u* and at step 195 as vk* all k EDs issued.

[0049] While the exemplary procedure described above uses an initialization that differs from the conventional, high-performance, highly complex SDR method and the initially described, low-complexity, lower-performance RQ method, it is possible to use any suitable initialization. This is due to the separation of the refinement process from the initialization, which provides greater freedom in the design of the complete process.

[0050] The following section discusses a second variant of the novel approach concerning the acceleration of iteration convergence, which is used below for comparison purposes.

[0051] This variant belongs to a BO (Bayesian optimization)-based hyperparameter tuning for the proximal gradient (PG - proximal gradient) method to find the optimal combiner u*.

[0052] As shown above for the exemplary CCP- and PG-based method 100, it is possible to design the AirComp precoder and combiner based solely on mathematical operations of low complexity. Since, in the case of the PG, the convergence property and the performance of the converged solution depend heavily on the value of hyperparameters, i.e., the step size α(1),…,α(ImaxCCP), To support this, all hyperparameters must be carefully determined. Classically, hyperparameter tuning can be performed by a complete grid search over all possible combinations of parameters. However, the complexity of a grid search is comparatively high, not only because the algorithm must be executed to evaluate each hyperparameter combination, but also because the number of possible hyperparameter combinations increases with the number of iterations of the outer loop. ImaxCCP. To achieve meaningful hyperparameter matching with limited search effort, a BO-based offline hyperparameter matching is proposed.

[0053] In AirComp, as can be seen from equation (7) and the above-presented procedure 100 for designing the UF precoder and combiner, hyperparameter tuning can be viewed as the following MSE minimization problem. minimizeq δ(u;η) subject to u←method 100(H,q) q=[α(1),…,α(ImaxCCP)] where q∈ℝ1×ImaxCCP denotes the vector of the collected hyperparameters.

[0054] Unfortunately, this problem cannot be solved analytically, and, as mentioned previously, a grid-based full search is computationally too expensive for practical use. To tune the hyperparameters with a reasonable level of complexity, a BO-based approach is proposed. BO is a black-box optimization procedure that efficiently searches for the optimal hyperparameters based on the evaluation results and Gaussian process regression (GPR). In particular, this variant discusses using BO to model the objective function δ(u;η) as a Gaussian process (GP), such that, after an initial dataset has been created, this procedure iterates as follows. (i) Modeling the objective function by GPR, (ii) Hyperparameter selection based on the acquisition function and (iii) Performance evaluation.

[0055] In this process, before the actual modeling of the objective function, the data set to be used in the BO procedure must be initialized. This is done by selecting the first data set for parameter search. IinitBO The hyperparameter vectors were created by random sampling from the feasible range, resulting in the following set. Qinit=∪i=1IinitBO{q(i)}, where q(i)∈[0,1]1×ImaxCCP.

[0056] Next, the value of the objective function, i.e., an observation, is determined using the procedure described above and the inputs created above. Qinite calculated.

[0057] To find a robust hyperparameter set opposite the statistically random channel matrix H, this step involves averaging. ImaxChan Consideration of observations on random channel conversions.

[0058] For an input q (i) With the j-th channel conversion, their MSE performance can be evaluated by θj(q(i))=δ(u←method 100(H(j),q(i));η).

[0059] After the MSE values ​​for ImaxChan Based on evaluations of various channel implementations, the average MSE can be obtained by θ(q(i))=1ImaxChan∑j=1ImaxChanθj(q(i)).

[0060] For further user-friendliness, the chained averaged MSE vector of all hyperparameters can be displayed in a set Q with cardinality |Q|=IQ as follows θ(Q)=[θ(q˜(1)),…,θ(q˜(IQ))]T, Q=∪i=1IQ{q˜(i)}, where q˜(i)∈[0,1]1×ImaxCCP The i-th hyperparameters are designated.

[0061] Then the first data set can be created as follows. DBO(0)=∪i=1IinitBO{[q(i),θ(q(i))]}

[0062] Now the actual function modeling based on GPR can be performed. In this step, the average MSE for unknown inputs is modeled based on the current data set and GPR, using a non-parametric regression procedure that assumes the modeling target follows a GP. It is assumed that parameter matching via BO with a total of ImaxBO Iterations are performed. At the i B In the -th iteration, the task at this step is to estimate the probability distribution of the averaged MSE vector. θ(Qtest(iB)) through a given set of test hyperparameters Qtest(iB)=∪i=1ItestBO{q¯i(iB)} where q¯i(iB)∈[0,1]1×ImaxCCP Test hyperparameters obtained by sampling are as in equation (28).

[0063] At the i BThe -th attempt will determine the joint distribution of the averaged MSE in the current data set. DBO(iB−1) and the expected observation, which corresponds to the given test hyperparameter set Qtest(iB) corresponds to the following multidimensional normal distribution [θ(Q(iB−1))θ(Qtest(iB))]∼N(μ,ϒ), where μ=[μ(Q(iB−1))μ(Qtest(iB))], ϒ=[∑Ψ(Q(iB−1),Qtest(iB))Ψ(Q(iB−1),Qtest(iB))TΨ(Qtest(iB),Qtest(iB))], where the covariance matrix Y, the matrix Σ can be given as follows ∑=Ψ(Q(iB−1),Q(iB−1))+σε2IIQ(iB−1) where σε2 denotes the variance for the observational noise, while the hyperparameter set for the past evaluation is given by Q(iB−1)=∪i=1IinitBO+iB−1{q(i)}.

[0064] Without loss of generality, the kernel matrix can be divided between hyperparameter sets of arbitrary cardinalities Q and Q̃, which consist of Q=∪i=1Q{qi∈[0,1]1×ImaxCCP}, and Q˜=∪j=1Q˜{q˜j∈[0,1]1×ImaxCCP} are defined as Ψ(Q,Q˜)∈ℝQ×Q˜.

