Localisation method and apparatus implementing the method

Abstract

A method of locating objects emitting wireless signals, in an environment having at least two receivers adapted to determine RSSI values of the object's wireless signals is presented. The receivers have known positions in the environment and are communicatively connected to a common computing unit. The method comprises, at the common computing unit, determining an optimised RSSI value-based coordinate mapping function for the environment under consideration of shadowing noise, using the known positions of the receivers and of training objects. The RSSI values for the objects and from each of the receivers are aggregated. The aggregated RSSI values are arranged into a received signal power vector containing, for each individual object, the sums of the RSSI values determined at all receivers. For each object, vectors containing the RSSI values received by the receivers are mapped onto x and y coordinates of the environment using the optimised mapping function.

Description

[0001] 202304050 -1- LOCALISATION METHOD AND APPARATUS IMPLEMENTING THE METHOD FIELD OF THE INVENTION The present invention relates to the field of object localisation using wireless signals emitted by the objects, in particular wireless communication signals. NOTATIONS Throughout this specification, bold symbols represent vectors or matrices. Scalar values are denoted herein by lowercase letters in italics, as in x. Superscripts T and H, respectively denote the transpose and complex conjugate transpose of a vector or matrix. BACKGROUND Wireless localisation technology has gained great attention and development in the last few decades, inter alia in connection with autonomous vehicles, where knowing a vehicle’s location is essential for safe and efficient operation. Localisation information can further be used for tracking vehicles and predicting their paths, which may be useful for collision prevention or detection. Other uses of wireless localisation include near-field radio frequency identification (RFID) positioning and Internet-of-Things (IoT) sensor networks in smart factories, homes and beyond. Traditional localisation techniques have typically relied on global navigation satellite systems (GNSS) for the satellite-based geolocation and time information, but such methods exhibit poor power-efficiency, precision, latency, and robustness in dense urban scenarios, especially for the expected requirements of beyond fifth- generation (B5G) communication applications. Therefore, alternative positioning systems have been investigated to utilise wireless signals from a more proximate environment, such as wireless sensors, devices, and access points, such systems using captured signal metrics such as time of arrival (ToA), angle of arrival (AoA), or received signal strength information (RSSI) values for estimating the respective locations of the signal source. Due to the distributed nature of such localisation scenarios, the required methods 202304050 -2- generally must resolve multidimensional, multivariate optimisation problems based on the signal metrics received at the sensors, where the position information of the signal sources, or targets, is the solution. In order to alleviate such challenging optimisation problem, recently proposed methods leverage machine learning (ML) techniques. For example, in "Autonomous 3d UAV localisation using Taylor series linearized TDoA-based approach with machine learning algorithms," 202213th International Conference on Information and Communication Technology Convergence (ICTC), 2022, pp.783-785, V. Tilwari and S. Pack, suggest using a supervised ML method to evaluate Taylor series-linearised time difference of arrival (TDoA) measurements to localise autonomous unmanned aerial vehicles. In "A deep learning based AoA estimation method in NLOS environments," 2021 IEEE Globecom Workshops (GC Wkshps), 2021, pp.1-6, Y. M. T. Wang and Y. Shen use a deep residual network for evaluating AoA measurements for a single sensor non-line-of-sight (NLOS) indoor environment. A method to incorporate heterogeneous sensor information was proposed by N. T. A. M. I. Al Hajri and R. M. Shubair, in "Indoor localisation for IoT using adaptive feature selection: A cascaded machine learning approach," IEEE Antennas and Wireless Propagation Letters vol.18, no.11, pp.2306-2310, Nov. 2019. There, a K-nearest-neighbour (KNN) algorithm is used to adaptively select and combine the various signal radio frequency (RF) features for indoor localisation. An increasingly popular technique is to only operate with RSSI values at the sensors, with the advantageous trait that the encoded information of the wireless signal itself is not relevant, such that it can be used for other applications such as communication or channel estimation. Further, there is no need to embed specific information for the purpose of localisation into the wireless signals. For example, in "RSSI-based multiple sources localisation with unknown log-normal shadow fading,” arXiv:2110.10435v1, 2021, Y. Chu, W. Guo, K. You, L. Zhao, T. Peng, and W. Wang suggest detecting RSSI values at distributed sensors and use this 202304050 -3- information to localise the signal-emitting targets on a discrete grid. The positions of the targets are initially estimated via a sparse dictionary updating and a K-means clustering, and are iteratively refined by a dynamic update of the dictionary. While the prior art methods provide viable solutions for specific environments and scenarios, they provide sub-optimal accuracy in the presence of noise and are mostly suitable for limited indoor environments only. SUMMARY OF THE INVENTION It is, therefore, desirable to provide a localisation method that exhibits improved robustness and accuracy in noisy environments, and an apparatus implementing the method. This need is addressed by the method of claim 1, the receiver of claim 6, the common compute unit of claim 8 and the computer program product of claim 9. A corresponding computer-readable storage medium is presented in claim 10. Embodiments and developments of the method and apparatus, respectively, are provided in the respective dependent claims. In accordance with a first aspect of the invention a noise-robust localisation method is presented, for estimating the two-dimensional (2D) locations of one or more objects using exclusively RSSI values of wireless signals emitted by the objects that are received at distributed receivers. In particular, for improving the estimation robustness against noise, an optimised Gaussian process regression (GPR) model is proposed and trained with a mini-batch stochastic gradient descent (SGD) scheme using noisy RSSI data from prior training locations, and with gradients given in closed form. Further, a pair of robust marginalisation procedures for the estimation of target locations is provided. The invention will be described in the following assuming a multi-user (MU) distributed massive multiple-input multiple-output (DM-MIMO) scenario, consisting of K single-antenna transmit devices, also referred to as objects or targets, which are served by M single-antenna remote radio heads (RRHs), also referred to as receivers or sensors, the receivers being