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IEEE The use of the Wiener–Khinchin theorem in the reverberation chamber reveals the relationships between a number of important parameters—the coherence bandwidth and the Q-factor measured in the time domain, the coherence time and the Q-factor measured in the frequency domain, the K-factor and the Doppler spectrum, and the K-factor and the total scattering cross section. The lower bound of the average K-factor is also given. Different physical quantities, which share similar mathematical insights, are unified. Analytical derivations are given, and results are validated by measurements.

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.

A discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise is developed. Based on the discrete version of Wiener-Khinchin theorem, the theoretical discrete Chebyshev spectrum for the Markov random process is calculated. It is characterized by two parameters: the dispersion and the relaxation frequency (or relaxation time). The noise of corrosion process and the noise of recording equipment are measured. Using the theoretical Chebyshev spectrum, the Markov parameters were found both for the noise of the corrosion process and for the noise of the measuring equipment.

This chapter presents the principles of dynamic analysis of structures subjected to the variation in time service loading. To illustrate, the single-degree-of-freedom behavior under cyclic excitation in the range of frequencies is described as introductory to dynamic analysis of structures. The principles of spectral analysis and evaluation of response of dynamic structures to spectral excitation (Wiener–Khinchin theorem) are discussed.

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.

Integrals with respect to stationary random measures are considered. A central limit theorem for such integrals is proved. The results are applied to obtain a functional central limit theorem for transformed solutions of the Burgers equation with random initial data.

which means that the autocorrelation function of a signal corresponds to the power spectrum of the signal, is very important in signal processing, spectroscopy and telecommunications engineering. However, because of needs for some relatively expensive equipments such as a correlator and the signal processing system, its demonstration in most undergraduate class is not easy so far. Recently, digital oscilloscopes whose functions can be replaced foresaid equipments are marketed with development of digital engineering. In this paper, a simple demonstration of the theorem is given by a digital storage oscilloscope and a personal computer with its theoretical background. The reason that deals again with this theorem which has been introduced in 1930 is that it has been not well informed yet to us and theoretical background of the demonstration is directly introduced from its driving process. Through deriving process of the theorem, some extended physical meanings of the impedance, power, power factor, Wiener spectrum, linear system response and, furthermore, basic idea of the Planck's quantization in the black body theory reveal themselves naturally. Hence it can be referred to lectures in general physics, modern physics, spectroscopy and material characterization experiment.Keywords: Wiener-Khinchin theorem, Digital oscilloscope, Demonstration, Wiener spectrumOCIS codes: (000.2060) Education; (000.2190) Experimental physics; (000.2658) Fundamental tests

The last chapter has illustrated the usefulness of power spectra to describe the frequency characteristics of stochastic processes. In this chapter, we generalize the power spectrum to characterize the frequency-dependent relation between two stochastic processes. This leads us to define first the cross-spectrum of two stochastic processes and then their coherence. Next, we tackle the problem of estimating numerically power and cross-spectra from experimental data. §20.2 makes some basic preliminary observations on the properties of estimates arising from random data samples. §20.3 then tackles the numerical power spectrum estimation problem.

Design of strong symmetric cipher systems requires that the underlying cryptographic Boolean function meet specific security requirements. Some of the required security criteria can be measured with the help of the Autocorrelation function as a tool, while other criteria can be measured using the Walsh transform as a tool. The connection between the Walsh transform and the Autocorrelation function is given by the well known Wiener-Khintchine theorem. In this paper, we present an analysis of the Autocorrelation function from the Haar spectral domain. We start by presenting a brief review on Boolean functions and the Autocorrelation function. Then we exploit the analogy between the Haar and Walsh in deriving the Haar general representation of the Autocorrelation function. The derivations are carried out in two ways namely, in terms of individual spectral coefficients, and based on zones within the spectrum. The main contribution of the paper is the establishment of the link between the Haar transform and the Wiener-Khintchine theorem. This is done by deducing the connection between the Haar transform, the Autocorrelation, and the Walsh power spectrum for an arbitrary Boolean function. In the process we show that, the same characteristics of the Wiener-Khintchine theorem holds locally within the Haar spectral zones, instead of globally as with the Walsh domain. The Haar general representations of Autocorrelation function are given for arbitrary Boolean functions in general and Bent Boolean functions in particular. Finally, we present a conclusion of the work with a summary of findings and future work.

We prove a theorem on the equivalence of some properties of a random field defined in terms of sample functions. We apply this theorem for studying generalized random fields.