[0065] Assuming automatic relevance determination (ARD), Matérn 5 / 2 Kernel, the (i,j)-th element of the kernel matrix, denoted by Ψ(Q,Q˜)(i,j) calculated as follows Ψ(Q,Q˜)(i,j)=γ02⋅(1+5⋅ri,j+5⋅ri,j23)⋅exp(−5⋅ri,j), where the distorted distance among the hyperparameters is given by ri,j=‖diag(γ1−2,…,γImaxCCP−2)⋅(qi−q˜j)‖2, where γ0,…,γImaxCCP trainable parameters.

[0066] Knowing the current amount of data DBO(iB−1) and the method of greatest plausibility (MLE - maximum likelihood estimation) can be used to determine the mean µ and the covariance matrix Ψ¯(Qtest(iB)) the conditional probability distribution for the desired maximum MSE can be derived as follows: μ¯(Qtest(iB))=μ(Qtest(iB))+Ψ(Q(iB−1),Qtest(iB))T∑−1θz(iB), Ψ¯(Qtest(iB))=Ψ(Qtest(iB),Qtest(iB))−Θz(iB)T∑−1Θz(iB), where θz(iB)=(θ(Q(iB−1))−μ(Q(iB−1))), Θz(iB)=Ψ(Q(iB−1),Qtest(iB)).

[0067] Since the previous mean of the averaged MSE was used to test the hyperparameters μ((Qtest(iB))) Since it cannot be observed, it is approximated by the arithmetic mean of the observed values, i.e. μ(Qtest(iB))≈∑i=1|Q(iB−1)|θ(q(i))|Q(iB−1)|1|Qtest(iB)|.

[0068] It is noted that the parameters γ0,...,γICCP and σ ∈, which are used in the calculation of equation (39), can be trained by maximizing the log-likelihood function. log p(θ(Q(iB−1))|σε,γ0,...,γImaxCCP)=−12log det(Σ)−|DBO(iB−1)|2log 2π−12(θz(iB))T⋅Σ−1⋅θz(iB).

[0069] This function can be maximized, for example, using gradient-based methods such as the quasi-Newton method or the Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm with limited memory.

[0070] The following section describes how to find 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 performance. When measured by the expected improvement (EI), as described, for example, by M. Balandat, B. Karrer, DR Jiang, S. Daulton, B. Letham, AG Wilson, and E. Bakshy in "BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization," Advances in Neural Information Processing Systems 33, 2020, [Online], available at: http: / / arxiv.org / abs / 1910.06403, the acquisition function for an input q can be expressed as follows. v(q)=E[max(−θ(q)+θmin(iB−1),0)], where θmin(iB−1) the minimum MSE in the current data set DBO(iB−1) designated.

[0071] After v(q) for all q∈Qtest(iB) Once the evaluation has been completed, the next hyperparameter can be determined as q(iB)=argmaxq∈Qtest(iB) v(q).

[0072] After the average MSE is calculated with the updated input, i.e. θ(q(iB)), The data set will be used to assess performance during the performance evaluation step. DBO(iB) Updated via DBO(iB)←DBO(iB−1)∪{[q(iB),θ(q(iB))]}.

[0073] Finally, after functional modeling, the steps of hyperparameter selection and performance evaluation are carried out. ImaxBO Male, how the tuned hyperparameters can be obtained, iterated through qopt=argmin θ(q).q∈Q(ImaxBO)

[0074] It is noted that the index `max` can be replaced accordingly if a termination criterion other than a maximum iteration count is used. Alternative or additional termination criteria may include, among others, the convergence of a suitable parameter or such a parameter reaching a predefined threshold, such as the expected improvement falling below a predetermined value.

[0075] The entire procedure of tuning the hyperparameters via BO can be shown below and exemplified in Fig. 4 are summarized.

[0076] The inputs to procedure 200 are the maximum iteration counts. ImaxINN,ImaxCCP,ImaxBO,ImaxChan, the first BO value IinitBO and the BO test result ItestBO. It is noted that further initialization values, e.g., for alternative or additional termination criteria, can be provided. The output of procedure 200 is the optimized version. qopt∈ℝImaxCCP×1.

[0077] An initialization phase of procedure 200 includes sampling a set of initial parameters. Qinite according to equation (27) at step 210, the evaluation of the first average MSE performance θ(q∈Qinit) according to equation (30) at step 220 and the initialization of the data set DBO(0) According to equation (32) at step 230. The actual iterative BO hyperparameter tuning begins at step 240 with the modeling of the function. Function modeling includes tuning the trainable parameters. γ0,…,γImaxCCP and the variance for the observational noise σ ∈by maximizing equation (42). Next, in step 250, the set of text inputs is Qtest(iB) selected according to equation (28). Now, in step 260, the mean can be calculated. μ¯(Qtest(iB)) and the covariance matrix Ψ¯(Qtest(iB)) The conditional probability distribution for the desired average MSE is calculated according to equation (39a) or (39b). Then, in step 270, the hyperparameters for the current iteration are selected, first calculating an acquisition function according to equation (43) and then... q(iB) is determined according to equation (44). In step 280, the average MSE power θ for the input vector is determined. q(iB) the collected hyperparameters of the current iteration are determined according to equation (30) and the data set DBO(iB) The current iteration is updated according to equation (45). Then, at step 290, the procedure checks whether a termination criterion has been met, e.g., whether a predetermined number of iterations have been performed. If the result is negative (the "No" branch of step 290), the procedure returns to step 240 and executes the next iteration. If the result is positive (the "Yes" branch of step 290), the procedure determines the optimally tuned hyperparameters according to equation (46) at step 295 and outputs them.

[0078] BO-based hyperparameter tuning of the hyperparameters discussed above provides an efficient way to observe well-designed hyperparameters with low evaluation overhead, resulting in a reduction or minimization of the expected MSE and thus faster convergence. However, the performance of BO-based searches can degrade with increasing dimension of the hyperparameter vector. Furthermore, the computational complexity of the PG-based scheme can be problematic, as its poor convergence behavior necessitates a high number of inner loop iterations to yield a good solution, even with well-designed hyperparameters.

[0079] A further development of this approach addresses this issue by applying convergence acceleration, compared to conventional methods, to obtain an exact solution with fewer hyperparameters and inner loop iterations. An exemplary suitable method, the application of which is proposed here, is adaptive moment estimation, also known as Adam, which is an algorithm for gradient-based optimization of stochastic objective functions. Details of Adam are provided, among others, 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.