centrally connected to a computing unit 202304050 -4- (CU) via error-free fronthaul links with infinite rate, as schematically depicted in figure 1. The two-dimensional positions of the ^-th receiver and the ^-th objects, with ^ ∈ ^ ≜ {1, ⋯ , ^} and k ∈ ^ ≜ {1, ⋯ , K}, are respectively described by the 2D coordinate vectors where xSmN, ySNm and xTG k , yTGk are respectively the x- and y-coordinates of the m-th receiver and k-th object. Following the above, the Euclidean distance between the m-th receiver and k-th object is given by The received signal vector at the m-th receiver over T consecutive transmission instances is given by where ℎ^^∈ ℂ is the channel gain between the m-th receiver and the k-th object, assumed to remain constant during T transmissions, ^^ ∈ ℂ^×^are respectively the transmit power and the arbitrary transmit symbol vector from the k-th object, and ^^ is the additive white Gaussian noise (AWGN) received at the m-th receiver. In this specification it is assumed that the transmit powers of all objects are uniform, such that ^^= ^, ∀^ ∈ ^. The flat-fading uplink channel gain coefficient ℎ^^in equation (3) is further modelled by where ^^^∼ ^ ^(0,1) is the small-scale fading factor, and ^^^is the large-scale fading factor given by 202304050 -5- with ^^∈ ℝ representing the reference path loss coefficient, i.e., the path loss at a reference distance, and ^ ∈ ℝ representing the path loss exponent determined by environmental assumptions. ^^^∈ ℝ and ^^^ are respectively the distance as defined in equation (2) and the channel gain due to shadowing of the path between the m-th receiver and the k-th object. In light of the above, the RSSI value of the received signal of each k-th object at the m-th receiver is obtained by which can be estimated – either semi-blindly, by leveraging short orthogonal pilot sequences together with independent payload data, as shown by K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, in “Machine learning methods for RSSI-based user positioning in distributed massive MIMO,” IEEE Trans. on Wireless Communications, vol.17, no.12, 2018, or by H. Q. Ngo et al., in “Cell-free massive MIMO versus small cells,” IEEE Transactions on Wireless Communications, vol.16, no.3, 2017; or blindly, by exploiting the sparsity resulting from intermittent user activity, such that only a relatively small random subset of transmitters share the channel at each transmission instance, as shown by H. Djelouat, M. Leinonen, and M. Juntti, in “Joint estimation of clustered user activity and correlated channels with unknown covariance in mmtc,” ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp.1–5, based on the relation ^ ^^diag where W ≜ and pm≜ [pm1…pmK]Trespectively denote a matrix collecting the transmit signals and the powers from all K transmitters at the m-th sensor. 202304050 -6- pmkcan be equivalently expressed in decibel (dB) scale as sensitivity. Consequently, the RSSI values from all objects and from all M receivers are aggregated, at the CU, and stacked into a vector (in dB ) as The multi-object problem presented above will require a suitable solution for obtaining a viable implementation. In accordance with the present invention an ML- based solution to the multi-object localisation problem is presented, which effectively translates to an optimisation problem over the ^ pairs of real-valued 2D coordinates, given the aggregated RSSI values at the ^ RRHs and the known positions of the ^ RRHs. Specifically, the proposed solution is elaborated in three subsections: - a Gaussian process regression (GPR)-based coordinate model of the objects, - a training method via mini-batch SGD with noisy training data, and - a localisation process for approximating the unknown coordinates using GPR methods. Note that some aspects of the GPR-based coordinate model, the SGD training process and the GPR localisation process are discussed in German patent application no.102023204636.9 filed for the same applicant and naming the same inventors, which application is hereby incorporated in its totality by reference. Not further that the method proposed herein may be used with perfect training data or noisy training data. 202304050 -7- In the following section the GPR-based coordinate model will be discussed, considering two arbitrary functions fx(⋅) and fy(⋅) that map the received signal power vector of any k-th object to its x- and y-coordinates, respectively, i.e., ^ = ^ ^ , and ^ ^^(^)^^= ^^(^^)∀^ ∈ ^, (10) where it is implied that, if the functions fx(⋅) and fy(⋅) are known at the CU, the positions of any object can be obtained, when the aggregated RSSI values are known. Remark: As the expressions are identical for the x-coordinate and the y-coordinate functions and derivations, the subscripts (⋅)xand (⋅)yare omitted for conciseness from this point onward, and the following expressions are assumed to apply identically for both x- and y-coordinates unless stated otherwise. The letter c may occasionally be used to indicate either subscript, interchangeably. In the proposed method, GPR is utilised to model the coordinate mapping functions, where it is first assumed that the target function is drawn from a user- defined Gaussian process (GP) prior, i.e., ^(⋅)∼ ^ ^ ^(0, ^), (11) where ^ ^(0, ^) denotes the GP prior with zero mean and covariance matrix ^ ∈ ℝ^×^, whose element at the k-th row and q-th column is given by the covariance function ^^^^, ^^^, dependent on the RSSI values between the objects k and q. To accurately model the relationship of the RSSI values and the coordinates, the covariance function must be designed to carefully capture the covariance between any two objects in the region of interest (ROI) of the considered system. The proposed method follows the covariance function proposed in "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.), 202304050 -8- which is designed to capture both stationary and non-stationary transmitter object pairs, as where ^^∈ ℝ^×^and ^^∈ ℝ^×^are respectively the RSSI vectors of the k-th and the q-th object, ^^^^∈ ℝ is the variance of the measurement error, ^^^is the indicator variable taking on the value 1 if ^ = ^ and ^, and the parameters ^, ^, ^ are the weight parameters to be learned, with ^ ≜ diag where ^^∈ ℝ ∀ ^ ∈ ^ are the weights corresponding to each RSSI vector in the exponential term of the covariance function defined in equation (12). In light of the above, it can be seen that in hand of the RSSI vectors yielding to the covariance matrix, the actual coordinate value can be obtained by evaluating the GP ^(⋅) in equation (11). This requires the information of the ^ + 2 unknown weight parameters ⋯ , ^^, ^ of the covariance function, whose optimisation is described in this section. For convenience the vector collecting the M + 2 unknown weight parameters of the covariance matrix is defined as Further, the sets of coordinates c ∈ ℝK×1corresponding to K training locations, and the associated set of training RSSI vectors pk ∈ ℝM×1, with k = 1, …, K are defined. An according training covariance matrix ∈ ℝK×Kassociated with the K training locations and corresponding RSSI values and the cross-covariance matrix Σ ∈ ℝK×K between the K training and the K target locations, constructed from the corresponding sets of RSSI values, are likewise defined. 