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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.

© 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.

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.

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)

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.