[0080] Here, the first and second moments of the gradient of the objective function are used according to the Adam scheme, thereby reducing the number of hyperparameters and thus improving the convergence behavior. Adam, as applied here, also increases numerical stability. For this purpose, the i i -th iteration of the vector of the first moment m1 ∈ ℂ N×\1 as a moving average of the previous first moment and the gradient of the objective function u (ii) with exponential decay rates β 1= [0,1) calculated as m1(ii)=β1m1(ii−1)+(1−β1)⋅u(ii−1)(1−β1ii).

[0081] Similarly, for the i i -th iteration of the second moment m2 ∈ ℝ as a moving average of the previous second moment and the squared gradient ||u (ii-1) || 2 calculated using exponential decay rates. β2 = [0,1) m2(ii)=β2m2(ii−1)+(1−β2)⋅‖u(ii−1)‖2(1−β2ii).

[0082] By incorporating the two types of moments derived above, the i i -th update given in Adam as u(ii)=prox(u(ii−1)−α⋅m1(ii)m2(ii)+ε), where ε = 10 -6 refers to a small-value scaler that was introduced to avoid numerical problems.

[0083] Adam's accelerated PG descent for MSE minimization can be demonstrated as shown below and as exemplified in Fig. Figure 5 summarizes the process. In this method, the three hyperparameters, i.e., α, β1, and β2, are used over different outer loops, so that the number of hyperparameters remains at three even when the number of outer loops is changed.

[0084] The inputs to procedure 300 are the channel matrix H ∈ ℂ N×K , the maximum iteration counts ImaxINN and ImaxCCP The three hyperparameters α, β1, and β2 and the scaler ε. It is noted that further initialization values, e.g., for alternative or additional termination criteria, can be provided. The output of procedure 300 is the optimized combination vector u* ∈ ℂ N×1 and the optimized precoding scalars vk*∈ℂ,∀k.

[0085] After receiving the inputs at step 310, the initialization point u is reached at step 320. (0) calculated using equation (19). The refinement to obtain the optimized output values ​​is then carried out in nested iteration loops, starting with step 330 in the outer iteration loop, where a k and b k via equation (24) using u (0) The moments m1 and m2 are calculated as the starting point for the optimization. The calculation is carried out in step 340. m1(0)=0N×1 or m2(0)=0 initialized. The inner iteration loop, an Adam iteration, begins at step 350, where the first moment m1(ii) updated via equation (47a), the second moment m2(ii) is updated via equation (47b) and u (ii) The value is updated via equation (48). In step 360, it is checked whether a termination criterion is met for the inner iteration loop. The termination criterion can be the reaching of a predefined number. ImaxCCP The termination criteria may include iterations or convergence to a stable value, or similar conditions. Termination upon convergence can be triggered, for example, when the difference or gradient in the output values ​​between a predetermined number of consecutive iterations falls below a specified value. Other convergence criteria are also conceivable. If the termination criterion is not met (the "No" branch of step 360), the procedure returns to step 350 for the next iteration. Otherwise (the "Yes" branch of step 360), the last updated value replaces the previous one at step 370. (ii) the inner loop in the exemplary case u(ImaxINN) the previous initialization point, e.g. u (0)For the first iteration, the procedure checks at step 380 whether a termination criterion for the outer loop is met. If negative (“No” branch at step 380), the procedure returns to step 330 for the next iteration. Otherwise (“Yes” branch at step 380), the last output of the inner loop is output at step 390 as the optimal combiner u*, and based on this, the optimal precoders are determined at step 400. vk* for all k EDs calculated via equation (5) and using u* and at step 410 as vk* issued.

[0086] It is noted that three hyperparameters tuned using procedure 300 can also be tuned using procedure 200, thereby replacing procedure 100 with procedure 300 and increasing the hyperparameter dimension ImaxCCP is set to three, or rather... γ0,…,γImaxCCP is changed to γ0, ..., γ3.

[0087] In the following section, a precoder and a combiner design using the proposed low-complexity refinement are compared with the conventional alternatives presented in "A uniform-forcing transceiver design for over-the-air function computation" (full citation above) and by K. Ando and GTF de Abreu in "Low-complexity and high-performance combiners for over-the-air computing" (full citation above). The comparisons consider not only the MSE performance, as determined by equation (7), but also the class of computational complexity. For the sake of practicality, the initialization via equation (17) will be referred to as RQ, and the initialization via equation (19) will be referred to as MRQ.

[0088] Since all techniques to be compared use the same UF precoder design, it is sufficient to consider only the effort required to design the combiners, i.e., the complexity class.

[0089] First, the complexity of designing the combiner is assessed using the best available initialization and the novel, low-complexity refinement, as well as the use of the PG-based variant, as summarized in Procedure 100. For initialization according to equation (19), the complexity class is... O(KN2), based on the calculation of the covariance matrix HH H and their maximum eigenvector. If one considers the complexity of the inner loop per iteration over i iFocusing on the specifics, the required calculations only concern the gradient and the proximal operation as in equation (25). Since there is no calculation to obtain the gradient via equation (25a), the complexity of the gradient descent per iteration is O(N) for (1 - α) · u (ii) For the proximal operation in equation (26b) O(NK) Calculations required.

[0090] In the outer loop, indexed by i c , concern the additional calculations per iteration, i.e., excluding the calculations of the inner loop, the calculation of a k and b k according to equation (24). The complexity class for creating a k For all k according to equation (24c), O(KN) is the complexity class for calculating b. k According to equation (24d), the class is O(NK).

[0091] For the accelerated refinement using Adam, presented with reference to Procedure 300, the required computational complexity is increased by calculating the first and second moments according to equation (47). When calculating both the first and second moments, the additional complexity per inner loop is that of the class O(N). Thus, the additional complexity of a PG accelerated by Adam is negligible in terms of complexity class, compared to the complexity of the other algorithmic components already mentioned.