202304050 -9- The joint distribution of the coordinate vectors of the training and target locations, according to a conventional GP approach, is given by, where the k-th row and the k-th column of ^ ∈ ℝK×Kis given by ϕ(pk, pk), as obtained by equation (12). In possession of the joint distribution as per equation (15), the conditional distribution of c is given by ℙ(c |c, p1,…, pK, p1,…,pK, ^) = ^ (µ, C)) (16) where the conditional mean vector µ ∈ ℝK×1and covariance matrix C ∈ ℝK ×Kis obtained via µ = ΣΨ-1c ∈ ℝK×1(17a) Finally, the marginal distribution of the individual object coordinate can be obtained as ℙ(ck |c, where ck and vk are respectively the marginal mean and variance of the estimate of the coordinate of the k-th target, given by where [ · and [ · ]k,irespectively denote the k-th element of a vector and the matrix element at the k-th row and i-th column. Since ck is Gaussian distributed, the marginalised mean ck is directly the maximum-a-posteriori estimate of xk, and hence the final predicted coordinate of the k-th object. It can be inferred from the 202304050 -10- procedure described above that the performance of the GPR-based RSSI localisation algorithm is highly dependent on the accuracy of the covariance matrix model given in equation (13), which in turn is fundamentally determined by the parameter vector θ. In other words, a core step of the method is to optimally determine θ, given a certain amount of training data. The authors of "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.) propose to use a maximum likelihood fitting of the distribution of the training coordinates c, given the corresponding set of K RSSI vectors pk, assumed to be free of errors. Such an assumption is not only virtually impossible to meet in practice, as assuming that the RSSI value can be obtained without any errors implies the need for pilot signals of either extremely high power or in extremely large numbers, which is already idealistic in the case of a single transmitter-sensor pair, let alone in multi-target-multi-sensor and multi-path settings, but also a cause of poor performance in real-life applications, since the inevitable presence of errors in RSSI values collected is not mitigated by design. Embodiments of the present invention addresses this unrealistic assumption by proposing a noise-robust training solution for determining the parameter vector θ, which lends both feasibility and robustness even to the known approach, significantly improving the multi-object localisation problem described before. The noise-robust training solution effectively translates into an optimisation problem over the K-pairs of real-valued 2D coordinates, given the aggregated RSSI values at the M receivers and the known positions of the M receivers, which is solved via SGD. These embodiments of the proposed method accommodate for random shadowing noise in the real RSSI values by assuming such random shadowing noise also in the training data. The training RSSI vectors used for obtaining the optimal parameter vector θ are thus also modelled as 202304050 -11- This is equivalent to treating the coordinates c of the training points not as deterministic, but rather as random variables with a distribution that, under the GPR model can be assumed to be approximated by P^c ∣ p1,⋯,pK,θ^ ≈ ^ (0K×1,Ψ) where the elements of the covariance matrix Ψ ∈ ℝK×Kare determined via equation (12), that is with ^( ⋅,⋅∣ ^ ) denoting the covariance function subject to the specific weight parameters ^. Given the distribution in equation (21), the optimal weight parameters θ can then be determined as the solution of the maximum likelihood problem where the objective g(θ; p1, …, pK) is implicitly defined, which for the purpose of training is a function of the parameter vector θ, as highlighted by the notation, and which in view of the Gaussian approximation in equation (21), can be modelled as ^^^; p , ⋯ , p ^ ^ = log ^(2^K^ ^^^^ ^ K ^ ) |Ψ| + ^ c Ψ c, (24) where | · | denotes the determinant of the argument matrix. It is evident from equations (12), (22) and (24), that g(θ; p1, …, pK) is not convex on the optimisation variable θ. In fact, the determinant and inversion operations onto Ψ, and the non- linear dependence of the latter on the parameters βmgathered in the matrix B, 202304050 -12- make the problem highly intractable from the perspective of optimisation theory. The authors of "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.) turn their attention to other matters and leave the issue largely unaddressed, limiting the discussion to citing gradient-based methods and classic literature on the subject. As mentioned before, in accordance with the invention this issue is addressed via an SGD approach, e.g., as discussed by by S. Ruder, in "An overview of gradient descent optimisation algorithms,“ arXiv preprint arXiv:1609.04747, 2016. To this end, first, consider the partial derivative of g(θ; p1, …, pK) with respect to α, which is given by Next, the matrix derivative identities are leveraged to simplify the expression in equation (25) to Observing that the partial derivatives of equation (24) with respect to and ^ are identical in form to the expression in equation (27), details of the derivations corresponding to the latter parameters are omitted here and only expressions for each element of the partial derivatives of Ψ with respect to ^, βmand ^ are provided, which are respectively given by 202304050 -13- where [p̅ki]m is the m-th element of p̅ki ≜ (pk−pi ) ∈ ℝM×1, and the notation [⋅]k,idenotes the matrix element at the k-th row and i-th column. The gradient of the objective function in equation (24) can then finally be put together, yielding such that the optimal parameter vector θ can be optimised via gradient descent (GD), according to the update equation where θ(i-1)and θ(i)are the are the parameter vectors at the (i−1)-th and i-th SGD iterations, respectively, while λ is the learning rate, which for convergence guarantees is bounded by the Lipschitz condition with L denoting the Lipschitz constant, given by ^ = max eig((^^^)^⋅ ^^^). (31b) While the training procedure described above may be performed over a large data set, consisting of multiple snapshots of data collected for a large number of training locations, this may result in a highly complex training algorithm, since the complexity of the training algorithm is fundamentally determined by the inversion of the covariance matrix ^, as is evident from equation (29). 