We consider the spatial power spectra associated with fluctuations of quadratic operators in field theory, such as quantum stress tensor components. We show that the power spectrum can be negative, in contrast to most fluctuation phenomena where the Wiener-Khinchine theorem requires a positive power spectrum. We show why the usual argument for positivity fails in this case, and discuss the physical interpretation of negative power spectra. Possible applications to cosmology are discussed., 4 pages, 1 figure

In this paper, some new results are given for the common random solution for a class of random operator equations which generalize several results in [4], [5] and [6] in Banach space. On the other hand, Altman’s inequality is also extending into the type of the determinant form. And comparing some solution for several examples, main results are theorem 2.3, theorem 3.3-3.4, theorem 4.1 and theorem 4.3.

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin’s theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

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.

International audience; We prove a weighted Almost Sure Limit Theorem in the setting of random iterative models. This theorem generalizes previous results obtained for sequences of normalized partial sums and some other classes of random sequences.

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 $$sigma(e^At) setminus 0 = overlinee^sigma(A)t setminus 0 .$$

Motivated by the works of Delsarte and Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed. The authors thank the referee for her/his valuable suggestions and comments which have contributed to improve the final form of this paper. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014 and partially supported by the Brazilian Government through the CNPq under the project 470019/2013-1 and the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The work of the second and third authors is partially supported by Dirección General de Programas y Transferencia de Conocimiento, Ministerio de Ciencia e Innovación of Spain under the project MTM2011–28781.

We derive a generalization of the Wiener-Khinchin theorem for nonstationary processes by introducing a time-dependent spectral density that is related to the time-averaged power. We use the nonstationary theorem to investigate aging processes with asymptotically scale-invariant correlation functions. As an application, we analyze the power spectrum of three paradigmatic models of anomalous diffusion: scaled Brownian motion, fractional Brownian motion and diffusion in a logarithmic potential. We moreover elucidate how the nonstationarity of generic subdiffusive processes is related to the infrared catastrophe of 1/f-noise., 7 pages, 2 figures (including supplemental material)

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

In this chapter we prove the spectral theorem for normal operators and study the corresponding spectral measures in some detail. In particular, we introduce the maximal spectral type and the multiplicity function yielding together a complete isomorphism invariant for unitary operators. Based on these fundamental results we interpret various mixing properties in spectral terms and explain systems with discrete spectrum from a point of view different from the one taken in Chapter 17 In addition, a number of examples illuminate the fundamental ideas of the spectral theory of dynamical systems. Readers interested in details and in the more advanced theory can consult Queffelec (1987), Nadkarni (1998b), Lemanczyk (1996), Katok and Thouvenot (2006), Lemanczyk (2009), and the multitude of further references therein.

In this paper, the idea of random selection in the theorem on gambling system is extended to Markov chains, by using the notion likelihood ratio and an analytic technique. A strong limit theorem on the relative frequency of ordered couple under random selection is established.

Proof of the central limit theorem for the sample mean can be obtained in the case of independent and identically distributed randomvariables through the moment generating function under some simplifying assumptions. The case for the central limit theorem for the sample mean from finite populations under simple random sample without replacement, the parallel to the simplest case in the standard framework, is not as simple. Most sampling textbooks avoid any technical discussion of the finite population central limit theorem so that there is little appreciation for the conditions that underlay the theorem. Most often the theorem is illustrated with a simulation study; but this can only hint at the technicalities of the theorem. Now, new software has been developed that allows for automatic calculation of moments and cumulants of estimators used in survey sampling, as well as automatic derivation of unbiased or consistent estimators. This software is used to examine the assumptions behind the applications of...

The purpose of this paper is to study the long-time behaviour of the partial autocorrelation function of a stationary process. Let {Xn} = {Xn : n ∈ Z} be a real, zero-mean, weakly stationary process, defined on a probability space (Ω,F , P ), which we shall simply call a stationary process . Throughout this paper, we assume that {Xn} is purely nondeterministic (see §2). The autocovariance function γ(·) of {Xn} is defined by

In this paper we study the limiting behavior of sums of dependent random variables under a strong mixing condition. We obtain conditions for which an analog of the Baum-Katz theorem holds and cite an example showing their optimality.

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.

We extend Szego's eigenvalue distribution theorem to block Toeplitz matrices. A sequence of such matrices arises from the autocorrelation of a 2-D discrete random process. We show that the eigenvalues of the matrix sequence are asymptotically distributed like the samples of the random process' 2-D power spectrum.

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.

A widely acclaimed achievement of probability has been its success in approximating the distributions of arbitrarily complicated random variables in terms of a rather small number of ‘universal’ distributions. Central limit theorem, which proves convergence to the Gaussian law, is the best known among this type of results. However, for practical purposes, it is much more important to know how accurate such an approximation is, and this is of course a more difficult question to answer. For instance, central limit theorem was known already around 1715 (and in full generality by 1900), whereas the corresponding approximation theorem of Berry and Esseen was only proved in 1941. Stein’s method, introduced in 1972 in [ 60], offers a general means of solving such problems.

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.