Search

The Wiener-Khinchin theorem for the Fourier-Laplace transformation (WKT-FLT) provides a robust method to obtain the single-side Fourier transforms of arbitrary time-domain relaxation functions (or autocorrelation functions). Moreover, by combining an on-the-fly algorithm with the WKT-FLT, the numerical calculations of various complex spectroscopic data in a wide frequency range become significantly more efficient. However, the discretized WKT-FLT equation, obtained simply by replacing the integrations with the discrete summations, always produces two artifacts in the frequency-domain relaxation function. In addition, the artifacts become more apparent in the frequency-domain response function converted from the relaxation function. We find the sources of these artifacts that are associated with the discretization of the WKT-FLT equation. Taking these sources into account, we derive discretized WKT-FLT equations designated for both the frequency-domain relaxation and response functions with the artifacts removed. The use of the discretized WKT-FLT equations with the on-the-fly algorithm is illustrated by a flow chart. We also give application examples for the wave-vector-dependent dynamic susceptibility in an isotropic amorphous polyethylene and the frequency-domain response functions of the orientation vectors in an $n$-alkane crystal.

Considering that fractional correlation function and the fractional power spectral density, for $\alpha$-stationary random signals, form a fractional Fourier transform pair. We present an interpolation formula to estimate a random signal from a temporal random series, based on the fractional sampling theorem for $\alpha$ -bandlimited random signals. Furthermore, by establishing the relationship between the sampling theorem and the von Neumann ergodic theorem, the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach. Thus, the validity of the sampling theorem for random signals is closely linked to an ergodic hypothesis in the mean sense.

© 2017 The Authors. Published by Elsevier Ltd. In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform.

This work derives a version of Isserlis’ theorem for the specific case of four mixed-Gaussian random variables. The theorem is then used to derive an expression for the auto-bispectral density for quadratically nonlinear systems driven with mixed-Gaussian iid noise.

The Wiener–Khinchin theorem states that the power spectrum is the Fourier transform of the autocovariance function. One form of the autocovariance function can be obtained through recurrence quantification. We show that the advantage of defining the autocorrelation function with recurrences can demonstrate higher dimensional dynamics.

We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 < x < l$ and $t > 0$ $$ rm(H) quad left beginarrayl displaystyle partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + C(x) beginpmatrix u(t,x) v(t,x) endpmatrix = 0, displaystyle d over dt left [ v(t,l) - D u(t,l) right ] = F u(t,cdot) + G v(t,cdot) , displaystyle u(t,0) = E v(t,0), endarray right . $$ where $u(t,x) in C^n_1$, $v(t,x) in C^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ is a diagonal matrix of functions $k_i in C^1left( [0,l], R right)$, $k_i(x) > 0$ for $i = 1, dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $mathrm(H)$ which generates $C_0$ semigroup $e^At$ on $L^2 times C^n_2$. Our main result is the following spectral mapping theorem $$sigma(e^At) setminus 0 = overlinee^sigma(A)t setminus 0 .$$

We present the anisotropic version of the classical Alon–Boppana theorem on the asymptotic spectrum of random walks on infinite families of graphs. The relations to the Grigorchuk–Żuk theory—on the space of graphs with uniformly bounded degree and the continuity of the spectral measure of Markov operators related to the Alon–Boppana theorem and the Burger–Greenberg–Serre theorem—are also indicated.

The classical Wiener-Khinchin theorem (WKT), which can extract spectral information by classical interferometers through Fourier transform, is a fundamental theorem used in many disciplines. However, there is still need for a quantum version of WKT, which could connect correlated biphoton spectral information by quantum interferometers. Here, we extend the classical WKT to its quantum counterpart, i.e., extended WKT (e-WKT), which is based on two-photon quantum interferometry. According to the e-WKT, the difference-frequency distribution of the biphoton wavefunctions can be extracted by applying a Fourier transform on the time-domain Hong-Ou-Mandel interference (HOMI) patterns, while the sum-frequency distribution can be extracted by applying a Fourier transform on the time-domain NOON state interference (NOONI) patterns. We also experimentally verified the WKT and e-WKT in a Mach-Zehnder interference (MZI), a HOMI and a NOONI. This theorem can be directly applied to quantum spectroscopy, where the spectral correlation information of biphotons can be obtained from time-domain quantum interferences by Fourier transform. This may open a new pathway for the study of light-matter interaction at the single photon level., 13 pages, 5 figures

The aim of this paper is to prove a flxed point theorem via contraction mappings of a pair of weakly increasing mappings using an altering function in a partially ordered complete separable metric spaces. Our theorem is useful to determine a large of nonlinear problems. We discuss the existence of a common solution to a system of nonlinear random integral equations.