202304050 -14- The present invention thus further proposes using a minibatch SGD-based variation of the general training scheme outlined above, which is described in the following section. The proposed minibatch SGD method, where the total training data is split into a number of minibatches of a size S for multiple applications of the SGD process, results in an implementation that has a significantly lower complexity than implementations for large data sets. First, a training data cube ^∈ℝM×K×Sis defined that comprises S noisy snapshots of each and all RSSI vectors mini- batch ^ ∈ ℝ^×K˜⊂ ^ consists of a number K˜ ≪ (s˜) ^ K of training RSSI vectors p k˜ , selected randomly from ^, with equal probability and mutually exclusively, such that k˜ ∈ {1, ⋯ , K˜} and S˜ ∈ {1, ⋯ ,S}. For notational convenience, the training locations are relabelled corresponding to a given b-th minibatch as c˜b= [x˜1,⋯,x˜K˜], and the corresponding RSSI vectors as p˜k˜, such that the minibatch can be described as ^b= ^p˜1,⋯,p˜K˜^. Next, consider the covariance matrix with structure similar to that of equation (22), but constructed only with respect to the training location vector c˜b, i.e. Then, under the SGD method, the update of the parameter vector θ for the minibatch is obtained by the corresponding variations of equations (29) and (31), namely 202304050 -15- ⊤ -1 0 ≤ λb≤ ^max eig ^^Ψ-1 b ^ ⋅Ψ-1 b ^^ (34b) An exemplary data cube is illustrated in figure 2. The exemplary data cube ^ corresponds to a case where the number of receivers is M = 5, the number of training locations is K = 7, the total number of snapshots of RSSI vectors per training location is S = 8. An example of a mini batch ^b with K˜ = 4 is illustrated by the grey columns, which exemplary mini batch comprises a random selection of RSSI vectors corresponding to the training locations c˜ ={x1,x2,x4,x7}, such that k˜ = {1, 2, 4, 7}, respectively taken at snapshots s˜={2, 8, 5, 8}, yielding A schematic block diagram of the parameter ^ optimisation method via mini-batch SGD using the data cube mini-batch embodiment is given in figure 3. The optimisation method is further illustrated in the schematic flow diagram of figure 4. It is reminded that the method is to be run independently for each coordinate ^ and ^, such that its execution yields the optimised, i.e., trained, parameters ^∗^and ^^∗. Inputs to the training process are the data cube ^∈ℝM×K×Swith S snapshots of each and all RSSI vectors p(s) k , with k ∈ {1, …, K} and s ∈ {1, …, S}, the number B and size K˜ of mini batches, the number ^ (^) ^^^of SGD iterations per mini batch, and an initial parameter vector ^, which are received in step 110. In step 120, a mini batch ^b∈ℝM×K˜⊂ ^ is taken from the data cube ^, each mini-batch comprising a number K˜ ≪ K mutually exclusive training RSSI vectors p˜ (s) k˜ = pk with k˜ ∈ {1, … , K˜}. The mini batch is fed to a looped calculation process in step 130. In step 140 the covariance matrix is constructed as per equation (32), and in step 150 gradients ∇^^ (^; p1, …, pK) are computed via equation (33). In step 160 ^ is updated as per equation (34). Computing the gradients and updating the intermediate optimised values is iteratively repeated until a termination criterion is 202304050 -16- met, which is checked in step 170. If the termination criterion is not met, “no”- branch of step 170, the next iteration is carried out. Suitable termination criteria comprise, inter alia, a predetermined maximum number of iterations ^ (^) ^^^or a convergence of the intermediate optimised values ^ below a predetermined threshold. The convergence criterion may also comprise that such convergence is stable over a predetermined number of subsequent iterations. Once it is determined, in step 170, that the termination criterion is met, “yes”-branch of step 170, step 180 checks if the last one of the B mini batches has been fed to the calculation process. If not, “no”-branch of step 180, the process returns to step 120 and is repeated. If all mini batches have been fed to the calculation process, “yes”- branch of step 180, the training phase is completed, and the optimised values for the weight parameters ^∗are output in step 190. Next, the RSSI-based localisation via GPR will be described. In possession of a parameter vector θ obtained through the training discussed above, the RSSI-based localisation algorithm via GPR as discussed in "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.) reduces to evaluating equation (19a) for each k-th target, given the corresponding RSSI measurements ^^obtained online, the noise-free training RSSI vectors pkobtained offline, and the associated covariance matrix Ψ. Besides using an optimised parameter vector θ* obtained through the ‘noisy’ training approach described above in some embodiments, another distinction between the proposed robust method and the known approaches is that instead of noise-free training RSSI vectors an entire data-cube ^ containing multiple snapshots of training RSSI vectors p(s) k is used, such that different alternatives exist to computing the GPR-based location estimates. One alternative is, for instance, to treat each RSSI vector p(s) k in the data cube ^ as a distinct noise-free data point and construct the corresponding KS ^ KS covariance matrix 202304050 -17- where Then, stacking the training location vector c onto itself S times, which can be concisely represented via the Kronecker product 1S⊗ c, the object locations can be obtained from the following variation of equation (19a) The complexity of evaluating equation (37) is of order ^((KS)^), which becomes quickly prohibitive as the number of training points K and / or snapshots S grows, such that this approach, described here only for completeness, is not recommended and will not be pursued any further hereafter. A lower-complexity alternative to the latter approach is obtained by performing location estimation independently for each snapshot of the data-cube using the covariance matrix from the corresponding vectors, and averaging the results afterwards, which will be hereafter referred to as the ‘estimate-then-average’ (EA) approach and can be concisely expressed as The approach summarised by equation (38) requires the inversion of S snapshot (s,s) K ^ K covariance matrices Ψ˜ , such that its complexity is of order ^^S ⋅ K^^ which, although substantially lower than that of equation (37), may still be too high for large S. 202304050 -18- In that case, a further alternative of yet lower complexity - of order ^^K^^ - which will be hereafter referred to as the ‘average-then-estimate’ (AE) approach, utilises the sample covariance matrix and determines the location estimates via where the expectation ^S[⋅] is taken over the S snapshots of data in the data cube ^. Here, first all covariance matrices are constructed, and the average thereof is used for the location estimation. Likewise, the RSSI vectors are averaged prior to determining the location based thereon. The proposed RSSI-based localisation scheme via the AE-GPR and EA-GPR approaches are summarised in the form of pseudo-code hereunder, separated into alternatives 2a) and 2b), respectively. Input: Data cube ^ ∈ ℝ^×K×Swith S snapshots of each and all RSSI vectors p(s) k , with k ∈ {1, ⋯ ,K} and s ∈ {1, ⋯ Measured RSSI values ^^from all objects; and optimised parameter vector θ* . Output: Estimated target x-, y-coordinate vector ^(^^)or ^(^^). Repeat ∀k ∈ ^ (s,s) 1: Compute all snapshot covariance matrices Ψ˜ , ∀s via equation (36); 2a) EA localisation approach: 2: Compute and return the object location estimates ^(^^) ^ via equation (38); 2b) AE localisation approach: 202304050 -19- 2: Compute the sample covariance matrix Ψ˜ via equation (39); 3: Compute and return the object location estimates ^(^^) ^ via equation (40); In the following section, the effectiveness of the proposed SGD-based robust training mechanism, and of the corresponding RSSI-based robust localisation method via EA / AE-GPR are evaluated via computer