Modelling of radio-frequency communication channels using Volterra model

Teaching Aids for Modeling and Control of Hybrid Systems (CAMCHS)

This paper presents developed engineering software tools used for nonparametric identification of nonlinear dynamical systems based on Volterra models. The polyharmonic test sequences with different amplitudes are used for test procedures in frequency domain. The wavelets application allowed enlarging computational stability of the identification method for measurement noises filtering of received responses and characteristics of the system being identified. The methodology proposed in pervious works and developed software tools applied for construction communication channel models of different orders.

This article presents a theoretical framework for the identification of nonlinear dynamic MISO systems with orthonormal base function models on fundamental basics of the Volterra theory. In the past the Volterra theory was used for the identification of nonlinear dynamic SISO systems (e.g. Hammerstein and Wiener models). In principle it is possible to extend these approaches to systems with more than one input (i.e. MISO systems). In former times this failed due to a lack of computational performance. In this paper an approach is presented which allows the identification of arbitrary coupled Hammerstein and Wiener models with multiple inputs. Some fundamental considerations for the identification of MISO systems based on arbitrary coupled Hammerstein models are made and extended to MISO systems based on arbitrary coupled Wiener models. This extended Volterra theory results in equations where the unknown parameters can be separated of the input values in a linear manner. In order to approximate the truncated impulse responses of the linear dynamic systems and to reduce the number of unknown parameters orthonormal base functions (OBFs) are introduced. As an example the proposed identification method is applied to a MISO system based on forward and backward coupled Hammerstein and Wiener models.

A structural-multipath approach has been developed for constructing a spatiotemporal model of a single-mode decameter communication channel, which makes it possible to determine the two-frequency spatial correlation function of a single-mode decameter communication channel and to establish the dependences of frequency and spatial correlation intervals of fading on the operating frequency, diffuse ionosphere parameters, and radio link geometry. The method for constructing this model of a communication channel using the structural multipath approach includes four stages: 1) development of a multibeam spatiotemporal decameter model of a communication channel; 2) development of a radiophysical space-time model of radio wave propagation in a decameter radio link, taking into account the influence of inhomogeneities (diffusion) of the ionosphere based on the method of smooth perturbations; 3) identification of multipath and radiophysical models to determine the two-frequency spatial correlation function of the decameter of the communication channel; 4) determining the dependence of frequency and spatial correlation intervals of fading in a single-mode decameter communication channel on the choice of operating frequency, diffuse ionosphere parameters, and radio link geometry.

The new criterion for selecting the frequencies of the test polyharmonic signals is developed. It allows uniquely filtering the values of multidimensional transfer functions — Fourier-images of Volterra kernel from the partial component of the response of a nonlinear system. It is shown that this criterion significantly weakens the known limitations on the choice of frequencies and, as a result, reduces the number of interpolations during the restoration of the transfer function, and, the more significant, the higher the order of estimated transfer function.

A method for identifying nonlinear dynamic systems with noise is proposed by using probabilistic universal learning networks (PrULNs). PrULNs are extensions of universal learning networks (ULNs). ULNs form a superset of neural networks and were proposed to provide a universal framework for modeling and control of nonlinear large-scale complex systems. But the ULN does not provide any stochastic characteristics of the signals propagating through it. The PrULNs are equipped with machinery to calculate stochastic properties of signals and to train network parameters so that the signals behave with the pre-specified stochastic properties. On the other hand it is generally recognized that there exists an overfitting problem when identification of nonlinear dynamic systems with noise is done by neural networks. In this paper, it is shown from simulation results of identification of a nonlinear robot dynamics that PrULNs are useful for avoiding the overfitting.

Discovering nonlinear PDEs with sparse identification of nonlinear dynamical systems (SINDy) is hindered by high dimensionality, noise, and expensive data acquisition. We propose the greedy sampling neural network for sparse identification of nonlinear PDEs (GN-SINDy), a three–stage framework that integrates strategic sampling, differentiable surrogate modelling, and sparse equation discovery. First, a two–way Q-DEIM–based greedy strategy selects maximally informative space–time samples from snapshot data, drastically reducing data requirements. Second, a deep neural network (DNN) is trained as a differentiable surrogate of the solution field, enabling noise–robust analytic derivatives via automatic differentiation. Third, sparse regression with sparsity–promoting estimators [Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016a). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937; Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614.] is applied to recover the governing PDE. Building on the DeepMoD paradigm, GN-SINDy embeds greedy sampling into data acquisition and stabilises coefficient estimation through neural–enhanced differentiation. We analyze noise robustness, structural stability, and QR–based sampling strategies to guide sampler and hyperparameter selection. Experiments on Burgers', Allen–Cahn, and Korteweg–de Vries equations show that GN-SINDy reliably recovers governing PDEs using under 1 % of the data, outperforming DeepMoD in efficiency, support recovery, and robustness to noise.

A procedure based on the use of artificial neural networks for the identification of nonlinear dynamic systems is developed and applied to the damped Duffing oscillator under deterministic excitation. The “generalization” ability of neural networks is invoked to predict the response of the same nonlinear oscillator under stochastic excitations of differing magnitude. The analogy between the neural network approach and a qualitatively similar nonparametric identification technique previously developed by the authors is illustrated. Some of the computational aspects of identification by neural networks, as well as their fault-tolerant nature, are discussed. It is shown that neural networks provide high-fidelity mathematical models of structure-unknown nonlinear systems encountered in the applied mechanics field.

Nonlinear dynamic systems are characterized with uncertainties in terms of structure and parameters. These uncertainties cannot be described by deterministic models. The modelling and identification of nonlinear dynamic systems through the measured experimental data is a problem in engineering and technical processes. Therefore, field of system identification have become an important area of research. Fuzzy technology is an effective tool for dealing with complex nonlinear processes that are characterized with uncertain factors. In this paper, a novel approach based on Local Linear method learning in dynamical filter weights neurons for the identification of non-linear dynamic systems is presented. The fuzzy wavelet neural network combines wavelet theory with fuzzy logic and neural networks. Learning fuzzy rules and parameter update in fuzzy wavelet neural network is based on gradient decent method. The proposed approach is said to be Fuzzy Local Linear Wavelet Neural Network based model. It has been explained through examples. The structure is tested for the identification with both wavelet neural network and Fuzzy Local Linear Wavelet Neural Network that shows the comparative performance.KeywordsSystem IdentificationNon-linear SystemWavelet Neural NetworkFuzzy Wavelet Neural NetworkLocal Linear Wavelet Neural NetworkFuzzy Local Linear Wavelet Neural Network

The paper demonstrates that radial basis function network (RBFN) with adaptive centers and width can be used effectively for identification of nonlinear dynamic system. The proposed RBFN is trained by hybrid learning algorithm, which uses conjugate gradient optimization algorithm to obtain the center and width of each radial basis function and the least squares method to obtain the weights. To avoid capturing a local optimum, regularization error energy function is used and the centers of basis functions are initialized using a fuzzy C-means clustering method. Simulation results reveal that the identification schemes based on RBFN gives considerably better performance and show faster learning in comparison to previous methods

In this paper, modeling and identification of nonlinear dynamic systems using neuro-fractional Hammerstein model are considered. The proposed model consists of the neural networks (NNs) as the nonlinear subsystem and the fractional-order state space (FSS) as the linear subsystem. The identification procedure consists of a hybrid frequency-/time-domain approach based on the input–output data acquired from the system. First in the frequency domain, the fractional order and fractional degree of the FSS subsystem are determined offline using an iterative linear optimization algorithm. Then, in the time domain, the state-space matrices of the FSS as well as parameters of the NN are estimated using Lyapunov stability theory. Moreover, in order to use only the input–output data from the system, a fractional-order linear observer based on auxiliary model idea is utilized to estimate the system states. The convergence and stability analysis of the proposed method are provided. Simulating and experimental examples show superior performance of the proposed method as compared with the Hammerstein models reported in the literature.

This paper compares radial basis function networks for identification of nonlinear dynamic systems with classical methods derived from the Volterra series. The performance of these different approaches, such as Hammerstein, Wiener and NDE models, is analysed. Since the centres and variances of the Gaussian radial basis functions will be fixed before learning and only the weights are learned, a linear optimization problem arises. Therefore training the network and parameter estimation becomes comparable in computational effort. It is shown that the classical methods can compete or even perform better than the neural network, if the assumptions for the structure are valid. However, in practical applications when the structure is not known the radial basis function network performs much better than the classical methods.

The integrated sensing and communication (ISAC) refers to the integration of radio sensing and wireless communications to realize the multiplexing of space, time, and frequency resources. In the sixth generation (6G) wireless networks, ISAC is considered as one of the most promising technologies. In this paper, a novel ISAC channel model combining forward and backward scattering is proposed. In addition to the non-stationarity caused by motions, the correlations between sensing and communication channels are investigated. The channel characteristics of sensing such as forward scattering and backward scattering are introduced into the communication channel model. Utilizing the correlations between sensing and communication channels, the communication channel model is divided into line-of-sight (LOS), forward scattering, and backward scattering components. These three components are summed according to probability weighting to obtain a more accurate channel model for sensing assisted communication systems. Moreover, important statistical properties of the proposed ISAC channel model are derived and simulated. The analytical and simulation results match well, demonstrating the correctness of derivations and simulations. Some derived/simulated statistical properties are verified by corresponding measurement data, which indicates the utility of the proposed ISAC channel model.

Structure Identification of Continuous-Time Block-Oriented Nonlinear Systems in the Frequency Domain

In this paper, advanced concepts for the identification of complex nonlinear systems are discussed. Three major problems are addressed: The nonlinearity of the system, noise in the data upon which the model has to be built, and the potential to incorporate qualitative and quantitative prior knowledge about the system. As an integrated solution approach, local model networks (LMNs) with appropriate parameter estimation schemes are proposed. LMNs generally offer a versatile structure for the identification of nonlinear dynamic systems. In order to account for a realistic situation when noise is present both in input and output data, an equality constrained generalised total least squares algorithm for the local model parameter estimation of the LMN is presented; the incorporation of equality constraints allows to mathematically enforce desired system properties. As an application and benchmark problem, the vertical dynamics of a vehicle is considered. After training the LMN on a rough road, excellent predictions of the behaviour of the vehicle at crossing a single obstacle are obtained, thus proving the effectiveness of the proposed algorithm. It is illustrated how both the application of a proper parameter estimation scheme and the integration of system constraints systematically improve the performance of the model.