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## Scattering target identification based on radial basis function artificial neural networks in the presence of non-stationary noise

The paper deals with the radar target discrimination problem performed on complex radar images. The approach based on radial basis function (RBF) artificial neural network (ANN) is proposed for the identification of point scatterers placed within a radar image. The renewed concept of simple adaptive units as the foundation for network assembling allows one to design an ANN-based feature extraction scheme for the two-dimensional signal processing. It was shown that ANN implementing RBF neural processing units could be applied for the identification of radar targets described by the set of separated scatterers, even in cases where the relative distance between the scatterers is comparable to or less than the effective width of each scatterer. The obtained results indicate a high accuracy estimation of separate scatterer centers in the presence of noise which is not limited to the stationary case but supposed to be cyclostationary. It was also shown that the parameters describing the coordinates of scattering centers could be successfully extracted from the trained ANN after about one hundred epochs spent on ANN training process, which is carried out by means of modified gradient descent method. The main result is to demonstrate the possibility of using neural networks to automatically analyze radar images, which is an integral part of a set of tasks that form the target recognition problem. The proposed algorithm implements an approach of identification systems made using a neural network training procedures.

This chapter presents the algorithm for estimating spectral correlation function of a random process that is a valid bi-frequency description of the probabilistic properties of any wide-sense cyclostationary process and relates to its cyclic autocorrelation function via Fourier transform. The key point of the algorithm is that it is based on the two-dimensional discrete Fourier transform of the sample dyadic correlation function weighted by the two-dimensional windowing function, which is rectangular in the direction orthogonal to the current-time axis shape. The dedicated mathematical software libraries implementing fast Fourier transform, which is typically used for image processing, achieve higher performance in comparison with other algorithms involving spectra accumulation. The signal containing a pulse sequence with random amplitudes masked by the additive stationary white Gaussian noise is used in numerical simulation to provide an example of the spectral correlation function estimation procedure and obtain results demonstrating the effectiveness of the proposed algorithm.

This paper presents an approach to estimating delays of the cyclostationary signal propagating through tracks of the data bus of a printed circuit board (PCB). The signal path is described by the series of delays estimated using cyclostationary characteristics in the frequency domain. A brief overview of cyclostationarity phenomena is given alongside with the designed practical algorithm performing wideband estimation of the spectral correlation density of the signal. Delay estimations obtained with the proposed approach based on cyclic spectral correlation function are much closer to the values evaluated geometrically on planar scheme than delays calculated by means of well-known generalized cross-correlation algorithm (GCC). The proposed algorithm is verified by the results of experiments featuring FPGA board.

The angular coordinates of the pulse source are determined by comparing the signals received simultaneously on several channels. To solve this issue, the application of neural networks is highly important. In this article, the application of the artificial neural network (ANN) approach to the task of target localization is discussed. The research was performed on the basis of a feature extraction technique executed by a discrete cosine transform, which allowed to obtain a compact representation of the signal energy subjected to digital processing. The author defines the angle-of-arrival estimation scheme based on time difference of arrival estimators and the particular problem of estimating constant delays as informative parameters embedded into received signals that are noisy and damped copies of the reference signal. The adaptive element framework is used for synthesis of the feedforward ANN which is fed with the reduced set of the most sensitive discrete cosine transform (DCT) coefficients, which provide a concise representation of the first-order cyclostationary random process. The investigation on the delay estimation accuracy has been carried out to evaluate the performance of the ANNs with different size of their hidden layer and various numbers of the DCT coefficients at their input. It has been proved that five DCT coefficients are enough for the discrimination of the phase shift in the whole range. In turn, it results in the reliable delay estimation produced by the trained ANN that contains eight neurons in its hidden layer.

Introduced in this paper, the averaged absolute spectral correlation density estimator (AASCD) is based on time-smoothing algorithm of estimating spectral correlation density (SCD) using a long-time sample of discrete signal. The key feature of proposed estimator is compressing the cyclic frequency dimension of SCD estimation to make the matrix containing the SCD estimation cells balanced in both dimensions. The compressing is performed by means of averaging the absolute values of SCD in blocks of neighboring cyclic frequencies which prevents any components of a signal possessing wide-sense cyclostationarity from leaving out of the coverage of the estimator. The application of the proposed algorithm to the cyclic property analysis of the baseband signal in a digital interface measured by an oscilloscope allowed to visualize the estimator over bifrequency plane “frequency–cyclic frequency” via a color plot. Further searching regions with relatively high level of correlation for fine peaks gave an opportunity to estimate the exact values of cyclic frequencies inherent to the process under investigation.

The paper deals with cyclostationarity as a natural extension of stationarity as the key property in designing the widely-used models of random processes. The comparative example of two processes, one is wide-sense stationary and the other is wide-sense cyclostationary, is given in the paper and reveals the lack of the conventional stationary description based on one-dimensional autocorrelation functions. It is shown that two significantly different random processes appear to be characterized by exactly the same autocorrelation function while their two-dimensional autocorrelation functions provide outlook where the difference between processes of two above-mentioned classes becomes much clearer. More concise representation by expanding the two-dimensional autocorrelation function to its Fourier series where the cyclic frequency appears as the transform parameter is illustrated. The closed-form expression for the components of the cyclic autocorrelation function is also given for the random process which is an infinite pulse train made of rectangular pulses with randomly varying amplitudes.

In this paper, we study scalar multivariate non-stationary subdivision

schemes with integer dilation matrix M and present a unifying, general approach

for checking their convergence and for determining their Hölder regularity (latter in

the case M = mI,m ≥ 2). The combination of the concepts of asymptotic similarity

and approximate sum rules allows us to link stationary and non-stationary settings

and to employ recent advances in methods for exact computation of the joint spectral

radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity

of the generalized Daubechies wavelets. We illustrate our results with several

examples.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables