相关论文: Nonlinear Spectral Characterization of Discrete Da…

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier…

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These…

Given an $n$-length input signal $\mbf{x}$, it is well known that its Discrete Fourier Transform (DFT), $\mbf{X}$, can be computed in $O(n \log n)$ complexity using a Fast Fourier Transform (FFT). If the spectrum $\mbf{X}$ is exactly…

In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the $\theta$-translation operator, the translation operator associated with the FrFT. Subsequently,…

For any finite group $G$, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to $G$, using $O(|G|^{\omega/2 + \epsilon})$ operations, for any $\epsilon > 0$. Here, $\omega$ is the exponent…

This paper introduces a couple of new time-frequency transforms, designed to adapt their scale to specific features of the analyzed function. Such an adaptation is implemented via so-called focus functions, which control the window scale as…

In this paper, a novel decomposition method for non-stationary and nonlinear signals is proposed. This method is inspired by the adaptive wavelet filter bank of the empirical wavelet transform (EWT) and Fourier intrinsic band functions…

Digital filters for recursively computing the discrete Fourier transform (DFT) and estimating the frequency spectrum of sampled signals are examined, with an emphasis on magnitude-response and numerical stability. In this tutorial-style…

We design a two-dimensional signal constellation based on the exact periodic inverse nonlinear Fourier transform. Feasibility of continuous transmission with periodic signals is experimentally demonstrated over more than 2000 km.

The paper is devoted to the development of the octonion Fourier transform (OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a continuation and generalization of earlier work by Blaszczyk and Snopek, where they proved…

The singular values of convolutional mappings encode interesting spectral properties, which can be used, e.g., to improve generalization and robustness of convolutional neural networks as well as to facilitate model compression. However,…

Time-frequency representations (TFRs) of signals, such as the windowed Fourier transform (WFT), wavelet transform (WT) and their synchrosqueezed variants (SWFT, SWT), provide powerful analysis tools. However, there are many important issues…

The state-of-the-art automotive radars employ multidimensional discrete Fourier transforms (DFT) in order to estimate various target parameters. The DFT is implemented using the fast Fourier transform (FFT), at sample and computational…

Nonlinear adaptive filtering allows for modeling of some additional aspects of a general system and usually relies on highly complex algorithms, such as those based on the Volterra series. Through the use of the Kronecker product and some…

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective…

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on…

The aim of this paper is to study the normal forms of nonautonomous differential systems. For doing so, we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy,…

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods…

The metabolic process in a cell is modeled with the use of the Fourier transformation. The histograms of the invariant measures of chaotic attractors are constructed. In particular, a scenario of adaptation of the metabolic process under a…

The significance of the broken ray transform (BRT) is due to its occurrence in a number of modalities spanning optical, x-ray, and nuclear imaging. When data are indexed by the scatter location, the BRT is both linear and shift invariant.…