The dynamics of the wave packet generated by a short-pulse laser can be characterized by an autocorrelation function. The autocorrelation function is an experimentally measurable quantity. This paper shows that the autocorrelation function can be written as a sum of modified Gaussian functions that correspond to the closed orbits of the system. The autocorrelation function can be calculated by using existing algorithms in closed-orbit theory. As an application of the theory, the autocorrelation function is obtained for a system that consists of an H[sup [minus]] ion in a static electric field.

We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

Consistency issues related to autocorrelation estimation for Gaussian processes with mixed spectra are clarified. The sample autocorrelation is known not to be consistent when the spectrum contains spectral atoms. This fact is verified by computing explicitly its mean-square limit in terms of the random measure assigned to the atoms of the process. The alternative estimator constructed from the zero-crossing rate is likewise not consistent in general. However, it is consistent for a spectrum supported at a single frequency, or a spectrum for which the ‘signal’ and ‘noise’ have the exact same first-order autocorrelation. This is proved, using recent results from multiple Wiener-Ito integral expansions for level-crossing counts, by direct computation of the spectrum of the zero-crossing indicator process when the underlying process has a mixed spectrum. In general, regardless of the spectral type, the asymptotic zero-crossing rate admits values between the lowest and highest positive frequencies with probability one. For band-limited processes, this fact provides an easy way to assess the precision of functions of the zero-crossing rate.

We are now in a good position to state and prove the spectral theorem for selfadjoint and compact operators on Hilbert spaces.

The Spectral Theorem is a landmark in the theory of operators on Hilbert space, providing a full statement about the nature and structure of normal operators. Normal operators play a central role in operator theory; they will be defined in Section 6.1 below. It is customary to say that the Spectral Theorem can be applied to answer essentially all questions on normal operators. This indeed is the case as far as “essentially all” means “almost all” or “all the principal”: there exist open questions on normal operators. First we consider the class of normal operators and its relatives (predecessors and successors). Next, the notion of spectrum of an operator acting on a complex Banach space is introduced. The Spectral Theorem for compact normal operators is fully investigated, yielding the concept of diagonalization. The Spectral Theorem for plain normal operators needs measure theory. We would not dare to relegate measure theory to an appendix just to support a proper proof of the Spectral Theorem for plain normal operators. Instead we assume just once, in the very last section of this book, that the reader has some familiarity with measure theory, just enough to grasp the statement of the Spectral Theorem for plain normal operators after having proved it for compact normal operators.

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.

In analyzing neural data, we often have to deal with quantities that fluctuate randomly in time. For example, an intracellular recording of the membrane potential of a neuron in vivo or a sequence of extracellular action potentials. Variables that fluctuate randomly in time are called stochastic processes or random functions. In this chapter, we define two types of stochastic processes that are used, respectively, to describe continuous variables, such as random potential fluctuations, and events, like random sequences of action potentials. We have already encountered the simplest such process in previous chapters in the form of the homogeneous Poisson and gamma renewal processes. Next, we introduce some of the tools used to study the properties of stochastic processes. In particular, we will see that the Fourier transform allows us to characterize the frequency content of stochastic processes through spectral analysis.

We generalize the concept of the autocorrelation function and give the generalization of the Wiener-Khinchin theorem. A full generalization is presented where both the autocorrelation function and power spectral density are defined in terms of a general basis set. In addition, we present a partial generalization where the density is the Fourier transform of the characteristic function but the characteristic function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.