[0092] From the above, the total complexity class for the combiner design proposed in procedures 100 and 300 is O(KN2+ImaxCCP⋅(KN+ImaxINN⋅KN))

[0093] The following determines the complexity of the two conventional combiner designs mentioned above for comparison purposes, starting with the conventional combiner design with SDR initialization 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 above). Assuming that the PCA method according to equation (13), used to extract the largest eigenvector of the SDR solution U, is implemented based on the efficient power-method algorithm, as presented in "Matrix Computations" (full citation above), the computational complexity is that of the class O(N2). In the SDR optimization problem for obtaining the solution matrix U, as represented by equation (12), the computational complexity is that of the class O((K+N2)3.5). Thus, the complexity class for the first combiner of this conventional combiner design is O((K+N2)3.5).

[0094] The complexity of the refinement via an iterative SCA is governed by the effort required for the optimization problem, as shown in equation (11), the worst complexity of which is given by the number of iterations I opt , the number of scalar constraints K cst and the number of variables N var is determined. As shown by S. Boyd and L. Vandenberghe in “Convex Optimisation”, Cambridge University Press, March 2004, the number of iterations I is opt , which is required to solve a convex optimization problem, Iopt=⌈log(Kcstξ1)log(ξ4)⌉⋅(Kcst(ξ4−1−log(ξ4))ξ2+ξ3), where ξ1,...,ξ4 denote constants related to a numerical tolerance and a convergence rate of the solver.

[0095] As shown by C. Roos, T. Terlaky and JP Vial in “Interior point methods for linear optimisation”, 2nd ed. New York: Springer, 2006, the complexity class associated with solving a convex optimization with N can be increased with each iteration. var variables are linked to at least O(Nvar3) can be estimated. From equation (11), the number of constraints and the number of optimization variables are each given as Kcst=K, and Nvar=N, so that the estimated complexity class for solving the optimization, as represented by equation (11), O(KN3log(K)) amounts.

[0096] Since each loop of an SCA calculation, as shown in equation (11a), has a complexity class of O(KN), the total complexity class of the first conventional combiner design method under consideration here is estimated to be O((K+N2)3.5+ImaxSCA⋅(KN+KN3log(K))) where ImaxSCA refers to the number of SCA-based refinements.

[0097] In the second conventional combiner design, referred to here as RQ / SCA, the effort required to obtain the first combiner corresponds to u (0) those of calculating the load-sensitive approximation, as shown in equation (15b), and normalizing the latter according to equation (17), which has a complexity class of O(KN). Since the SCA-based refinement in the second conventional combiner design is the same as in the first conventional combiner design, the total complexity class can be estimated as follows. O((KN)+ImaxSCA⋅(KN+KN3log(K)))

[0098] Table I shows an overview of the complexity classes of the conventional methods and the variants of the proposed methods presented here. Table I: Computational Complexity of AirComp Combiners Proceedings Complexity class Conventional SDR / SCA O((K+N2)3.5+ImaxSCA⋅(KN+KN3log(K))) Conventional RQ / SCA O((KN)+ImaxSCA⋅(KN+KN3log(K))) Proposed procedure 100 PG Proposed procedure 300Adam O(KN2+ImaxCCP⋅(KN+ImaxINN⋅KN)) \

[0099] The following section provides a numerical assessment of the convergence behavior of the proposed methods with respect to MSE. The following system parameters are used for the computer simulation: N = 10, K = 5, 10, 15. ImaxCCP=10, ImaxINN=10.60. The signal-to-noise ratio (SNR) is defined as PK / σ2≜20[dB]. In the calculation of the two methods 100 and 300, each hyperparameter is adjusted according to method 200 by BO. An overview of the setup is shown in Table II. Table II: Structure of the BO-based hyperparameter tuning definition Function / Parameter Acquisition function Eq. for Expected Improvement (43) Kernel function ARD Matérn 5 / 2 Kernel Eq. (38) Number of first observations IinitBO=2∼7 Total number of observations IinitBO+ImaxBO=20 Number of random channels ImaxChan=1000 Number of test hyperparameters ItestBO=10000

[0100] The Fig. 6 and Fig.Figure 7 shows the convergence behavior of the mean MSE achieved by the proposed methods 100 and 300 as a function of the total iterations, where the hyperparameters are determined by BO using method 200. To highlight the performance improvement achieved by the BO-based hyperparameter tuning, the performance of method 100 with a grid-search-based hyperparameter tuning is also shown. For a fair comparison of a grid-search-based and a BO-based tuning, the number of grid points to search is shown. ImaxGrid=20 The parameters are set and the same parameters for the outer loop are used in all compared approaches, i.e. a (ic) = α, ∀i c .

[0101] During grid search, the average MSE performance, as determined by the parameter α according to equation (30), is calculated for 20 grid points uniformly located in the area [10-3 ,1] at ImaxChan Random channel conversions were taken and evaluated using the hyperparameters that achieve the lowest MSE as solutions. For hyperparameter tuning using methods 100 and 300, the permissible ranges of each hyperparameter are defined as follows: α(ic)∈[10−3,1],∀ic and α∈[10−3,1],β1∈[10−3,1],β2∈[10−3,1]

[0102] Fig.Figure 8 shows the effect of system load, specifically the number of EDs K compared to the number of antennas N, on MSE performance. In particular, it is evident that the MSE value increases with K for a given N, while the convergence of the proposed designs is unaffected by the load conditions. It is noted that the MSE values ​​are normalized by K, resulting in a normalized mean square error (NMSE). The normalized MSE results are the same at K=5 (unloaded) and K=10 (loaded), while the results at K=15 (overloaded) show a higher NMSE.

[0103] The performance gains achieved by the variants using BO-based hyperparameter tuning and Adam-based acceleration are clearly evident, confirming that BO enables effective hyperparameter design even with a limited total number of observations. Furthermore, Adam acceleration allows for a reduction in the number of hyperparameters, further reducing the computational complexity of BO-based tuning and enabling more accurate searches.

[0104] Having observed that MSE performance can be improved by BO and Adam, the following section presents a performance-complexity trade-off for the number of iterations of the inner loop. ImaxINN evaluated below.

