Wiener-Khinchine theorem

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

In order to be able to take Aco small, one has to choose T large, so that many values n fall inside the interval Aco. The answer to our question is provided by the Wiener-Khinchin theorem, which states that S(co) is the cosine transform of the autocorrelation function ... 

Exercise. Alternative forms of the Wiener-Khinchin theorem (3.4) are... 

Note that as a consequence of the fact that the process is not stationary 0 this autocorrelation function does not depend on xl — x2 alone, and even contains the total length L. Hence the Wiener-Khinchin theorem does not apply directly yet a similar calculation of the Fourier coefficients An yields < An) = 0 and... 

The Wiener-Khinchin theorem is a special case of Bocliner theorem applicable to time averages of stationary stochastic variables. Bochner s theorem enables the Wiener-Khinchin theorem to be applied to ensemble averaged time-correlation functions in quantum mechanics where it is difficult to think of properties as stochastic processes. 

Relation to a Noise Current. The fluctuating polarization results from a movement of charge in the dielectric medium. In a short-drcuited condenser, the polarization changes induce a corresponding current /(/) in the external drcuit. The mean square current G y) per unit frequency bandwidth at frequency v can be related to a series of observations of /(/) by the important Wiener-Khinchin theorem... 

This is the Wiener-Khinchin theorem where the spectral density is the Fourier transform of the autocorrelation function. 

This is the velocity correlation function as obtained by the Wiener-Khinchin theorem. The above arguments may be extended to multidimensional systems, as discussed in some detail by Wang and Uhlenbeck [10] and by McConnell [57] in the context of rotational Brownian motion. 

We shall allude to the Wiener-Khinchin theorem later in this appendix in our discussion of linear response theory. 

D = mfikT) which we obtained earlier using the Wiener-Khinchin theorem and the Langevin equation. This procedure may be also applied for many dimensions as in the following example. 

The reader will recall, from Appendix A, the Wiener-Khinchin theorem namely, the a.c.f. and the spectral density are each other s Fourier cosine transform. Thus... 

Qualitative justification for the highly random nature of the power spectrum in a chaotic system is readily given. That is, IT[co, x(0)] is a function of a single trajectory. Since a single trajectory is known to depend sensitively on initial conditions, any such function is expected to show such extreme sensitivity as well. Nonetheless, it is necessary to reconcile this result with standard textbook statements28 of the Wiener-Khinchin theorem,29 which equate the power spectrum and spectral density S(co, M) ... 

According to the Wiener-Khinchine theorem, the power spectrum of a fluctuating quantity is given by the (time) Fourier transform of the autocorrelation function. So for the bending mode one derives from (7.4), for... 

Lampard, D. G. (1954). Generalization of the Wiener-Khinchine theorem to nonstationary processes. J. Appl. Phys., 25, 802-3. 

Theorem 5.3 Wiener-Khinchin Theorem. The autocovariance and the spectral density function are related as follows (Khintchine 1934) ... 

Wiener-Khinchin theorem of statistical mechanics, the Fourier transform of the autocorrelation function of a fluctuating quantity is the power spectrum of the fluctuations, where the power at a given frequency is the mean square amplitude of the fluctuations at that frequency. To convert Mnmjkit) to a suitable function of frequency, Redfield [19] defined the spectral density function, J mjk(,co)> ... 

K. Ito, S. Qoi, Y. Kinura, and R. Hayakawa. New measurement method for the quasielastic light scattering with the sinusoidal electric field by use of an extended Wiener-Khinchin theorem. J. Chem. Phys., 101 (1994), 4463-4465. 

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