simulations and comparisons against the known approach presented in "Machine learning methods for RSSI- based user positioning in distributed massive MIMO” (ibid.). Figure 5 shows a comparison of the average convergence behaviour as a function of gradient descent epochs, of the proposed mini-batch SGD-based robust training algorithm with different mini-batch sizes K˜. In order to keep the comparison fair in terms of the total amount of training data used to optimise the parameter vector θ, the number of mini-batches B is varied so that K˜ ⋅ B is the same for all curves. The results show that smaller mini-batch sizes lead to slower convergence but also to lower points of the objective function, while larger mini-batches yield faster convergence but to higher local minima, which is a typical trade-off of convergence speed and optimality found in minibatch-based SGD schemes. Next, the proposed EA / AE-GPR-based localisation methods are evaluated, and their performances are compared with that of the known RSSI-based GPR scheme presented in "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.). The considered scenario is a street intersection containing ^ = 3 targets, with a ROI of 100 m × 100 m, serviced by ^ = 56 sensors placed along the roadsides. Training is conducted over a grid of ^ = 204 training locations distributed in a regular grid within the area, as illustrated in figure 6. Environment parameters such as the reference path loss coefficient and exponent are set according to the 3GPP Urban Micro propagation model described in 3GPP, "Evolved Universal Terrestrial Radio Access (E-UTRA); Further advancements for E-UTRA physical layer aspects," 3rd Generation Partnership Project (3GPP), Technical Report (TR) 36.814, 032017 , version 9.2.0. [Online]. Available: 202304050 -20- https: / / portal.3gpp.org / desktopmodules / Specifications / SpecificationDetails.aspx?sp ecificationId =2493. Radio parameters such as transmit / noise powers and radio sensitivity are set according to the LTE standard, e.g., as discussed by J. Salo, M. Nur-Alam, and K.- K. Chang, in "Practical introduction to LTE radio planning," 2010, and finally SGD parameters are set in light of the results of figure 5, which have shown that the optimised objective function reaches lower convergence points with minibatches of smaller sizes. The path-loss parameters ^^, and ^ are defined by the 3GPP Urban Micro propagation model as defined in 3GPP, "Evolved Universal Terrestrial Radio Access (E-UTRA); Further advancements for E-UTRA physical layer aspects," (ibid.), and the transmit power ^ of 21dBM(125 mW) is defined by LTE standards, e.g., as discussed in "Practical introduction to LTE radio planning“ (ibid.). In addition, the noise power in the system is defined as −107.5dBm with a receiver sensitivity of −106.5dBm, meaning that if the RSSI at a receiver is below the receiver sensitivity, the signal is assumed to have noise power. These parameters are summarised in the table below: Parameter Value ^^= −47.5 dB, Path-loss parameters 0, if ^^^< 10 m (3GPP UMi) ^ = ^ 2, if 10 m ≤ ^^^≤ 45 m 6.7, otherwise ^^^= 21dBm(125 mW) Radio parameters (LTE standard) ^^^^^= −107.5dBm ^^^= −106.5dBm ^˜ = {5,10,20,50} SGD parameters ^ ⋅ ^˜ = 1000 ^ = 200, ^ (^) ^^^= 100 Figure 6 shows the estimates obtained from multiple runs of the proposed method and the known approach. Measurement noise variance is = 1, and the proposed 202304050 -21- AE-GPR algorithm employs a parameter vector θ∗ trained as hereinbefore with mini-batches of size K˜ = 5 and B = 200. To the advantage of the known method used as reference for this comparison, and for the sake of visibility, the proposed method is represented only by the lower- complexity robust AE-GPR alternative which, as shall been shown later, has slightly worse performance than the EA-GPR alternative. It can be seen from the cloud of points representing the location estimates that the proposed robust approach is indeed superior to the known approach. A more quantitative assessment of the gains achieved by the method proposed herein in relation to the known method can be had by comparing the performances in terms of the location root mean square error (RMSE), defined as (41) where xT ^ ^ rue k , yTruek ^ denotes the true coordinates of the k-th target, while and Est kE^ ^x , ystk ^ are the corresponding estimates obtained by the respective algorithm. For the sake of providing a lower-bounding reference, results corresponding to the location estimates c(GA)obtained via a genie-aided (GA)-GPR scheme are also included, which are computed via the expression where pˆ k are noise-free RSSI vectors generated via 202304050 -22- is the corresponding noise-free covariance matrix constructed using the optimised parameter vector θ* obtained with the proposed noise-robust minibatch training method described above, that is The results are shown in figure 7, where figure 7 a) shows the results for small mini batches (K˜ = 5), and figure 7 b) shows the results for large mini-batches (K˜ = 50). A wide gap between the proposed robust EA / AE-GPR methods and the known method presented in "Machine learning methods for RSSI-based user positioning in distributed massive MIMO” (ibid.) is visible. Note also that the proposed alternative of lowest computational complexity, namely the AE-GPR scheme, is slightly outperformed by the slightly more computationally expensive proposed EA- GPR method. It is also obvious that both the AE-GPR and EA-GPR robust proposed algorithms achieve RMSE performances that come very close to that of the (unfeasible) genie- aided lower-bounding reference. Since the GA-GPR reference scheme likewise employs the parameter vector θ* obtained with the noise-robust training method proposed herein, the results indicate that most of the gains achieved are in fact due to the proposed SGD-based training approach, with the remaining gap between the EA / AE-GPR methods and the GA-GPR reference due merely to the available amount of data. The latter finding, namely, that both the AE-GPR and EA-GPR robust proposed algorithms asymptotically approach the same performance of the genie-aided reference is corroborated by the results of figure 7 b), which show that indeed the performances of these schemes are very similar when larger mini-batch sizes K˜ are employed. To further evaluate the proposed methods, the CDF curves of the RMSE of location estimates are given in figure 8, for K˜ = 5 in figure 8 a) and K˜ = 50 in figure 202304050 -23- 8 b), and with a noise variance = 1. It is emphasised that this choice of is to the disadvantage of the proposed methods, since the gains of the latter over the known method were shown in figure 7 to increase with that parameter. Still, the results clearly show that the improvement achieved with the proposed method is very significant not only in average terms, but rather for the entire range of RMSE values. It is particularly remarkable that with larger minibatch sizes the performance of the proposed methods at the low RMSE range overlaps with that of the ideal GA reference. Finally, the impact of the number of sensors ^ onto the performance of the GPR- based methods is assessed. To that end, in order to isolate the effect of ^ from other factors such as the