[0105] Since a sensible design of the maximum number of iterations ImaxINN To achieve a sufficiently accurate solution with low complexity, the performance loss ratio with regard to a relative share of an averaged MSE is absolutely necessary. Fig. Figure 8 illustrates the convex optimization problem, as represented by equation (24), which can be solved using newer convex optimization 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[δ(u←method 300;η)]E[δ(u←solution of eq. (24);η)].

[0106] Fig. Figure 8 shows that the MSE performance loss is significant for a smaller number of inner loop iterations and can be improved by increasing the number of inner loop iterations. Specifically, it is found that the gap in MSE performance loss for different system load conditions decreases as the number of inner loop iterations increases. An MSE performance loss of approximately 1% is assumed to be acceptable for the lower complexity design, thus increasing the number of inner loop iterations. ImaxINN=60 can be used for further comparisons.

[0107] The following section compares the performance of the proposed methods with conventional approaches. The first comparison uses a median of the achievable MSE performance via random channel conversion with varying numbers of EDs and a corresponding quantitative complexity class, as described in the Fig. 9 and Fig. 10 illustrated.

[0108] Out of Fig. It is evident from section 9 that the refinement phase is fundamental for achieving good performance and that all three refinement procedures are equivalent with regard to the median MSE achieved.

[0109] Out of Fig.10 again shows 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 most powerful method among the high-performance alternatives with a lower complexity than the conventional techniques used for comparison.

[0110] In the quantitative assessment, which is described in the Fig. 9 and Fig. As shown in Figure 10, the median MSE performance for the proposed conventional high-complexity design and for the conventional low-complexity design for the case K=10 with a refinement of 5.05×10 -2 , 4.86×10 -2 or 5.13×10 -2 , while the complexity classes 6.2×10 4 , 1.5×10 7 or 2.6×10 5In other words, compared to the conventional, highly complex design, the proposed combiner offers a computational complexity reduced by approximately 230 times, while the loss in MSE performance is only a factor of 1.9×10. -3 On the other hand, compared to the conventional low-complexity design, the proposed combiner offers a computational complexity reduced by approximately 3.7 times, while providing an additional MSE performance gain of 7.8×10 -4 provides.

[0111] In summary, the following points were made: Fig. 9 and Fig. It can be concluded from section 10 that investing computing power during initialization, as in the case of the conventional SDR-based approach, is not as good a strategy as investing computing resources for a better refinement phase. This conclusion is supported by the findings in the Fig.The results shown in Figures 11 to 13 further substantiate this, comparing the cumulative distribution functions (CDFs) of the MSE achieved by conventional methods and the proposed methods under different system utilization conditions. If 90 percent of the MSE performance is taken as a reference value in the case of full utilization, then... Fig. 5, for example, shows that the proposed combiner design, the conventional high-complexity combiner design, and the conventional low-complexity combiner design, all with refinement (black markings), have MSEs of 6.37×10 -2 , 5.93×10 -2 or 6.38×10 -2 This results in a MSE performance gap between the proposed and the conventional design of only 4.4×10 -4 or 1.7×10 -3In other words, the proposed method essentially performs the same as conventional schemes with SCA-based refinement, although the latter techniques are more complex.

[0112] Based on the preceding description, a method for determining operating parameters is described according to a first aspect of the present invention. vk*,u* The respective pre-encoder phases of several transmitters or a combiner phase of a receiver, configured for over-the-air computing, are presented. The operating parameters vk*,u* are designed to minimize the average MSE between the objective function and the reconstructed over-the-air computing function. The procedure involves determining a first combiner u (0) and the iterative refinement of the combiner u (ii)by minimizing the square of the Euclidean vector norm of the combiner ||u|| 2 , until a termination criterion is satisfied, thereby obtaining an optimized combiner u*. The first combiner can be based on an aggregated channel matrix H containing all channel vectors h kThe minimization in the refinement phase is based on the constraint that, in each iteration, the Euclidean vector norm of the product of the rearranged channel matrix and the combiner of the respective iteration must be equal to or greater than 1 for each of the plurality of transmitters. According to the present invention, the iterative refinement comprises approximating the constraint based on the product of the derivative of the original constraint and its solution from the previous iteration. The resulting inequality, which represents the constraint, must not exceed the negative value of the Euclidean vector norm of the product of the rearranged channel matrix and the combiner of the respective iteration minus 1.Furthermore, the minimization goal represented by the approximation is solved according to the invention by applying a PG function. Based on the previously optimized combiner u* and while simultaneously ensuring that the square |v. k | 2 If the precoding scalar of each transmitter does not exceed a predetermined power value P, the procedure calculates optimized precoder parameters. vk* for all transmitters. The operating parameters calculated in this way vk*,u* are provided to the senders or the receiver to combine and pre-code pre-processed data to be transmitted in the senders, or received signals representing transmitted data from multiple senders, in the receiver.

[0113] Determining the first combiner u (0) can determine the maximum eigenvector of the product of the aggregated channel matrix H and its associated transposition HH include.

[0114] Hyperparameter tuning can be applied to further reduce the computational complexity of the iterative refinement step. Hyperparameter tuning can be performed, for example, according to the procedure described in section 200.

[0115] Hyperparameter reduction can be applied to further reduce the computational complexity of the iterative refinement step. Hyperparameter reduction can be performed, for example, according to the Adam-based method 300 described herein.

[0116] According to a second aspect of the present invention, a transmitter configured for wireless data transmission according to an over-the-air computing process is presented. The transmitter comprises an antenna, a circuit arrangement 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 perform a method comprising: - Receiving optimized precoder parameters vk*, which are determined by the method according to the first aspect of the invention, and of data to be transmitted, - Preprocessing the data to be transmitted for over-the-air computing according to a target function, - Pre-coding the pre-processed data using the received pre-coder parameters vk* and - Transmitting the pre-coded, pre-processed data via an antenna of the transmitter.