We generalize the Wiener-Khinchin theorem. A full generalization is presented where both the autocorrelation function and power spectral density are defined in terms of a general basis set. In addition, we present a partial generalization where the density is the Fourier transform of the autocorrelation function but the autocorrelation function is defined in terms of an arbitrary basis set. Both the deterministic and random cases are considered.

In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.

The spectral characterization of second-order (or wide-sense) cyclostationarity gives rise to a generalization of the Wiener relation between the power spectral density and the autocorrelation associated with second-order stationary time-series. This generalization, called the cyclic Wiener relation, is a Fourier transform relation between the spectral autocorrelation function and the cyclic temporal autocorrelation function, both defined in terms of time averages on a single time-series. The spectral characterization is generalized from second-order cyclostationarity to n-th order cyclostationarity for n=2,3,4,5,. . ., and some basic properties of the generalised spectral characterization are presented. These include a further generalization of the Wiener relation, called the n-th order cyclic Wiener relation, which relates the n-th order joint cyclic temporal moment function to the n-th order joint spectral moment function. >

Equilibrium fluctuations of dynamic quantities and the associated spectral amplitudes can be regarded as stationary functions and orthogonal measures, respectively, with values in a Hilbert space. The mathematical basis of this description and of the equivalent probabilistic description is re-examined. The latter is in terms of stationary random functions and orthogonal random measures, with real or complex values in the classical case, and values in the Hilbert space of state vectors in quantum mechanics. In either case, the Wiener-Khinchin theorem assures that self-correlation functions can be represented as Fourier transforms of non-negative measures—the squared norms of the amplitudes. The role of the theorem in linear response theory is discussed.

This paper deals with a new method for the design and implementation of a gaussian pseudorandom generator with a specific power spectral density or autocorrelation function. The method is based on the central limit theorem. The mathematical analysis of the generator, which consists of a digital white-noise generator and correlation-shaping transversal filter, is given. Computer simulation and experimental results show that the required process is generated with high accuracy.

It is shown in this note that if Q is an algebra of uniformly bounded mean-square continuous real-valued random functions indexed in a compact set T, containing all bounded random variables and separating points of T (i.e., given t1 and t2 in T, there is a random function Xt in Q such that , then given any mean square continuous random function, there is a sequence in Q converging in mean square to the given random function uniformly on T.

The study of Pick functions has long been associated with certain problems in analysis, in particular the Moment Problem and the theory of the spectral representation of self-adjoint operators in Hilbert space. The traditional proof of the spectral theorem deduces it from the integral representation of Pick functions given in Chapter II, and this argument has the advantage that the self-adjoint operator under study need not be supposed continuous. This chapter is devoted to that proof.

A classical theorem of statistical optics, the van Cittert-Zernike theorem, is generalized to pulse-echo ultrasound. This theorem fully describes the second-order statistics of the spatial fluctuations (the spatial covariance) of the field produced by an incoherent source. As a random scattering medium is insonified, it behaves as an incoherent source. The van Cittert-Zernike theorem can thus predict the spatial covariance of the pressure field backscattered by a random medium. Experimental results obtained with a linear array are in good agreement with theoretical expectations. The implications of this theorem in speckle reduction and in focusing in nonhomogeneous media are discussed. >

It is shown that for a random function of two or three dimensions and the case where statistical properties are isotropic in space, the Wiener-Khintchine theorem takes on the form of a Fourier-Bessel transform instead of a one-dimensional Fourier transform.

The essential concepts and equations associated with the spectral analysis of stationary random processes are the power spectral density, the autocorrelation function, their mutual relationship known as the Wiener‐Khintchine theorem, and the power spectrum input‐response relationship for linear systems. These concepts and equations can be extended in a natural and unified way to nonstationary random processes. To provide these extensions, it is shown that two time‐dependent power spectrum definitions are required—both of which reduce to the same definition in the case of stationary processes. One of these spectrum definitions is essentially equivalent to the class of time‐dependent spectra obtained by narrow‐band filtering; hence, this definition possesses a clear physical interpretation. However, it does not possess a simple input‐response relationship for linear systems. The second spectrum definition is the Fourier transform of a time‐dependent autocorrelation function. Realizations of this second spec...