geometry of the scenario, the schemes are tested under a highly symmetric single-target scenario as depicted in figure 9, where the target is located at the centre of the ROI, and the sensors are positioned symmetrically in the surroundings, as they if they were installed on the roadsides of a crossing. The results, given in terms of the average RMSE as a function of the number of sensors ^, are shown in figure 10, and again demonstrate that the proposed methods not only consistently outperforms the known method, but also tend to approach the GA performance - in particular the EA-GPR scheme - as the number of sensors increases. The mini-batch SGD-based noise-robust training procedure used for optimizing the parameters of a GPR model for the RSSI-based multi-target localisation method, along with either of the alternative AE and EA marginalisation procedures for the computation of corresponding location estimates presented herein, significantly outperform the best known alternative methods that utilise a similar GPR-based method, while approaching the performance of an genie-aided (GA) scheme that employs the proposed training method combined with an unfeasible marginalisation procedure based on the true covariance matrix of the RSSI data. Note that the GA scheme is unfeasible as it assumes an infinite amount of data. 202304050 -24- It is noted that, although the invention has been described hereinbefore using the noise-robust training, i.e., assuming noisy training data, the AE and EA localisation approach can also be used assuming ideal, noise-free training data. In light of the foregoing discussion, in accordance with a first aspect of the present invention a method of locating one or more objects emitting wireless signals, in an environment having at least two receivers adapted to at least determine RSSI values of the one or more object’s wireless signals is presented. The at least two receivers have known positions in the environment and are communicatively connected to a common computing unit. The method comprises, at the common computing unit, determining an optimised RSSI value-based coordinate mapping function for the environment under consideration of RSSI training values, using the known positions of the receivers and of training objects. The training objects and their signals may be simulated. The method further comprises aggregating RSSI values for each of the one or more objects and from each of the two or more receivers, and arranging the aggregated RSSI values into received signal power vectors representing the RSS values of all of the one or more objects as determined in each of the receivers. In accordance with the invention determining the optimised RSSI value-based coordinate mapping function comprises establishing RSSI training values between each receiver from a first plurality of receivers at known locations and each training transmitter from a second plurality of known training transmitter locations, multiple values being established for each pair of receiver and training transmitter and arranged as a succession of snapshots representing, for each training location, training RSSI vectors p(s) k whose elements are the training RSSI values between each receiver and the corresponding training location. Finally, for determining the location of the objects, the method comprises constructing, for all snapshots, covariance matrices for all training RSSI vectors p(s) k of a respective snapshot. Then, in a first one of two alternatives, the method comprises mapping, for each object and for each snapshot, vectors containing the RSSI values received by the two or more receivers onto x- and y- coordinates of the environment using the respective covariance matrices , and calculating the average or the median from all resulting locations. In a second one of the two alternatives the method comprises calculating an average or a median 202304050 -25- for each element of the plurality of covariance matrices (^), and mapping, for each object, vectors containing the RSSI values received by the two or more receivers (400) onto x- and y-coordinates of the environment using the resulting averaged or median expectation covariance matrix. In one or more embodiments of the method, an optimised parameter vector θ*for the coordinate mapping function is derived by a Gaussian Process using the mean and / or covariance of the RSS values of the training objects as input. The Gaussian Process may comprise a gradient descent process. In embodiments of the method, determining the coordinate mapping function comprises determining a conditional distribution of the coordinates of the training transmitters and the actual transmitters, and determining a marginal distribution of the coordinates therefrom using weight parameters determined in a training phase. In one or more embodiments of the method the optimised parameter vector θ*for the coordinate mapping function is determined by iteratively feeding small batches of training data sets containing training data of the training objects to the gradient descent process. In one or more embodiments of the method the RSSI values of the training objects are arranged in a data-cube ^ containing multiple snapshots of training RSSI vectors p(s) k , out of which the training vectors are chosen arbitrarily. In one or more embodiments of the method the wireless signals are wireless communication signals and the at last two receivers are adapted to determine identities of the one or more objects transmitted in the respective wireless communication signals, and to transmit the identities to the common computing unit along with the respective RSSI values. In accordance with a second aspect of the present invention a receiver of a wireless communication system configured for use with the method presented hereinbefore is presented. The receiver comprises one or more software and / or hardware blocks configured for receiving a wireless signal from one or more 202304050 -26- objects emitting such signals, for determining at least an RSSI value for the received signals, and for transmitting the signals to a common computing unit. The receiver comprises one or more antennas and associated receiving circuitry for receiving wireless signals from one or more objects, one or more microprocessors, volatile and non-volatile memory, and an interface for communicatively coupling with a common computing unit. The elements or components of receiver are coupled or communicatively connected through one or more communication lines or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure the one or more microprocessors to control the software or hardware blocks or modules, or the combination thereof, to at least determine RSSI values of wireless signals of one or more objects and to communicate at least the RSSI values to the common computing unit. In one or more embodiments the non-volatile memory of the receiver stores computer program instructions which configure the receiver to determine an identifier of the one or more objects and to communicate the identifier along with the RSSI values to the common computing unit. In accordance with a third aspect of the present invention a common compute unit of a wireless communication system comprising two or more receivers in accordance with the second aspect of the invention is presented. The common compute unit comprises one or more microprocessors, volatile and non-volatile memory, and an interface for communicatively coupling with two or more receivers. The elements or components of common compute unit are coupled or communicatively connected through one or more communication lines or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure the common compute unit (500) to execute the method in accordance with the first aspect of the invention. The methods described hereinbefore may be represented by computer program instructions. Accordingly, a computer program product comprises computer program instructions which, when executed by a microprocessor of a receiver in 202304050 -27- accordance with the second aspect of the invention, cause the microprocessor and / or control hardware blocks, modules or components of the receiver to at least determine RSSI values of wireless signals of one or more objects and to communicate the RSSI values to a common compute unit. When executed by a microprocessor of a common compute unit in accordance with the third aspect of the invention, the computer program instructions cause the microprocessor and / or control hardware blocks, modules or components of the common compute unit to execute embodiments of the method in accordance with the first aspect of the invention. 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 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. In accordance with a further aspect of the invention a wireless communication system is presented. The wireless communication system comprises two or more receivers in accordance with the second aspect of the invention, which are communicatively connected to a common compute unit in accordance with the third aspect of the invention. The method and apparatus proposed herein provide a basis for a robust object localisation based on RSSI values of wireless signals emitted from the objects, in particular vehicles – autonomous or under human control – in noisy environments. The improved performance in the presence of noise is achieved, inter alia, by performing training using noisy training signals. The method and apparatus proposed herein can be used in any wireless communication system environment, in particular in 6G communication systems and beyond, in indoor and outdoor environments, including industrial scenarios. 202304050 -28- Incorporating the variance of the shadowing noise in the training phase in addition to the deterministic, i.e., noise-free, training RSSI values yields a more noise- robust ML prediction model than prior art methods that do not incorporate the noise in the training phase. The derived closed-form expressions of the Hessian of the objective optimised during the training stage reduces the complexity of implementation by avoiding numerical evaluation of matrix derivatives. The noise-robust RSSI-based localisation method via GPR brings robustness into the training and to the computation of covariance matrices the of location estimates. BRIEF DESCRIPTION OF THE DRAWING The figures in the attached drawing are used for detailing aspects of the present invention. In the drawing Fig.1 shows a schematic representation of a system in accordance with the invention comprising a common computing unit, multiple receivers and multiple objects, Fig.2 shows an exemplary data cube ^, Fig.3 shows a schematic block diagram of the noise-robust parameter ^ optimisation method via mini-batch SGD, Fig.4 shows a schematic flow diagram of a method of obtaining an optimised noise-robust parameter θ∗ via mini-batch SGD, Fig.5 shows a comparison of the average convergence behaviour as a function of gradient descent epochs, of the proposed mini-batch SGD-based robust training algorithm with different mini-batch sizes K˜, Fig.6 shows a comparison of location estimates obtained from multiple runs of the method in accordance with the invention and a known approach, Fig.7 shows a comparison of the RSME for the proposed localisation methods and a known method for various minibatch sizes over the noise variance, Fig.8 shows comparison of the CDF curves of the RSME of location estimates for different methods and different parameters, 202304050 -29- Fig.9 shows idealised scenarios for determining the impact of the number of receivers on the performance of the proposes method, Fig.10 shows a comparison of the average RMSE as a function of the number of sensors ^ for different methods, Fig.11 shows a schematic block diagram of the method of locating one or more objects emitting wireless signals in accordance with the invention, Fig.12 shows an exemplary schematic block diagram of a receiver in accordance with the invention, and Fig.13 shows an exemplary schematic block diagram of a common computing unit in accordance with the invention. In the figures, identical or similar elements may be referenced using the same reference designators. DETAILED DESCRIPTION OF EMBODIMENTS Figures 1 to 10 have been described further above and will not be discussed again. Figure 11 shows a schematic block and flow diagram of the main steps of the localisation process in accordance with the present invention. The inputs to the process are the measured RSSI values of the objects, the coordinates of the sensors or receivers and the training points, the optimised parameters and the values from the data-cube. The inputs are used for computing a covariance matrix for the respective sample, which is then input to a conditioning and marginalisation step, which ultimately outputs the estimated coordinates of the object. Figure 12 shows an exemplary schematic block diagram of a receiver 400 in accordance with the present invention. The receiver 400 comprises one or more antennas 402 and associated wireless interface circuitry 456, adapted to receive wireless signals from one or more objects, one or more microprocessors 450, volatile memory 452, non-volatile memory 454, and a communication interface 404 for communicating with a common compute unit 500. The aforementioned elements are communicatively connected via one or more signal or data lines or buses 458. The non-volatile memory 454 stores computer program instructions which, when executed by the microprocessor 450, configure the one or more 202304050 -30- microprocessors to control the software or hardware blocks or modules, or the combination thereof, of the receiver 400 to at least determine RSSI values of wireless signals of one or more objects and to communicate at least the RSSI values to the common computing unit 500. Figure 13 shows an exemplary schematic block diagram of a common compute unit 500 in accordance with the present invention. The common compute unit 500 comprises one or more microprocessors 450, volatile memory 452, non-volatile memory 454, and a communication interface 404 for communicating with two or more receivers 400. The aforementioned elements are communicatively connected via one or more signal or data lines or buses 458. The non-volatile memory 454 stores computer program instructions which, when executed by the microprocessor 450, cause the common compute unit 500 to execute the method according to the invention as presented hereinbefore.