[0117] According to a third aspect of the present invention, a receiver for wirelessly receiving signals representing data processed according to an over-the-air computing process is presented. The receiver comprises at least one antenna, a circuit arrangement 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 perform a method comprising: - Receiving an optimized combiner u*, which is determined via the method according to the first aspect of the invention, and signals representing data transmitted by two or more transmitters, - Combining the received signals representing data, using the received optimized combiner u* and - Outputting the combined signals that correspond to an output of the objective function determined by preprocessing in the transmitters.

[0118] Receiving optimized pre-encoder parameters vk* or an optimized combiner u* in the sender or receiver can include receiving the information from a remote computing unit or from a local computing unit or a local computing process that performs the method according to the first aspect of the invention.

[0119] According to a fourth aspect of the invention, a method for over-the-air computing in a system comprising two or more transmitters wirelessly connected to a receiver is proposed. The method includes receiving optimized pre-encoder parameters at each transmitter. vk*, which are determined by the method according to the first aspect of the invention, and of the data to be transmitted. The data to be transmitted is then preprocessed for over-the-air computing according to a target function. The preprocessed data is used with the received pre-encoder parameters. vk* The pre-coded and pre-processed data is pre-coded before being transmitted via an antenna of the transmitter. At the receiver, the method comprises receiving an optimized combiner u*, which is determined by the method according to the first aspect of the invention, and signals representing data transmitted by two or more transmitters. The received signals representing data are combined using the received optimized combiner u* before outputting the combined signals, which correspond to an output of the target function determined by preprocessing in the transmitters.

[0120] Two or more transmitters according to the second aspect of the invention and at least one receiver according to the third aspect of the invention can form a system configured to perform the method according to the fourth aspect for implementing over-the-air computing.

[0121] In one or more embodiments of the transmitter and / or receiver, the circuit arrangement for processing radio frequency signals comprises a low-noise amplifier and / or a mixer configured to provide a representation of a received signal at an intermediate frequency. The mixer preferably uses the same oscillator signal as a transmitter located at the same position as the receiver in the wireless device.

[0122] As a person skilled in the art would understand, aspects of the embodiments can be designed as a system, a device, a process, or a software product. Accordingly, embodiments can take the form of an embodiment implemented entirely in hardware, an embodiment implemented entirely in software, including firmware, resident software, microcode, etc., or an embodiment that combines software and hardware aspects.

[0123] For example, the disclosed embodiments can be implemented as a hardware circuit, comprising specifically adapted circuits with a very large-scale integration (VLSI) level, or gate arrays, commercially available semiconductors such as logic chips, transistors, or other discrete components. The disclosed embodiments can also be implemented in programmable hardware devices, such as field-programmable gate arrays, programmable logic arrays, programmable logic devices, or the like. As another example, the disclosed embodiments can include one or more physical or logical blocks of executable code, which may be organized, for example, as an object, a procedure, or a function.

[0124] The method presented above can be represented by computer program instructions. Accordingly, according to a further aspect of the invention, a computer program product comprises computer program instructions which, when executed by a microprocessor of a wireless device according to the second aspect of the invention, cause the microprocessor to execute methods according to the first aspect of the present invention and to control hardware and / or software blocks or modules of the wireless device accordingly.

[0125] Computer program instructions or code for performing operations for embodiments can consist of any number of lines and can 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 programming language "C" or the like, and / or machine languages ​​such as assembly languages. The code can be executed entirely on the user's computer, partially on the user's computer as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on the remote computer or server.In the latter scenario, the remote computer can be connected to the user's computer through any type of network, including a local area network (LAN), a wireless LAN (WLAN), or a wide area network (WAN), or the connection can be made to an external computer (for example, via the Internet using an Internet Service Provider (ISP)).

[0126] Computer program instructions can be stored or transmitted on a computer-readable medium or data carrier. The medium or data carrier can be tangible or physically embodied, for example, in the form of a hard drive, solid-state storage device, flash memory device, or the like. However, the medium or data carrier can also comprise a modulated electromagnetic, electrical, or optical signal that is received by the computer via a suitable receiver and transmitted to and stored in the computer's memory.

[0127] The described features, structures, or characteristics of the embodiments can be combined in any suitable way. Numerous specific details are provided in this description, such as examples of programming, software modules, user selections, network transmissions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of the embodiments. However, those skilled in the art will recognize that embodiments can be implemented without one or more of the specific details or with different methods, components, materials, and so forth. In other cases, known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment.References throughout this description to "a (single) embodiment," "an embodiment," or similar phrases mean that a specific feature, structure, or property described in connection with the embodiment is included in at least one embodiment. Thus, the occurrence of the expressions "in a (single) embodiment," "in an embodiment," and similar phrases throughout this description may always refer to the same embodiment, but not necessarily, and they mean "one or more, but not all, embodiments" unless expressly stated otherwise. The terms "containing," "comprising," "having," and their variants mean "containing, but not limited to," unless expressly stated otherwise.An enumerated list of elements does not imply that any or all of the elements are mutually exclusive, unless explicitly stated otherwise. The terms "a," "an," and "the" also refer to "one or more," unless explicitly stated otherwise.

[0128] Where aspects of the embodiments in this description are described with reference to schematic flowcharts and / or schematic block diagrams of processes, devices, systems, and program products according to embodiments, it is understood that each block of the schematic flowcharts and / or schematic block diagrams, as well as combinations of blocks in the schematic flowcharts and / or schematic block diagrams, may be implemented by code. This code may be provided to a processor of a general-purpose computer, a specialized computer, or other programmable data processing device to create a machine such that, by the instructions executed through the processor of the computer or other programmable data processing device, means for implementing the functions / operations specified in the flowcharts and / or block diagrams are generated.

[0129] It is noted that in some implementations or embodiments, the functions mentioned in the exemplary embodiments shown in the figures may occur in a different order than that shown in the figures. For example, two consecutively shown blocks may in fact be executed essentially simultaneously, or the blocks may sometimes be executed in reverse order, depending on the functionality involved. Other steps and procedures are conceivable that are equivalent in function, logic, or effect to one or more of the blocks or sections thereof shown in the figures.

[0130] The present invention proposes a low-complexity combiner design for minimizing mean square effects (MSE) in AirComp systems. The novel refinement process can be used in conjunction with sophisticated RQ-based initialization and can be further enhanced with BO-based hyperparameter tuning and Adam-based acceleration. The novel refinement is obtained from a CCP-based reformulation of the mean square effect minimization problem, which can be solved with low complexity using a PG method. Numerical results show that embodiments and variants of the proposed design can achieve MSE performance equivalent to the best currently known conventional alternatives, while being approximately 200 times less complex than the most efficient conventional scheme and approximately 4 times less complex than its low-complexity conventional counterpart. BRIEF DESCRIPTION OF THE DRAWING

[0131] The invention is described in the following section with reference to the drawings. It shows: Fig. 1 a simplified representation of an AirComp system, Fig. 2. An exemplary simplified block diagram of an AirComp transmitter and an AirComp receiver, Fig. 3 a flowchart of an exemplary method according to the invention, Fig. 4 a flowchart of a first embodiment of the exemplary method according to the invention for accelerating the convergence of the iteration, Fig. 5 a flowchart of a second embodiment of the exemplary method according to the invention for accelerating the convergence of the iteration, Fig. 6 a representation of the MSE convergence behavior of the proposed method and variants thereof for a small number of iterations of the inner loop, Fig. 7 a representation of the MSE convergence behavior of the proposed method and variants thereof for a high number of iterations of the inner loop, Fig. 8. An illustration of the effect of system utilization on MSE performance for the proposed method and variants thereof over the number of iterations of the inner loop. Fig. 9 a comparison of the MSE performance of the proposed method with conventional approaches via system utilization, Fig. 10. A comparison of the computational complexity of the proposed method with conventional approaches via system utilization, Fig. 11 a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for an underutilized system, Fig.12 a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for a fully utilized system, Fig. 13 a comparison of the CDF of the MSE between conventional methods and embodiments of the proposed method for an overloaded system, Fig. 14 an exemplary block diagram of a transmitter according to the invention, Fig. 15 an exemplary block diagram of a receiver according to the invention and Fig. 16 an exemplary and schematic flowchart of a method for over-the-air computing according to a fourth aspect of the invention.

[0132] The figures may refer to identical or similar elements using the same reference symbols. DETAILED DESCRIPTION OF EXECUTION FORMS

[0133] The Fig.Items 1 to 13 have already been described above; they will not be discussed again.

[0134] Fig. Figure 14 shows an exemplary block diagram of a wireless transmitter 500 according to the invention. The wireless receiver 500 comprises at least one antenna 502, a circuit arrangement 504 for processing radio frequency signals, one or more microprocessors 505, volatile memory 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 the following: - Receiving an optimized combiner u*, which is determined via method 100 according to the first aspect of the invention, and of data to be transmitted, - Preprocessing the data to be transmitted for over-the-air computing according to a target function, - Pre-coding the pre-processed data using the received pre-encoder parameters vk* and - Transmitting the pre-coded, pre-processed data via an antenna of the transmitter.

[0135] Fig.Figure 15 shows an exemplary block diagram of a wireless receiver 600 according to the invention. The wireless receiver 600 comprises at least one antenna 602, a circuit arrangement 604 for processing radio frequency signals, one or more microprocessors 606, volatile memory 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 perform a method comprising the following: - Receiving an optimized combiner u*, determined via the method 100 according to the first aspect of the invention, and signals representing data transmitted by two or more transmitters 500, - Combining the received signals representing data, using the received optimized combiner u* and - Outputting the combined signals that correspond to an output of the objective function determined by preprocessing in the transmitters.

[0136] Fig. Figure 16 shows an exemplary and schematic flowchart of a method 700 for over-the-air computing according to the fourth aspect of the invention. The method comprises receiving optimized pre-coder parameters at two or more transmitters 500 in step 710. vk*, which are determined via the method 100 according to the first aspect of the invention, and of data to be transmitted (not shown in the figure). The two or more transmitters 500 preprocess the data to be transmitted for over-the-air computing in step 720 according to a target function and precode the preprocessed data in step 730 using the received precoder parameters. vk* Before the transmission of the pre-coded and pre-processed data via an antenna 502 of the transmitter 500 at step 740. The transmission is indicated by the dashed zigzag arrows. The method comprises, at the receiver 600 at step 750, receiving an optimized combiner u*, which is determined via the method 100 according to the first aspect of the invention, and signals representing data transmitted by two or more transmitters 500, represented by the dashed zigzag arrows. At step 760, the method comprises combining the received signals representing data, using the optimized combiner u*, before outputting, at step 780, the combined signals, which correspond to an output of the target function determined by preprocessing in the transmitters 500. REFERENCE SIGN LIST (PART OF THE DESCRIPTION) 100 procedures 101 iterative refinement 110 Receiving input 120 Calculating the initialization point 130 Calculating a k and b k 140 Update u (ii) 150 Is the termination criterion of the inner loop met? 160 Replacing the previous initialization point 170 Is the termination criterion of the outer loop met? 180 Output of the optimized combiner 190 Calculating the optimal precoder 195 Outputting the optimal precoder 200 procedures 210 Key points for initial parameters 220 Evaluating an initial MSE performance 230 Initializing a data set 240 Modeling a Function 250 Selecting an entered test set 260 Calculating an average MSE and a mean variance 270 Selecting Hyperparameters 280 Determining the MSE performance and updating the data set of the current iteration 290 Termination criterion fulfilled? 295 Determining and outputting optimized hyperparameters 300 procedures 310 Receiving input 320 Calculating the initialization point 330 Calculating a k and b k 340 Initializing the moments m1 and m2 350 Updating the first and second moments m1 and m2 Has the 360° termination criterion of the inner loop been met? 370 Replacing the previous initialization point 380 Is the termination criterion of the outer loop met? 390 Outputting an optimal combiner u* 400 Calculating optimal vk* 410 Outputting optimal vk* 500 transmitters 550 Pre-processing 560 Pre-coding 600 receivers 650 Combi-design 660 Combiners 670 processing 700 procedures 710 Receive optimized parameters and data 720 Pre-processing 730 Precoding 740 Transferred 750 received optimized parameters and signals 760 Combine Spend 770 QUOTES INCLUDED IN THE DESCRIPTION