[0002] 202304050 -31- LIST OF REFERENCE NUMERALS (PART OF THE DESCRIPTION) 400 receiver 454 non-volatile memory 402 antenna 456 wireless interface circuitry 404 interface 458 signal / data connection / bus 450 microprocessor 10 500 common compute unit / CU 452 volatile memory

Claims

202304050 -32- CLAIMS 1. A method of locating one or more objects emitting wireless signals, in an environment having at least two receivers (400) adapted to at least determine RSSI values of the one or more object’s wireless signals, the at least two receivers (400) having known positions in the environment and being communicatively connected to a common computing unit (500), the method comprising, at the common computing unit (500): - determining an optimised RSSI value-based coordinate mapping function for the environment under consideration of RSSI training values, using the known positions of the receivers (400) and of training objects, - aggregating RSSI values for each of the one or more objects and from each of the two or more receivers (400), - arranging the aggregated RSSI values into received signal power vectors representing the RSSI values of all of the one or more objects as determined in each of the receivers (400), wherein determining the optimised RSSI value-based coordinate mapping function comprises: - establishing RSSI training values between each receiver from a first plurality of receivers at known locations and each training transmitter from a second plurality of known training transmitter locations, multiple values being established for each pair of receiver and training transmitter and arranged as a succession of snapshots representing, for each training location, training RSSI vectors (p(s) k ) whose elements are the training RSSI values between each receiver and the corresponding training location, wherein the method further comprises - constructing, for all snapshots, covariance matrices (^˜(^)) for all training RSSI vectors (p(s) k ) of a respective snapshot, and - mapping, for each object and for each snapshot, vectors containing the RSSI values received by the two or more receivers (400) onto x- and y- coordinates of the environment using the respective covariance matrix (^˜(^)), and202304050 -33- - calculating the average or the median from all resulting locations, or - calculating an average or a median for each element of the plurality of covariance matrices (^˜(^)), and - mapping, for each object, vectors containing the RSSI values received by the two or more receivers (400) onto x- and y-coordinates of the environment using the resulting averaged or median expectation covariance matrix.

2. The method of claim 1, wherein an optimised parameter vector (θ*) for the coordinate mapping function is derived by a Gaussian Process using the mean and / or covariance of the RSSI values of the training objects as input.

3. The method of claim 2, wherein the optimised parameter vector (θ*) for the coordinate mapping function is determined by iteratively feeding small batches of training data sets containing training data of the training objects to the gradient descent process.

4. The method of claim 2 or 3, wherein the RSSI values of the training objects are arranged in a data-cube (^) containing multiple snapshots of training RSSI vectors (p(s) k ), out of which the training vectors are chosen arbitrarily.

5. The method of one or more of claims 1 to 4, wherein the wireless signals are wireless communication signals and wherein the at last two receivers (400) are adapted to determine identities of the one or more objects transmitted in the respective wireless communication signals, and to transmit the identities to the common computing unit (500) along with the respective RSSI values.

6. A receiver (400) of a wireless communication system comprising one or more antennas (402) and associated receiving circuitry (456), one or more microprocessors (450), volatile (452) and non-volatile memory (454), and an interface (404) for communicatively coupling with a common computing unit (500), the elements or components of the receiver (400) being communicatively connected via one or more signal and / or data lines and / or202304050 -34- buses (458), wherein the non-volatile memory (454) stores computer program instructions which, when executed by the one or more microprocessors (450), configure the receiver (400) to at least determine RSSI values of wireless signals of one or more objects and to communicate at least the RSSI values to the common computing unit (500).

7. The receiver of claim 6, wherein the computer program instructions, when executed by the microprocessor (450), configure the receiver (400) to determine an identifier of the one or more objects and to communicate the identifier along with the RSSI values to the common compute unit (500).

8. A common compute unit (500) of a wireless communication system comprising two or more receivers (400) in accordance with claim 6 or 7, the common compute unit (500) comprising one or more microprocessors (450), volatile (452) and non-volatile memory (454), and an interface (404) for communicatively coupling with two or more receivers (400), the elements or components of the common compute unit (500) being communicatively connected via one or more signal and / or data lines and / or buses (458), wherein the non-volatile memory (454) stores computer program instructions which, when executed by the one or more microprocessors (450), configure the common compute unit (500) to execute the method of one or more of claims 1 to 5.

9. Computer program product comprising computer program instructions which, when executed by a microprocessor of a receiver (400) in accordance with claim 6 or 7, cause the microprocessor and / or control hardware blocks, modules or components of the receiver (400) to at least determine RSSI values of wireless signals of one or more objects and to communicate the RSSI values to a common computing unit (500) in accordance with claim 8 or, when executed by a microprocessor of a common compute unit (500) in accordance with claim 8, cause the microprocessor and / or control hardware blocks, modules or components of the common compute unit (500) to execute the method of one or more of claims 1 to 5.202304050 -35- 10. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 9.

11. A wireless communication system comprising two or more receivers (400) in accordance with claims 6 or 7, which are communicatively connected to a common compute unit (500) in accordance with claim 8.

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