[0000] This list of documents cited by the applicant was automatically generated and is included solely for the reader's convenience. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions. Cited non-patent literature

[0000] W. Liu, X. Zang, Y. Li und B. Vucetic in „Over-the-Air Computation Systems: Optimisation, Analysis and Scaling Laws“, IEEE Trans. Wireless Comm., Band 19, Nr. 8, S. 5488-5502, 2020 untersucht und später von T. Qin, W. Liu, B. Vucetic und Y. Li in „Over-the-Air Computation via Broadband Channels“, IEEE Wireless Commun. Lett., Band 10, Nr. 10, S. 2150-2154, 2021

[0009] L. Chen, X. Qin und G. Wei in „A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation“, IEEE Wireless Commun. Lett., Band. 7, Nr. 6, S. 942-945, 2018

[0011] H. Zhu, Y. Shi und 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, S. 61-65

[0011] K. Ando und 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, S. 126-130

[0011] M. Balandat, B. Karrer, D. R. Jiang, S. Daulton, B. Letham, A. G. Wilson und E. Bakshy in „BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization“, Advances in Neural Information Processing Systems 33, 2020, [Online], verfügbar unter: http: / / arxiv.org / abs / 1910.06403

[0070] S. Boyd und L. Vandenberghe in „Convex Optimisation“, Cambridge University Press, März 2004

[0094] C. Roos, T. Terlaky und J. P. Vial in „Interior point methods for linear optimisation“, 2nd ed. New York: Springer, 2006

[0095] 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

[0105]

Claims

[1] Method (100) for determining operating parameters of respective pre-encoder phases of a plurality of transmitters (500) or a combiner phase of a receiver (600) configured for over-the-air computing, wherein the operating parameters (vk*,u*) are designed to minimize the mean square error (MSE) between the objective function and the reconstructed over-the-air computing function, the procedure comprising the following: - Determining (130) a first combiner (u (0) ), - iterative refinement (101) of the combiner (u (ii) ) by minimizing the square of the Euclidean vector norm of the combiner (||u|| 2), while in each iteration the Euclidean vector norm of the product of the rearranged channel matrix and the combiner of the respective iteration is kept equal to or greater than 1 for each of the plurality of transmitters (500) until a termination criterion is satisfied, thereby obtaining an optimized combiner (u*), - Calculating (190) optimized precoder parameters (vk*) for all transmitters (500) based on the previously optimized combiner (u*), while ensuring that the square (|v k | 2 ) the precoding scalars of each transmitter (500) does not exceed a predetermined power value (P), wherein the iterative refinement (101) comprises: - Approximating, for all of the plurality of transmitters (500), the respective constraint, i.e., the requirement that the Euclidean vector norm of the product of the rearranged channel matrix and the combiner of the respective iteration must be equal to or greater than 1, based on the product of the derivative of the original constraint and the solution of the previous iteration thereof, and - Applying a proximate gradient (PG) function to solve the minimization goal represented by the approximation. [2] Wireless transmitter (500) comprising an antenna (502), a circuit arrangement (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 perform a method comprising: - Receiving (710) optimized precoder parameters (vk*), which are determined according to the method (100) of claim 1, and of data to be transmitted, - Preprocessing (720) the data to be transmitted for over-the-air computing according to an objective function, - Pre-coding (730) the pre-processed data using the received pre-coder parameters (vk*) and - Transmitting (740) the pre-coded pre-processed data via an antenna (502) of the transmitter (500). [3] Wireless receiver (600) comprising at least one antenna (602), a circuit arrangement (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 perform a method comprising: - Receiving (750) an optimized combiner (u*) determined according to the method (100) of claim 1, and signals representing data transmitted by two or more transmitters (500), - Combining (760) the received signals representing data, using the received optimized combiner (u*) and - Output (780) of the combined signals corresponding to an output of the objective function determined by preprocessing (720) in the transmitters (500). [4] Wireless transmitter (500) according to claim 2 or wireless receiver according to claim 3, wherein the receiving (710; 750) of optimized precoder parameters (vk*) and / or an optimized combiner (u*) comprising receiving this information from a remote computing unit or from a local computing unit or local computing process that performs the method according to claim 1. [5] Method (700) for 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) optimized precoder parameters (vk*), which are determined according to the method (100) of claim 1, and of data to be transmitted, - Preprocessing (720) the data to be transmitted for over-the-air computing according to an objective function, - Pre-coding (730) the pre-processed data using the received pre-coder parameters (vk*) and - Transmitting (740) the pre-coded and pre-processed data via an antenna (502) of the transmitter (500), and that at the recipient (600) includes the following: - Receiving (750) an optimized combiner (u*) determined according to the method (100) of claim 1, and signals representing data transmitted by two or more transmitters (500), - Combining (760) the received signals representing data, using the received optimized combiner (u*) and - Output (770) of the combined signals corresponding to an output of the objective function determined by preprocessing (720) in the transmitters (500). [6] Wireless communication system comprising two or more transmitters (500) according to claim 2 and at least one receiver (600) according to claim 3, wherein the system is configured to implement the method according to claim 5. [7] Computer program product, comprising computer program instructions which, - if they are executed by a computer, configure the computer to execute the method according to 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) to implement or perform a method comprising the following: - Receiving (710) optimized precoder parameters (vk*), which are determined according to the method (100) of claim 1, and of data to be transmitted, - Preprocessing (720) the data to be transmitted for over-the-air computing according to an objective function, - Pre-coding (730) the pre-processed data using the received pre-coder parameters (vk*), and - Transmitting (740) the pre-coded 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) to implement or perform a method comprising the following: - Receiving (750) an optimized combiner (u*) determined according to the method (100) of claim 1, and signals representing data transmitted by two or more transmitters (500), - Combining (760) the received signals representing data, using the received optimized combiner (u*) and - Output (770) of the combined signals corresponding to an output of the objective function determined by preprocessing (720) in the transmitters (500). [8] Computer-readable medium or data carrier that transmits or stores the computer program product according to claim 7 in a retrievable manner.

Images